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+widely used in downstream NLP tasks via task- +specific fine-tuning. +Recently, an array of +Parameter-Efficient Fine-Tuning (PEFT) meth- +ods have also achieved strong task perfor- +mance while updating a much smaller num- +ber of parameters compared to full model tun- +ing. +However, it is non-trivial to make in- +formed per-task design choices (i.e., to create +PEFT configurations) concerning the selection +of PEFT architectures and modules, the num- +ber of tunable parameters, and even the lay- +ers in which the PEFT modules are inserted. +Consequently, it is highly likely that the cur- +rent, manually set PEFT configurations might +be suboptimal for many tasks from the perspec- +tive of the performance-to-efficiency trade-off. +To address the core question of the PEFT con- +figuration selection that aims to control and +maximise the balance between performance +and parameter efficiency, we first define a rich +configuration search space spanning multiple +representative PEFT modules along with finer- +grained configuration decisions over the mod- +ules (e.g., parameter budget, insertion layer). +We then propose AUTOPEFT, a novel frame- +work to traverse this configuration space: it +automatically configures multiple PEFT mod- +ules via high-dimensional Bayesian optimisa- +tion. We show the resource scalability and task +transferability of AUTOPEFT-found configu- +rations, outperforming existing PEFT methods +on average on the standard GLUE benchmark +while conducting the configuration search on +a single task. The per-task AUTOPEFT-based +configuration search even outperforms full- +model tuning. +1 +Introduction and Motivation +Pretrained language models (PLM) are used in +downstream tasks via the standard transfer learning +*Equal contribution. +Code is available at https:// +github.com/cambridgeltl/autopeft +100 +101 +Fine-tuned Parameters (%) +80 +81 +82 +83 +84 +Average Score +Pfeiffer +UniPELT +MAM +AdaMix +Prefix +LoRA +Parallel +AutoPEFT +Full Model FT +Figure 1: +The performance of +AUTOPEFT-found +PEFT configurations compared to other standard PEFT +methods and full model FT on the GLUE bench- +mark (Wang et al., 2018). We report the average score +for each method by taking the mean of metrics for 8 +GLUE tasks. The dashed horizontal bar (Full Model +FT) indicates the full-model FT that updates 100% of +parameters, and our approach aims to learn the best +trade-off configuration between task performance and +parameter efficiency. +paradigm, where they get fine-tuned for particu- +lar tasks (Devlin et al., 2019; Liu et al., 2019b). +This achieves state-of-the-art results in a wide spec- +trum of NLP tasks, becoming a prevalent modelling +paradigm in NLP (Raffel et al., 2020). Fine-tuning +the PLMs typically requires a full update of their +original parameters (i.e., the so-called full-model +fine-tuning (FT)); however, this is (i) computation- +ally expensive and also (ii) storage-wise expensive +as it requires saving a separate full model copy +for each task-tuned model. With the ever-growing +size of the PLMs (Brown et al., 2020; Sanh et al., +2022), the cost of full model FT becomes a major +bottleneck, due to its increasing demands as well +as computational (time and space) non-efficiency. +Parameter-Efficient Fine-Tuning (PEFT) deliv- +ers a solution for alleviating the issues with full- +model FT (Houlsby et al., 2019). By freezing the +majority of pretrained weights of PLMs, PEFT ap- +proaches only update a small portion of parameters +arXiv:2301.12132v1 [cs.CL] 28 Jan 2023 + +for efficiently adapting the PLM to a new down- +stream task. Recent studies have shown that PEFT +can achieve competitive task performance while be- +ing modular, adaptable, and preventing catastrophic +forgetting in comparison to traditional FT (Wang +et al., 2022). +Recent developments have created diverse PEFT +modules with distinctive characteristics (Pfeiffer +et al., 2020b; Li and Liang, 2021), with one of +the two main aims in focus: 1) improve task perfor- +mance over other PEFT approaches while maintain- +ing the same parameter budget as the competitor +PEFT methods; or 2) maintain task performance +while reducing the parameter budget needed. Exist- +ing PEFT modules, optimising for one of the two +aims, have been successfully applied to transfer +learning tasks (Chen et al., 2022b; Pfeiffer et al., +2022). However, different tasks, with different +complexity, show distinct sensitivity to the allo- +cated parameter budget and even to the chosen +PEFT approach (He et al., 2022). At the same +time, most PEFT applications are limited to a sin- +gle PEFT architecture (e.g., serial adapters, prefix- +tuning) with fixed decisions on its components (e.g., +hidden size dimensionality, insertion layers) result- +ing in potentially suboptimal PEFT configurations +across many tasks. Therefore, in this work, we +propose a new, versatile and unified framework +that automatically searches for improved and task- +adapted PEFT configurations, aiming to effectively +balance between the two (often colliding goals) +of (i) improving performance and (ii) keeping the +desired low parameter budget for PEFT. +While recent research has started exploring more +dynamic PEFT configurations, the prior studies +remain limited across several dimensions, includ- +ing how they define the configuration search space. +Namely, they typically focus only on a single PEFT +architecture (e.g., adapters) or their simple combi- +nations, or a single property (e.g., insertion layers – +where to insert the module); see a short overview +later in §2. Here, we propose a unified and more +comprehensive framework for improved configu- +ration search. It covers multiple standard PEFT +modules (1. serial adapters, 2. parallel adapters, +3. prefix-tuning), combined with the critical pa- +rameter budget-related decisions: the size of each +constituent module and the insertion layers for the +modules. +Our defined comprehensive search space is huge; +as a consequence, traversing it effectively and effi- +ciently is extremely challenging. To enable search +over the large configuration space, we thus propose +the AUTOPEFT framework. It automatically con- +figures multiple PEFT modules along with their +efficiency-oriented design decisions, relying on a +high-dimensional Bayesian optimisation (BO) ap- +proach. Crucially, within the search space, we pro- +pose a multi-objective optimisation which learns +to simultaneously balance between maximising the +searched configurations’ task performance and pa- +rameter efficiency. +We conduct extensive experiments on the stan- +dard GLUE benchmark (Wang et al., 2018). We +first study the transferability of the AUTOPEFT- +searched architecture by running AUTOPEFT on a +single task, followed by transferring the found ar- +chitecture to other tasks. Experimental results show +that this architecture can outperform existing PEFT +baselines while achieving on-par performance to +the standard full-model FT, relying only on 1.4% +of the original trainable parameters. Further slight +gains can be achieved via a computationally more +expensive approach, where we run AUTOPEFT per +each single task to find a task-adapted PEFT config- +uration. As demonstrated in Figure 1, AUTOPEFT +is able to find configurations that offer a solid trade- +off between task performance and parameter effi- +ciency, even outperforming full-model FT. We also +provide ablation studies over the search space, vali- +dating that the AUTOPEFT framework is versatile +and portable to different search spaces. +Contributions. 1) We propose a large and com- +prehensive search space of PEFT configurations, +which integrates three representative PEFT mod- +ules, the tunable number of parameters of each +module, and the binary decisions concerning Trans- +former layers for inserting these modules. 2) We +propose a novel AUTOPEFT framework with high- +dimensional Bayesian optimisation that can auto- +matically and feasibly search for the effective PEFT +configuration in terms of both task performance +and parameter efficiency. 3) We demonstrate that +the AUTOPEFT-found configurations can not only +reduce the parameter budget but also outperform +existing PEFT modules while being transferable +across tasks. The AUTOPEFT framework can also +be easily extended to other and new PEFT modules. +2 +Related Work +Parameter-Efficient Fine-Tuning. +Standard +PEFT methods can be divided into two main + +groups. 1) Some methods fine-tune a small por- +tion of pretrained parameters (Zhao et al., 2020; +Guo et al., 2021). For instance, Ben Zaken et al. +(2022) propose to fine-tune the PLM’s bias terms, +while Sung et al. (2021) and Ansell et al. (2022) +fine-tune sparse subnetworks withing the original +PLM for a particular task. 2) Other methods fine- +tune an additional set of parameters (Liu et al., +2022). Since there is no interference with the pre- +trained parameters, this class of PEFT modules, be- +sides offering strong task performance, is arguably +more modular; we thus focus on this class of PEFT +methods in this work. The original adapter mod- +ules (Houlsby et al., 2019; Pfeiffer et al., 2020b) +have a bottleneck serial architecture which can be +inserted into every Transformer layer, see Figure 2. +LoRA (Hu et al., 2022a) assumes the low-rank +intrinsic dimensionality of the target task and per- +forms low-rank updates (Mahabadi et al., 2021). +Li and Liang (2021) propose the Prefix-Tuning +method that appends a learnable vector to the at- +tention heads at each Transformer layer. Similarly, +prompt-tuning (Lester et al., 2021) only appends +this vector to the input embedding. UniPELT (Mao +et al., 2022) integrates multiple PEFT modules with +a dynamic gating mechanism. He et al. (2022) +provide a unified formulation of existing PEFT +modules and propose a parallel adapter module, +along with a combined ‘Mix-and-Match Adapter +(MAM)’ architecture that blends parallel adapters +and prefix-tuning. Wang et al. (2022) propose the +mixture-of-adaptations (AdaMix) combined archi- +tecture that leverages weight averaging for a mix- +ture of adapters. +Optimising Parameter Efficiency in PEFT. Re- +cent work further aims to optimise the parameter +efficiency of existing PEFT modules while main- +taining task performance. The standard approach +is to insert (typically serial) adapters into all Trans- +former layers, which still requires a sizeable pa- +rameter budget. Rücklé et al. (2021) address this +question by performing random dropout of adapters +from lower-level layers, displaying only a small de- +crease in task performance. Adaptable Adapters +(AA) (Moosavi et al., 2022) generalise this idea +by learning gates that switch on or off adapters +in particular Transformer layers. Neural Architec- +ture Search (NAS) methods aim to automate the +design of neural net architectures themselves, and +NAS has seen great advances recently, with per- +formance often surpassing human expert-designed +architectures in various tasks (Zoph and Le, 2017; +Ren et al., 2021; Elsken et al., 2019). Concerning +NLP tasks and PEFT, Hu et al. (2022b) propose +S3PET, which adapts Differentiable Architecture +Search (DARTS) (Liu et al., 2019a) to learn the po- +sitions for inserting the PEFT modules. This work +is closest in spirit to ours. +Our method, discussed in detail in §3, offers +a spectrum of advantages over S3PET and other +related PEFT work. Relying on multi-objective +optimisation, unlike S3PET, we can automatically +discover a family of configurations at different pa- +rameter efficiency levels in a single search run, ef- +fectively balancing between task performance and +parameter efficiency, without the need to set the +‘parameter budget’ in advance; similarly, we en- +able an automatic search over multiple constituent +modules over the desirable range of parameter bud- +get and effective layers, whereas previous work +can only support one architecture per each search +run. Further, previous work indicated that weight- +sharing NAS such as DARTS may suffer with the +reliability of prediction (White et al., 2021b), and +its success often hinges heavily on the design of +the actual search space (Li and Talwalkar, 2019; +Ru et al., 2020; Dong and Yang, 2020; Yang et al., +2020). We mitigate those issues with our design of +AUTOPEFT. Finally, while weight-sharing NAS is +arguably more computationally efficient, through +combining the use of low-fidelity performance pre- +dictors and the strong transferability of the configu- +rations found across tasks, AUTOPEFT can also be +made very computationally efficient in discovering +effective PEFT configurations. We further discuss +this in §3 and demonstrate empirically in §5. +3 +AUTOPEFT Framework +We start by designing a large configuration space, +providing the motivation behind each decision to +include a particular module and its components +into the configuration space, along with a mathe- +matical formulation. We then propose AUTOPEFT, +a novel framework to search over this challenging +configuration space. It automatically configures +(components of) multiple PEFT modules via high- +dimensional Bayesian optimisation. +PEFT Configuration Search Space. The search +space is an influential factor in the performance +of any search algorithm. In order to simultane- +ously maximise task performance along with pa- +rameter efficiency, it is necessary to first define a + +‘parameter-reducible’ search space, where each di- +mension within the space potentially contributes +to reducing the parameter budget. Similarly, each +dimension might potentially bring positive impact +to the task performance without introducing redun- +dancy in the space (Wan et al., 2022). Therefore, +we propose the search space with representative +PEFT modules, as follows, spanning a plethora of +(non-redundant) configurations, as also shown in +Figure 2. +PEFT Modules. We include three distinctive PEFT +designs to efficiently adapt different forwarding +stages of hidden states in the PLM layers. We +combine Serial Adapters (SA), Parallel Adapters +(PA), and Prefix-Tuning (PT) as the three represen- +tative modules in the search space, where the PT +module adapts the multi-head attention layer, and +SA and PA interact with the FFN layer (Figure 2). +Each configuration makes a decision on the PEFT +modules in the insertion layer: all of them can be +‘turned’ on or off. We combine this binary decision +with the actual non-binary decision on the module +size (see next), so that the value of 0 in fact denotes +the absence of the modules in the layer(s). +Size. Previous studies show that PEFT methods +are highly sensitive to the number of tunable pa- +rameters: adaptively setting their capacity in ac- +cordance with the target task is then essential for +achieving good performance (Chen et al., 2022a). +The number of tunable parameters is dependent on +each particular module. The additional parameters +introduced by both SA and PA are dominated by +their bottleneck dimension D. Similarly, the size +of the PT module is defined by its prefix length +LPT. Thus, we define a binary logarithmic search +scale for the respective discrete sets DSA, DPA, +and LPT, spanning the values from 0 (absence of +the module) to Dh where Dh is the dimensionality +of the PLM (e.g., Dh=768 for BERTbase). +Insertion Layers. Prior work has also shown that +different layers in the PLMs store different se- +mantic information (Vuli´c et al., 2020), where the +higher layers produce more task-specific and con- +textualized representations (Tenney et al., 2019). +Therefore, as another configuration dimension, we +aim to search for the minimal number and the ac- +tual position of layers in which to insert the PEFT +modules. We define a binary ‘insertion’ decision at +each layer li. +Combining PEFT Modules. The SA module and +the PA module share a bottleneck architecture. The +Feed Forward +LayerNorm +Multi-Head Attention +LayerNorm +Prefix-Tuning +Serial +Parallel +PEFT Layer +PEFT Layer +Layer +PEFT Layer +Layer +PEFT Layer +Trainable +Search +Frozen +Figure 2: Illustration of the main components of our +configuration search space, traversed via AUTOPEFT. +AUTOPEFT configures the selected Transformer layers +with PEFT modules, where the activation of each sub- +module is controlled by the learned size of each sub- +module. See also Table 4 in the appendix. +SA receives hidden states from the FFN output as +its inputs, adapting it with a down-projection ma- +trix W down +SA +∈ RDh×DSA, followed by a non-linear +activation function, and then an up-projection ma- +trix W up +SA ∈ RDSA×Dh: +fSA(h) = ReLU(hW down +SA +)W up +SA. +(1) +PA, on the other hand, receives its inputs from +hidden states before the FFN layer with the same +formulation: +fPA(x) = ReLU(xW down +PA +)W up +PA. +(2) +Therefore, it is able to act in parallel with the SA +without interference. Note that the FFN hidden +states h = F(x) contain the task-specific bias +learned in its pretrained weights. Therefore, by +combining SA with PA, the following composition +of functions is achieved: +fSAPA(x) =ReLU(F(x)W down +SA +)W up +SA ++ReLU(xW down +PA +)W up +PA. +(3) +The final composition should provide an effective +adaptation to both bias-influence hidden states and +the original inputs before the pretrained FFN layer.1 +Further, applying PEFT modules to interact both +with FFNs and multi-head attention should have a +positive impact on task performance (Mao et al., +1The PA module also acts as the low-rank reparametriza- +tion of the learned SA together with the frozen FFN layer to +further match the intrinsic dimensionality of the target task. + +Query new configuration by +Suggested +config in +AutoPEFT +search +space +Performance & +Parameter Efficiency +Evaluate +Parameter-Efficient Fine-Tuning + + +Multi-Objective Bayesian Optimisation +Serial +Parallel +Prefix +Config +Layer Configuration +maximising the acquisition function +GP Surrogate +Figure 3: Illustration of the AUTOPEFT framework: to search for optimal architectures in the defined configu- +ration space, AUTOPEFT uses a multi-objective BO agent, which trains on previous observations of the PEFT +configuration vector and its performance (e.g., accuracy – obtained by fine-tuning the language model with the +PEFT configuration) and cost (e.g., number of parameters). The BO agent then suggests new configurations, and +the algorithm continues iteratively until convergence. +2022; He et al., 2022). PT learns two prefix vectors, +Pk and Pv ∈ RLPT×Dh, that are concatenated with +the original multi-head attention’s key and value +vectors, which efficiently adapts the multi-head +attention layer to fit the target task. We thus finally +combine the SA and the PA (i.e., SAPA from above) +with PT. +In sum, the overview of the dimensions spanning +the final configuration space is provided in Figure 2 +and Table 4. The combination of the different ‘con- +figuration dimensions’ outlined above gives rise to +a total of e.g., 5,451,776 possible configurations +with BERTbase and ∼ 3×1010 configurations with +RoBERTalarge (i.e., the number of configurations is +2|l|×|DSA|×|DPA|×|LPT|). While a large search +space is crucial for expressiveness and to ensure +that good-performing configurations are contained, +it also increases the difficulty for search strategies +to both navigate the search space well while re- +maining sample- and thus computationally efficient. +Furthermore, in the PEFT setting, we are also often +interested in discovering a family of configurations +that trade off between performance and efficiency +for general application in various scenarios with +different resource constraints, thus giving rise to a +multi-objective optimisation problem where we si- +multaneously aim to maximise performance while +minimising costs. In what follows, we propose a +search framework that satisfies all those criteria. +AUTOPEFT via Multi-Objective Bayesian Opti- +misation. Formally, denoting the full AUTOPEFT +search space as A and a single configuration a ∈ A +with trainable weights W, without loss of gener- +ality, assuming our objective is to maximise (i) a +performance metric f(a, W) (e.g., the accuracy +on the dev set) and to (ii) minimise a cost metric +g(a) (e.g., the number of parameters in a), a search +method aims to solve the bi-level, bi-objective op- +timisation problem: +max +a∈A +� +f(a, W ∗), −g(a) +� +; +s.t.W ∗ = arg min +W Ltrain(a, W), +(4) +where the inner loop optimisation problem is the op- +timisation of the configuration weights achieved by +fine-tuning the configuration a itself over the train +loss Ltrain. Given the bi-objective nature of the +problem, there is in general no single maximiser of +Eq. (4) but a set of non-dominated Pareto-optimal +configurations A∗ = {a∗ +1, ..., a∗ +|A∗|}. +To address these challenges in this work, we +adopt a Bayesian optimisation (BO) approach, il- +lustrated in Figure 3. BO is a sample-efficient, +zeroth-order model-based sequential optimisation +algorithm (Garnett, 2023) with proven successes +in NAS and automated machine learning in gen- +eral (Snoek et al., 2012; White et al., 2021a; Ru +et al., 2021; Kandasamy et al., 2018). BO is partic- +ularly popular in the multi-objective setups where +one is interested in recovering a Pareto front where +it is less straightforward to apply methods such as +differentiable / one-shot architecture search meth- +ods that are typically used to discover a single best- +performing configuration (Eriksson et al., 2021; +Izquierdo et al., 2021). BO consists of a surro- +gate model, usually a Gaussian Process (GP) that +sequentially approximates the objective function +based on the observations so far, and an acquisition +function, which balances between exploitation (i.e., +regions in the search space with high perceived + +Mateérn kernel +Samplesfrompriordistribution +3 +1 +0 +-1 +-2 +Sampled function #1 +Sampled function #2 +Sampled function #3 +Sampled function #4 +Samplesfromposteriordistribution +Sampled function #5 +3 +Mean +± 1 std. dev. +2 +Observations +1 +-1 +-2 +-3 +0 +2 +3 +5value) and exploration (i.e., regions that have not +been visited before). It is optimised at each iter- +ation to actively select the next configuration to +evaluate. For a detailed overview of BO, we refer +the readers to Frazier (2018). +While vanilla BO methods are better-suited +in modestly-dimensioned and continuous prob- +lems, our current setup instead features a high- +dimensional and combinatorial search space. Here, +performance of non-parametric methods such as +GP-based BO tend to suffer due to the exponen- +tially exploding volume of space the surrogate +needs to model as dimensionality increases. For- +tunately, recent advances in search methods have +allowed us to address these challenges effectively. +Specifically, we adopt the SAAS-GP (Eriksson and +Jankowiak, 2021) model as the surrogate function: +on a high level, SAAS-GP (1) places a relatively +strong regularising half-Cauchy prior on the model +lengthscales (which dictate the perceived impor- +tance of search dimensions to the objective func- +tion value) to induce sparsity and (2) approximately +marginalises over model hyperparameters via a +No-U-Turn Monte Carlo sampler (Hoffman et al., +2014) to reduce overfitting in high dimensions. We +argue that both are appealing in our setup, while the +benefit of (2) in our setup is self-evident, (1) also ef- +fectively places a prior to encode our belief that in +spite of the high nominal complexity search space, +the effective dimensionality of the problem should +be much lower – this is appropriate in our setup, +as although we have a nominally high dimensions, +consistent to previous findings in NAS (Wan et al., +2022), we do expect a few disproportionately in- +fluential key dimensions (although we do not have +information on which a priori – this is meant to be +discovered by the BO algorithm). +For the acquisition function, +we use the +noisy expected hypervolume improvement (NE- +HVI) (Daulton et al., 2021), which is suitable for +the setup described in Eq. 4. Lastly, while BO is +sample-efficient, it may still require 100-200 eval- +uations of different configurations in the search +space to sufficiently explore the search space; to +make sure the search remains cost-efficient, during +search we also adopt low-fidelity approximations +commonly employed in NAS: at the search stage, +for a configuration a, instead of evaluating the ob- +jective f(a, W) defined in Eq. 4 in full, we only +fine-tune the a using a smaller computational bud- +get – for example, if a complete fine-tuning takes +100% of training data, at search time we are able +to only fine-tune with 1% of training data and use +the accuracy after that as a lower-cost proxy to the +accuracy after full-length FT, the latter of which is +significantly more expensive to obtain. Therefore, +when we are facing high-resource tasks, fine-tuning +the full training resources is only performed once +at evaluation time after the Pareto-optimal configu- +rations are finalised. Other low-cost proxies such +as training for fewer number of epochs than full +FT are also compatible but not used in the present +work. +4 +Experimental Setup +Evaluation Data. We follow prior PEFT research +and base our evaluation on the standard GLUE +benchmark. +We include 4 types of text classi- +fication tasks, including linguistic acceptability: +CoLA; similarity and paraphrase: STS-B, MRPC, +QQP; sentiment analysis: SST-2; natural language +inference: RTE, QNLI, MNLI. We exclude WNLI +following previous work (Houlsby et al., 2019; +Mao et al., 2022). +Baselines. We compare the performance of the +AUTOPEFT-found configurations to the standard +full model FT and each individual PEFT module +(SA, PA, PT) from the search space used in their +default setup from respective original work. We +also compare with the LoRA module, to provide +a comparison to low-rank decomposition methods. +In order to provide comparisons with recently pro- +posed methods that also integrate multiple PEFT +modules (see §2), we further include the UniPELT +and the MAM adapter in their default settings. We +reproduce AdaMix for a comparison to a mixture +of homogeneous adaptations. In ablations on inser- +tion layers, we also include the Adaptable Adapter +(AA) as a baseline that proposes a differentiable +gate learning method to select the insertion layer +for PEFT modules (i.e., serial adapters originally). +Implementation Details. +Following previous +work on the GLUE benchmark, we report the best +GLUE dev set performance (Ben Zaken et al., +2022) and use 20 training epochs with an early +stopping scheme of 10 epochs for all tasks. We +use AdapterHub (Pfeiffer et al., 2020a) as the code- +base and conduct extensive experiments with the +uncased BERTbase (Devlin et al., 2019) as the main +backbone model. +We report main experiments +with the mean and standard deviation over 5 dif- +ferent random seeds. Experimental results using + +Method +#Param. +RTE +MRPC +STS-B +CoLA +SST-2 +QNLI +QQP +MNLI +Avg. +Fine-tune +100% +71.121.46 +85.741.75 +89.000.45 +59.320.62 +92.570.24 +91.500.08 +91.520.04 +84.430.22 +83.15 +Prefix +0.17% +70.540.49 +85.930.89 +88.760.15 +58.881.15 +91.930.45 +90.760.14 +89.120.07 +82.780.16 +82.33 +LoRA +0.27% +65.851.49 +84.461.04 +88.730.08 +57.580.78 +92.060.38 +90.620.22 +89.41 0.04 +83.000.07 +81.46 +Serial +0.81% +68.011.34 +84.750.45 +88.610.11 +59.730.62 +91.930.33 +91.060.12 +90.520.05 +84.180.22 +82.35 +AdaMix +0.81% +70.110.62 +86.861.12 +89.120.11 +59.111.00 +92.060.22 +91.520.15 +90.220.04 +84.250.14 +82.91 +UniPELT +1.25% +67.071.82 +84.220.78 +88.840.11 +60.130.46 +92.520.24 +91.090.13 +90.690.11 +84.280.18 +82.35 +Parallel +6.46% +68.523.44 +86.520.96 +88.900.28 +58.721.69 +92.130.35 +90.830.22 +90.740.08 +73.9319.24 +81.29 +MAM +6.97% +69.101.76 +87.160.74 +89.010.48 +47.8723.97 +83.9416.52 +90.850.22 +90.760.05 +83.310.17 +80.25 +AUTOPEFTRTE +S +0.06% +69.680.76 +85.540.78 +88.780.18 +56.830.54 +91.930.34 +90.810.18 +88.510.05 +82.260.11 +81.79 +AUTOPEFTMNLI +S +0.30% +69.770.47 +85.730.61 +88.780.17 +57.501.79 +91.880.32 +91.120.13 +89.900.05 +83.920.10 +82.32 +AUTOPEFTRTE +M +1.42% +72.350.84 +86.130.62 +89.060.09 +60.231.00 +92.110.23 +91.000.09 +90.640.07 +84.010.21 +83.19 +AUTOPEFTRTE +L +6.60% +71.701.18 +86.620.65 +89.190.13 +59.440.75 +92.410.28 +91.090.12 +90.790.06 +83.910.14 +83.14 +AUTOPEFTtask +Avg. +1.40% +72.350.94 +87.450.87 +89.170.00 +60.921.47 +92.110.25 +91.120.13 +90.640.05 +84.010.10 +83.47 +Table 1: Results on the GLUE benchmark with BERTbase, where tasks are ordered in ascending order of the +training resources. We conduct three groups of task transferability experiments on RTE and one resource scalability +experiment on MNLI. We report the average fine-tuned parameters of per-task AUTOPEFT, where we conduct +additional per-task searches on MRPC, STS-B, and CoLA, and take best-found configurations for the remaining +tasks. We report Spearman’s Correlation for STS-B, Matthew’s Correlation for CoLA, and accuracy for all other +tasks, where we report the matched accuracy for MNLI. The percentage of parameters is computed as a ratio of +the number of additional parameters to the pretrained parameters. We reproduce all baselines and report the mean +and standard deviation of all results for 5 random seeds. The best, second-best, and third-best results are marked +in bold fonts and ranked by colour. +RoBERTalarge (Liu et al., 2019b) show findings +that are consistent to the ones BERTbase, and are +included in Table 3 in the appendix. We report +the setup for each PEFT module and the detailed +training scheme in §A. +5 +Results and Discussion +Transferability of Configurations across Tasks. +The main results are summarized in Table 1. First, +we analyze task transferability of AUTOPEFT- +found configurations by running AUTOPEFT on +the most low-resource and challenging task, RTE, +followed by transferring the three best AUTOPEFT- +found configurations to other tasks. +First, we +note that the parameter budget of the configura- +tion AUTOPEFTRTE +M +is only 1.42%, while it shows +considerable average gains over all the PEFT base- +lines on the RTE task, by a margin of at least 2%. +The AUTOPEFT-found configuration also outper- +forms the full-model FT baseline on the RTE task +by more than 1%. These results indicate the ef- +fectiveness of the AUTOPEFT framework in opti- +mising both task performance and parameter effi- +ciency. Transferring the RTE-based configurations +to other tasks, we find that strong performance is +maintained across the target tasks, with more bene- +fits on the medium-resource tasks (MRPC, STS-B, +CoLA), but the configuration remains competitive +also for higher-resource tasks (e.g., QQP, MNLI). +10 +2 +10 +1 +100 +Fine-tuned Parameters (%) +62.5 +65.0 +67.5 +70.0 +72.5 +75.0 +Task Score +RTE +10 +2 +10 +1 +100 +Fine-tuned Parameters (%) +75 +80 +85 + +MRPC +Serial +Parallel +Prefix +LoRA +AutoPEFT +Figure 4: The Pareto front of the AUTOPEFT on tasks +RTE and MRPC compared to baselines with BERTbase +in various settings of parameter budgets. +We report +the single-seed task score for each task following the +settings in Table 1. The plots for STS-B, and CoLA, +showing the same trends, are in Appendix §B. +When we assign a large parameter budget to +the potential configurations, AUTOPEFTRTE +L +also +shows a stronger transfer performance in high- +resource tasks. This indicates that, as expected, +the parameter capacity of the configuration is an +important factor in transfer learning (Chen et al., +2022a). On average, the AUTOPEFTRTE +M +configura- +tion shows a comparable fine-tuning performance, +83.19, to the full model FT, 83.15, by only updating +1.42% of parameters. With strong transferability +across similar tasks, AUTOPEFT provides distinct +advantages in parameter efficiency; the search al- +gorithm itself coupled with transfer becomes more + +sample-efficient within limited training resources. +Resource Scalability and Efficiency. +We next +‘stress-test’ the ability of AUTOPEFT in a more +challenging scenario with limited task training +data, carrying out an experiment on the most high- +resource MNLI task using only a small set of its +training data. We randomly sample 1% of the orig- +inal MNLI training data to train AUTOPEFT, and +retain using the original dev set for evaluation.2 +We report AUTOPEFTMNLI +S +in Table 1 as the best- +found configuration in this low-resource setting. It +requires only 0.30% of fine-tuned parameters and +shows the strong MNLI performance of 83.92%. +In another efficiency-oriented test, we conduct con- +figuration transfer in a radically parameter-efficient +setup (training on the full RTE training set but with +reduced parameter budget, and then transferring to +other tasks; AUTOPEFTRTE +S +in Table 1). The main +finding is that, while performance does decrease +slightly as expected, strong task performance can +still be achieved even with the parameter budget of +0.06% within this very efficient setup. +Per-Task Configuration Search. Finally, we con- +duct full-resource per-task AUTOPEFT searches, +which naturally come with increased computational +costs, for RTE, MRPC, STS-B, and CoLA, and +then, for efficiency reasons, port the small set of +best configurations to the remaining high-resource +tasks: SST-2, QNLI, QQP, MNLI. In addition +to the peak score on RTE, we observe gains on +MRPC (87.16% to 87.45%) and CoLA (60.13% +to 60.92%) over the best-performing PEFT base- +lines. We also observe gains over the transferred +configuration AUTOPEFTRTE +M . One interpretation +of the results is that AUTOPEFT is strong at match- +ing the intrinsic dimensionality of the low-resource +downstream task to the capacity (i.e., parameter +budget) of the PEFT modules, whereas full model +FT performs better in high-resource scenarios, giv- +ing the largest capacity to capture the informa- +tion in high-resource tasks.3 However, the per- +task AUTOPEFTtask variant outperforms even full +model FT by 0.3% while its parameter budget is +only 1.4% of the full model per task. +Analysing the ‘Behaviour’ of Bayesian Optimi- +2With this setup, we effectively save 99% of training re- +sources and the search framework becomes extremely fast +even for high-resource datasets. +3Due to the richness of training resources in high-resource +datasets, the results in these tasks are mostly saturated. Pre- +vious work shows that PEFT methods can only reach on-par +performance to full model FT on those tasks. +10 +2 +10 +1 +100 +101 +Fine-tuned Parameters (%) +60 +65 +70 +75 +Accuracy (%) +Initialisation +Random Search +AutoPEFT +Figure 5: The distribution of AUTOPEFT-found config- +urations compared to the random search on RTE with +a single random seed. We initialise the AUTOPEFT +search with 100 runs of random sampling for initial ex- +plorations in the search space. We then conduct 100 +runs of the AUTOPEFT with Bayesian optimisation. +sation. Figure 5 shows the distribution of AU- +TOPEFT-found configurations when we conduct +its search experiment on RTE. Due to the greedy +nature of our predefined acquisition function, we +enforce the initialisation of our algorithm with a +wide exploration of potential configurations. In +the subsequent AUTOPEFT runs, it starts exploit- +ing the best-found configurations while optimising +towards the region with improved parameter effi- +ciency, whereas the random search baseline keeps +obtaining inefficient configurations in a lottery- +ticket manner in the expensive region of param- +eters. It is observable that AUTOPEFT exploits the +region with roughly 1.4% of parameters and finds +configurations with further enhanced task perfor- +mance from 74.4% to 75.1% of accuracy, which +is also the architecture AUTOPEFTRTE +M +with the +strongest transferability across tasks. We also in- +clude the best-found architecture within the initiali- +sation stage as the AUTOPEFTRTE +L +, and our trans- +ferability experiments show that the AUTOPEFT- +found architecture is more robust to the random +initialisation of the neural network, outperforming +the best random search baseline in the searched +task by 0.7% with 5.2% less parameter cost. +Ablation of the Configuration Space. To pro- +vide a finer-grained analysis of factors that bring +positive impact to AUTOPEFT, we ablate the AU- +TOPEFT search space from the full configuration +space: 1) to the basic enumeration of the bottleneck +size DSA of the SA only (the ‘SA’ space). We then +include the Transformer layer and the SA size to- +gether into the search space (the ‘SA-Layer’ space) +to validate the usefulness of using layer selection +as one configuration dimension. We can then also + +Method +#Layers +Size DSA +RTE Accu- +racy (%) +Serial Adapter +24 +64 +72.560.76 +Adaptable Adapter +13 +128 +73.360.80 +AdapterDrop +13 +128 +73.501.40 +AUTOPEFTSA +Layer +10 +128 +73.860.94 +Table 2: The results of AUTOPEFT to layer selection +baselines with the same parameter budget on BERTlarge. +We report the Pfeiffer adapter for all 24 layers. We in- +clude the specialised AdapterDrop (Rücklé et al., 2021) +that inserts SA for the last 13 layers. We report the +AAuni architecture (Moosavi et al., 2022) without its ra- +tional activation function with 13 selected layers. We +run our AUTOPEFT with the comparable search space +of 24 layers and the size of the Pfeiffer adapter. +expand the search space by adding another module +(e.g., PA yields the ‘SA-PA-Layer’ space). Figure 6 +plots the performance over the ‘ablated’ configura- +tion spaces and over different parameter budgets. +Several key findings emerge. First, combining mul- +tiple single PEFT modules has a positive impact +on AUTOPEFT in general (cf., full AUTOPEFT +versus ’SA-PA-Layer’ versus ’SA-Layer’). Rely- +ing on layer selection also brings benefits (cf., ’SA’ +versus ’SA-Layer’). The comparison also indicates +that leaving out some Transformer layers while +increasing the capacity of the PEFT module is a +straightforward method to improve the parameter +efficiency and task performance of the PEFT mod- +ule within a fixed parameter budget. Figure 6 sug- +gests that AUTOPEFT can effectively operate over +configuration spaces of different ‘granularity’. +We analyse the impact of each single PEFT mod- +ule in more detail in Appendix §B. +Layer Selection. +To further compare different +layer selection approaches, we conduct a controlled +experiment with the SA module on BERTlarge (24 +Transformer layers) under a predefined parameter +budget. In Table 2, the simple AdapterDrop ap- +proach simply drops the adapters for the first 11 +layers while doubling their bottleneck sizes, im- +proving the RTE result by roughly 1%. Within +the same architecture, we include the Adaptable +Adapter with selected layers from switch learning, +which has 3 and 10 layers from the first 12 and +the other 12 layers, respectively. We show that AU- +TOPEFT outperforms all existing layer selection +baselines by learning less activated adapter layers, +leading to better parameter efficiency (12.5% fewer +parameters in relative terms) and higher task perfor- +mance. It indicates that selecting the best insertion +0 +2 +4 +6 +Fine-tuned Parameters (%) +65.0 +67.5 +70.0 +72.5 +75.0 +Accuracy (%) +SA +SA-Layer +SA-PA-Layer +PA-PT-Layer +AutoPEFT +Figure 6: The performance of AUTOPEFT with ab- +lation of search space on RTE with a single random +seed on BERTbase. The SA results refer to the Pfeiffer +adapter (Pfeiffer et al., 2020b) with an enumeration of +its bottleneck size. For other search spaces, we report +the Pareto front of AUTOPEFT-found configurations, +where SA-PA-PT-Layer forms the search space of AU- +TOPEFT. +layer is non-trivial, and AUTOPEFT can learn the +correlation between layers. +6 +Conclusion +We proposed AUTOPEFT, a novel search frame- +work for automatically configuring various PEFT +modules in selective layers of pretrained language +models. AUTOPEFT searches the optimal architec- +ture via Bayesian optimisation with iterative evalu- +ation and predicting the desired architecture given +the configuration search space. The proposed multi- +objective optimisation can produce a Pareto front +of candidate architectures by simultaneously max- +imising the model performance and parameter effi- +ciency. We demonstrated that AUTOPEFT-found +architectures offer an effective trade-off between +task performance and parameter efficiency, outper- +forming a variety of PEFT baselines. +Limitations +The proposed AUTOPEFT method is relatively +expensive since it requires iterative optimisation +by learning to optimise each explored configura- +tion. While all intermediate configurations can +be skipped without laying a burden on the final +storage space, the intermediate computation cost +becomes the main bottleneck of this approach. In +this work, we alleviated this problem by (i) con- +ducting the search with 1% of training resources +for large datasets, and (ii) configuration transfer +from low-resource tasks. The search itself can be +seen as a one-time cost yielding a ‘permanent’ well- +performing and shareable configuration for particu- + +lar tasks. We plan to delve deeper into the related +efficiency and computational tractability aspects in +future work. +We have conducted extensive experiments on +the search space that contains three representative +PEFT modules. The AUTOPEFT framework is +decoupled from the actual single PEFT modules: +with further PEFT developments and new PEFT +approaches, those may also get integrated into the +AUTOPEFT framework in future work. +Acknowledgements +Xingchen Wan is supported by the Clarendon +Scholarship at University of Oxford. The work +has been supported in part by a personal Royal So- +ciety University Research Fellowship (no 221137; +2022-) awarded to Ivan Vuli´c. +References +Alan Ansell, Edoardo Ponti, Anna Korhonen, and Ivan +Vuli´c. 2022. +Composable sparse fine-tuning for +cross-lingual transfer. +In Proceedings of the 60th +Annual Meeting of the Association for Computa- +tional Linguistics (Volume 1: Long Papers), pages +1778–1796, Dublin, Ireland. Association for Com- +putational Linguistics. +Maximilian Balandat, Brian Karrer, Daniel Jiang, +Samuel Daulton, Ben Letham, Andrew G Wilson, +and Eytan Bakshy. 2020. +Botorch: A framework +for efficient monte-carlo bayesian optimization. Ad- +vances in neural information processing systems, +33:21524–21538. +Elad Ben Zaken, Yoav Goldberg, and Shauli Ravfogel. +2022. BitFit: Simple parameter-efficient fine-tuning +for transformer-based masked language-models. In +Proceedings of the 60th Annual Meeting of the As- +sociation for Computational Linguistics (Volume 2: +Short Papers), pages 1–9, Dublin, Ireland. Associa- +tion for Computational Linguistics. +Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie +Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind +Neelakantan, Pranav Shyam, Girish Sastry, Amanda +Askell, +Sandhini Agarwal, +Ariel Herbert-Voss, +Gretchen Krueger, Tom Henighan, Rewon Child, +Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, +Clemens Winter, Christopher Hesse, Mark Chen, +Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin +Chess, Jack Clark, Christopher Berner, Sam Mc- +Candlish, Alec Radford, Ilya Sutskever, and Dario +Amodei. 2020. Language models are few-shot learn- +ers. In Advances in Neural Information Processing +Systems 33: Annual Conference on Neural Informa- +tion Processing Systems 2020, NeurIPS 2020, De- +cember 6-12, 2020, virtual. +Guanzheng Chen, Fangyu Liu, Zaiqiao Meng, and +Shangsong Liang. 2022a. +Revisiting parameter- +efficient tuning: Are we really there yet? +In Pro- +ceedings of the 2022 Conference on Empirical Meth- +ods in Natural Language Processing, pages 2612– +2626, Abu Dhabi, United Arab Emirates. Associa- +tion for Computational Linguistics. +Shoufa Chen, Chongjian Ge, Zhan Tong, Jiangliu +Wang, Yibing Song, Jue Wang, and Ping Luo. 2022b. +Adaptformer: Adapting vision transformers for scal- +able visual recognition. In Advances in Neural In- +formation Processing Systems. +Samuel Daulton, Maximilian Balandat, and Eytan Bak- +shy. 2021. +Parallel bayesian optimization of mul- +tiple noisy objectives with expected hypervolume +improvement. In Advances in Neural Information +Processing Systems 34: Annual Conference on Neu- +ral Information Processing Systems 2021, NeurIPS +2021, December 6-14, 2021, virtual, pages 2187– +2200. +Jacob Devlin, Ming-Wei Chang, Kenton Lee, and +Kristina Toutanova. 2019. +BERT: Pre-training of +deep bidirectional transformers for language under- +standing. +In Proceedings of the 2019 Conference +of the North American Chapter of the Association +for Computational Linguistics: Human Language +Technologies, Volume 1 (Long and Short Papers), +pages 4171–4186, Minneapolis, Minnesota. Associ- +ation for Computational Linguistics. +Xuanyi Dong and Yi Yang. 2020. Nas-bench-201: Ex- +tending the scope of reproducible neural architecture +search. In 8th International Conference on Learning +Representations, ICLR 2020, Addis Ababa, Ethiopia, +April 26-30, 2020. +Thomas Elsken, Jan Hendrik Metzen, and Frank Hutter. +2019. Neural architecture search: A survey. The +Journal of Machine Learning Research, 20(1):1997– +2017. +David Eriksson, Pierce I-Jen Chuang, Samuel Daulton, +Peng Xia, Akshat Shrivastava, Arun Babu, Shicong +Zhao, Ahmed A Aly, Ganesh Venkatesh, and Max- +imilian Balandat. 2021. +Latency-aware neural ar- +chitecture search with multi-objective bayesian op- +timization. In 8th ICML Workshop on Automated +Machine Learning (AutoML). +David Eriksson and Martin Jankowiak. 2021. +High- +dimensional bayesian optimization with sparse axis- +aligned subspaces. In Uncertainty in Artificial Intel- +ligence, pages 493–503. PMLR. +Peter I. Frazier. 2018. A tutorial on bayesian optimiza- +tion. CoRR, abs/1807.02811. +Roman Garnett. 2023. Bayesian Optimization. Cam- +bridge University Press. +Demi Guo, Alexander Rush, and Yoon Kim. 2021. +Parameter-efficient transfer learning with diff prun- +ing. In Proceedings of the 59th Annual Meeting of + +the Association for Computational Linguistics and +the 11th International Joint Conference on Natu- +ral Language Processing (Volume 1: Long Papers), +pages 4884–4896, Online. Association for Computa- +tional Linguistics. +Junxian He, Chunting Zhou, Xuezhe Ma, Taylor Berg- +Kirkpatrick, and Graham Neubig. 2022. Towards a +unified view of parameter-efficient transfer learning. +In The Tenth International Conference on Learning +Representations, ICLR 2022, Virtual Event, April 25- +29, 2022. +Matthew D Hoffman, Andrew Gelman, et al. 2014. +The no-u-turn sampler: +adaptively setting path +lengths in hamiltonian monte carlo. J. Mach. Learn. +Res., 15(1):1593–1623. +Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, +Bruna Morrone, Quentin de Laroussilhe, Andrea +Gesmundo, Mona Attariyan, and Sylvain Gelly. +2019. Parameter-efficient transfer learning for NLP. +In Proceedings of the 36th International Conference +on Machine Learning, ICML 2019, 9-15 June 2019, +Long Beach, California, USA, pages 2790–2799. +Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan +Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, +and Weizhu Chen. 2022a. Lora: Low-rank adapta- +tion of large language models. In The Tenth Inter- +national Conference on Learning Representations, +ICLR 2022, Virtual Event, April 25-29, 2022. +Shengding Hu, Zhen Zhang, Ning Ding, Yadao Wang, +Yasheng Wang, Zhiyuan Liu, and Maosong Sun. +2022b. +Sparse structure search for delta tuning. +In Advances in Neural Information Processing Sys- +tems. +Sergio Izquierdo, Julia Guerrero-Viu, Sven Hauns, +Guilherme +Miotto, +Simon +Schrodi, +André +Biedenkapp, Thomas Elsken, Difan Deng, Marius +Lindauer, and Frank Hutter. 2021. +Bag of base- +lines for multi-objective joint neural architecture +search and hyperparameter optimization. +In 8th +ICML Workshop on Automated Machine Learning +(AutoML). +Kirthevasan Kandasamy, +Willie Neiswanger, +Jeff +Schneider, Barnabás Póczos, and Eric P. Xing. 2018. +Neural architecture search with bayesian optimisa- +tion and optimal transport. In Advances in Neural +Information Processing Systems 31: Annual Con- +ference on Neural Information Processing Systems +2018, NeurIPS 2018, December 3-8, 2018, Mon- +tréal, Canada, pages 2020–2029. +Brian Lester, Rami Al-Rfou, and Noah Constant. 2021. +The power of scale for parameter-efficient prompt +tuning. In Proceedings of the 2021 Conference on +Empirical Methods in Natural Language Processing, +pages 3045–3059, Online and Punta Cana, Domini- +can Republic. Association for Computational Lin- +guistics. +Liam Li and Ameet Talwalkar. 2019. Random search +and reproducibility for neural architecture search. In +Proceedings of the Thirty-Fifth Conference on Un- +certainty in Artificial Intelligence, UAI 2019, Tel +Aviv, Israel, July 22-25, 2019, pages 367–377. +Xiang Lisa Li and Percy Liang. 2021. Prefix-tuning: +Optimizing continuous prompts for generation. In +Proceedings of the 59th Annual Meeting of the +Association for Computational Linguistics and the +11th International Joint Conference on Natural Lan- +guage Processing (Volume 1: Long Papers), pages +4582–4597, Online. Association for Computational +Linguistics. +Hanxiao Liu, Karen Simonyan, and Yiming Yang. +2019a. DARTS: differentiable architecture search. +In 7th International Conference on Learning Repre- +sentations, ICLR 2019, New Orleans, LA, USA, May +6-9, 2019. +Haokun Liu, Derek Tam, Muqeeth Mohammed, Jay +Mohta, Tenghao Huang, Mohit Bansal, and Colin +Raffel. 2022. +Few-shot parameter-efficient fine- +tuning is better and cheaper than in-context learning. +In Advances in Neural Information Processing Sys- +tems. +Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- +dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, +Luke Zettlemoyer, and Veselin Stoyanov. 2019b. +Roberta: A robustly optimized BERT pretraining ap- +proach. CoRR, abs/1907.11692. +Rabeeh Karimi Mahabadi, James Henderson, and Se- +bastian Ruder. 2021. Compacter: Efficient low-rank +hypercomplex adapter layers. In Advances in Neu- +ral Information Processing Systems 34: Annual Con- +ference on Neural Information Processing Systems +2021, NeurIPS 2021, December 6-14, 2021, virtual, +pages 1022–1035. +Yuning Mao, Lambert Mathias, Rui Hou, Amjad Alma- +hairi, Hao Ma, Jiawei Han, Scott Yih, and Madian +Khabsa. 2022. UniPELT: A unified framework for +parameter-efficient language model tuning. In Pro- +ceedings of the 60th Annual Meeting of the Associa- +tion for Computational Linguistics (Volume 1: Long +Papers), pages 6253–6264, Dublin, Ireland. Associ- +ation for Computational Linguistics. +Nafise Moosavi, Quentin Delfosse, Kristian Kersting, +and Iryna Gurevych. 2022. Adaptable adapters. In +Proceedings of the 2022 Conference of the North +American Chapter of the Association for Computa- +tional Linguistics: Human Language Technologies, +pages 3742–3753, Seattle, United States. Associa- +tion for Computational Linguistics. +Jonas Pfeiffer, Naman Goyal, Xi Lin, Xian Li, James +Cross, Sebastian Riedel, and Mikel Artetxe. 2022. +Lifting the curse of multilinguality by pre-training +modular transformers. In Proceedings of the 2022 +Conference of the North American Chapter of the +Association for Computational Linguistics: Human + +Language Technologies, pages 3479–3495, Seattle, +United States. Association for Computational Lin- +guistics. +Jonas Pfeiffer, Andreas Rücklé, Clifton Poth, Aish- +warya +Kamath, +Ivan +Vuli´c, +Sebastian +Ruder, +Kyunghyun Cho, +and Iryna Gurevych. 2020a. +AdapterHub: A framework for adapting transform- +ers. In Proceedings of the 2020 Conference on Em- +pirical Methods in Natural Language Processing: +System Demonstrations, pages 46–54, Online. Asso- +ciation for Computational Linguistics. +Jonas Pfeiffer, Ivan Vuli´c, Iryna Gurevych, and Se- +bastian Ruder. 2020b. MAD-X: An Adapter-Based +Framework for Multi-Task Cross-Lingual Transfer. +In Proceedings of the 2020 Conference on Empirical +Methods in Natural Language Processing (EMNLP), +pages 7654–7673, Online. Association for Computa- +tional Linguistics. +Colin Raffel, Noam Shazeer, Adam Roberts, Katherine +Lee, Sharan Narang, Michael Matena, Yanqi Zhou, +Wei Li, and Peter J. Liu. 2020. Exploring the limits +of transfer learning with a unified text-to-text trans- +former. J. Mach. Learn. Res., 21:140:1–140:67. +Pengzhen Ren, Yun Xiao, Xiaojun Chang, Po-Yao +Huang, Zhihui Li, Xiaojiang Chen, and Xin Wang. +2021. A comprehensive survey of neural architec- +ture search: Challenges and solutions. ACM Com- +puting Surveys (CSUR), 54(4):1–34. +Bin Xin Ru, Xingchen Wan, Xiaowen Dong, and +Michael A. Osborne. 2021. +Interpretable neural +architecture search via bayesian optimisation with +weisfeiler-lehman kernels. In 9th International Con- +ference on Learning Representations, ICLR 2021, +Virtual Event, Austria, May 3-7, 2021. +Robin Ru, Pedro M. Esperança, and Fabio Maria Car- +lucci. 2020. Neural architecture generator optimiza- +tion. In Advances in Neural Information Processing +Systems 33: Annual Conference on Neural Informa- +tion Processing Systems 2020, NeurIPS 2020, De- +cember 6-12, 2020, virtual. +Andreas Rücklé, +Gregor Geigle, +Max Glockner, +Tilman Beck, Jonas Pfeiffer, Nils Reimers, and Iryna +Gurevych. 2021. AdapterDrop: On the efficiency +of adapters in transformers. In Proceedings of the +2021 Conference on Empirical Methods in Natural +Language Processing, pages 7930–7946, Online and +Punta Cana, Dominican Republic. Association for +Computational Linguistics. +Victor Sanh, Albert Webson, Colin Raffel, Stephen +Bach, Lintang Sutawika, Zaid Alyafeai, Antoine +Chaffin, Arnaud Stiegler, Arun Raja, Manan Dey, +M Saiful Bari, +Canwen Xu, +Urmish Thakker, +Shanya Sharma Sharma, Eliza Szczechla, Taewoon +Kim, Gunjan Chhablani, Nihal V. Nayak, De- +bajyoti Datta, Jonathan Chang, Mike Tian-Jian +Jiang, Han Wang, Matteo Manica, Sheng Shen, +Zheng Xin Yong, Harshit Pandey, Rachel Bawden, +Thomas Wang, Trishala Neeraj, Jos Rozen, Ab- +heesht Sharma, Andrea Santilli, Thibault Févry, Ja- +son Alan Fries, Ryan Teehan, Teven Le Scao, Stella +Biderman, Leo Gao, Thomas Wolf, and Alexan- +der M. Rush. 2022. +Multitask prompted training +enables zero-shot task generalization. In The Tenth +International Conference on Learning Representa- +tions, ICLR 2022, Virtual Event, April 25-29, 2022. +Jasper Snoek, Hugo Larochelle, and Ryan P. Adams. +2012. Practical bayesian optimization of machine +learning algorithms. In Advances in Neural Infor- +mation Processing Systems 25: 26th Annual Con- +ference on Neural Information Processing Systems +2012. Proceedings of a meeting held December 3- +6, 2012, Lake Tahoe, Nevada, United States, pages +2960–2968. +Yi-Lin Sung, Varun Nair, and Colin Raffel. 2021. +Training neural networks with fixed sparse masks. +In Advances in Neural Information Processing Sys- +tems 34: Annual Conference on Neural Information +Processing Systems 2021, NeurIPS 2021, December +6-14, 2021, virtual, pages 24193–24205. +Ian Tenney, Dipanjan Das, and Ellie Pavlick. 2019. +BERT rediscovers the classical NLP pipeline. +In +Proceedings of the 57th Annual Meeting of the Asso- +ciation for Computational Linguistics, pages 4593– +4601, Florence, Italy. Association for Computational +Linguistics. +Ivan Vuli´c, Edoardo Maria Ponti, Robert Litschko, +Goran Glavaš, and Anna Korhonen. 2020. Probing +pretrained language models for lexical semantics. In +Proceedings of the 2020 Conference on Empirical +Methods in Natural Language Processing (EMNLP), +pages 7222–7240, Online. Association for Computa- +tional Linguistics. +Xingchen Wan, +Vu Nguyen, +Huong Ha, +Binxin +Ru, Cong Lu, and Michael A Osborne. 2021. +Think global and act local: Bayesian optimisation +over high-dimensional categorical and mixed search +spaces. +In International Conference on Machine +Learning, pages 10663–10674. PMLR. +Xingchen Wan, Binxin Ru, Pedro M. Esperança, and +Zhenguo Li. 2022. On redundancy and diversity in +cell-based neural architecture search. In The Tenth +International Conference on Learning Representa- +tions, ICLR 2022, Virtual Event, April 25-29, 2022. +OpenReview.net. +Alex Wang, Amanpreet Singh, Julian Michael, Fe- +lix Hill, Omer Levy, and Samuel Bowman. 2018. +GLUE: A multi-task benchmark and analysis plat- +form for natural language understanding. +In Pro- +ceedings of the 2018 EMNLP Workshop Black- +boxNLP: Analyzing and Interpreting Neural Net- +works for NLP, pages 353–355, Brussels, Belgium. +Association for Computational Linguistics. +Yaqing Wang, Sahaj Agarwal, Subhabrata Mukherjee, +Xiaodong Liu, Jing Gao, Ahmed Hassan Awadal- + +lah, and Jianfeng Gao. 2022. +Adamix: Mixture- +of-adaptations for parameter-efficient model tuning. +In Proceedings of the 2022 Conference on Empiri- +cal Methods in Natural Language Processing, pages +5744–5760, Abu Dhabi, United Arab Emirates. As- +sociation for Computational Linguistics. +Colin White, Willie Neiswanger, and Yash Savani. +2021a. BANANAS: bayesian optimization with neu- +ral architectures for neural architecture search. In +Thirty-Fifth AAAI Conference on Artificial Intelli- +gence, AAAI 2021, Thirty-Third Conference on In- +novative Applications of Artificial Intelligence, IAAI +2021, The Eleventh Symposium on Educational Ad- +vances in Artificial Intelligence, EAAI 2021, Virtual +Event, February 2-9, 2021, pages 10293–10301. +Colin White, Arber Zela, Robin Ru, Yang Liu, and +Frank Hutter. 2021b. +How powerful are perfor- +mance predictors in neural architecture search? In +Advances in Neural Information Processing Systems +34: Annual Conference on Neural Information Pro- +cessing Systems 2021, NeurIPS 2021, December 6- +14, 2021, virtual, pages 28454–28469. +Antoine Yang, Pedro M. Esperança, and Fabio Maria +Carlucci. 2020. NAS evaluation is frustratingly hard. +In 8th International Conference on Learning Repre- +sentations, ICLR 2020, Addis Ababa, Ethiopia, April +26-30, 2020. +Mengjie Zhao, Tao Lin, Fei Mi, Martin Jaggi, and Hin- +rich Schütze. 2020. +Masking as an efficient alter- +native to finetuning for pretrained language models. +In Proceedings of the 2020 Conference on Empirical +Methods in Natural Language Processing (EMNLP), +pages 2226–2241, Online. Association for Computa- +tional Linguistics. +Barret Zoph and Quoc V. Le. 2017. Neural architec- +ture search with reinforcement learning. In 5th Inter- +national Conference on Learning Representations, +ICLR 2017, Toulon, France, April 24-26, 2017, Con- +ference Track Proceedings. +A +Supplemental Material: Technical +Details +PEFT Modules: Architectures and Setup. +We +implement the serial adapter architecture (SA) fol- +lowing the setup of Pfeiffer et al. (2020b). The +parallel adapter (PA) architecture is the same as the +one proposed by He et al. (2022), where a scaling +factor of 4 is implemented in all PA experiments. +The prefix-tuning (PT) architecture has an interme- +diate MLP with a bottleneck size of 800, which is +trained the same way as in the original wor (Li and +Liang, 2021). We also use the default setting for +LoRA with a scaling of 8 and a rank of 8. We re- +produce the experimental results with the reported +setup of the MAM adapter He et al. (2022) and +UniPELT (Mao et al., 2022). We reproduce the +AdaMix results with the reported hyperparameter +setup from the original work (Wang et al., 2022) +in 20 epochs. In the experiments of Figure 4, we +control the bottleneck size DSA and DPA for SA +and PA baselines, respectively, while keeping other +setups unchanged to discover their performance +across the parameter budget. Similarly, we control +the prefix length LPT for prefix-tuning and the rank +r of LoRA without changing other setups. +Training +Details. +Following previous work +(Pfeiffer et al., 2020b), we use a recommended +learning rate of 1e-4 for all PEFT experiments. +In RoBERTalarge experiments, we report the +RTE results with a learning rate of 2e-5 for +AUTOPEFTMRPC and AUTOPEFTCoLA; 1e-4 for +AUTOPEFTRTE. We use the learning rate of 2e- +5 for full model FT according to Mao et al. (2022). +We use the batch size of 32 and 16 for all BERT and +RoBERTa experiments, respectively. The optimiser +settings for each PEFT module follow the default +settings in AdapterHub (Pfeiffer et al., 2020a). +AUTOPEFT Search Setup. +We implement the +BO algorithm in BoTorch (Balandat et al., 2020). +We use the Matern 5/2 kernel as the covariance +function, and for the Monte Carlo sampling settings +of SAAS-BO (Eriksson and Jankowiak, 2021), we +use a warm-up step of 256, the number of samples +to retain as 128, and thinning as 16. For the opti- +misation of the acquisition function, to adapt to the +discrete setup, we use a local search method sim- +ilar to previous literature involving similar setup +(Wan et al., 2021; Eriksson et al., 2021): at each +search iteration (after the initial randomly sampled +points), we collect the Pareto-optimal architectures +up to this point. From this collection of Pareto- +optimal architectures, we perform a local search by +evaluating the acquisition function values of their +neighbours, and move the current point to a neigh- +bour with a higher acquisition function value and +this process is repeated until convergence (which is +a local minimum in terms of acquisition function), +or 100 evaluations in acquisition function value are +reached. At each search iteration, we restart this +process 10 times and select the top candidate for +the query (in this case, fine-tuning) for the next +iteration. For all BO experiments, we use 200 total +evaluations; given the noisy nature of the problem, +we use a relatively large number of random initiali- +sation points (100) to ensure that the search results + +10 +2 +10 +1 +100 +Fine-tuned Parameters (%) +87.0 +87.5 +88.0 +88.5 +89.0 +89.5 +Task Score +STS-B +10 +2 +10 +1 +100 +Fine-tuned Parameters (%) +50.0 +52.5 +55.0 +57.5 +60.0 +62.5 + +CoLA +Serial +Parallel +Prefix +LoRA +AutoPEFT +Figure 7: The Pareto front of the AUTOPEFT frame- +work on tasks STS-B and CoLA compared to baselines +with BERTbase in various settings of parameter budgets. +We report the single-seed task score for each task fol- +lowing the settings in Table 1. +are not overly sensitive to initialisation. We use the +same hyperparameter settings as described for all +experiments conducted in this paper. +Calculation of Fine-tuned Parameters. +The +uncased BERTbase model (109M) has 12 Trans- +former layers with a hidden dimension size of +768. The uncased BERTlarge model (335M) and +RoBERTalarge (355M) both have 24 layers with a +hidden dimension size of 1, 024. For both SA and +PA, their fine-tuned parameters are computed by +2 × Dadapter × Dh × |l|, where Dh represents the +corresponding hidden dimension of the selected +model, and |l| refers to the total selected number +of insertion layers. Similarly, we calculate the fine- +tuned parameters of PT by 2 × LPT × Dh × |l|. +Thus, the number of fine-tuned parameters of the +AUTOPEFT-found configurations is a summation +of individual PEFT modules’ parameters. We re- +port the default fine-tuned parameters for the re- +maining PEFT modules as defined in their original +papers. +B +Search Space and Discovered +Architectures +Impact of Single PEFT Modules within AU- +TOPEFT and Other Side Analyses. +We pro- +vide a more detailed analysis of the behaviour +of AUTOPEFT by inspecting the Pareto front of +AUTOPEFT-found configurations when we ablate +each PEFT module into the search space, as plot- +ted in Figure 6. After combining the serial adapter +with the parallel adapter, the upper bound of perfor- +mance is improved by more than 1%. We consider +the gain here leverages the capacity of multiple het- +erogeneous PEFT modules as a mixture-of-experts +while providing a more efficient adaptation by up- +dating both bias-influenced hidden states and the +original states according to Eq. 3. We recall that +prefix-tuning stabilises its learning with an interme- +diate reparametrization network, which is dropped +in the inference stage. Therefore, at the cost of +the increased training parameters, prefix-tuning +is one of the most parameter-efficient approaches. +Consequently, we notice that incorporating prefix- +tuning into the search space further improves the +overall parameter efficiency (4% to 1.4%) of the +AUTOPEFT-found configuration. Due to the pa- +rameter efficiency of each single PEFT module, +it also explains the distribution of the parameter +budget for each PEFT module in the learned con- +figurations. We also analyse the learned config- +urations in terms of the selected layers over dif- +ferent parameter scales in Table 5. They show a +common trend in selecting the higher Transformer +layers to insert the PEFT modules, which coincides +with previous findings that the higher layer con- +tains richer task-specific representations, and intro- +ducing PEFT modules to these layers is more effi- +cient than other layers. With the AUTOPEFT-found +configurations reported in Table 5, we hope future +PEFT research and applications can benefit from +the architecture design similar to AUTOPEFTRTE +M +that we find the most transferable across tasks. + +Method +#Param. +RTE +MRPC +STS-B +CoLA +SST-2 +QNLI +Avg. +Fine-tune† +100% +86.6 +90.9 +92.4 +68.0 +96.4 +94.7 +88.2 +LoRA‡ +0.22% +85.2 +90.2 +92.3 +68.2 +96.2 +94.8 +87.8 +Serial +0.89% +84.8 +90.2 +92.0 +66.8 +96.3 +94.7 +87.5 +AUTOPEFTRTE +S +0.03% +88.1 +89.5 +92.3 +62.7 +96.0 +94.6 +87.2 +AUTOPEFTMRPC +S +0.25% +86.6 +92.2 +92.2 +66.6 +96.2 +94.6 +88.1 +AUTOPEFTCoLA +M +2.36% +85.9 +90.0 +91.8 +70.6 +96.8 +94.6 +88.3 +AUTOPEFTRTE +L +9.41% +89.5 +88.5 +91.6 +65.6 +95.9 +94.6 +87.6 +AUTOPEFTtask +Avg. +0.88% +88.1 +92.2 +92.4 +70.6 +96.8 +94.6 +89.1 +Table 3: Experimental results on the GLUE benchmark with RoBERTalarge. We report the full model fine-tuning† +results from Liu et al. (2019b) with Pearson correlation for STS-B and Matthew’s correlation for CoLA. We +include the LoRA‡ module performance from Hu et al. (2022a). We report single-seed results for the experiments +and exclude QQP and MNLI tasks due to the large computation cost of RoBERTalarge. Similar to Table 1, we +conduct per-task search experiments on RTE, MRPC, STS-B, and CoLA, transferring best-found configurations +to the remaining tasks. In addition to the transfer experiment from RTE, we also report transfer performance +from MRPC and CoLA tasks with significantly different parameter budgets. All reported results are from the +configurations listed in Table 7. The best, second-best, and third-best results are marked in bold fonts and ranked +by colour. +Model +Insertion Layer {li} +Module +Size +BERTbase +1, 2, 3, 4, 5, 6, +7, 8, 9, 10, 11, 12 +Serial Adapter DSA +0, 1, 3, 6, 12, 24, 48, 96, 192, 384, 768 +Parallel Adapter DPA +0, 1, 3, 6, 12, 24, 48, 96, 192, 384, 768 +Prefix-Tuning LPT +0, 1, 3, 6, 12, 24, 48, 96, 192, 384, 768 +BERT/RoBERTalarge +1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, +13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 +Serial Adapter DSA +0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 +Parallel Adapter DPA +0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 +Prefix-Tuning LPT +0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 +Table 4: The search space of the AUTOPEFT. Each insertion layer has a Boolean decision for inserting the PEFT +modules. The 0 size of submodules indicates that we exclude the corresponding submodule from the configuration. +The total number of configurations for BERTbase: 212 × 11 × 11 × 11 ≈ 5 × 106 and for BERT/RoBERTalarge: +224 × 12 × 12 × 12 ≈ 3 × 1010. + +Task +#Param. +Search Space +Configuration +Submodule +Configuration +RTE +0.06% +Layer li +3, 4, 6, +8, 9, 11 +Serial Adapter DSA +3 +Parallel Adapter DPA +1 +Prefix-Tuning LPT +3 +RTE +1.42% +Layer li +2, 5, 6, +7, 8, 9, 10 +Serial Adapter DSA +96 +Parallel Adapter DPA +48 +Prefix-Tuning LPT +1 +RTE +6.60% +Layer li +3, 4, 6, +7, 8, 9, 10 +Serial Adapter DSA +384 +Parallel Adapter DPA +192 +Prefix-Tuning LPT +96 +MRPC +3.86% +Layer li +2, 3, 6, +7, 9, 10, 11 +Serial Adapter DSA +6 +Parallel Adapter DPA +384 +Prefix-Tuning LPT +3 +STS-B +1.06% +Layer li +2, 5, +7, 8, 9, 11 +Serial Adapter DSA +96 +Parallel Adapter DPA +6 +Prefix-Tuning LPT +24 +CoLA +0.29% +Layer li +3, 4, +8, 9, 10 +Serial Adapter DSA +12 +Parallel Adapter DPA +24 +Prefix-Tuning LPT +6 +MNLI +0.30% +Layer li +3, 6, +7, 8, 9, 11, 12 +Serial Adapter DSA +24 +Parallel Adapter DPA +6 +Prefix-Tuning LPT +1 +Table 5: The AUTOPEFT-found configurations reported in Table 1 using BERTbase. The average of fine-tuned +parameters (%) of AUTOPEFTtask +Avg. is calculated by (1.42+3.86+1.06+0.29+1.42+0.30+1.42+1.42)/8 = 1.40, +where we transfer the best-found AUTOPEFTRTE +M +to SST-2, QQP, and MNLI as their best per-task configurations +for achieving the best trade-off between task performance and efficiency. +Task +#Param. +Search Space +Configuration +Submodule +Configuration +RTE +0.78% +Layer li +2, 6, 8, 11, 14, 15, 16, 17, 21, 23 +Serial Adapter DSA +128 +Table 6: The AUTOPEFT-found configurations reported in Table 2 using BERTlarge. +Task +#Param. +Search Space +Configuration +Submodule +Configuration +RTE +0.03% +Layer li +6, 10, +14, 15, 18, 19, 21, 23 +Serial Adapter DSA +2 +Parallel Adapter DPA +4 +Prefix-Tuning LPT +1 +RTE +9.41% +Layer li +1, 2, 3, 4, 5, 7, 11, 12, +14, 15, 17, 19, 20, 21, 23 +Serial Adapter DSA +64 +Parallel Adapter DPA +1 +Prefix-Tuning LPT +1024 +MRPC +0.25% +Layer li +1, 2, 4, 5, 6, 8, 9, 10, 11, +13, 14, 16, 17, 21, 22, 23, 24 +Serial Adapter DSA +8 +Parallel Adapter DPA +2 +Prefix-Tuning LPT +16 +STS-B +0.25% +Layer li +1, 2, 4, 5, 6, 7, 8, 9, 10, 11, +13, 14, 16, 17, 21, 22, 24 +Serial Adapter DSA +8 +Parallel Adapter DPA +2 +Prefix-Tuning LPT +16 +CoLA +2.36% +Layer li +1, 5, 6, 8, 9, 10, +13, 14, 15, 19, 21, 22, 23, 24 +Serial Adapter DSA +256 +Parallel Adapter DPA +32 +Prefix-Tuning LPT +4 +Table 7: The AUTOPEFT-found configurations reported in Table 3 using RoBERTalarge. The average of fine-tuned +parameters (%) of AUTOPEFTtask +Avg. is calculated by (0.03 + 0.25 + 0.25 + 2.36 + 2.36 + 0.03)/6 = 0.88, where we +transfer the best-found AUTOPEFTCoLA +M +to SST-2 and AUTOPEFTRTE +S +to QNLI as their best per-task configurations +for achieving the best trade-off between performance and efficiency. + diff --git a/0NFLT4oBgHgl3EQfoi-S/content/tmp_files/load_file.txt b/0NFLT4oBgHgl3EQfoi-S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b677f6fed58c402042d8910cc3b05af8cd32b971 --- /dev/null +++ b/0NFLT4oBgHgl3EQfoi-S/content/tmp_files/load_file.txt @@ -0,0 +1,1213 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf,len=1212 +page_content='AUTOPEFT: Automatic Configuration Search for Parameter-Efficient Fine-Tuning Han Zhou1,* Xingchen Wan2,* Ivan Vuli´c1 Anna Korhonen1 1Language Technology Lab, University of Cambridge 2Machine Learning Research Group, University of Oxford {hz416, iv250, alk23}@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='uk xwan@robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='uk Abstract Large pretrained language models have been widely used in downstream NLP tasks via task- specific fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Recently, an array of Parameter-Efficient Fine-Tuning (PEFT) meth- ods have also achieved strong task perfor- mance while updating a much smaller num- ber of parameters compared to full model tun- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' However, it is non-trivial to make in- formed per-task design choices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', to create PEFT configurations) concerning the selection of PEFT architectures and modules, the num- ber of tunable parameters, and even the lay- ers in which the PEFT modules are inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Consequently, it is highly likely that the cur- rent, manually set PEFT configurations might be suboptimal for many tasks from the perspec- tive of the performance-to-efficiency trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' To address the core question of the PEFT con- figuration selection that aims to control and maximise the balance between performance and parameter efficiency, we first define a rich configuration search space spanning multiple representative PEFT modules along with finer- grained configuration decisions over the mod- ules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', parameter budget, insertion layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We then propose AUTOPEFT, a novel frame- work to traverse this configuration space: it automatically configures multiple PEFT mod- ules via high-dimensional Bayesian optimisa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We show the resource scalability and task transferability of AUTOPEFT-found configu- rations, outperforming existing PEFT methods on average on the standard GLUE benchmark while conducting the configuration search on a single task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The per-task AUTOPEFT-based configuration search even outperforms full- model tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 1 Introduction and Motivation Pretrained language models (PLM) are used in downstream tasks via the standard transfer learning Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Code is available at https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='com/cambridgeltl/autopeft 100 101 Fine-tuned Parameters (%) 80 81 82 83 84 Average Score Pfeiffer UniPELT MAM AdaMix Prefix LoRA Parallel AutoPEFT Full Model FT Figure 1: The performance of AUTOPEFT-found PEFT configurations compared to other standard PEFT methods and full model FT on the GLUE bench- mark (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the average score for each method by taking the mean of metrics for 8 GLUE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The dashed horizontal bar (Full Model FT) indicates the full-model FT that updates 100% of parameters, and our approach aims to learn the best trade-off configuration between task performance and parameter efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' paradigm, where they get fine-tuned for particu- lar tasks (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' This achieves state-of-the-art results in a wide spec- trum of NLP tasks, becoming a prevalent modelling paradigm in NLP (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Fine-tuning the PLMs typically requires a full update of their original parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', the so-called full-model fine-tuning (FT));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' however, this is (i) computation- ally expensive and also (ii) storage-wise expensive as it requires saving a separate full model copy for each task-tuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' With the ever-growing size of the PLMs (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022), the cost of full model FT becomes a major bottleneck, due to its increasing demands as well as computational (time and space) non-efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Parameter-Efficient Fine-Tuning (PEFT) deliv- ers a solution for alleviating the issues with full- model FT (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' By freezing the majority of pretrained weights of PLMs, PEFT ap- proaches only update a small portion of parameters arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='12132v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='CL] 28 Jan 2023 for efficiently adapting the PLM to a new down- stream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Recent studies have shown that PEFT can achieve competitive task performance while be- ing modular, adaptable, and preventing catastrophic forgetting in comparison to traditional FT (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Recent developments have created diverse PEFT modules with distinctive characteristics (Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Li and Liang, 2021), with one of the two main aims in focus: 1) improve task perfor- mance over other PEFT approaches while maintain- ing the same parameter budget as the competitor PEFT methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' or 2) maintain task performance while reducing the parameter budget needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Exist- ing PEFT modules, optimising for one of the two aims, have been successfully applied to transfer learning tasks (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' However, different tasks, with different complexity, show distinct sensitivity to the allo- cated parameter budget and even to the chosen PEFT approach (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' At the same time, most PEFT applications are limited to a sin- gle PEFT architecture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', serial adapters, prefix- tuning) with fixed decisions on its components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', hidden size dimensionality, insertion layers) result- ing in potentially suboptimal PEFT configurations across many tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Therefore, in this work, we propose a new, versatile and unified framework that automatically searches for improved and task- adapted PEFT configurations, aiming to effectively balance between the two (often colliding goals) of (i) improving performance and (ii) keeping the desired low parameter budget for PEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' While recent research has started exploring more dynamic PEFT configurations, the prior studies remain limited across several dimensions, includ- ing how they define the configuration search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Namely, they typically focus only on a single PEFT architecture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', adapters) or their simple combi- nations, or a single property (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', insertion layers – where to insert the module);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' see a short overview later in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Here, we propose a unified and more comprehensive framework for improved configu- ration search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' It covers multiple standard PEFT modules (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' serial adapters, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' parallel adapters, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' prefix-tuning), combined with the critical pa- rameter budget-related decisions: the size of each constituent module and the insertion layers for the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Our defined comprehensive search space is huge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' as a consequence, traversing it effectively and effi- ciently is extremely challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' To enable search over the large configuration space, we thus propose the AUTOPEFT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' It automatically con- figures multiple PEFT modules along with their efficiency-oriented design decisions, relying on a high-dimensional Bayesian optimisation (BO) ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Crucially, within the search space, we pro- pose a multi-objective optimisation which learns to simultaneously balance between maximising the searched configurations’ task performance and pa- rameter efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We conduct extensive experiments on the stan- dard GLUE benchmark (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We first study the transferability of the AUTOPEFT- searched architecture by running AUTOPEFT on a single task, followed by transferring the found ar- chitecture to other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Experimental results show that this architecture can outperform existing PEFT baselines while achieving on-par performance to the standard full-model FT, relying only on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4% of the original trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Further slight gains can be achieved via a computationally more expensive approach, where we run AUTOPEFT per each single task to find a task-adapted PEFT config- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' As demonstrated in Figure 1, AUTOPEFT is able to find configurations that offer a solid trade- off between task performance and parameter effi- ciency, even outperforming full-model FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We also provide ablation studies over the search space, vali- dating that the AUTOPEFT framework is versatile and portable to different search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 1) We propose a large and com- prehensive search space of PEFT configurations, which integrates three representative PEFT mod- ules, the tunable number of parameters of each module, and the binary decisions concerning Trans- former layers for inserting these modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2) We propose a novel AUTOPEFT framework with high- dimensional Bayesian optimisation that can auto- matically and feasibly search for the effective PEFT configuration in terms of both task performance and parameter efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 3) We demonstrate that the AUTOPEFT-found configurations can not only reduce the parameter budget but also outperform existing PEFT modules while being transferable across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The AUTOPEFT framework can also be easily extended to other and new PEFT modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2 Related Work Parameter-Efficient Fine-Tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Standard PEFT methods can be divided into two main groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 1) Some methods fine-tune a small por- tion of pretrained parameters (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For instance, Ben Zaken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022) propose to fine-tune the PLM’s bias terms, while Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2021) and Ansell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022) fine-tune sparse subnetworks withing the original PLM for a particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2) Other methods fine- tune an additional set of parameters (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Since there is no interference with the pre- trained parameters, this class of PEFT modules, be- sides offering strong task performance, is arguably more modular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' we thus focus on this class of PEFT methods in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The original adapter mod- ules (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020b) have a bottleneck serial architecture which can be inserted into every Transformer layer, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' LoRA (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022a) assumes the low-rank intrinsic dimensionality of the target task and per- forms low-rank updates (Mahabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Li and Liang (2021) propose the Prefix-Tuning method that appends a learnable vector to the at- tention heads at each Transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Similarly, prompt-tuning (Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021) only appends this vector to the input embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' UniPELT (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022) integrates multiple PEFT modules with a dynamic gating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022) provide a unified formulation of existing PEFT modules and propose a parallel adapter module, along with a combined ‘Mix-and-Match Adapter (MAM)’ architecture that blends parallel adapters and prefix-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022) propose the mixture-of-adaptations (AdaMix) combined archi- tecture that leverages weight averaging for a mix- ture of adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Optimising Parameter Efficiency in PEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Re- cent work further aims to optimise the parameter efficiency of existing PEFT modules while main- taining task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The standard approach is to insert (typically serial) adapters into all Trans- former layers, which still requires a sizeable pa- rameter budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Rücklé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2021) address this question by performing random dropout of adapters from lower-level layers, displaying only a small de- crease in task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Adaptable Adapters (AA) (Moosavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022) generalise this idea by learning gates that switch on or off adapters in particular Transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Neural Architec- ture Search (NAS) methods aim to automate the design of neural net architectures themselves, and NAS has seen great advances recently, with per- formance often surpassing human expert-designed architectures in various tasks (Zoph and Le, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Elsken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Concerning NLP tasks and PEFT, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022b) propose S3PET, which adapts Differentiable Architecture Search (DARTS) (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019a) to learn the po- sitions for inserting the PEFT modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' This work is closest in spirit to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Our method, discussed in detail in §3, offers a spectrum of advantages over S3PET and other related PEFT work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Relying on multi-objective optimisation, unlike S3PET, we can automatically discover a family of configurations at different pa- rameter efficiency levels in a single search run, ef- fectively balancing between task performance and parameter efficiency, without the need to set the ‘parameter budget’ in advance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' similarly, we en- able an automatic search over multiple constituent modules over the desirable range of parameter bud- get and effective layers, whereas previous work can only support one architecture per each search run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Further, previous work indicated that weight- sharing NAS such as DARTS may suffer with the reliability of prediction (White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021b), and its success often hinges heavily on the design of the actual search space (Li and Talwalkar, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Dong and Yang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We mitigate those issues with our design of AUTOPEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Finally, while weight-sharing NAS is arguably more computationally efficient, through combining the use of low-fidelity performance pre- dictors and the strong transferability of the configu- rations found across tasks, AUTOPEFT can also be made very computationally efficient in discovering effective PEFT configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We further discuss this in §3 and demonstrate empirically in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 3 AUTOPEFT Framework We start by designing a large configuration space, providing the motivation behind each decision to include a particular module and its components into the configuration space, along with a mathe- matical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We then propose AUTOPEFT, a novel framework to search over this challenging configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' It automatically configures (components of) multiple PEFT modules via high- dimensional Bayesian optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' PEFT Configuration Search Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The search space is an influential factor in the performance of any search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In order to simultane- ously maximise task performance along with pa- rameter efficiency, it is necessary to first define a ‘parameter-reducible’ search space, where each di- mension within the space potentially contributes to reducing the parameter budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Similarly, each dimension might potentially bring positive impact to the task performance without introducing redun- dancy in the space (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Therefore, we propose the search space with representative PEFT modules, as follows, spanning a plethora of (non-redundant) configurations, as also shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' PEFT Modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We include three distinctive PEFT designs to efficiently adapt different forwarding stages of hidden states in the PLM layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We combine Serial Adapters (SA), Parallel Adapters (PA), and Prefix-Tuning (PT) as the three represen- tative modules in the search space, where the PT module adapts the multi-head attention layer, and SA and PA interact with the FFN layer (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Each configuration makes a decision on the PEFT modules in the insertion layer: all of them can be ‘turned’ on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We combine this binary decision with the actual non-binary decision on the module size (see next), so that the value of 0 in fact denotes the absence of the modules in the layer(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Previous studies show that PEFT methods are highly sensitive to the number of tunable pa- rameters: adaptively setting their capacity in ac- cordance with the target task is then essential for achieving good performance (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The number of tunable parameters is dependent on each particular module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The additional parameters introduced by both SA and PA are dominated by their bottleneck dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Similarly, the size of the PT module is defined by its prefix length LPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Thus, we define a binary logarithmic search scale for the respective discrete sets DSA, DPA, and LPT, spanning the values from 0 (absence of the module) to Dh where Dh is the dimensionality of the PLM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', Dh=768 for BERTbase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Insertion Layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Prior work has also shown that different layers in the PLMs store different se- mantic information (Vuli´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020), where the higher layers produce more task-specific and con- textualized representations (Tenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Therefore, as another configuration dimension, we aim to search for the minimal number and the ac- tual position of layers in which to insert the PEFT modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We define a binary ‘insertion’ decision at each layer li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Combining PEFT Modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The SA module and the PA module share a bottleneck architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The Feed Forward LayerNorm Multi-Head Attention LayerNorm Prefix-Tuning Serial Parallel PEFT Layer PEFT Layer Layer PEFT Layer Layer PEFT Layer Trainable Search Frozen Figure 2: Illustration of the main components of our configuration search space, traversed via AUTOPEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AUTOPEFT configures the selected Transformer layers with PEFT modules, where the activation of each sub- module is controlled by the learned size of each sub- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' See also Table 4 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' SA receives hidden states from the FFN output as its inputs, adapting it with a down-projection ma- trix W down SA ∈ RDh×DSA, followed by a non-linear activation function, and then an up-projection ma- trix W up SA ∈ RDSA×Dh: fSA(h) = ReLU(hW down SA )W up SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (1) PA, on the other hand, receives its inputs from hidden states before the FFN layer with the same formulation: fPA(x) = ReLU(xW down PA )W up PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2) Therefore, it is able to act in parallel with the SA without interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Note that the FFN hidden states h = F(x) contain the task-specific bias learned in its pretrained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Therefore, by combining SA with PA, the following composition of functions is achieved: fSAPA(x) =ReLU(F(x)W down SA )W up SA +ReLU(xW down PA )W up PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (3) The final composition should provide an effective adaptation to both bias-influence hidden states and the original inputs before the pretrained FFN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='1 Further, applying PEFT modules to interact both with FFNs and multi-head attention should have a positive impact on task performance (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 1The PA module also acts as the low-rank reparametriza- tion of the learned SA together with the frozen FFN layer to further match the intrinsic dimensionality of the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Query new configuration by Suggested config in AutoPEFT search space Performance & Parameter Efficiency Evaluate Parameter-Efficient Fine-Tuning Multi-Objective Bayesian Optimisation Serial Parallel Prefix Config Layer Configuration maximising the acquisition function GP Surrogate Figure 3: Illustration of the AUTOPEFT framework: to search for optimal architectures in the defined configu- ration space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AUTOPEFT uses a multi-objective BO agent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' which trains on previous observations of the PEFT configuration vector and its performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', accuracy – obtained by fine-tuning the language model with the PEFT configuration) and cost (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', number of parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The BO agent then suggests new configurations, and the algorithm continues iteratively until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' PT learns two prefix vectors, Pk and Pv ∈ RLPT×Dh, that are concatenated with the original multi-head attention’s key and value vectors, which efficiently adapts the multi-head attention layer to fit the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We thus finally combine the SA and the PA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', SAPA from above) with PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In sum, the overview of the dimensions spanning the final configuration space is provided in Figure 2 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The combination of the different ‘con- figuration dimensions’ outlined above gives rise to a total of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 5,451,776 possible configurations with BERTbase and ∼ 3×1010 configurations with RoBERTalarge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', the number of configurations is 2|l|×|DSA|×|DPA|×|LPT|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' While a large search space is crucial for expressiveness and to ensure that good-performing configurations are contained, it also increases the difficulty for search strategies to both navigate the search space well while re- maining sample- and thus computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Furthermore, in the PEFT setting, we are also often interested in discovering a family of configurations that trade off between performance and efficiency for general application in various scenarios with different resource constraints, thus giving rise to a multi-objective optimisation problem where we si- multaneously aim to maximise performance while minimising costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In what follows, we propose a search framework that satisfies all those criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AUTOPEFT via Multi-Objective Bayesian Opti- misation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Formally, denoting the full AUTOPEFT search space as A and a single configuration a ∈ A with trainable weights W, without loss of gener- ality, assuming our objective is to maximise (i) a performance metric f(a, W) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', the accuracy on the dev set) and to (ii) minimise a cost metric g(a) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', the number of parameters in a), a search method aims to solve the bi-level, bi-objective op- timisation problem: max a∈A � f(a, W ∗), −g(a) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='W ∗ = arg min W Ltrain(a, W), (4) where the inner loop optimisation problem is the op- timisation of the configuration weights achieved by fine-tuning the configuration a itself over the train loss Ltrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Given the bi-objective nature of the problem, there is in general no single maximiser of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (4) but a set of non-dominated Pareto-optimal configurations A∗ = {a∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', a∗ |A∗|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' To address these challenges in this work, we adopt a Bayesian optimisation (BO) approach, il- lustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' BO is a sample-efficient, zeroth-order model-based sequential optimisation algorithm (Garnett, 2023) with proven successes in NAS and automated machine learning in gen- eral (Snoek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Kandasamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' BO is partic- ularly popular in the multi-objective setups where one is interested in recovering a Pareto front where it is less straightforward to apply methods such as differentiable / one-shot architecture search meth- ods that are typically used to discover a single best- performing configuration (Eriksson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Izquierdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' BO consists of a surro- gate model, usually a Gaussian Process (GP) that sequentially approximates the objective function based on the observations so far, and an acquisition function, which balances between exploitation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', regions in the search space with high perceived Mateérn kernel Samplesfrompriordistribution 3 1 0 1 2 Sampled function #1 Sampled function #2 Sampled function #3 Sampled function #4 Samplesfromposteriordistribution Sampled function #5 3 Mean ± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2 Observations 1 1 2 3 0 2 3 5value) and exploration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', regions that have not been visited before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' It is optimised at each iter- ation to actively select the next configuration to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For a detailed overview of BO, we refer the readers to Frazier (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' While vanilla BO methods are better-suited in modestly-dimensioned and continuous prob- lems, our current setup instead features a high- dimensional and combinatorial search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Here, performance of non-parametric methods such as GP-based BO tend to suffer due to the exponen- tially exploding volume of space the surrogate needs to model as dimensionality increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For- tunately, recent advances in search methods have allowed us to address these challenges effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Specifically, we adopt the SAAS-GP (Eriksson and Jankowiak, 2021) model as the surrogate function: on a high level, SAAS-GP (1) places a relatively strong regularising half-Cauchy prior on the model lengthscales (which dictate the perceived impor- tance of search dimensions to the objective func- tion value) to induce sparsity and (2) approximately marginalises over model hyperparameters via a No-U-Turn Monte Carlo sampler (Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2014) to reduce overfitting in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We argue that both are appealing in our setup, while the benefit of (2) in our setup is self-evident, (1) also ef- fectively places a prior to encode our belief that in spite of the high nominal complexity search space, the effective dimensionality of the problem should be much lower – this is appropriate in our setup, as although we have a nominally high dimensions, consistent to previous findings in NAS (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022), we do expect a few disproportionately in- fluential key dimensions (although we do not have information on which a priori – this is meant to be discovered by the BO algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For the acquisition function, we use the noisy expected hypervolume improvement (NE- HVI) (Daulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021), which is suitable for the setup described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Lastly, while BO is sample-efficient, it may still require 100-200 eval- uations of different configurations in the search space to sufficiently explore the search space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' to make sure the search remains cost-efficient, during search we also adopt low-fidelity approximations commonly employed in NAS: at the search stage, for a configuration a, instead of evaluating the ob- jective f(a, W) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 4 in full, we only fine-tune the a using a smaller computational bud- get – for example, if a complete fine-tuning takes 100% of training data, at search time we are able to only fine-tune with 1% of training data and use the accuracy after that as a lower-cost proxy to the accuracy after full-length FT, the latter of which is significantly more expensive to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Therefore, when we are facing high-resource tasks, fine-tuning the full training resources is only performed once at evaluation time after the Pareto-optimal configu- rations are finalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Other low-cost proxies such as training for fewer number of epochs than full FT are also compatible but not used in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 4 Experimental Setup Evaluation Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We follow prior PEFT research and base our evaluation on the standard GLUE benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We include 4 types of text classi- fication tasks, including linguistic acceptability: CoLA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' similarity and paraphrase: STS-B, MRPC, QQP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' sentiment analysis: SST-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' natural language inference: RTE, QNLI, MNLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We exclude WNLI following previous work (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We compare the performance of the AUTOPEFT-found configurations to the standard full model FT and each individual PEFT module (SA, PA, PT) from the search space used in their default setup from respective original work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We also compare with the LoRA module, to provide a comparison to low-rank decomposition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In order to provide comparisons with recently pro- posed methods that also integrate multiple PEFT modules (see §2), we further include the UniPELT and the MAM adapter in their default settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We reproduce AdaMix for a comparison to a mixture of homogeneous adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In ablations on inser- tion layers, we also include the Adaptable Adapter (AA) as a baseline that proposes a differentiable gate learning method to select the insertion layer for PEFT modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', serial adapters originally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Following previous work on the GLUE benchmark, we report the best GLUE dev set performance (Ben Zaken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022) and use 20 training epochs with an early stopping scheme of 10 epochs for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We use AdapterHub (Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020a) as the code- base and conduct extensive experiments with the uncased BERTbase (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019) as the main backbone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report main experiments with the mean and standard deviation over 5 dif- ferent random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Experimental results using Method #Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' RTE MRPC STS-B CoLA SST-2 QNLI QQP MNLI Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Fine-tune 100% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='121.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='75 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='28 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='12 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='790.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='06 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='14 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='14 AUTOPEFTtask Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='40% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='94 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='87 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='00 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='47 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='25 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='13 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='05 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='10 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='47 Table 1: Results on the GLUE benchmark with BERTbase, where tasks are ordered in ascending order of the training resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We conduct three groups of task transferability experiments on RTE and one resource scalability experiment on MNLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the average fine-tuned parameters of per-task AUTOPEFT, where we conduct additional per-task searches on MRPC, STS-B, and CoLA, and take best-found configurations for the remaining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report Spearman’s Correlation for STS-B, Matthew’s Correlation for CoLA, and accuracy for all other tasks, where we report the matched accuracy for MNLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The percentage of parameters is computed as a ratio of the number of additional parameters to the pretrained parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We reproduce all baselines and report the mean and standard deviation of all results for 5 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The best, second-best, and third-best results are marked in bold fonts and ranked by colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' RoBERTalarge (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2019b) show findings that are consistent to the ones BERTbase, and are included in Table 3 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the setup for each PEFT module and the detailed training scheme in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 5 Results and Discussion Transferability of Configurations across Tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The main results are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' First, we analyze task transferability of AUTOPEFT- found configurations by running AUTOPEFT on the most low-resource and challenging task, RTE, followed by transferring the three best AUTOPEFT- found configurations to other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' First, we note that the parameter budget of the configura- tion AUTOPEFTRTE M is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='42%, while it shows considerable average gains over all the PEFT base- lines on the RTE task, by a margin of at least 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The AUTOPEFT-found configuration also outper- forms the full-model FT baseline on the RTE task by more than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' These results indicate the ef- fectiveness of the AUTOPEFT framework in opti- mising both task performance and parameter effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Transferring the RTE-based configurations to other tasks, we find that strong performance is maintained across the target tasks, with more bene- fits on the medium-resource tasks (MRPC, STS-B, CoLA), but the configuration remains competitive also for higher-resource tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', QQP, MNLI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 10 2 10 1 100 Fine-tuned Parameters (%) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 Task Score RTE 10 2 10 1 100 Fine-tuned Parameters (%) 75 80 85 MRPC Serial Parallel Prefix LoRA AutoPEFT Figure 4: The Pareto front of the AUTOPEFT on tasks RTE and MRPC compared to baselines with BERTbase in various settings of parameter budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the single-seed task score for each task following the settings in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The plots for STS-B, and CoLA, showing the same trends, are in Appendix §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' When we assign a large parameter budget to the potential configurations, AUTOPEFTRTE L also shows a stronger transfer performance in high- resource tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' This indicates that, as expected, the parameter capacity of the configuration is an important factor in transfer learning (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' On average, the AUTOPEFTRTE M configura- tion shows a comparable fine-tuning performance, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='19, to the full model FT, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='15, by only updating 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='42% of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' With strong transferability across similar tasks, AUTOPEFT provides distinct advantages in parameter efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' the search al- gorithm itself coupled with transfer becomes more sample-efficient within limited training resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Resource Scalability and Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We next ‘stress-test’ the ability of AUTOPEFT in a more challenging scenario with limited task training data, carrying out an experiment on the most high- resource MNLI task using only a small set of its training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We randomly sample 1% of the orig- inal MNLI training data to train AUTOPEFT, and retain using the original dev set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 We report AUTOPEFTMNLI S in Table 1 as the best- found configuration in this low-resource setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' It requires only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='30% of fine-tuned parameters and shows the strong MNLI performance of 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='92%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In another efficiency-oriented test, we conduct con- figuration transfer in a radically parameter-efficient setup (training on the full RTE training set but with reduced parameter budget, and then transferring to other tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AUTOPEFTRTE S in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The main finding is that, while performance does decrease slightly as expected, strong task performance can still be achieved even with the parameter budget of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='06% within this very efficient setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Per-Task Configuration Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Finally, we con- duct full-resource per-task AUTOPEFT searches, which naturally come with increased computational costs, for RTE, MRPC, STS-B, and CoLA, and then, for efficiency reasons, port the small set of best configurations to the remaining high-resource tasks: SST-2, QNLI, QQP, MNLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In addition to the peak score on RTE, we observe gains on MRPC (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='16% to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='45%) and CoLA (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='13% to 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='92%) over the best-performing PEFT base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We also observe gains over the transferred configuration AUTOPEFTRTE M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' One interpretation of the results is that AUTOPEFT is strong at match- ing the intrinsic dimensionality of the low-resource downstream task to the capacity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', parameter budget) of the PEFT modules, whereas full model FT performs better in high-resource scenarios, giv- ing the largest capacity to capture the informa- tion in high-resource tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='3 However, the per- task AUTOPEFTtask variant outperforms even full model FT by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='3% while its parameter budget is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4% of the full model per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Analysing the ‘Behaviour’ of Bayesian Optimi- 2With this setup, we effectively save 99% of training re- sources and the search framework becomes extremely fast even for high-resource datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 3Due to the richness of training resources in high-resource datasets, the results in these tasks are mostly saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Pre- vious work shows that PEFT methods can only reach on-par performance to full model FT on those tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 10 2 10 1 100 101 Fine-tuned Parameters (%) 60 65 70 75 Accuracy (%) Initialisation Random Search AutoPEFT Figure 5: The distribution of AUTOPEFT-found config- urations compared to the random search on RTE with a single random seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We initialise the AUTOPEFT search with 100 runs of random sampling for initial ex- plorations in the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We then conduct 100 runs of the AUTOPEFT with Bayesian optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' sation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Figure 5 shows the distribution of AU- TOPEFT-found configurations when we conduct its search experiment on RTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Due to the greedy nature of our predefined acquisition function, we enforce the initialisation of our algorithm with a wide exploration of potential configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In the subsequent AUTOPEFT runs, it starts exploit- ing the best-found configurations while optimising towards the region with improved parameter effi- ciency, whereas the random search baseline keeps obtaining inefficient configurations in a lottery- ticket manner in the expensive region of param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' It is observable that AUTOPEFT exploits the region with roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4% of parameters and finds configurations with further enhanced task perfor- mance from 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4% to 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='1% of accuracy, which is also the architecture AUTOPEFTRTE M with the strongest transferability across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We also in- clude the best-found architecture within the initiali- sation stage as the AUTOPEFTRTE L , and our trans- ferability experiments show that the AUTOPEFT- found architecture is more robust to the random initialisation of the neural network, outperforming the best random search baseline in the searched task by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='7% with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2% less parameter cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ablation of the Configuration Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' To pro- vide a finer-grained analysis of factors that bring positive impact to AUTOPEFT, we ablate the AU- TOPEFT search space from the full configuration space: 1) to the basic enumeration of the bottleneck size DSA of the SA only (the ‘SA’ space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We then include the Transformer layer and the SA size to- gether into the search space (the ‘SA-Layer’ space) to validate the usefulness of using layer selection as one configuration dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We can then also Method #Layers Size DSA RTE Accu- racy (%) Serial Adapter 24 64 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='76 Adaptable Adapter 13 128 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='80 AdapterDrop 13 128 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='40 AUTOPEFTSA Layer 10 128 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='94 Table 2: The results of AUTOPEFT to layer selection baselines with the same parameter budget on BERTlarge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the Pfeiffer adapter for all 24 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We in- clude the specialised AdapterDrop (Rücklé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021) that inserts SA for the last 13 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the AAuni architecture (Moosavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022) without its ra- tional activation function with 13 selected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We run our AUTOPEFT with the comparable search space of 24 layers and the size of the Pfeiffer adapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' expand the search space by adding another module (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', PA yields the ‘SA-PA-Layer’ space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Figure 6 plots the performance over the ‘ablated’ configura- tion spaces and over different parameter budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Several key findings emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' First, combining mul- tiple single PEFT modules has a positive impact on AUTOPEFT in general (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', full AUTOPEFT versus ’SA-PA-Layer’ versus ’SA-Layer’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Rely- ing on layer selection also brings benefits (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', ’SA’ versus ’SA-Layer’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The comparison also indicates that leaving out some Transformer layers while increasing the capacity of the PEFT module is a straightforward method to improve the parameter efficiency and task performance of the PEFT mod- ule within a fixed parameter budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Figure 6 sug- gests that AUTOPEFT can effectively operate over configuration spaces of different ‘granularity’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We analyse the impact of each single PEFT mod- ule in more detail in Appendix §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Layer Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' To further compare different layer selection approaches, we conduct a controlled experiment with the SA module on BERTlarge (24 Transformer layers) under a predefined parameter budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Table 2, the simple AdapterDrop ap- proach simply drops the adapters for the first 11 layers while doubling their bottleneck sizes, im- proving the RTE result by roughly 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Within the same architecture, we include the Adaptable Adapter with selected layers from switch learning, which has 3 and 10 layers from the first 12 and the other 12 layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We show that AU- TOPEFT outperforms all existing layer selection baselines by learning less activated adapter layers, leading to better parameter efficiency (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5% fewer parameters in relative terms) and higher task perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' It indicates that selecting the best insertion 0 2 4 6 Fine-tuned Parameters (%) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 Accuracy (%) SA SA-Layer SA-PA-Layer PA-PT-Layer AutoPEFT Figure 6: The performance of AUTOPEFT with ab- lation of search space on RTE with a single random seed on BERTbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The SA results refer to the Pfeiffer adapter (Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020b) with an enumeration of its bottleneck size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For other search spaces, we report the Pareto front of AUTOPEFT-found configurations, where SA-PA-PT-Layer forms the search space of AU- TOPEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' layer is non-trivial, and AUTOPEFT can learn the correlation between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 6 Conclusion We proposed AUTOPEFT, a novel search frame- work for automatically configuring various PEFT modules in selective layers of pretrained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AUTOPEFT searches the optimal architec- ture via Bayesian optimisation with iterative evalu- ation and predicting the desired architecture given the configuration search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The proposed multi- objective optimisation can produce a Pareto front of candidate architectures by simultaneously max- imising the model performance and parameter effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We demonstrated that AUTOPEFT-found architectures offer an effective trade-off between task performance and parameter efficiency, outper- forming a variety of PEFT baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Limitations The proposed AUTOPEFT method is relatively expensive since it requires iterative optimisation by learning to optimise each explored configura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' While all intermediate configurations can be skipped without laying a burden on the final storage space, the intermediate computation cost becomes the main bottleneck of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In this work, we alleviated this problem by (i) con- ducting the search with 1% of training resources for large datasets, and (ii) configuration transfer from low-resource tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The search itself can be seen as a one-time cost yielding a ‘permanent’ well- performing and shareable configuration for particu- lar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We plan to delve deeper into the related efficiency and computational tractability aspects in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We have conducted extensive experiments on the search space that contains three representative PEFT modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The AUTOPEFT framework is decoupled from the actual single PEFT modules: with further PEFT developments and new PEFT approaches, those may also get integrated into the AUTOPEFT framework in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Acknowledgements Xingchen Wan is supported by the Clarendon Scholarship at University of Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The work has been supported in part by a personal Royal So- ciety University Research Fellowship (no 221137;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022-) awarded to Ivan Vuli´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' References Alan Ansell, Edoardo Ponti, Anna Korhonen, and Ivan Vuli´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Composable sparse fine-tuning for cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the Association for Computa- tional Linguistics (Volume 1: Long Papers), pages 1778–1796, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Com- putational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Maximilian Balandat, Brian Karrer, Daniel Jiang, Samuel Daulton, Ben Letham, Andrew G Wilson, and Eytan Bakshy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Botorch: A framework for efficient monte-carlo bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ad- vances in neural information processing systems, 33:21524–21538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Elad Ben Zaken, Yoav Goldberg, and Shauli Ravfogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' BitFit: Simple parameter-efficient fine-tuning for transformer-based masked language-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the As- sociation for Computational Linguistics (Volume 2: Short Papers), pages 1–9, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Associa- tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Tom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam Mc- Candlish, Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Language models are few-shot learn- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Informa- tion Processing Systems 2020, NeurIPS 2020, De- cember 6-12, 2020, virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Guanzheng Chen, Fangyu Liu, Zaiqiao Meng, and Shangsong Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Revisiting parameter- efficient tuning: Are we really there yet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Pro- ceedings of the 2022 Conference on Empirical Meth- ods in Natural Language Processing, pages 2612– 2626, Abu Dhabi, United Arab Emirates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Associa- tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Shoufa Chen, Chongjian Ge, Zhan Tong, Jiangliu Wang, Yibing Song, Jue Wang, and Ping Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Adaptformer: Adapting vision transformers for scal- able visual recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural In- formation Processing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Samuel Daulton, Maximilian Balandat, and Eytan Bak- shy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Parallel bayesian optimization of mul- tiple noisy objectives with expected hypervolume improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 34: Annual Conference on Neu- ral Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pages 2187– 2200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' BERT: Pre-training of deep bidirectional transformers for language under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171–4186, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Xuanyi Dong and Yi Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Nas-bench-201: Ex- tending the scope of reproducible neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Thomas Elsken, Jan Hendrik Metzen, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Neural architecture search: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The Journal of Machine Learning Research, 20(1):1997– 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' David Eriksson, Pierce I-Jen Chuang, Samuel Daulton, Peng Xia, Akshat Shrivastava, Arun Babu, Shicong Zhao, Ahmed A Aly, Ganesh Venkatesh, and Max- imilian Balandat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Latency-aware neural ar- chitecture search with multi-objective bayesian op- timization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In 8th ICML Workshop on Automated Machine Learning (AutoML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' David Eriksson and Martin Jankowiak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' High- dimensional bayesian optimization with sparse axis- aligned subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Uncertainty in Artificial Intel- ligence, pages 493–503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Peter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Frazier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' A tutorial on bayesian optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' CoRR, abs/1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='02811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Roman Garnett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Bayesian Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Cam- bridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Demi Guo, Alexander Rush, and Yoon Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Parameter-efficient transfer learning with diff prun- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natu- ral Language Processing (Volume 1: Long Papers), pages 4884–4896, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Junxian He, Chunting Zhou, Xuezhe Ma, Taylor Berg- Kirkpatrick, and Graham Neubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Towards a unified view of parameter-efficient transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25- 29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Matthew D Hoffman, Andrew Gelman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The no-u-turn sampler: adaptively setting path lengths in hamiltonian monte carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 15(1):1593–1623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, Bruna Morrone, Quentin de Laroussilhe, Andrea Gesmundo, Mona Attariyan, and Sylvain Gelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Parameter-efficient transfer learning for NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, pages 2790–2799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Edward J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Lora: Low-rank adapta- tion of large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In The Tenth Inter- national Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Shengding Hu, Zhen Zhang, Ning Ding, Yadao Wang, Yasheng Wang, Zhiyuan Liu, and Maosong Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Sparse structure search for delta tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Sergio Izquierdo, Julia Guerrero-Viu, Sven Hauns, Guilherme Miotto, Simon Schrodi, André Biedenkapp, Thomas Elsken, Difan Deng, Marius Lindauer, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Bag of base- lines for multi-objective joint neural architecture search and hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In 8th ICML Workshop on Automated Machine Learning (AutoML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Kirthevasan Kandasamy, Willie Neiswanger, Jeff Schneider, Barnabás Póczos, and Eric P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Neural architecture search with bayesian optimisa- tion and optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 31: Annual Con- ference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Mon- tréal, Canada, pages 2020–2029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Brian Lester, Rami Al-Rfou, and Noah Constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The power of scale for parameter-efficient prompt tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 3045–3059, Online and Punta Cana, Domini- can Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computational Lin- guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Liam Li and Ameet Talwalkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Random search and reproducibility for neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the Thirty-Fifth Conference on Un- certainty in Artificial Intelligence, UAI 2019, Tel Aviv, Israel, July 22-25, 2019, pages 367–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Xiang Lisa Li and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Prefix-tuning: Optimizing continuous prompts for generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Lan- guage Processing (Volume 1: Long Papers), pages 4582–4597, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Hanxiao Liu, Karen Simonyan, and Yiming Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' DARTS: differentiable architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In 7th International Conference on Learning Repre- sentations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Haokun Liu, Derek Tam, Muqeeth Mohammed, Jay Mohta, Tenghao Huang, Mohit Bansal, and Colin Raffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Few-shot parameter-efficient fine- tuning is better and cheaper than in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Roberta: A robustly optimized BERT pretraining ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' CoRR, abs/1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='11692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Rabeeh Karimi Mahabadi, James Henderson, and Se- bastian Ruder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Compacter: Efficient low-rank hypercomplex adapter layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neu- ral Information Processing Systems 34: Annual Con- ference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pages 1022–1035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Yuning Mao, Lambert Mathias, Rui Hou, Amjad Alma- hairi, Hao Ma, Jiawei Han, Scott Yih, and Madian Khabsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' UniPELT: A unified framework for parameter-efficient language model tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Pro- ceedings of the 60th Annual Meeting of the Associa- tion for Computational Linguistics (Volume 1: Long Papers), pages 6253–6264, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Nafise Moosavi, Quentin Delfosse, Kristian Kersting, and Iryna Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Adaptable adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computa- tional Linguistics: Human Language Technologies, pages 3742–3753, Seattle, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Associa- tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Jonas Pfeiffer, Naman Goyal, Xi Lin, Xian Li, James Cross, Sebastian Riedel, and Mikel Artetxe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Lifting the curse of multilinguality by pre-training modular transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 3479–3495, Seattle, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computational Lin- guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Jonas Pfeiffer, Andreas Rücklé, Clifton Poth, Aish- warya Kamath, Ivan Vuli´c, Sebastian Ruder, Kyunghyun Cho, and Iryna Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AdapterHub: A framework for adapting transform- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Em- pirical Methods in Natural Language Processing: System Demonstrations, pages 46–54, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Asso- ciation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Jonas Pfeiffer, Ivan Vuli´c, Iryna Gurevych, and Se- bastian Ruder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' MAD-X: An Adapter-Based Framework for Multi-Task Cross-Lingual Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 7654–7673, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Exploring the limits of transfer learning with a unified text-to-text trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 21:140:1–140:67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Pengzhen Ren, Yun Xiao, Xiaojun Chang, Po-Yao Huang, Zhihui Li, Xiaojiang Chen, and Xin Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' A comprehensive survey of neural architec- ture search: Challenges and solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' ACM Com- puting Surveys (CSUR), 54(4):1–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Bin Xin Ru, Xingchen Wan, Xiaowen Dong, and Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Osborne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Interpretable neural architecture search via bayesian optimisation with weisfeiler-lehman kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In 9th International Con- ference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Robin Ru, Pedro M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Esperança, and Fabio Maria Car- lucci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Neural architecture generator optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Informa- tion Processing Systems 2020, NeurIPS 2020, De- cember 6-12, 2020, virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Andreas Rücklé, Gregor Geigle, Max Glockner, Tilman Beck, Jonas Pfeiffer, Nils Reimers, and Iryna Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AdapterDrop: On the efficiency of adapters in transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 7930–7946, Online and Punta Cana, Dominican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Victor Sanh, Albert Webson, Colin Raffel, Stephen Bach, Lintang Sutawika, Zaid Alyafeai, Antoine Chaffin, Arnaud Stiegler, Arun Raja, Manan Dey, M Saiful Bari, Canwen Xu, Urmish Thakker, Shanya Sharma Sharma, Eliza Szczechla, Taewoon Kim, Gunjan Chhablani, Nihal V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Nayak, De- bajyoti Datta, Jonathan Chang, Mike Tian-Jian Jiang, Han Wang, Matteo Manica, Sheng Shen, Zheng Xin Yong, Harshit Pandey, Rachel Bawden, Thomas Wang, Trishala Neeraj, Jos Rozen, Ab- heesht Sharma, Andrea Santilli, Thibault Févry, Ja- son Alan Fries, Ryan Teehan, Teven Le Scao, Stella Biderman, Leo Gao, Thomas Wolf, and Alexan- der M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Multitask prompted training enables zero-shot task generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In The Tenth International Conference on Learning Representa- tions, ICLR 2022, Virtual Event, April 25-29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Jasper Snoek, Hugo Larochelle, and Ryan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Practical bayesian optimization of machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Infor- mation Processing Systems 25: 26th Annual Con- ference on Neural Information Processing Systems 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Proceedings of a meeting held December 3- 6, 2012, Lake Tahoe, Nevada, United States, pages 2960–2968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Yi-Lin Sung, Varun Nair, and Colin Raffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Training neural networks with fixed sparse masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Sys- tems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pages 24193–24205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ian Tenney, Dipanjan Das, and Ellie Pavlick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' BERT rediscovers the classical NLP pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Asso- ciation for Computational Linguistics, pages 4593– 4601, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Ivan Vuli´c, Edoardo Maria Ponti, Robert Litschko, Goran Glavaš, and Anna Korhonen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Probing pretrained language models for lexical semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 7222–7240, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Xingchen Wan, Vu Nguyen, Huong Ha, Binxin Ru, Cong Lu, and Michael A Osborne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Think global and act local: Bayesian optimisation over high-dimensional categorical and mixed search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 10663–10674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Xingchen Wan, Binxin Ru, Pedro M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Esperança, and Zhenguo Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' On redundancy and diversity in cell-based neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In The Tenth International Conference on Learning Representa- tions, ICLR 2022, Virtual Event, April 25-29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Alex Wang, Amanpreet Singh, Julian Michael, Fe- lix Hill, Omer Levy, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' GLUE: A multi-task benchmark and analysis plat- form for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Pro- ceedings of the 2018 EMNLP Workshop Black- boxNLP: Analyzing and Interpreting Neural Net- works for NLP, pages 353–355, Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Yaqing Wang, Sahaj Agarwal, Subhabrata Mukherjee, Xiaodong Liu, Jing Gao, Ahmed Hassan Awadal- lah, and Jianfeng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Adamix: Mixture- of-adaptations for parameter-efficient model tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2022 Conference on Empiri- cal Methods in Natural Language Processing, pages 5744–5760, Abu Dhabi, United Arab Emirates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' As- sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Colin White, Willie Neiswanger, and Yash Savani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' BANANAS: bayesian optimization with neu- ral architectures for neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Thirty-Fifth AAAI Conference on Artificial Intelli- gence, AAAI 2021, Thirty-Third Conference on In- novative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Ad- vances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2-9, 2021, pages 10293–10301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Colin White, Arber Zela, Robin Ru, Yang Liu, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' How powerful are perfor- mance predictors in neural architecture search?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Pro- cessing Systems 2021, NeurIPS 2021, December 6- 14, 2021, virtual, pages 28454–28469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Antoine Yang, Pedro M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Esperança, and Fabio Maria Carlucci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' NAS evaluation is frustratingly hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In 8th International Conference on Learning Repre- sentations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Mengjie Zhao, Tao Lin, Fei Mi, Martin Jaggi, and Hin- rich Schütze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Masking as an efficient alter- native to finetuning for pretrained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 2226–2241, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Barret Zoph and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Neural architec- ture search with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In 5th Inter- national Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Con- ference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' A Supplemental Material: Technical Details PEFT Modules: Architectures and Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We implement the serial adapter architecture (SA) fol- lowing the setup of Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The parallel adapter (PA) architecture is the same as the one proposed by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022), where a scaling factor of 4 is implemented in all PA experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The prefix-tuning (PT) architecture has an interme- diate MLP with a bottleneck size of 800, which is trained the same way as in the original wor (Li and Liang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We also use the default setting for LoRA with a scaling of 8 and a rank of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We re- produce the experimental results with the reported setup of the MAM adapter He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022) and UniPELT (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We reproduce the AdaMix results with the reported hyperparameter setup from the original work (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2022) in 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In the experiments of Figure 4, we control the bottleneck size DSA and DPA for SA and PA baselines, respectively, while keeping other setups unchanged to discover their performance across the parameter budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Similarly, we control the prefix length LPT for prefix-tuning and the rank r of LoRA without changing other setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Training Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Following previous work (Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020b), we use a recommended learning rate of 1e-4 for all PEFT experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In RoBERTalarge experiments, we report the RTE results with a learning rate of 2e-5 for AUTOPEFTMRPC and AUTOPEFTCoLA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 1e-4 for AUTOPEFTRTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We use the learning rate of 2e- 5 for full model FT according to Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We use the batch size of 32 and 16 for all BERT and RoBERTa experiments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The optimiser settings for each PEFT module follow the default settings in AdapterHub (Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' AUTOPEFT Search Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We implement the BO algorithm in BoTorch (Balandat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We use the Matern 5/2 kernel as the covariance function, and for the Monte Carlo sampling settings of SAAS-BO (Eriksson and Jankowiak, 2021), we use a warm-up step of 256, the number of samples to retain as 128, and thinning as 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For the opti- misation of the acquisition function, to adapt to the discrete setup, we use a local search method sim- ilar to previous literature involving similar setup (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Eriksson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=', 2021): at each search iteration (after the initial randomly sampled points), we collect the Pareto-optimal architectures up to this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' From this collection of Pareto- optimal architectures, we perform a local search by evaluating the acquisition function values of their neighbours, and move the current point to a neigh- bour with a higher acquisition function value and this process is repeated until convergence (which is a local minimum in terms of acquisition function), or 100 evaluations in acquisition function value are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' At each search iteration, we restart this process 10 times and select the top candidate for the query (in this case, fine-tuning) for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For all BO experiments, we use 200 total evaluations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' given the noisy nature of the problem, we use a relatively large number of random initiali- sation points (100) to ensure that the search results 10 2 10 1 100 Fine-tuned Parameters (%) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 Task Score STS-B 10 2 10 1 100 Fine-tuned Parameters (%) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 CoLA Serial Parallel Prefix LoRA AutoPEFT Figure 7: The Pareto front of the AUTOPEFT frame- work on tasks STS-B and CoLA compared to baselines with BERTbase in various settings of parameter budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the single-seed task score for each task fol- lowing the settings in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' are not overly sensitive to initialisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We use the same hyperparameter settings as described for all experiments conducted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Calculation of Fine-tuned Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The uncased BERTbase model (109M) has 12 Trans- former layers with a hidden dimension size of 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The uncased BERTlarge model (335M) and RoBERTalarge (355M) both have 24 layers with a hidden dimension size of 1, 024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' For both SA and PA, their fine-tuned parameters are computed by 2 × Dadapter × Dh × |l|, where Dh represents the corresponding hidden dimension of the selected model, and |l| refers to the total selected number of insertion layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Similarly, we calculate the fine- tuned parameters of PT by 2 × LPT × Dh × |l|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Thus, the number of fine-tuned parameters of the AUTOPEFT-found configurations is a summation of individual PEFT modules’ parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We re- port the default fine-tuned parameters for the re- maining PEFT modules as defined in their original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' B Search Space and Discovered Architectures Impact of Single PEFT Modules within AU- TOPEFT and Other Side Analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We pro- vide a more detailed analysis of the behaviour of AUTOPEFT by inspecting the Pareto front of AUTOPEFT-found configurations when we ablate each PEFT module into the search space, as plot- ted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' After combining the serial adapter with the parallel adapter, the upper bound of perfor- mance is improved by more than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We consider the gain here leverages the capacity of multiple het- erogeneous PEFT modules as a mixture-of-experts while providing a more efficient adaptation by up- dating both bias-influenced hidden states and the original states according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We recall that prefix-tuning stabilises its learning with an interme- diate reparametrization network, which is dropped in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Therefore, at the cost of the increased training parameters, prefix-tuning is one of the most parameter-efficient approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Consequently, we notice that incorporating prefix- tuning into the search space further improves the overall parameter efficiency (4% to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4%) of the AUTOPEFT-found configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Due to the pa- rameter efficiency of each single PEFT module, it also explains the distribution of the parameter budget for each PEFT module in the learned con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We also analyse the learned config- urations in terms of the selected layers over dif- ferent parameter scales in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' They show a common trend in selecting the higher Transformer layers to insert the PEFT modules, which coincides with previous findings that the higher layer con- tains richer task-specific representations, and intro- ducing PEFT modules to these layers is more effi- cient than other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' With the AUTOPEFT-found configurations reported in Table 5, we hope future PEFT research and applications can benefit from the architecture design similar to AUTOPEFTRTE M that we find the most transferable across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Method #Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' RTE MRPC STS-B CoLA SST-2 QNLI Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Fine-tune† 100% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='9 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 LoRA‡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='22% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='8 Serial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='89% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='8 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 AUTOPEFTRTE S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='03% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 AUTOPEFTMRPC S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='25% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='1 AUTOPEFTCoLA M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='36% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='3 AUTOPEFTRTE L 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='41% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='9 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 AUTOPEFTtask Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='88% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='1 Table 3: Experimental results on the GLUE benchmark with RoBERTalarge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report the full model fine-tuning† results from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2019b) with Pearson correlation for STS-B and Matthew’s correlation for CoLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We include the LoRA‡ module performance from Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' We report single-seed results for the experiments and exclude QQP and MNLI tasks due to the large computation cost of RoBERTalarge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Similar to Table 1, we conduct per-task search experiments on RTE, MRPC, STS-B, and CoLA, transferring best-found configurations to the remaining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' In addition to the transfer experiment from RTE, we also report transfer performance from MRPC and CoLA tasks with significantly different parameter budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' All reported results are from the configurations listed in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The best, second-best, and third-best results are marked in bold fonts and ranked by colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Model Insertion Layer {li} Module Size BERTbase 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 12 Serial Adapter DSA 0,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' 1024 Table 4: The search space of the AUTOPEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Each insertion layer has a Boolean decision for inserting the PEFT modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The 0 size of submodules indicates that we exclude the corresponding submodule from the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The total number of configurations for BERTbase: 212 × 11 × 11 × 11 ≈ 5 × 106 and for BERT/RoBERTalarge: 224 × 12 × 12 × 12 ≈ 3 × 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Task #Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Search Space Configuration Submodule Configuration RTE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='06% Layer li 3, 4, 6, 8, 9, 11 Serial Adapter DSA 3 Parallel Adapter DPA 1 Prefix-Tuning LPT 3 RTE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='42% Layer li 2, 5, 6, 7, 8, 9, 10 Serial Adapter DSA 96 Parallel Adapter DPA 48 Prefix-Tuning LPT 1 RTE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='60% Layer li 3, 4, 6, 7, 8, 9, 10 Serial Adapter DSA 384 Parallel Adapter DPA 192 Prefix-Tuning LPT 96 MRPC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='86% Layer li 2, 3, 6, 7, 9, 10, 11 Serial Adapter DSA 6 Parallel Adapter DPA 384 Prefix-Tuning LPT 3 STS-B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='06% Layer li 2, 5, 7, 8, 9, 11 Serial Adapter DSA 96 Parallel Adapter DPA 6 Prefix-Tuning LPT 24 CoLA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='29% Layer li 3, 4, 8, 9, 10 Serial Adapter DSA 12 Parallel Adapter DPA 24 Prefix-Tuning LPT 6 MNLI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='30% Layer li 3, 6, 7, 8, 9, 11, 12 Serial Adapter DSA 24 Parallel Adapter DPA 6 Prefix-Tuning LPT 1 Table 5: The AUTOPEFT-found configurations reported in Table 1 using BERTbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The average of fine-tuned parameters (%) of AUTOPEFTtask Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' is calculated by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='42+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='86+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='29+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='30+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='42+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='42)/8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='40, where we transfer the best-found AUTOPEFTRTE M to SST-2, QQP, and MNLI as their best per-task configurations for achieving the best trade-off between task performance and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Task #Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Search Space Configuration Submodule Configuration RTE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='78% Layer li 2, 6, 8, 11, 14, 15, 16, 17, 21, 23 Serial Adapter DSA 128 Table 6: The AUTOPEFT-found configurations reported in Table 2 using BERTlarge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Task #Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' Search Space Configuration Submodule Configuration RTE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='03% Layer li 6, 10, 14, 15, 18, 19, 21, 23 Serial Adapter DSA 2 Parallel Adapter DPA 4 Prefix-Tuning LPT 1 RTE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='41% Layer li 1, 2, 3, 4, 5, 7, 11, 12, 14, 15, 17, 19, 20, 21, 23 Serial Adapter DSA 64 Parallel Adapter DPA 1 Prefix-Tuning LPT 1024 MRPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='25% Layer li 1, 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 17, 21, 22, 23, 24 Serial Adapter DSA 8 Parallel Adapter DPA 2 Prefix-Tuning LPT 16 STS-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='25% Layer li 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 21, 22, 24 Serial Adapter DSA 8 Parallel Adapter DPA 2 Prefix-Tuning LPT 16 CoLA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='36% Layer li 1, 5, 6, 8, 9, 10, 13, 14, 15, 19, 21, 22, 23, 24 Serial Adapter DSA 256 Parallel Adapter DPA 32 Prefix-Tuning LPT 4 Table 7: The AUTOPEFT-found configurations reported in Table 3 using RoBERTalarge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' The average of fine-tuned parameters (%) of AUTOPEFTtask Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content=' is calculated by (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='03 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='25 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='25 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='36 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='36 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='03)/6 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFLT4oBgHgl3EQfoi-S/content/2301.12132v1.pdf'} +page_content='88, where we transfer the best-found AUTOPEFTCoLA M to SST-2 and AUTOPEFTRTE S to QNLI as their best per-task configurations for achieving the best trade-off between performance 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To address this challenge, we propose the robust +multilabel Takagi-Sugeno-Kang fuzzy system (R-MLTSK-FS) +with three mechanisms. First, we design a soft label learning mech- +anism to reduce the effect of label noise by explicitly measuring the +interactions between labels, which is also the basis of the other two +mechanisms. Second, the rule-based TSK FS is used as the base +model to efficiently model the inference relationship between fea- +tures and soft labels in a more transparent way than many existing +multilabel models. Third, to further improve the performance of +multilabel learning, we build a correlation enhancement learning +mechanism based on the soft label space and the fuzzy feature +space.1Extensive experiments are conducted to demonstrate the +superiority of the proposed method. + +Index Terms—Multilabel classification, label correlation, model +transparency, label noise. +I. INTRODUCTION +ULTILABEL learning concerns instances that can be asso- +ciated with more than one labels. For example, an article +can be labeled as being related to “politics”, “culture” and “re- +ligion” at the same time; and a travel photo can be given the +labels “beach”, “sunrise”, “sail” and “tourist” simultaneously +because of the presence of the corresponding objects. For mul- +tilabel learning, label correlation learning, model transparency +and robustness against label noise are essential. Constructing +the correlation between labels is the basic work to improve the +performance of multilabel learning [1, 2]. A transparent struc- +ture is important to enhance the interpretability of multilabel +learning [3]. And robustness against label noise enhances the +effectiveness in practical applications under noisy environment +[4]. +For label correlation learning, existing multilabel methods +are mainly based on first-order [5], second-order [6] and high- +order [7] strategies to consider the correlation between labels. + +This work was supported in part by the National key R & D plan under Grant +(2022YFE0112400), the NSFC under Grant 62176105, the Six Talent Peaks +Project in Jiangsu Province under Grant XYDXX-056, the Hong Kong Re- +search Grants Council (PolyU 152006/19E), the Project of Strategic Importance +of the Hong Kong Polytechnic University (1-ZE1V) and the Postgraduate Re- +search & Practice innovation Program of Jiangsu Province under Grant +KYCX22_2313. (Corresponding author: Zhaohong Deng). +Q. Lou, S. Wang are with the School of Artificial Intelligence and Computer +Science, Jiangnan University and Jiangsu Key Laboratory of Digital Design and +Software Technology, Wuxi 214122, China, and Q. Lou is with the Centre for +First-order methods ignore label correlation and adopt label- +by-label approach for multilabel learning. For example, sparse +weighted instance-based multilabel (SWIM) realizes the mul- +tilabel learning only based on the association between instances +[8]. Second-order methods build the pairwise relationship be- +tween labels. For example, labels related to the sample are +ranked before labels unrelated to the sample [9]. Multilabel +learning with global and local label correlation (GLOCAL) de- +composes the Laplacian matrix to indirectly learn the correla- +tion between any two labels [10]. High-order methods construct +the correlation between multiple labels simultaneously. For ex- +ample, cross-coupling aggregation (COCOA) first models the +correlation between random label pairs and then aggregates +their learning effects [11]. Multilabel classification with label- +specific features and label-specific classifiers (MLC-LFLC) in- +troduces the sparse learning to analyze the dependency between +a single label and other labels [12]. +For model transparency in multilabel learning, existing work +is mainly based on rules or logical inference to achieve trans- +parency [13]. For example, hierarchical multilabel classifica- +tion with a genetic algorithm (HMC-GA) [14] utilizes the ge- +netic algorithm to induce classification rules for protein func- +tion prediction which belongs to hierarchical multilabel learn- +ing. The gradient-weighted class activation mapping (Grad- +CAM) is used in [15] to realize the inferential interpretation for +predicted label results. The causal discovery is exploited in [16] +to analyze the specific features of a label. The multilabel Tak- +agi-Sugeno-Kang fuzzy system (TSK FS), i.e., ML-TSK FS [17] +offers good transparency through fuzzy rule-based structure and +fuzzy inference. Among the above existing multilabel methods, +ML-TSK FS has shown more promising performance because +it realizes the complete inference process from feature to label. +For robustness against label noise, much work has been stud- +ied because of the urgent need of practical application [18, 19]. +For example, class-conditional multilabel noise (CCMN) [20] +designs two unbiased estimators with error bounds to reduce the +influence of label noise. Multilabel noise robust collaborative +learning (RCML) [21] employs the group lasso to detect noisy +Smart Health, and the School of Nursing, the Hong Kong Polytechnic Univer- +sity, Hong Kong. (e-mail: 6171610005@stu.jiangnan.edu.cn; wxwangst@ali- +yun.com). +Z. Deng is with the School of Artificial Intelligence and Computer Science, +Jiangnan University, Wuxi 214122, China, and Key Laboratory of Computa- +tional Neuroscience and Brain-Inspired Intelligence (LCNBI) and ZJLab, +Shanghai 200433, China. (e-mail: dengzhaohong@jiangnan.edu.cn). +K.S. Choi is with the Centre for Smart Health, Hong Kong Polytechnic Uni- +versity. (e-mail: kschoi@ieee.org). +A Robust Multilabel Method Integrating Rule-based +Transparent Model, Soft Label Correlation Learning +and Label Noise Resistance +Qiongdan Lou, Zhaohong Deng, Senior Member, IEEE, Kup-Sze Choi, Shitong Wang +M + +labels. Partial multilabel learning with noisy label identification +(PML-NI) [22] builds the feature-induce noise term to identify +noisy labels. Multilabel iterated learning (MILe) [23] strength- +ens learning bottleneck for successive generations of teacher +and student networks to improve the robustness against label +noise. Different from removing noisy labels directly, noisy la- +bel tolerated partial multilabel learning (NATAL) [24] reduces +the impact of noisy labels by assuming that the label infor- +mation is precise and feature information is inadequate. +The above related work indicates that the importance of label +correlation, model transparency and robustness against noisy +labels has received extensive attention. However, such desira- +ble characteristics are still rarely studied simultaneously in mul- +tilabel learning. Therefore, it is necessary to further study the +multilabel method with transparency, label correlation learning +ability and robustness to noise labels. +Based on the above analysis, we aim to develop a multilabel +learning method with strong fuzzy inference ability and label +correlation learning ability, even under the influence of noisy +labels. To achieve the goal need, a robust multilabel learning +classifier, called robust multilabel Takagi-Sugeno-Kang fuzzy +system (R-MLTSK-FS), is proposed by developing three ena- +bling mechanisms. The first mechanism concerns soft label +learning. The R-MLTSK-FS maps the original label matrix to +the soft label space where each soft label is affected by all the +original labels. The mechanism thus reduces the influence of +label noise in the original label space, and is the basis of the +other two mechanisms. The second mechanism concerns the +construction of soft multilabel loss function. In R-MLTSK-FS, +the “IF-THEN” rule-based TSK FS is used to model the infer- +ence between the inputs and outputs. Specifically, multi-output +TSK FS is employed in this paper. The IF-part of a multi-output +TSK FS is leveraged to transform the original feature matrix +into the fuzzy feature space; the THEN-part is used to imple- +ment the inference between inputs and outputs; and the regres- +sion loss is constructed based on the TSK FS and soft label +learning. The adoption of TSK FS is advantageous in that the +rule-based TSK FS makes the proposed R-MLTSK-FS more +transparent than traditional models. The third mechanism con- +cerns correlation enhancement learning. The mechanism estab- +lishes associations between any two soft labels and their corre- +sponding fuzzy discriminative features, which can effectively +improve the performance of R-MLTSK-FS. +The main contributions of this paper are summarized as fol- +lows: +(1) A soft label learning mechanism is constructed to explic- +itly measure the interaction between the labels and reduce the +influence of label noise. +(2) A soft multilabel loss function is constructed based on +soft labels and TSK FS to improve the efficiency and transpar- +ency of the learning process of R-MLTSK-FS. +(3) A correlation enhancement learning mechanism based on +soft label space and fuzzy feature space is built to further en- +hance the learning ability of R-MLTSK-FS. +(4) Extensive experiments are conducted using 10 bench- +mark multilabel datasets and 3 synthetic multilabel datasets to +compare with 8 methods. Comprehensive evaluations are +carried out by conducting classification performance evaluation, +robustness analysis, effectiveness analysis of soft label learning +and correlation enhancement learning, parameter analysis, con- +vergence analysis, and statistical analysis. +The rest of this paper is organized as follows. Section II re- +views the concepts of multilabel learning, and the traditional +TSK FS. Section III gives details of the proposed method. Ex- +tensive experimental analyses are presented and discussed in +Section IV. Finally, Section V summarizes the paper. +II. BACKGROUND KNOWLEDGE +In this section, the problem statement of the multilabel learn- +ing research concerned in the study is given, followed by the +review of traditional TSK FS. +A. Problem Statement +Let 𝒳 ∈ ℛ𝐷 and 𝒴 ∈ ℛ𝐿 be a D-dimensional feature +space and an L-dimensional label space respectively. 𝒟 = +{(𝒙𝑖, 𝒚𝑖)}𝑖=1 +𝑁 + is the training set with N samples. 𝑿 = +[𝒙1, 𝒙2, … , 𝒙𝑁] ∈ ℛ𝐷×𝑁 is the input matrix, and +𝒀 = +[𝒚1, 𝒚2, … , 𝒚𝑁] ∈ ℛ𝐿×𝑁 is the output matrix. In multilabel +learning, the label of an instance 𝒙𝑖 = [𝑥𝑖1, 𝑥𝑖2, … , 𝑥𝑖𝐷]T is +given by a vector 𝒚𝑖 = [𝑦𝑖1, 𝑦𝑖2, … , 𝑦𝑖𝐿]T. If 𝒙𝑖 is related to +the jth label, then 𝑦𝑖𝑗 = 1, otherwise, 𝑦𝑖𝑗 = 0. The aim of this +study is to find a robust mapping function 𝑓: 𝒳 → 𝒴 that can +reduce the influence of label noise and effectively predict the +label vector for a new instance on the basis of transparent infer- +ence rules. +B. TSK Fuzzy System +TSK FS is a classical inference model based on fuzzy rules +with superior interpretability (transparency) and learning ability. +It has been successfully applied in different areas, e.g., transfer +learning [25, 26], multiview learning [27], multitask learning +[28] and others [29, 30, 31, 32]. For a classical TSK FS with K +rules, the kth rule can be expressed as follows: +IF: 𝑥1 𝑖𝑠 𝐴1 +𝑘 ∧ 𝑥2 𝑖𝑠 𝐴2 +𝑘 ∧ … ∧ 𝑥𝐷 𝑖𝑠 𝐴𝐷 +𝑘, +THEN: 𝑓𝑘(𝒙) = 𝑐0 +𝑘 + 𝑐1 +𝑘𝑥1 + ⋯ + 𝑐𝐷 +𝑘𝑥𝐷, +𝑘 = 1, 2, … , 𝐾 +(1) +where D is the feature dimension, and 𝑓𝑘(𝒙) is the output of +instance 𝒙 on the kth rule. 𝐴𝑑 +𝑘 (𝑑 = 1, 2, … , 𝐷) in IF-part +represents the antecedent fuzzy set, which can be described by +membership functions. 𝑐𝑑 +𝑘 in THEN-part is the consequent pa- +rameter. +Depending on application scenarios, different membership +functions can be chosen for the antecedent fuzzy sets. Gaussian +function, which is commonly used, is adopted in this paper and +the corresponding membership function associated with 𝐴𝑑 +𝑘 +can be expressed as follows: +𝜇𝐴𝑑 +𝑘(𝑥𝑑) = exp {− +1 +2 ( +𝑥𝑑−𝑚𝑑 +𝑘 +𝛿𝑑 +𝑘 +)2} +(2) +where 𝑚𝑑 +𝑘 and 𝛿𝑑 +𝑘 can be obtained using different methods. +In the absence of domain knowledge, data-driven methods are +usually utilized to estimate 𝑚𝑑 +𝑘 and 𝛿𝑑 +𝑘. For example, the Var- +Part clustering has been used for this purpose [33]. It is insen- +sitive to the parameters and is therefore beneficial in terms of + +stability and practicability. Hence, the Var-Part clustering is +used in this study. +For TSK FS, the firing strength of instance 𝒙 on the kth rule +can be computed as follows: +𝜇𝑘(𝒙) = ∏ +𝜇𝐴𝑑 +𝑘(𝑥𝑑) +𝐷 +𝑑=1 + +(3) +𝜇̃𝑘(𝒙) = 𝜇𝑘(𝒙) ∑ +𝜇𝑘′(𝒙) +𝐾 +𝑘′=1 +⁄ + +(4) +where Eq. (4) is the normalized form of Eq. (3). +Finally, the output of TSK FS for instance 𝒙 can be ex- +pressed as +𝑦 = 𝑓(𝒙) = ∑ +𝜇̃𝑘(𝒙)𝑓𝑘(𝒙) +𝐾 +𝑘=1 + +(5) +In fact, Eq. (5) can also be expressed as a linear model in a new +fuzzy feature space, that is, +𝑦 = 𝑓(𝒙) = 𝒄T𝒙𝑔 +(6) +where +𝒙𝑒 = [1, 𝒙T]T ∈ ℛ(𝐷+1)×1 +(7) +𝒙̃𝑘 = 𝜇̃𝑘(𝒙)𝒙𝑒 ∈ ℛ(𝐷+1)×1 +(8) +𝒙𝑔 = [(𝒙̃1)T, (𝒙̃2)T, … , (𝒙̃𝐾)T]T ∈ ℛ𝐾(𝐷+1)×1 +(9) +𝒄𝑘 = [𝑐0 +𝑘, 𝑐1 +𝑘, … , 𝑐𝐷 +𝑘]T ∈ ℛ(𝐷+1)×1 +(10) +𝒄 = [(𝒄1)T, (𝒄2)T, … , (𝒄𝐾)T]T ∈ ℛ𝐾(𝐷+1)×1 +(11) +Here, 𝒙𝑔 is the fuzzy representation of instance 𝒙 in a new +feature space generated by fuzzy rules. 𝒄 is the consequent pa- +rameter vector of all the rules, which can be optimized by solv- +ing the linear model in Eq. (6). +III. PROPOSED METHOD: R-MLTSK-FS +A. System Architecture +The architecture of the R-MLTSK-FS proposed in this study +is shown in Fig. 1. It aims to provide a robust multilabel model +with fuzzy inference ability, label correlation learning ability +and resistance against noisy labels. R-MLTSK-FS contains +three mechanisms for soft label learning, soft multilabel loss +function construction and correlation enhancement learning, re- +spectively. + + + +Fig. 1 The architecture of the proposed R-MLTSK-FS. + +The first mechanism, soft label learning, maps the original +labels to soft label space by linear transformation. Each soft la- +bel in the soft label space is associated with all the original la- +bels, which reduces the influence of label noise in the original +label space. It is the basis of the other two mechanisms. The +second mechanism, i.e., soft multilabel loss function construc- +tion, leverages the IF-part of the TSK FS to transform the orig- +inal features into the fuzzy feature space, uses the THEN-part +of the TSK FS to complete the inference between inputs and +outputs, and then constructs the regression function between the +fuzzy feature space and the soft label space. Rule-based TSK +FS makes R-MLTSK-FS transparent in modeling inference re- +lationship between features and labels. The third mechanism, +correlation enhancement learning, implements label correlation +learning by establishing associations between any two soft la- +bels and their corresponding fuzzy discriminative features. This +mechanism further enhances the learning ability of R-MLTSK- +FS. +The details of R-MLTSK-FS are expanded in the following +three sections. The learning criteria of R-MLTSK-FS is intro- +duced in Section III-B. The optimization process and the algo- +rithm description are given in Section III-C, and the computa- +tional complexity is analyzed in Section III-D. +B. Learning Criteria of R-MLTSK-FS +According to the analysis in Section III-A, the multilabel +learning problem in this paper can be expressed as the following +optimization objective criteria: +min +𝜙1,𝜙2 𝛽 ∙ 𝑆𝑜𝑓_𝑙𝑎𝑏(𝒀|𝜙1) + 𝑆𝑜𝑓_𝑙𝑜𝑠(𝒀, 𝑿|𝜙1, 𝜙2) + +𝛾 ∙ 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) +(12) +The first term represents soft label learning, where 𝜙1 trans- +forms the original labels to the soft labels. The second term rep- +resents soft multilabel loss function construction, where 𝜙2 is +used to predict the labels from the original feature space to the +soft label space. The third term represents correlation enhance- +ment learning, which is used to measure the association be- +tween any two soft labels and their corresponding fuzzy dis- +criminative features. The hyperparameters β and γ are used to +balance the influences of different terms in Eq. (12). The solu- +tions of 𝜙1 and 𝜙2 can be obtained by optimizing Eq. (12). +The implementation of three terms is described below. +1) Soft Label Learning based on Original Label Space and +Soft Label Space +For the lth label 𝒀𝑙 ∈ ℛ1×𝑁 (1 ≤ 𝑙 ≤ 𝐿) (i.e., the lth row in +𝒀), the interference of its label noise can be reduced by consid- +ering the influence of all labels on 𝒀𝑙 comprehensively. Based +on this, for soft label learning, we assume that each label is as- +sociated with all the other original labels to some extent. The +learning process involves two steps. First, we construct the label +transformation 𝜙1 to effectively measure the interaction be- +tween the labels. 𝜙1 maps the output matrix 𝒀 explicitly +from the original label space to the soft label space. In the soft +label space, each soft label is associated with all the original +labels. The transformation function of 𝜙1 is defined as: +𝜙1(𝒀) = 𝑺𝒀 +(13) +where 𝑺 = [𝒔1, 𝒔2, … , 𝒔𝐿]T ∈ ℛ𝐿×𝐿 , and 𝒔𝑙 ∈ ℛ𝐿×1 (1 ≤ 𝑙 ≤ +𝐿) represents the influence weights of all the original labels on +the lth soft label. +Second, we preserve the expression consistency between the +soft labels and original labels to ensure the classification per- +formance. Therefore, the overall soft label learning is defined +as: +min +𝜙1 𝑆𝑜𝑓_𝑙𝑎𝑏(𝒀|𝜙1) = min +𝑺 ‖(𝒀 − 𝑺𝒀)T‖2,1 +(14) + +TSKFuzzy System +Soft Label Space +SoftLabel +Soft Multilabel Loss +Learning +Function Construction +Correlation +EnhancementLearningAlthough different regularization norms can be used in Eq. +(14), we choose the L2,1 norm for two reasons: (1) since L2,1 +norm has the characteristic of row sparsity, we can screen out +the original label subsets which have significant impact on the +corresponding soft label, (2) L2,1 norm is well-known for its +ability in robust group selection [34, 35, 36], which is helpful +to reduce the impact of label noise on soft label learning. +2) Soft Multilabel Loss Function Construction based on TSK +FS +Multilabel loss function can be constructed by employing an +evaluation metric as the multilabel objective function [37, 38], +or by using linear regression to derive the multilabel loss func- +tion [39, 40, 41]. Unlike these methods, we construct the loss +function using soft label learning and TSK FS, which essen- +tially constructs a rule-based transparent model that maps the +original feature space to the soft label space. The construction +of the soft multilabel loss function is divided into three steps. +First, the original feature matrix is transformed into the fuzzy +feature space through the IF-part of the fuzzy rules. Second, the +inference between inputs and outputs is completed through the +THEN-part of fuzzy rules. Third, the regression loss function is +constructed based on the fuzzy rules and soft labels. These de- +tails are as follows. +• +IF-part implementation of fuzzy rules. In the multi-out- +put TSK FS with K rules, the fuzzy feature matrix obtained +by 𝑿 using fuzzy rules is given by +𝑿𝑔 = [𝒙𝑔,1, 𝒙𝑔,2, … , 𝒙𝑔,𝑁] ∈ ℛ𝐾(𝐷+1)×𝑁 +(15) +where 𝒙𝑔,𝑖 (1 ≤ 𝑖 ≤ 𝑁) is mapped by the instance 𝒙𝑖 +through the IF-part of fuzzy rules, and it can be obtained +by Eqs. (2)-(4) and (7)-(9). +Compared with the original features, the rule-based +fuzzy features can empower R-MLTSK-FS to analyze the +implicit inference relationship between features and labels +[42], thereby strengthening the learning ability. +• +THEN-part adaptation of fuzzy rules. Based on Eq. (6), +the THEN-part of multi-output TSK FS is used to complete +the inference, i.e., +𝜙2(𝑿) = 𝑪𝑿𝑔 +(16) +where +𝑪 = [𝒄1, 𝒄2, … , 𝒄𝐿]T ∈ ℛ𝐿×𝐾(𝐷+1) +(17) +is composed of L consequent parameter vectors in THEN- +part. As defined in Eq. (11), 𝒄𝑙 ∈ ℛ𝐾(𝐷+1)×1 (1 ≤ 𝑙 ≤ 𝐿) +is the consequent parameter vector corresponding to the +lth-output in multi-output TSK FS and the lth soft label. +The main difference between multi-output TSK FS and sin- +gle-output TSK FS is that the consequent parameters of sin- +gle-output TSK FS are represented with a vector, whereas +the consequent parameters of multi-output TSK FS are rep- +resented by a matrix composed of multiple vectors. +• +Construction of regression loss. The loss function is a +fundamental part of the optimization objective for multila- +bel classification. In this paper, it is built based on soft label +learning and TSK FS. Combining Eqs. (13) and (16), we +construct the soft multilabel loss function as follows: +min +𝜙1,𝜙2 𝑆𝑜𝑓_𝑙𝑜𝑠(𝒀, 𝑿|𝜙1, 𝜙2) += min +𝑺,𝑪 ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 +2 +(18) +where α is a hyperparameter to balance the influence of the +soft multilabel loss function and the regularization term. +Taking the Frobenius norm ‖∙‖𝐹 as the regularization +term can not only reduce the risk of overfitting, but also +facilitate the solution of consequent parameter matrix 𝑪. +3) Correlation Enhancement Learning based on Soft Label +Space and Fuzzy Feature Space +Section I has clarified that mining the correlation information +between labels can effectively improve the performance of the +model. In this paper, we analyze the label correlation based on +the fact that the correlation between two labels is consistent +with the correlation between their discriminative features. For +example, there is an intersection between the labels “Cat” and +“Animal”, and then their discriminative features should par- +tially overlap. +Based on the above analysis, we utilize the correlation infor- +mation on the basis of soft label learning and fuzzy features as +follows: +min +𝜙1,𝜙2 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) += min +𝑺,𝑪 ∑ +∑ +‖(𝒔𝑖 +T𝒀 − 𝒔𝑗 +T𝒀)T‖ +2𝒄𝑖 +T𝒄𝑗 +𝐿 +𝑗=1 +𝐿 +𝑖=1 + +(19) +where 𝒔𝑙 +T𝒀 ∈ ℛ1×𝑁 (1 ≤ 𝑙 ≤ 𝐿) represents the lth soft label +vector corresponding to N samples. 𝒔𝑙 ∈ ℛ𝐿×1 represents the +influence weights of all original labels on the lth soft label. 𝒄𝑙 ∈ +ℛ𝐾(𝐷+1)×1 (1 ≤ 𝑙 ≤ 𝐿) is used to learn the discriminative fea- +tures from fuzzy feature space for the lth soft label. The larger +the difference between the ith and jth soft labels, the more sig- +nificant the difference between their fuzzy discriminative fea- +tures, and further, the smaller the value of 𝒄𝑖 +T𝒄𝑗. Further, Eq. +(19) can be expressed as: +min +𝜙1,𝜙2 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) += min +𝑺,𝑪 ∑ +∑ +‖(𝒔𝑖 +T𝒀 − 𝒔𝑗 +T𝒀)T‖ +2𝒄𝑖 +T𝒄𝑗 +𝐿 +𝑗=1 +𝐿 +𝑖=1 + += min +𝑺,𝑪 2Tr(𝒀T𝑺T𝑳𝑺𝒀) +(20) +where 𝑳 = 𝑫 − 𝑹, 𝑹 = 𝑪𝑪𝑇 ∈ ℛ𝐿×𝐿, 𝑫 ∈ ℛ𝐿×𝐿 is a diago- +nal matrix, and 𝐷𝑖𝑖 = ∑ +𝑅𝑖𝑗 +𝐿 +𝑗=1 +. +C. Complete Objective Function and its Optimization +By integrating Eqs. (14), (18) and (20), the multilabel learn- +ing problem in Eq. (12) is defined and the complete objective +function of R-MLTSK-FS is expressed as: +min +𝜙1,𝜙2 𝛽 ∙ 𝑆𝑜𝑓_𝑙𝑎𝑏(𝒀|𝜙1) + 𝑆𝑜𝑓_𝑙𝑜𝑠(𝒀, 𝑿|𝜙1, 𝜙2) + +𝛾 ∙ 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) += min +𝑺,𝑪 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + ‖(𝑺𝒀 − 𝑪𝑿𝑔) +T‖ +2,1 + +𝛼‖𝑪‖𝐹 +2 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) += min +𝑺,𝑪 ‖(𝑺𝒀 − 𝑪𝑿𝑔) +T‖ +2,1 + 𝛼‖𝑪‖𝐹 +2 + 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + +2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) + +(21) + +To optimize 𝑺 and 𝑪, we adopt the alternating direction +minimization strategy, where Eq. (21) is divided into two sub- +problems, namely, the 𝑺-subproblem and the 𝑪-subproblem. +The optimization processes are as follows. +1) 𝑺-Subproblem +By fixing 𝑪, the 𝑺-subproblem can be expressed as: +𝑺∗ = 𝑎𝑟𝑔𝑚𝑖𝑛𝑺 ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + +2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) +(22) +In Eq. (22), the Lagrange function for 𝑺 is +𝐿(𝑺) = ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + +2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) +(23) +Set the derivative of Eq. (23) with respect to 𝑺 to 0, i.e., + 𝜕𝐿(𝑺) 𝜕𝑺 +⁄ += 2𝑺𝒀𝑫𝑆1𝒀T − 2𝑪𝑿𝑔𝑫𝑆1𝒀T + 2𝛽𝑺𝒀𝑫𝑆2𝒀T +−2𝛽𝒀𝑫𝑆2𝒀T + 4𝛾𝑳𝑺𝒀𝒀T = 0 +(24) +where 𝑫𝑆1 ∈ ℛ𝑁×𝑁 and 𝑫𝑆2 ∈ ℛ𝑁×𝑁 are diagonal matrices, +and +𝐷𝑆1,𝑖𝑖 = 1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 +T‖) +⁄ +, +𝐷𝑆2,𝑖𝑖 = +1 (2‖(𝒀 − 𝑺𝒀)𝑖 +T‖) +⁄ +. (𝑨𝑖 +T represents the ith row in 𝑨T.) +Then, Eq. (24) can be re-expressed as +(2𝛾𝑳)𝑺 + 𝑺(𝒀𝑫𝑆1𝒀T(𝒀𝒀T)−1 + 𝛽𝒀𝑫𝑆2𝒀T(𝒀𝒀T)−1) += 𝑪𝑿𝑔𝑫𝑆1𝒀T(𝒀𝒀T)−1 + 𝛽𝒀𝑫𝑆2𝒀T(𝒀𝒀T)−1 +(25) +Eq. (25) is a classical optimization problem, i.e., the Sylvester +equation, which has been thoroughly studied [43, 44, 45]. +In general, for the Sylvester equation 𝑨𝑾 + 𝑾𝑩 = 𝒁 (𝑨 ∈ +ℛ𝑚×𝑚, 𝑩 ∈ ℛ𝑛×𝑛, 𝒁 ∈ ℛ𝑚×𝑛, 𝑾 ∈ ℛ𝑚×𝑛), the matrix 𝑾 +is the variable to be solved. The specific solution formula of 𝑾 +is as follows: +𝑾(: ) = (𝑰1⨂𝑨 + 𝑩T⨂𝑰2)−𝟏𝒁(: ) +(26) +where 𝑰1 ∈ ℛ𝑛×𝑛 and 𝑰2 ∈ ℛ𝑚×𝑚 are identity matrices, ⨂ +is the Kronecker tensor product, 𝒁(: ) ∈ ℛ𝑚𝑛×1 and 𝑾(: ) ∈ +ℛ𝑚𝑛×1 denote that the matrices 𝒁 and 𝑾 are single column +vectors. 𝑾(: ) can be reshaped to 𝑾∗ ∈ ℛ𝑚×𝑛, which is the +solution of 𝑨𝑾 + 𝑾𝑩 = 𝒁. For simplicity, the solution 𝑾∗ +is denoted as 𝑾∗ = 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝑨, 𝑩, 𝒁). +Therefore, the solution of Eq. (25) is +𝑺∗ = 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(2𝛾𝑳, 𝒀(𝑫𝑆1 + 𝛽𝑫𝑆2)𝒀T(𝒀𝒀T)−1, +(𝑪𝑿𝑔𝑫𝑆1 + 𝛽𝒀𝑫𝑆2)𝒀T(𝒀𝒀T)−1) +(27) +2) 𝑪-Subproblem +By fixing 𝑺, the 𝑪-subproblem can be expressed as: +𝑪∗ = 𝑎𝑟𝑔𝑚𝑖𝑛𝑪 ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 +2 + +2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) +(28) +In Eq. (28), the Lagrange function for 𝑪 is +𝐿(𝑪) = ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 +2 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) + = ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 +2 + 2𝛾Tr(𝒀T𝑺T(𝑫 − 𝑹)𝑺𝒀) += ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 +2 + 2𝛾Tr(𝒀T𝑺T(𝑪𝑪T𝟏𝟏T ∘ + 𝑰3 − 𝑪𝑪T)𝑺𝒀) + +(29) +where 𝟏 ∈ ℛ𝐿×1 is a column vector with all elements equal to +one. The symbol (∘) represents the Hadamard product. 𝑰3 ∈ +ℛ𝐿×𝐿 is the identity matrix. +Set the derivative of Eq. (29) with respect to 𝑪 to 0, i.e., + 𝜕𝐿(𝑪) 𝜕𝑪 +⁄ += 2𝑪𝑿𝑔𝑫𝐶𝑿𝑔 +T − 2𝑺𝒀𝑫𝐶𝑿𝑔 +T + 2𝛼𝑪 + +2𝛾(((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3)T𝟏𝟏T𝑪 + 𝟏𝟏T((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3)𝑪 − +2𝑺𝒀𝒀T𝑺T𝑪) = 0 + +(30) +where 𝑫𝐶 ∈ ℛ𝑁×𝑁 is a diagonal matrix, and 𝐷𝐶,𝑖𝑖 = +1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 +𝑇‖) +⁄ +. (𝑨𝑖 +T is the ith row of 𝑨T.) +Eq. (30) is also a Sylvester equation. Therefore, we can solve +𝑪 as follows: +𝑪∗ = 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝛼𝑰3 + 𝛾((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3) +T𝟏𝟏T + +𝛾𝟏𝟏T((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3) − 2𝛾𝑺𝒀𝒀T𝑺T, 𝑿𝑔𝑫𝐶𝑿𝑔 +T, 𝑺𝒀𝑫𝐶𝑿𝑔 +T) + +(31) +When the optimal 𝑺∗ and 𝑪∗ are obtained, the prediction +output of the test instance 𝒙′ (i.e., 𝒚′ = [𝑦1 +′, … , 𝑦𝐿 +′]𝑇) can be +formulated as follows: +𝒚′ = 𝜑𝜏(𝑪∗𝒙𝑔 +′ ) +(32) +where 𝒙𝑔 +′ is the fuzzy feature representation of 𝒙′ through +fuzzy rules. It can be obtained from Eqs. (2)-(4) and (7)-(9). +𝜑𝜏(∙) is a threshold function to convert the continuous output +to the discrete output, and 𝜏 is the threshold. Therefore, for the +lth label 𝑦𝑙 +′ (1 ≤ 𝑙 ≤ 𝐿) in 𝒚′, its definition is +𝑦𝑙 +′ = {1, 𝑖𝑓 (𝑪∗𝒙𝑔 +′ )𝑙 > 𝜏 +0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 +(33) +where (𝑪∗𝒙𝑔 +′ )𝑙 is the lth element in (𝑪∗𝒙𝑔 +′ ). The value of 𝜏 +can be optimized by cross-validation. In this paper, we set it to +the fixed value of 0.5. +Based on the above analysis, the procedure of the proposed +R-MLTSK-FS is described in Algorithm I. +D. Computational Complexity Analysis +The computational complexity of R-MLTSK-FS is analyzed +according to the steps in Algorithm I, which is expressed using +the big-O notation. For step 1, the complexity of initialization +is 𝑂(1). For step 2, the computational complexity of trans- +forming 𝑿 into 𝑿𝑔 is 𝑂(2𝑁𝐾𝐷 + 2𝑁𝐾) . The computa- +tional complexity of step 4 is 𝑂(𝐿2𝑁 + 𝐿𝑁𝐾(𝐷 + 1)). For the +step 5, the computational complexity of 𝑻1 is 𝑂(2𝐿2𝑁 + +𝐿3 + 2𝐿2). For step 6, the computational complexity of 𝑻2 is +𝑂(𝑁2𝐾(𝐷 + 1) + 𝑁𝐾2(𝐷 + 1)2). For step 7, the computa- +tional complexity of calculating 𝑻3 is 𝑂(𝐿2𝑁 + 𝐿𝑁2 + +Algorithm I R-MLTSK-FS +Input: Input matrix 𝑿 ∈ ℛ𝐷×𝑁, output matrix 𝒀 ∈ ℛ𝐿×𝑁, rule number K, +trade-off parameters α, β and γ. +Procedure: +1: Initialize: +𝑺 = 𝟏𝐿×𝐿, 𝑪 = (1 𝐿 +⁄ )𝟏𝐿×𝐾(𝐷+1), 𝑫 = 𝟎𝐿×𝐿, 𝑫𝐶 = 𝟎𝑁×𝑁, 𝑫𝑆1 = +𝟎𝑁×𝑁, 𝑫𝑆2 = 𝟎𝑁×𝑁. +2: Transform 𝑿 into 𝑿𝑔 using Eqs. (2)-(4) and (7)-(9). +3: While not converged do +4: 𝐷𝐶,𝑖𝑖 = 1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 +𝑇‖) +⁄ +; +5: 𝑻1 ← 𝛼𝑰3 + 𝛾((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3)T𝟏𝟏T + 𝛾𝟏𝟏T((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3) − +2𝛾𝑺𝒀𝒀T𝑺T; +6: 𝑻2 ← 𝑿𝑔𝑫𝐶𝑿𝑔 +T; +7: 𝑻3 ← 𝑺𝒀𝑫𝐶𝑿𝑔 +T; + +8: 𝑪 ← 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝑻1, 𝑻2, 𝑻3); +9: 𝐷𝑆1,𝑖𝑖 = 1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 +T‖) +⁄ +; +10: 𝐷𝑆2,𝑖𝑖 = 1 (2‖(𝒀 − 𝑺𝒀)𝑖 +T‖) +⁄ +; +11: 𝑹 ← 𝑪𝑪T; +12: 𝐷𝑖𝑖 ← ∑ +𝑅𝑖𝑗 +𝐿 +𝑗=1 +; +13: 𝑳 = 𝑫 − 𝑹; +14: 𝑻4 ← 2γ𝑳; +15: 𝑻5 ← 𝒀(𝑫𝑆1 + 𝛽𝑫𝑆2)𝒀T(𝒀𝒀T)−1; +16: 𝑻6 ← (𝑪𝑿𝑔𝑫𝑆1 + 𝛽𝒀𝑫𝑆2)𝒀T(𝒀𝒀T)−1; +17: 𝑺 ← 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝑻4, 𝑻5, 𝑻6); +18: Check the convergence conditions; +19: End +Output: 𝑺, 𝑪. + +TABLE I +STATISTICS OF DATASETS + +Dataset +#Instance +#Feature +#Label +Arts +5000 +462 +26 +Birds +645 +260 +19 +CAL500 +502 +68 +174 +Corel5k +5000 +499 +374 +Flags +194 +19 +7 +Genbase +662 +1185 +27 +Medical +978 +1449 +45 +Mirflickr +25000 +150 +24 +Recreation +5000 +606 +22 +Science +5000 +743 +40 + +𝐿𝑁𝐾(𝐷 + 1)) . The computational complexity of step 8 is +𝑂(3𝐿4) . For step 9, the complexity of calculating 𝑫𝑆1 is +𝑂(𝐿2𝑁 + 𝐿𝑁𝐾(𝐷 + 1)). For step 10, the complexity of 𝑫𝑆2 is +𝑂(𝐿2𝑁). The complexity of step 11 is 𝑂(𝐿2𝐾(𝐷 + 1)). The +complexity of steps 12-14 is 𝑂(1). For step 15, the complexity +of 𝑻5 is 𝑂(𝐿𝑁2 + 𝐿2𝑁 + 𝐿3). The complexity of step 16 is +𝑂(𝐿𝑁𝐾(𝐷 + 1) + 𝐿𝑁2 + 𝐿2𝑁 + 𝐿3) . For step 17, the com- +plexity is 𝑂(3𝐿2𝐾2(𝐷 + 1)2). Hence, the overall complexity +of the whole algorithm is dominated by steps 6 and 16. Let 𝑎 = +max (𝐿, 𝐷, 𝐾) , 𝑏 = max (𝑁, 𝐾(𝐷 + 1)) . In general, 𝑎 ≪ 𝑏 . +Therefore, the maximum computational complexity of R- +MLTSK-FS is 𝑂(𝑎3 + 𝑏(2𝑎𝑏 + 𝑎2 + 2𝑏2)). +IV. EXPERIMENTAL ANALYSIS +Extensive experiments are conducted to fully assess the ef- +fectiveness of R-MLTSK-FS, including classification perfor- +mance evaluation, robustness analysis, effectiveness analysis of +soft label learning and correlation enhancement learning, pa- +rameter analysis, convergence analysis, and statistical analysis. +The datasets, evaluation metrics and the settings used in the ex- +periments are described below. +A. Datasets +We adopt 10 benchmark multilabel datasets to evaluate the +performance of R-MLTSK-FS. Table I shows the details of +these datasets, where #Instance, #Feature, and #Label denote +the instance number, the feature dimension, and the label space +dimension respectively. These datasets are available from the +Github1. +B. Evaluation Metrics +Let {(𝒙̃𝑖, 𝒚̃𝑖)|1 ≤ 𝑖 ≤ 𝑁𝑡} be a test set with 𝑁𝑡 samples, +𝒚̂𝑖 be the predicted labels of 𝒙̃𝑖, 𝑓(𝒙̃𝑖, 𝑙) be the continuous +output predicted by the multilabel method for the instance 𝒙̃𝑖 +on the lth label. The ranking function 𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙) is obtained +according +to +𝑓(𝒙̃𝑖, 𝑙) . +If +𝑓(𝒙̃𝑖, 𝑙) > 𝑓(𝒙̃𝑖, 𝑙′) , +then +𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙) < 𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙′). Let 𝐿𝒙𝑖 be the label set related to +𝒙̃𝑖, and 𝐿𝒙𝑖 is the complement of 𝐿𝒙𝑖. Based on the settings, +the four metrics below, commonly used in multilabel learning, +are employed in the experiments [46]. +(1) Average Precision (AP): It is the average proportion of +the related labels of an instance that are ranked lower than a +given label l. The larger the value of AP, the better the classifi- +cation performance. +AP = +1 +𝑁𝑡 ∑ +1 +|𝐿𝒙𝑖| ∑ +|{𝑙′ ∈ 𝐿𝒙𝑖|𝑓(𝒙̃𝑖, 𝑙′) ≥ 𝑓(𝒙̃𝑖, 𝑙)}| +𝑟𝑎𝑛𝑘(𝒙̃𝑖,𝑙) +𝑙∈𝐿𝒙𝑖 +𝑁𝑡 +𝑖=1 + (34) +(2) Hamming Loss (HL): It is the average proportion of an +instance that is predicted incorrectly. The smaller the value of +HL, the better the classification performance. +HL = +1 +𝑁𝑡 ∑ +|𝒚̃𝑖⨁𝒚̂𝑖| +𝐿 +𝑁𝑡 +𝑖=1 + +(35) +where ⨁ is the XOR operation. +(3) Ranking Loss (RL): It is the proportion of the related la- +bels that are ranked higher than the unrelated labels. The +smaller the value of RL, the better the classification perfor- +mance. +RL = +1 +𝑁𝑡 ∑ +|{(𝑙, 𝑙′)|𝑓(𝒙̃𝑖, 𝑙) ≤ 𝑓(𝒙̃𝑖, 𝑙′), (𝑙, 𝑙′) ∈ 𝐿𝒙𝑖 × 𝐿𝒙𝑖}| +|𝐿𝒙𝑖||𝐿𝒙𝑖| +𝑁𝑡 +𝑖=1 + + +(36) +(4) Coverage (CV): It is the average number of times that all +related labels of an instance are found. The smaller the value of +CV, the better the classification performance. +CV = +1 +𝑁𝑡 ∑ +max +𝑙∈𝐿𝒙𝑖 +𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙) − 1 +𝑁𝑡 +𝑖=1 + +(37) +C. Experimental Settings +In this paper, we employ eight methods for comparison, in- +cluding binary relevance (BR) [47], multilabel k-nearest neigh- +bor (MLkNN) [48], meta-label-specific features (MLSF) [49], +ML-TSK FS [17], classifier chains (CC) [50], random k-label- +sets (RAkEL) [51], correlated logistic models (CorrLog) [52] +and hybrid noise-oriented multilabel learning (HNOML) [53]. +These methods and the settings of the parameters for grid search +are described in Table II. We adopt the 5-fold cross-validation +strategy to evaluate the performance. + + + +1https://github.com/ZesenChen/multi-label-dataset and https://github.com/ +KKimura360/MLC_toolbox/tree/master/dataset/matfile + +TABLE II +DESCRIPTION OF METHODS + +Methods +Description +Parameter Setting +BR +This method is a first-order method. To improve the robustness, it introduces -in- +sensitive learning (a fuzzy method) by solving a system of linear inequalities +(LSSLI) [54] as the binary classifier. + +𝐶 = 2. ^(−5: 1: 5), +𝑀 = {2, 3, 4, 5, 6, 7, 8, 9}. +MLkNN +This method is a first-order method that predicts a new instance by maximizing the +posterior probability of each label. The number of nearest neighbors affects the ro- +bustness of the model to some extent. + +𝐾 = {1, 3, 5, 7, 9, 11, 13}, +𝑠 = {0.01, 0.03, 0.05, 0.07, 0.09}. +MLSF +This method is a second-order method. It improves the performance through meta- +label learning and specific feature selection. + +𝑘 = {2,4,6,8}, 𝜀 = {0.1,1,10}, +𝛼 = {0.1,0.5,0.9}, 𝛾 = {0.1,1,10}. +ML-TSK FS +This method is a second-order method that uses the correlation between any two la- +bels to improve performance. To realize the transparency, it uses fuzzy rules to +model the inference relationship between features and labels. This method does not +consider the influence of label noise. + +𝐾 = {2,3,4,5}, +𝛼 = {0.01,0.1,1,10,100}, +𝛽 = {0.01,0.1,1,10,100}. +CC +This method is a high-order method which adds the prediction result of the previous +label to the feature space to participate in the prediction of the next label. The -in- +sensitive learning (a fuzzy method) by solving a system of linear inequalities +(LSSLI) [54] is used as the binary classifier to improve the robustness. + +𝐶 = 2. ^(−5: 1: 5), +𝑀 = {2, 3, 4, 5, 6, 7, 8, 9}. +RAkEL +This method is a high-order method. In this method, the label space is randomly di- +vided into multiple label subspaces, and the prediction result of a label is associated +with other labels in the subspace. + +𝑘 = 𝑁./(12: −2: 2) (N is the instance number), +𝛼 = {0.1, 0.3, 0.5, 0.7, 0.9}. +CorrLog +This method is a high-order method. It achieves robustness by constructing the asso- +ciation between a label and all other labels. + +𝑟ℎ𝑜1 = {0.001, 0.003,0.005,0.007, 0.009, 0.01, +0.03,0.05,0.07,0.09,0.1,0.3,0.5,0.7,0.9}, +𝑟ℎ𝑜2 = {0.001, 0.005, 0.01, 0.05,0.1,0.5}. + +HNOML +This method is a high-order method. It designs a label enrichment matrix to improve +the robustness. + +𝛼 = {0.01,0.1,1,10}, +𝛽 = {0.01,0.1,1,10,100}, +𝛾 = {0.01,0.1,1,10}. + +R-MLTSK-FS +(ours) +The method proposed in this paper. It is a second-order method and achieves the +transparency and robustness against label noise through fuzzy rules, correlation en- +hancement learning, soft multilabel loss function construction, and soft label learn- +ing. +𝛼 = {0.001,0.005,0.01,0.05,0.1,0.5,1,5,10,50,100}, +𝛽 = {0.001,0.005,0.01,0.05,0.1,0.5,1,5,10,50,100}, +𝛾 = {0.001,0.005,0.01,0.05,0.1,0.5,1,5,10,50,100}, +𝑘 = {2,3}. + +D. Performance Analysis +1) Classification Performance Evaluation +To verify the effectiveness of R-MLTSK-FS, we compare the +R-MLTSK-FS with eight methods on 10 datasets. The experi- +mental results, expressed in terms of the mean and standard de- +viation (inside brackets) of the four metrics, are shown in Table +III. For each dataset, the best value of each metric is bold-faced. +We can see that compared to the eight methods, the overall per- +formance of R-MLTSK-FS is the best on all the metrics. This +is attributable to the three mechanisms introduced. + +2) Robustness Analysis +In order to verify the robustness of R-MLTSK-FS against la- +bel noise, we introduce label noise to the data and evaluate the +performance. Specifically, we randomly select 0%, 10%, 20%, +30% and 40% samples from the training set, and then create +noise by changing their related (unrelated) labels to unrelated +(related) ones. The 5-fold cross-validation strategy is adopted +in the experiment. Fig. 2 shows the experimental results, from +which the following findings are obtained: +(1) Despite the increase in the amount of noise in the experi- +ments, the proposed R-MLTSK-FS maintains outstanding clas- +sification performance, indicating the effectiveness of the three +mechanisms introduced in reducing the influence of label noise. +(2) Label noise has different effect on the comparison meth- +ods. For example, the performance of MLkNN in the presence +of label noise is unstable because the robustness of MLkNN +against noisy labels is affected by the number of nearest neigh- +bors. For RAkEL and CorrLog, their performance is unsatisfac- +tory since they ignore label noise in modeling the correlation +between labels. For ML-TSK FS, its overall robustness is infe- +rior to the proposed method as it also ignores the influence of +label noise in model training. + +3) Effectiveness Analysis of Soft Label Learning +To evaluate the effectiveness of R-MLTSK-FS in soft label +learning, we study the influence weights 𝑺 with three synthetic +multilabel datasets, namely Independence dataset, Equality da- +taset and Union dataset [55], each containing 1000 samples. +For each sample, the feature dimension is 20 and the label di- +mension is 5. Each feature in the synthetic datasets is normal- +ized in [0, 1]. +Each synthetic dataset has five labels, 𝒴1, …, 𝒴5. For the +first four labels, their logical relationships are designed as fol- +lows: +Independence dataset: The first four labels 𝒴1 , 𝒴2 , 𝒴3 +and 𝒴4 are independent of each other. +Equality dataset: 𝒴1 = 𝒴2 and 𝒴3 = 𝒴4 . That is, for a +sample (𝒙𝑖, 𝒚𝑖) (1 ≤ 𝑖 ≤ 1000), 𝑦𝑖1 = 𝑦𝑖2 and 𝑦𝑖3 = 𝑦𝑖4. +Union dataset: 𝒴1 = 𝒴2 ∨ 𝒴3 ∨ 𝒴4. That is, for a sample +(𝒙𝑖, 𝒚𝑖) (1 ≤ 𝑖 ≤ 1000), if 𝑦𝑖2 = 1 or 𝑦𝑖3 = 1 or 𝑦𝑖4 = 1, +then 𝑦𝑖1 = 1, otherwise, 𝑦𝑖1 = 0. + +TABLE III + +MEAN (SD) OF THE METRICS OF THE MULTILABEL CLASSIFICATION METHODS + +Datasets + +Methods + + +Met- +rics +BR +MLkNN +MLSF +ML-TSK FS +CC +RAkEL +CorrLog +HNOML +R-MLTSK-FS +Arts +AP +0.6270 +(0.0076) +0.5454 +(0.0082) +0.4977 +(0.0859) +0.6207 +(0.0141) +0.6164 +(0.0084) +0.2682 +(0.0285) +0.3646 +(0.0482) +0.6090 +(0.0082) +0.6289 +(0.0130) +HL +0.0902 +(0.0050) +0.0629 +(0.0007) +0.0604 +(0.0022) +0.0529 +(0.0019) +0.1025 +(0.0011) +0.1950 +(0.0092) +0.0597 +(0.0018) +0.0573 +(0.0009) +0.0546 +(0.0017) +RL +0.1266 +(0.0042) +0.1396 +(0.0028) +0.1257 +(0.0309) +0.1161 +(0.0039) +0.1300 +(0.0069) +0.4123 +(0.0325) +0.3865 +(0.0878) +0.1509 +(0.0052) +0.1118 +(0.0075) +CV +0.1965 +(0.0053) +0.1981 +(0.0036) +0.3047 +(0.0663) +0.1807 +(0.0083) +0.2054 +(0.0082) +0.8363 +(0.0369) +0.4724 +(0.0694) +0.2371 +(0.0045) +0.1720 +(0.0073) +Birds +AP +0.3422 +(0.0340) +0.2303 +(0.0185) +0.2712 +(0.0203) +0.3438 +(0.0347) +0.3360 +(0.0174) +0.3591 +(0.0319) +0.2124 +(0.0230) +0.3352 +(0.0325) +0.3694 +(0.0354) +HL +0.0556 +(0.0022) +0.0551 +(0.0058) +0.0648 +(0.0027) +0.0514 +(0.0038) +0.0545 +(0.0033) +0.0446 +(0.0032) +0.0451 +(0.0027) +0.0515 +(0.0065) +0.0430 +(0.0063) +RL +0.0983 +(0.0230) +0.1565 +(0.0127) +0.0807 +(0.0205) +0.0863 +(0.0221) +0.1097 +(0.0055) +0.6509 +(0.0634) +0.1611 +(0.0067) +0.0968 +(0.0215) +0.0710 +(0.0124) +CV +0.1311 +(0.0151) +0.1887 +(0.0203) +0.1699 +(0.0495) +0.1132 +(0.0315) +0.1445 +(0.0094) +0.7032 +(0.0364) +0.1939 +(0.0141) +0.1179 +(0.0188) +0.0957 +(0.0193) +CAL500 +AP +0.5048 +(0.0055) +0.4965 +(0.0037) +0.4906 +(0.0119) +0.5075 +(0.0104) +0.4541 +(0.0088) +0.2150 +(0.0047) +0.3108 +(0.0171) +0.4314 +(0.1844) +0.5153 +(0.0152) +HL +0.1447 +(0.0034) +0.1371 +(0.0031) +0.1368 +(0.0027) +0.1368 +(0.0027) +0.1442 +(0.0026) +0.1363 +(0.0036) +0.1371 +(0.0046) +0.1411 +(0.0072) +0.1358 +(0.0034) +RL +0.1879 +(0.0058) +0.1822 +(0.0043) +0.1780 +(0.0053) +0.1763 +(0.0035) +0.2515 +(0.0085) +0.6145 +(0.0161) +0.6750 +(0.1145) +0.1423 +(0.0797) +0.1744 +(0.0012) +CV +0.7656 +(0.0132) +0.7583 +(0.0122) +0.7600 +(0.0132) +0.7380 +(0.0091) +0.9085 +(0.0105) +0.7835 +(0.0264) +0.8722 +(0.0119) +0.7669 +(0.0579) +0.7348 +(0.0278) +Corel5k +AP +0.3044 +(0.0068) +0.2561 +(0.0077) +0.2134 +(0.0178) +0.3064 +(0.0003) +0.2639 +(0.0061) +0.0652 +(0.0032) +0.2079 +(0.0085) +0.2884 +(0.0105) +0.3070 +(0.0070) +HL +0.0094 +(0.0001) +0.0094 +(0.0001) +0.0094 +(0.0001) +0.0094 +(0.0003) +0.0094 +(0.0001) +0.0197 +(0.0002) +0.0094 +(0.0003) +0.0111 +(0.0006) +0.0094 +(0.0001) +RL +0.1649 +(0.0044) +0.1313 +(0.0040) +0.2591 +(0.0290) +0.1294 +(0.0047) +0.1784 +(0.0068) +0.5564 +(0.0279) +0.1432 +(0.0032) +0.1119 +(0.2279) +0.1092 +(0.0028) +CV +0.3852 +(0.0045) +0.3023 +(0.0059) +0.6994 +(0.0983) +0.3018 +(0.0108) +0.4288 +(0.0108) +0.5552 +(0.0167) +0.3207 +(0.0101) +0.3678 +(0.0092) +0.2600 +(0.0090) +Flags +AP +0.8101 +(0.0316) +0.8020 +(0.0415) +0.8163 +(0.0226) +0.8176 +(0.0118) +0.8076 +(0.0413) +0.6581 +(0.0544) +0.7704 +(0.0180) +0.8080 +(0.0110) +0.8209 +(0.0391) +HL +0.2796 +(0.0216) +0.3275 +(0.0272) +0.2768 +(0.0155) +0.2649 +(0.0254) +0.2711 +(0.0307) +0.2755 +(0.0323) +0.2856 +(0.0258) +0.2711 +(0.0124) +0.2647 +(0.0438) +RL +0.2155 +(0.0341) +0.2443 +(0.0374) +0.1374 +(0.0066) +0.2132 +(0.0173) +0.2340 +(0.0495) +0.6030 +(0.0419) +0.3566 +(0.0408) +0.2178 +(0.0159) +0.2054 +(0.0345) +CV +0.5523 +(0.0159) +0.5626 +(0.0198) +0.5524 +(0.0206) +0.5232 +(0.0127) +0.5553 +(0.0123) +0.8903 +(0.0252) +0.5486 +(0.0150) +0.5431 +(0.0341) +0.5318 +(0.0276) +Genbase +AP +0.9922 +(0.0067) +0.9910 +(0.0043) +0.9913 +(0.0051) +0.9968 +(0.0027) +0.9802 +(0.0181) +0.7784 +(0.0697) +0.9717 +(0.0097) +0.9941 +(0.0050) +0.9977 +(0.0031) +HL +0.0011 +(0.0006) +0.0016 +(0.0005) +0.0044 +(0.0016) +0.0015 +(0.0017) +0.0095 +(0.0033) +0.0022 +(0.0012) +0.0022 +(0.0007) +0.0020 +(0.0015) +0.0010 +(0.0012) +RL +0.0035 +(0.0049) +0.0061 +(0.0040) +0.0038 +(0.0026) +0.0011 +(0.0009) +0.0087 +(0.0081) +0.0242 +(0.0184) +0.0355 +(0.0095) +0.0006 +(0.0007) +0.0006 +(0.0005) +CV +0.0150 +(0.0061) +0.0192 +(0.0073) +0.0195 +(0.0073) +0.0105 +(0.0042) +0.0244 +(0.0154) +0.0588 +(0.0159) +0.0407 +(0.0063) +0.0126 +(0.0046) +0.0102 +(0.0021) +Medical +AP +0.8755 +(0.0266) +0.8067 +(0.0128) +0.8272 +(0.0250) +0.8959 +(0.0143) +0.8765 +(0.0307) +0.4443 +(0.0219) +0.7562 +(0.0181) +0.8761 +(0.0495) +0.8822 +(0.0150) +HL +0.0142 +(0.0018) +0.0156 +(0.0004) +0.0131 +(0.0012) +0.0107 +(0.0006) +0.0125 +(0.0014) +0.0109 +(0.0008) +0.0113 +(0.0007) +0.0213 +(0.0085) +0.0105 +(0.0019) +RL +0.0274 +(0.0147) +0.0430 +(0.0061) +0.0273 +(0.0038) +0.0371 +(0.0136) +0.0311 +(0.0175) +0.1079 +(0.0250) +0.2742 +(0.0258) +0.0232 +(0.0320) +0.0197 +(0.0039) +CV +0.0415 +(0.0186) +0.0629 +(0.0056) +0.0717 +(0.0082) +0.0363 +(0.0068) +0.0453 +(0.0226) +0.1394 +(0.0304) +0.1969 +(0.0280) +0.0357 +(0.0217) +0.0308 +(0.0105) +Mirflickr +AP +0.4540 +(0.0421) +0.5096 +(0.0028) +0.2906 +(0.0156) +0.5239 +(0.0045) +0.4703 +(0.0019) +0.2216 +(0.0030) +0.4779 +(0.0085) +0.5121 +(0.0084) +0.5246 +(0.0015) +HL +0.1528 +(0.0122) +0.1533 +(0.0006) +0.1543 +(0.0010) +0.1521 +(0.0005) +0.1588 +(0.0010) +0.2122 +(0.0030) +0.1548 +(0.0005) +0.1523 +(0.0022) +0.1521 +(0.0004) +RL +0.3218 +(0.0419) +0.2050 +(0.0027) +0.2616 +(0.0012) +0.1946 +(0.0015) +0.2444 +(0.0015) +0.5694 +(0.0087) +0.2146 +(0.0028) +0.2106 +(0.0097) +0.1929 +(0.0012) +CV +0.6120 +(0.0327) +0.4395 +(0.0045) +0.4703 +(0.0082) +0.4190 +(0.0031) +0.5314 +(0.0037) +0.9937 +(0.0021) +0.4495 +(0.0041) +0.4434 +(0.0043) +0.4182 +(0.0051) +Recreation +AP +0.6363 +(0.0151) +0.5333 +(0.0092) +0.4817 +(0.0426) +0.6362 +(0.0061) +0.6286 +(0.0152) +0.2922 +(0.0193) +0.2104 +(0.0247) +0.6062 +(0.0076) +0.6366 +(0.0058) +HL +0.0905 +(0.0014) +0.0647 +(0.0012) +0.0637 +(0.0014) +0.0592 +(0.0012) +0.0998 +(0.0019) +0.2923 +(0.0148) +0.0583 +(0.0010) +0.0563 +(0.0021) +0.0553 +(0.0017) +RL +0.1391 +(0.0082) +0.1640 +(0.0011) +0.1408 +(0.0410) +0.1297 +(0.0020) +0.1400 +(0.0083) +0.4073 +(0.0155) +0.4839 +(0.0119) +0.1989 +(0.0061) +0.1246 +(0.0058) +CV +0.1877 +(0.0117) +0.2035 +(0.0040) +0.3076 +(0.0867) +0.1697 +(0.0043) +0.1906 +(0.0125) +0.8912 +(0.0206) +0.4554 +(0.0240) +0.2545 +(0.0113) +0.1675 +(0.0054) +Science +AP +0.5983 +(0.0132) +0.5134 +(0.0119) +0.4461 +(0.0063) +0.5978 +(0.0217) +0.5861 +(0.0125) +0.2333 +(0.0115) +0.2492 +(0.0106) +0.5737 +(0.0144) +0.5984 +(0.0051) +HL +0.0526 +(0.0007) +0.0363 +(0.0006) +0.0343 +(0.0011) +0.0329 +(0.0004) +0.0603 +(0.0009) +0.1288 +(0.0087) +0.0370 +(0.0036) +0.0333 +(0.0004) +0.0324 +(0.0009) +RL +0.1140 +(0.0068) +0.1211 +(0.0046) +0.0990 +(0.0143) +0.0996 +(0.0072) +0.1128 +(0.0071) +0.3794 +(0.0352) +0.4989 +(0.1339) +0.1867 +(0.0086) +0.0976 +(0.0050) +CV +0.1596 +(0.0089) +0.1574 +(0.0050) +0.1823 +(0.0269) +0.1357 +(0.0088) +0.1620 +(0.0093) +0.7443 +(0.0366) +0.3614 +(0.0219) +0.2434 +(0.0061) +0.1321 +(0.0058) + + + + + + + + (a) Arts + (b) Birds + (c) CAL500 + (d) Corel5k + (e) Flags + + + + + + (f) Genbase + (g) Medical + (h) Mirflickr + (i) Recreation + (j) Science +Fig. 2 Performance in terms of AP on datasets with label noise. (Noise ratio is defined as the proportion of samples that are randomly selected from the training set +and their related (unrelated) labels are changed to unrelated (related) ones. The larger the value of AP, the better the classification performance.) + +TABLE IV +INFLUENCE WEIGHTS (S) OF ORIGINAL LABELS ON A SOFT LABEL IN INDEPENDENCE DATASET + + +original label 1 (𝒴1) +original label 2 (𝒴2) +original label 3 (𝒴3) +original label 4 (𝒴4) +original label 5 (𝒴5) +soft label 1 (𝒴1 +′) +0.2016 +0.0510 +0.0697 +0.0462 +0.0797 +soft label 2 (𝒴2 +′) +0.1409 +0.3149 +0.1921 +0.1552 +0.2182 +soft label 3 (𝒴3 +′) +0.2447 +0.2523 +0.4662 +0.2628 +0.3666 +soft label 4 (𝒴4 +′) +0.0031 +0.0051 +0.0053 +0.1191 +0.0061 +soft label 5 (𝒴5 +′) +0.1179 +0.1046 +0.1068 +0.1281 +0.2832 +N.B. 𝒴1, 𝒴2, 𝒴3 and 𝒴4 are independent. 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4). + +TABLE V +INFLUENCE WEIGHTS (S) OF ORIGINAL LABELS ON A SOFT LABEL IN EQUALITY DATASET + + +original label 1 (𝒴1) +original label 2 (𝒴2) +original label 3 (𝒴3) +original label 4 (𝒴4) +original label 5 (𝒴5) +soft label 1 (𝒴1 +′) +0.3645 +0.3645 +0.2252 +0.2252 +0.6172 +soft label 2 (𝒴2 +′) +0.3645 +0.3645 +0.2252 +0.2252 +0.6172 +soft label 3 (𝒴3 +′) +0.1900 +0.1900 +0.2456 +0.2456 +0.4350 +soft label 4 (𝒴4 +′) +0.1900 +0.1900 +0.2456 +0.2456 +0.4350 +soft label 5 (𝒴5 +′) +0.1252 +0.1252 +0.1260 +0.1260 +0.4480 +N.B. 𝒴1 = 𝒴2 and 𝒴3 = 𝒴4. 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4). + +TABLE VI +INFLUENCE WEIGHTS (S) OF ORIGINAL LABELS ON A SOFT LABEL IN UNION DATASET + + +original label 1 (𝒴1) +original label 2 (𝒴2) +original label 3 (𝒴3) +original label 4 (𝒴4) +original label 5 (𝒴5) +soft label 1 (𝒴1 +′) +0.2295 +0.0798 +0.0981 +0.1206 +0.2654 +soft label 2 (𝒴2 +′) +0.0791 +0.1529 +0.0363 +0.0551 +0.1327 +soft label 3 (𝒴3 +′) +0.1378 +0.0520 +0.1694 +0.1017 +0.2151 +soft label 4 (𝒴4 +′) +0.0077 +-0.0002 +0.0005 +0.0668 +0.0106 +soft label 5 (𝒴5 +′) +0.0649 +-0.0107 +-0.0264 +0.0351 +0.1057 +N.B. 𝒴1 = 𝒴2 ∨ 𝒴3 ∨ 𝒴4. 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4). + +The fifth label is mutually exclusive with the first four labels +(i.e., 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4) ). Specifically, +for a sample (𝒙𝑖, 𝒚𝑖) (1 ≤ 𝑖 ≤ 1000), if 𝑦𝑖1 = 0 and 𝑦𝑖2 = +0 and 𝑦𝑖3 = 0 and 𝑦𝑖4 = 0, then 𝑦𝑖5 = 1, otherwise, 𝑦𝑖5 = +0. +The learned influence weights 𝑺 for each of the three syn- +thetic datasets are shown in Tables IV-VI respectively. The fol- +lowing findings can be obtained from the tables: +(1) In Tables IV-VI, since the fifth label is mutually exclusive +with the first four labels (i.e., 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ +(¬𝒴3) ∧ (¬𝒴4)), reconstruction cannot be achieved with the +first four labels. From the results of influence weights in Tables +IV-VI, we can find that the influence of 𝒴5 on the soft label +𝒴5 +′ is most significant, whereas the influence of 𝒴1 ∼ 𝒴4 on +𝒴5 +′ is relatively small. +(2) In Table IV, the first four labels 𝒴1, 𝒴2, 𝒴3 and 𝒴4 +are independent of each other, and 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ + +(¬𝒴3) ∧ (¬𝒴4). The results of influence weights in Table IV +show that the effect of 𝒴1, 𝒴2, 𝒴3 and 𝒴4 on the soft labels +𝒴1 +′, 𝒴2 +′, 𝒴3 +′ and 𝒴4 +′, respectively, are significant. In addition, +the contribution of 𝒴5 to 𝒴1 +′, 𝒴2 +′, 𝒴3 +′ and 𝒴4 +′ is also obvi- +ous. +(3) In Table V, 𝒴1 = 𝒴2 , 𝒴3 = 𝒴4, and 𝒴5 = (¬𝒴1) ∧ +(¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4). The results of influence weights in +Table V reveal that 𝒴5 has a greater influence on the soft la- +bels 𝒴1 +′, 𝒴2 +′, 𝒴3 +′ and 𝒴4 +′. Meanwhile, it is obvious that 𝒴1 +and 𝒴2 have the same influence on 𝒴1 +′ (𝒴2 +′ ), and 𝒴3 and +𝒴4 have the same influence on 𝒴3 +′ (𝒴4 +′). +(4) In Table VI, 𝒴1 = 𝒴2 ∨ 𝒴3 ∨ 𝒴4 and 𝒴5 = (¬𝒴1) ∧ +(¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4) . From the results of influence +weights in Table VI, we can see that it is 𝒴1 and 𝒴5 that af- +fect the soft label 𝒴1 +′ significantly, and that the effect of 𝒴2 ∼ +𝒴4 on the soft label 𝒴1 +′ are similar. +The above findings are consistent with the logical relation- +ship we designed for the labels, which validates that the soft +label learning in R-MLTSK-FS is effective. + +4) Effectiveness Analysis of Correlation Enhancement Learn- +ing +In order to verify the effectiveness of the correlation en- +hancement learning mechanism in guiding the consequent vec- +tor optimization, we conduct correlation visualization experi- +ment on the Science dataset, where the dimension of label space +is 40. Specifically, the Pearson correlation coefficient is used to +measure the correlation between two vectors. The higher the +value of Pearson correlation coefficient, the stronger the corre- +lation between two vectors. Experimental results are shown in +Fig. 3, where Fig. 3(a) visualizes the correlation between any +two original labels, and Fig. 3(b) visualizes the correlation be- +tween any two optimized consequent vectors associated with +the corresponding labels. For an effective correlation enhance- +ment learning mechanism, the correlation coefficient between +two consequent vectors should be kept close to that between +their corresponding labels. + + + +(a) +(b) +Fig. 3 Visualization of label correlation learning on the Science dataset: (a) +visualization of the correlation coefficient between any two original label +vectors, and (b) visualization of the correlation coefficient between any two +consequent vectors associated with the corresponding labels. The higher the +value of correlation coefficient, the stronger the correlation between two +vectors. + +It is clear that there is little difference between Fig. 3(a) and +Fig. 3(b), indicating that the correlation between the labels can +closely guide the learning of the corresponding consequent vec- +tors, and demonstrating the effectiveness of the correlation en- +hancement learning mechanism. +5) Parameter Analysis +In this section, we analyze the influence of the hyperparam- +eters α, β, γ and K on the classification performance of R- +MLTSK-FS in terms of AP. In the analysis, we study the sensi- +tivity of the classification performance to a specific hyperpa- +rameter by keeping the other three fixed. For example, we fix +the values of β, γ and K, and adjust the value of α to analyze the +effect of α. The hyperparameters α, β and γ are varied within +{10-3, 10-2, 10-1, 100, 101, 102} and K is varied within {2, 3, 4, +5, 6, 7, 8, 9, 10}. The AP values of R-MLTSK-FS are obtained +with the 5-fold cross-validation strategy. + + + + (a) α + (b) β + + + (c) γ + (d) K + +Fig. 4 The influence of the hyperparameters (a) α, (b) β, (c) γ, +and (d) K on AP of the R-MLTSK-FS. + + +The experimental results are shown in Fig. 4, from which the +following observations are obtained: +(1) When α is in the range of (10-3, 100), the performance of +R-MLTSK-FS in terms of AP is stable for most datasets. In ad- +dition, AP decreases with increasing α for most datasets when +α is within (101, 102). For the CAL500 dataset, AP increases +with α. In general, R-MLTSK-FS is stable and can achieve op- +timal performance when α is in the range of (10-2, 100). +(2) In general, R-MLTSK-FS is sensitive to β when it is in +the range of (10-3, 100). It is stable and can reach an optimal AP +value for the 10 datasets when β is within (101, 102). +(3) For the hyperparameter γ, AP fluctuates in a similar way +for all the 10 datasets. In general, the performance of R- +MLTSK-FS is stable when γ is within (10-3, 10-1). The AP value +fluctuates significantly when γ is in the range of (10-1, 102), +while exhibiting a decreasing trend with increasing γ. In general, +optimal AP can be achieved for all the 10 datasets when γ is in +the range of (10-3, 10-1). +(4) The AP value for the 10 datasets fluctuates slightly with +increasing K. Optimal values of AP can be obtained when K is +within (4, 9). +According to the above analysis, it is necessary for R- +MLTSK-FS to adopt the grid search strategy and the cross-val- +idation strategy to get the optimal hyperparameters for different +datasets. + +6) Convergence Analysis +The Birds and Flags datasets are adopted in this part to inves- +tigate the convergence of the proposed method. The results are +shown in Fig. 5, where the vertical axis represents the absolute +value of the difference between the previous and the current +value of the objective function (denoted by df), and the hori- +zontal axis represents the number of iterations. It can be seen +from Fig. 5 that for the Birds and Flags datasets, R-MLTSK-FS +is convergent within 10 iterations. + + + + +(a) +(b) +Fig. 5 Convergence analysis for datasets (a) Birds and (b) Flags. + +7) Statistical Analysis +We employ the Friedman test and the Bonferroni-Dunn test +to evaluate the statistical significance of the difference observed +between the proposed R-MLTSK-FS and the eight comparison +methods [56]. The details are as follows. +(1) Friedman Test: Based on the experimental results in Ta- +ble III, we perform the Friedman test on the four metrics, i.e., +AP, HL, RL and CV. The null hypothesis is that there is no sig- +nificant difference between all the methods in terms of the four +metrics. For each metric, if the Friedman statistic FF is greater +than a critical value (i.e., 2.0698), the null hypothesis for that +metric is rejected, which means the difference is statistically +significant. The results of the Friedman test, corresponding to +the results in Table III, are shown in Table VII. It can be seen +from Table VII that the null hypotheses on AP, HL, RL and CV +are all rejected. This means that the differences in classification +performance of the nine methods are significant in terms of the +four metrics. Next, we conduct the post-hoc Bonferroni-Dunn +test to evaluate whether the difference in performance between +R-MLTSK-FS and the comparison methods is statistically sig- +nificant. + +TABLE VII +FRIEDMAN STATISTICS + +Evaluation met- +ric +FF +Critical value (α = 0.05) +AP +28.6045 +2.0698 +HL +6.6863 +RL +20.3718 +CV +26.6201 + +(2) Bonferroni-Dunn Test: According to the results in Fried- +man test, we conduct the post-hoc test based on the results of +AP, HL, RL and CV respectively, where R-MLTSK-FS is set +as the control method. First, we calculate the average rank of +the nine methods for each metric respectively. We also calcu- +late the critical difference (CD), which is a standard used for +evaluating the difference in average rank between the methods, +using the equation below: + + +(a) AP +(b) HL + + +(c) RL +(d) CV +Fig. 6 Comparison of R-MLTSK-FS (as control) with the other meth- +ods using the Bonferroni-Dunn test. The letter A refers to R-MLTSK- +FS, B to BR, C to MLkNN, D to MLSF, E to ML-TSK FS, F to CC, +G to RAkEL, H to CorrLog, and I to HNOML, respectively. + +CD = 𝑞𝛼√𝑛(𝑛 + 1) 6𝑀 +⁄ + +(38) +where n and M are the number of methods (n = 9) and the num- +ber of datasets (M = 10), respectively. With confidence level α += 0.05 and 𝑞𝛼 = 2.724, we have CD = 3.3362. +Fig. 6 gives the average rank of the nine methods, which are +shown on the horizontal line with ticks marking 1 to 9. The +smaller the average rank (i.e., closer to the right), the better the +method. As R-MLTSK-FS is at the rightmost position on the +horizontal line, for all the four metrics, it is the best among the +nine methods. A red line of length one CD is drawn from R- +MLTSK-FS to the left. For a method located within the span of +the red line, the difference in average rank between the method +and R-MLTSK-FS is less than one CD, indicating that the per- +formance difference between them is small. Otherwise, the dif- +ference is significant. The following conclusions can be drawn +from Fig. 6. Firstly, R-MLTSK-FS is superior to other methods +on the four metrics. Secondly, in general, the performance of +ML-TSK FS is the second best. Thirdly, the performance of +MLkNN, CC, RAkEL and CorrLog are significantly lower than +that of R-MLTSK-FS in terms of the four metrics. Fourthly, for +BR, MLSF and HNOML, their performance is mediocre. +V. CONCLUSION +The robust multilabel learning method R-MLTSK-FS with +strong fuzzy inference ability, label correlation learning ability +and robustness against noisy labels is proposed in this paper. +From the aspect of soft label learning, R-MLTSK-FS constructs +the soft label space to reduce the influence of label noise. From +the aspect of soft multilabel loss function construction, R- +MLTSK-FS utilizes the fuzzy rule-based TSK FS as a transpar- +ent model to build the inference relationship between input fea- +tures and soft labels, and then the loss function is constructed +based on TSK FS and soft labels to enhance model training. +From the aspect of correlation enhancement learning, R- +MLTSK-FS utilizes the correlation information between soft la- +bels to constrain the learning of model parameters and enhance +the learning ability. Experimental analyses on ten benchmark +multilabel datasets and three synthetic multilabel datasets show +the promising performance of R-MLTSK-FS. +Further research on R-MLTSK-FS will proceed along two +directions. First, we will reduce the complexity of soft label + +learning. Since R-MLTSK-FS considers all the original labels +for a soft label, which is computationally intensive, research +will be conducted to model with random label subsets for a soft +label to reduce the complexity. Second, we will simplify the +rule base of TSK FS. In R-MLTSK-FS, the fuzzy system trans- +forms all the original features into the fuzzy feature space. If the +dimension of the original feature space is large, the learning +speed of R-MLTSK-FS will be slow. Hence, a screening mech- +anism will be developed to identify representative subsets of the +original features to improve the learning efficiency. +REFERENCE +[1] W. W. Liu, H. B. Wang, X. B. Shen, and I. W. Tsang, "The emerging +trends of multi-label learning," IEEE Transactions on Pattern Analysis +and Machine Intelligence, vol. 44, no. 11, pp. 7955-7974, 2021. +[2] M. L. Zhang and Z. H. Zhou, "A review on multi-label learning +algorithms," IEEE Transactions on Knowledge and Data Engineering, vol. +26, no. 8, pp. 1819-1837, 2014. +[3] M. Monfort, B. Pan, K. Ramakrishnan, A. Andonian, B. A. McNamara, A. +Lascelles, Q. Fan, D. Gutfreund, R. Feris, and A. Oliva, "Multi-moments +in time: learning and interpreting models for multi-action video +understanding," IEEE Transactions on Pattern Analysis and Machine +Intelligence, vol. 44, no. 12, pp. 9434-9445, 2022. +[4] J. Speth and E. M. Hand, "Automated label noise identification for facial +attribute recognition," in Proc. the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2019, pp. 25-28. +[5] Q. W. Zhang, Y. Zhong, and M. L. Zhang, "Feature-induced labeling +information enrichment for multi-label learning," in Proc. the 32th AAAI +Conference on Artificial Intelligence, 2018, pp. 4446-4453. +[6] S. J. Huang, G. X. Li, W. Y. Huang, and S. Y. Li, "Incremental multi-label +learning with active queries," Journal of Computer Science and +Technology, vol. 35, no. 2, pp. 234-246, 2020. +[7] Q. Tan, G. Yu, J. Wang, C. Domeniconi, and X. Zhang, "Individuality-and +commonality-based multiview multilabel learning," IEEE Transactions on +Cybernetics, vol. 51, no. 3, pp. 1716-1727, 2019. +[8] H. Liu, X. Li, and S. Zhang, "Learning instance correlation functions for +multilabel classification," IEEE Transactions on Cybernetics, vol. 47, no. +2, pp. 499-510, 2017. +[9] J. Du and C. M. Vong, "Robust online multilabel learning under dynamic +changes in data distribution with labels," IEEE Transactions on +Cybernetics, vol. 50, no. 1, pp. 374-385, 2019. +[10] Y. Zhu, J. T. Kwok, and Z. H. Zhou, "Multi-label learning with global and +local label correlation," IEEE Transactions on Knowledge and Data +Engineering, vol. 30, no. 6, pp. 1081-1094, 2018. +[11] M. L. Zhang, Y. K. Li, H. Yang, and X. Y. Liu, "Towards class-imbalance +aware multi-label learning," IEEE Transactions on Cybernetics, vol. 52, +no. 6, pp. 4459-4471, 2020. +[12] J. Ma, H. Zhang, and T. W. Chow, "Multilabel classification with label- +specific features and classifiers: a coarse-and fine-tuned framework," +IEEE Transactions on Cybernetics, vol. 51, no. 2, pp. 1028-1042, 2019. +[13] E. Lughofer, "Evolving multi-label fuzzy classifier," Information Sciences, +vol. 597, pp. 1-23, 2022. +[14] R. Cerri, M. P. Basgalupp, R. C. Barros, and A. C.P.L.F Carvalho, +"Inducing hierarchical multi-label classification rules with genetic +algorithms," Applied Soft Computing Journal, vol. 77, pp. 584-604, 2019. +[15] H. Y. Jiang, J. Xu, R. J. Shi, K. Yang, D. D. Zhang, M. D. Gao, H. Ma, +and W. Qian, "A multi-label deep learning model with interpretable Grad- +CAM for diabetic retinopathy classification," in Proc. IEEE Engineering +in Medicine and Biology Society, 2020, pp. 1560-1563. +[16] J. Wang, Y. J. Lin, L. Z. Li, Y. A. Wang, M. Y. Xu, and J. K. Chen, "Multi- +label cause feature selection based on neighbourhood mutual information," +International Journal of Machine Learning and Cybernetics, vol. 13, no. +11, pp. 3509-3522, 2022. +[17] Q. Lou, Z. Deng, Z. Xiao, K. S. Choi, and S. Wang, "Multi-label Takagi- +Sugeno-Kang fuzzy system," IEEE Transactions on Fuzzy Systems, vol. +30, no. 9, pp. 3410-3425, 2021. +[18] E. Cole, O. M. Aodha, T. Lorieul, P. Perona, D. Morris, and N. Jojic, +"Multi-label learning from single positive labels," in Proc. the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, 2021, pp. 933- +942. +[19] M. Hu, H. Han, S. Shan, and X. Chen, "Weakly supervised image +classifiaction through noise regularization," in Proc. the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, 2019, pp. +11517-11525. +[20] M. K. Xie and S. J. Huang, "CCMN: a general framework for learning +with class-conditional multi-label noise," IEEE Transactions on Pattern +Analysis +and +Machine +Intelligence, +2022. +Doi: +10.1109/TPAMI.2022.3141240 +[21] A. K. Aksoy, M. Ravanbakhsh, and B. Demir, "Multi-label noise robust +collaborative learning for remote sensing image classification," IEEE +Transactions on Neural Networks and Learning Systems, 2022. Doi: +10.1109/TNNLS.2022.3209992 +[22] M. K. Xie and S. J. Huang, "Partial multi-label learning with noisy label +identification," IEEE Transactions on Pattern Analysis and Machine +Intelligence, vol. 44, no. 7, pp. 3676-3687, 2022. +[23] S. Rajeswar, P. Rodriguez, S. Singhal, D. Vazquez, and A. Courville, +"Multi-label iterated learning for image classification with label +ambiguity," in Proc. the IEEE/CVF Conference on Computer Vision and +Pattern Recognition, 2022, pp. 4783-4793. +[24] G. Lyu, S. Feng, and Y. Li, "Noisy label tolerance: a new perspective of +partial multi-label learning," Information Sciences, vol. 543, pp. 454-466, +2021. +[25] Z. Deng, P. Xu, L. Xie, K. S. Choi, and S. Wang, "Transductive joint- +knowledge-transfer TSK FS for recognition of epileptic EEG signals," +IEEE Transactions on Neural Systems and Rehabilitation Engineering, +vol. 26, no. 8, pp. 1481-1494, 2018. +[26] C. Yang, Z. Deng, K. S. Choi, and S. Wang, "Takagi-Sugeno-Kang +transfer learning fuzzy logic system for the adaptive recognition of +epileptic electroencephalogram signals," IEEE Transactions on Fuzzy +Systems, vol. 24, no. 5, pp. 1079-1094, 2016. +[27] T. Zhang, Z. Deng, D. Wu, and S. Wang, "Multiview fuzzy logic system +with the cooperation between visible and hidden views," IEEE +Transactions on Fuzzy Systems, vol. 27, no. 6, pp. 1162-1173, 2019. +[28] Y. Jiang, Z. Deng, K. S. Choi, F. L. Chung, and S. Wang, "A novel multi- +task TSK fuzzy classifier and its enhanced version for labeling-risk-aware +multi-task classification," Information Sciences, vol. 357, no. C, pp. 39-60, +2016. +[29] Y. Jiang, Z. Deng, F. L. Chung, G. Wang, P. Qian, K. S. Choi, et al., +"Recognition of epileptic EEG signals using a novel multiview TSK fuzzy +system," IEEE Transactions on Fuzzy Systems, vol. 25, no. 1, pp. 3-20, +2017. +[30] L. Kong, W. He, W. Yang, Q. Li, and O. Kaynak, "Fuzzy approximation- +based finite-time control for a robot with actuator saturation under time- +varying constraints of work space," IEEE Transactions on Cybernetics, +vol. 51, no. 10, pp. 4873-4884, 2020. +[31] Q. Liao and D. Sun, "Sparse and decoupling control strategies based on +Takagi–Sugeno fuzzy models," IEEE Transactions on Cybernetics, vol. +51, no. 2, pp. 947-960, 2019. +[32] S. C. Tong, Y. M. Li, G. Feng, and T. S. Li, "Observer-based adaptive +fuzzy backstepping dynamic surface control for a class of MIMO +nonlinear systems," IEEE Transactions on Systems, Man, and Cybernetics, +Part B (Cybernetics), vol. 41, no. 4, pp. 1124-1135, 2011. +[33] P. Xu, Z. Deng, J. Wang, Q. Zhang, K. S. Choi, and S. Wang, "Transfer +representation learning with TSK fuzzy system," IEEE Transactions on +Fuzzy Systems, vol. 29, no. 3, pp. 649-663, 2019. +[34] Y. Guo and W. Xue, "Probabilistic multi-label classification with sparse +feature learning," in Proc. the 23rd International Joint Conference on +Artificial Intelligence, 2013, pp. 1373-1379. +[35] Y. Yang, H. T. Shen, Z. Ma, Z. Huang, and X. Zhou, "L2,1-norm +regularized discriminative feature selection for unsupervised learning," in +Proc. the 22nd International Joint Conference on Artificial Intelligence, +2011, pp. 1589-1594. +[36] F. Nie, H. Huang, X. Cai, and C. H. Q. Ding, "Efficient and robust feature +selection via joint ℓ2, 1-norms minimization," in Proc. the 23rd +International Conference on Neural Information Processing Systems, +2010, pp. 1813-1821. +[37] M. L. Zhang and Z. H. Zhou, "Multilabel neural networks with +applications to functional genomics and text categorization," IEEE +Transactions on Knowledge and Data Engineering, vol. 18, no. 10, pp. +1338-1351, 2006. +[38] N. Li and Z. H. Zhou, "Selective ensemble of classifier chains," in +International Workshop on Multiple Classifier Systems, 2013, pp. 146-156. +[39] J. Huang, G. Li, Q. Huang, and X. Wu, "Learning label specific features +for multi-label classification," in Proc. IEEE International Conference on +Data Mining, 2015, pp. 181-190. + +[40] S. J. Huang, W. Gao, and Z. H. Zhou, "Fast multi-instance multi-label +learning," IEEE Transactions on Pattern Analysis and Machine +Intelligence, vol. 41, no. 11, pp. 2614-2627, 2018. +[41] S. Ji, L. Tang, S. Yu, and J. Ye, "Extracting shared subspace for multi- +label classification," in Proc. the 14th ACM SIGKDD International +Conference on Knowledge Discovery and Data Mining, 2008, pp. 381-389. +[42] S. Wang, F. L. Chung, H. B. Shen, and D. Hu, "Cascaded centralized TSK +fuzzy system: universal approximator and high interpretation," Applied +Soft Computing, vol. 5, no. 2, pp. 131-145, 2005. +[43] D. Y. Hu and L. Reichel, "Krylov-subspace methods for the sylvester +equation," Linear Algebra and its Applications, vol. 172, no. 15, pp. 283- +313, 1992. +[44] G. Chen, Y. Song, F. Wang, and C. Zhang, "Semi-supervised multi-label +learning by solving a sylvester equation," in Proc. the Siam International +Conference on Data Mining, 2008, pp. 410-419. +[45] D. C. Sorensen and A. C. Antoulas, "The sylvester equation and +approximate balanced reduction," in Proc. Linear Algebra and its +Applications, 2002, pp. 351-352. +[46] S. Sun and D. Zong, "LCBM: a multi-view probabilistic model for multi- +label classification," IEEE Transactions on Pattern Analysis and Machine +Intelligence, vol. 43, no. 8, pp. 2682-2696, 2020. +[47] O. Luaces, J. Díez, J. Barranquero, J. J. Coz, and A. Bahamonde, "Binary +relevance efficacy for multilabel classification," Progress in Artificial +Intelligence, vol. 1, no. 4, pp. 303-313, 2012. +[48] M. L. Zhang and Z. H. Zhou, "ML-KNN: a lazy learning approach to +multi-label learning," Pattern Recognition, vol. 40, no. 7, pp. 2038-2048, +2007. +[49] L. Sun, M. Kudo, and K. Kimura, "Multi-label classification with meta- +label-specific features," in Proc. the 23rd International Conference on +Pattern Recognition, 2016, pp. 1613-1618. +[50] J. Read, B. Pfahringer, G. Holmes, and E. Frank, "Classifier chains for +multi-label classification," Machine Learning, vol. 85, no. 3, pp. 333-359, +2011. +[51] Tsoumakas, Grigorios, Katakis, Ioannis, and Vlahavas, "Random k- +labelsets for multilabel classification," IEEE Transactions on Knowledge +and Data Engineering, vol. 23, no. 7, pp. 1079-1089, 2011. +[52] W. Bian, B. Xie, and D. Tao, "Corrlog: correlated logistic models for joint +prediction of multiple labels," in Proc. the 15th International Conference +on Artificial Intelligence and Statistics, 2012, pp. 109-117. +[53] C. Zhang, Z. Yu, H. Fu, P. Zhu, L. Chen, and Q. Hu, "Hybrid noise- +oriented multilabel learning," IEEE Transactions on Cybermetics, vol. 50, +no. 6, pp. 2837-2850, 2019. +[54] J. Łȩski, "Improving the generalization ability of neuro-fuzzy systems by +ε-insensitive learning," International Journal of Applied Mathematics and +Computer Science, vol. 12, no. 3, pp. 437-447, 2002. +[55] S. J. Huang, Y. Yu, and Z. H. Zhou, "Multi-label hypothesis reuse," in +Proc. the 18th ACM SIGKDD International Conference on Knowledge +Discovery and Data Mining, 2012, pp. 525-533. +[56] J. Demšar, "Statistical comparisons of classifiers over multiple data sets," +Journal of Machine Learning Research, vol. 7, no. 1, pp. 1-30, 2006. + + diff --git a/1tE1T4oBgHgl3EQflQSM/content/tmp_files/load_file.txt b/1tE1T4oBgHgl3EQflQSM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3786962f79631f95ec5001c7355861c3474fd330 --- /dev/null +++ b/1tE1T4oBgHgl3EQflQSM/content/tmp_files/load_file.txt @@ -0,0 +1,1831 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf,len=1830 +page_content='Abstract—Model transparency, label correlation learning and the robustness to label noise are crucial for multilabel learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' However, few existing methods study these three characteristics simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' To address this challenge, we propose the robust multilabel Takagi-Sugeno-Kang fuzzy system (R-MLTSK-FS) with three mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' First, we design a soft label learning mech- anism to reduce the effect of label noise by explicitly measuring the interactions between labels, which is also the basis of the other two mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Second, the rule-based TSK FS is used as the base model to efficiently model the inference relationship between fea- tures and soft labels in a more transparent way than many existing multilabel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Third, to further improve the performance of multilabel learning, we build a correlation enhancement learning mechanism based on the soft label space and the fuzzy feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1Extensive experiments are conducted to demonstrate the superiority of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Index Terms—Multilabel classification, label correlation, model transparency, label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' INTRODUCTION ULTILABEL learning concerns instances that can be asso- ciated with more than one labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, an article can be labeled as being related to “politics”, “culture” and “re- ligion” at the same time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' and a travel photo can be given the labels “beach”, “sunrise”, “sail” and “tourist” simultaneously because of the presence of the corresponding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For mul- tilabel learning, label correlation learning, model transparency and robustness against label noise are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Constructing the correlation between labels is the basic work to improve the performance of multilabel learning [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' A transparent struc- ture is important to enhance the interpretability of multilabel learning [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' And robustness against label noise enhances the effectiveness in practical applications under noisy environment [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For label correlation learning, existing multilabel methods are mainly based on first-order [5], second-order [6] and high- order [7] strategies to consider the correlation between labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' This work was supported in part by the National key R & D plan under Grant (2022YFE0112400), the NSFC under Grant 62176105, the Six Talent Peaks Project in Jiangsu Province under Grant XYDXX-056, the Hong Kong Re- search Grants Council (PolyU 152006/19E), the Project of Strategic Importance of the Hong Kong Polytechnic University (1-ZE1V) and the Postgraduate Re- search & Practice innovation Program of Jiangsu Province under Grant KYCX22_2313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (Corresponding author: Zhaohong Deng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang are with the School of Artificial Intelligence and Computer Science, Jiangnan University and Jiangsu Key Laboratory of Digital Design and Software Technology, Wuxi 214122, China, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lou is with the Centre for First-order methods ignore label correlation and adopt label- by-label approach for multilabel learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, sparse weighted instance-based multilabel (SWIM) realizes the mul- tilabel learning only based on the association between instances [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Second-order methods build the pairwise relationship be- tween labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, labels related to the sample are ranked before labels unrelated to the sample [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Multilabel learning with global and local label correlation (GLOCAL) de- composes the Laplacian matrix to indirectly learn the correla- tion between any two labels [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' High-order methods construct the correlation between multiple labels simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For ex- ample, cross-coupling aggregation (COCOA) first models the correlation between random label pairs and then aggregates their learning effects [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Multilabel classification with label- specific features and label-specific classifiers (MLC-LFLC) in- troduces the sparse learning to analyze the dependency between a single label and other labels [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For model transparency in multilabel learning, existing work is mainly based on rules or logical inference to achieve trans- parency [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, hierarchical multilabel classifica- tion with a genetic algorithm (HMC-GA) [14] utilizes the ge- netic algorithm to induce classification rules for protein func- tion prediction which belongs to hierarchical multilabel learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The gradient-weighted class activation mapping (Grad- CAM) is used in [15] to realize the inferential interpretation for predicted label results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The causal discovery is exploited in [16] to analyze the specific features of a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The multilabel Tak- agi-Sugeno-Kang fuzzy system (TSK FS), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', ML-TSK FS [17] offers good transparency through fuzzy rule-based structure and fuzzy inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Among the above existing multilabel methods, ML-TSK FS has shown more promising performance because it realizes the complete inference process from feature to label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For robustness against label noise, much work has been stud- ied because of the urgent need of practical application [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, class-conditional multilabel noise (CCMN) [20] designs two unbiased estimators with error bounds to reduce the influence of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Multilabel noise robust collaborative learning (RCML) [21] employs the group lasso to detect noisy Smart Health, and the School of Nursing, the Hong Kong Polytechnic Univer- sity, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (e-mail: 6171610005@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='jiangnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' wxwangst@ali- yun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng is with the School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi 214122, China, and Key Laboratory of Computa- tional Neuroscience and Brain-Inspired Intelligence (LCNBI) and ZJLab, Shanghai 200433, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (e-mail: dengzhaohong@jiangnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Choi is with the Centre for Smart Health, Hong Kong Polytechnic Uni- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (e-mail: kschoi@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' A Robust Multilabel Method Integrating Rule-based Transparent Model, Soft Label Correlation Learning and Label Noise Resistance Qiongdan Lou, Zhaohong Deng, Senior Member, IEEE, Kup-Sze Choi, Shitong Wang M labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Partial multilabel learning with noisy label identification (PML-NI) [22] builds the feature-induce noise term to identify noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Multilabel iterated learning (MILe) [23] strength- ens learning bottleneck for successive generations of teacher and student networks to improve the robustness against label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Different from removing noisy labels directly, noisy la- bel tolerated partial multilabel learning (NATAL) [24] reduces the impact of noisy labels by assuming that the label infor- mation is precise and feature information is inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The above related work indicates that the importance of label correlation, model transparency and robustness against noisy labels has received extensive attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' However, such desira- ble characteristics are still rarely studied simultaneously in mul- tilabel learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Therefore, it is necessary to further study the multilabel method with transparency, label correlation learning ability and robustness to noise labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Based on the above analysis, we aim to develop a multilabel learning method with strong fuzzy inference ability and label correlation learning ability, even under the influence of noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' To achieve the goal need, a robust multilabel learning classifier, called robust multilabel Takagi-Sugeno-Kang fuzzy system (R-MLTSK-FS), is proposed by developing three ena- bling mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The first mechanism concerns soft label learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The R-MLTSK-FS maps the original label matrix to the soft label space where each soft label is affected by all the original labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The mechanism thus reduces the influence of label noise in the original label space, and is the basis of the other two mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The second mechanism concerns the construction of soft multilabel loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In R-MLTSK-FS, the “IF-THEN” rule-based TSK FS is used to model the infer- ence between the inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Specifically, multi-output TSK FS is employed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The IF-part of a multi-output TSK FS is leveraged to transform the original feature matrix into the fuzzy feature space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the THEN-part is used to imple- ment the inference between inputs and outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' and the regres- sion loss is constructed based on the TSK FS and soft label learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The adoption of TSK FS is advantageous in that the rule-based TSK FS makes the proposed R-MLTSK-FS more transparent than traditional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The third mechanism con- cerns correlation enhancement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The mechanism estab- lishes associations between any two soft labels and their corre- sponding fuzzy discriminative features, which can effectively improve the performance of R-MLTSK-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The main contributions of this paper are summarized as fol- lows: (1) A soft label learning mechanism is constructed to explic- itly measure the interaction between the labels and reduce the influence of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (2) A soft multilabel loss function is constructed based on soft labels and TSK FS to improve the efficiency and transpar- ency of the learning process of R-MLTSK-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (3) A correlation enhancement learning mechanism based on soft label space and fuzzy feature space is built to further en- hance the learning ability of R-MLTSK-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (4) Extensive experiments are conducted using 10 bench- mark multilabel datasets and 3 synthetic multilabel datasets to compare with 8 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Comprehensive evaluations are carried out by conducting classification performance evaluation, robustness analysis, effectiveness analysis of soft label learning and correlation enhancement learning, parameter analysis, con- vergence analysis, and statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Section II re- views the concepts of multilabel learning, and the traditional TSK FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Section III gives details of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ex- tensive experimental analyses are presented and discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Finally, Section V summarizes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' BACKGROUND KNOWLEDGE In this section, the problem statement of the multilabel learn- ing research concerned in the study is given, followed by the review of traditional TSK FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Problem Statement Let 𝒳 ∈ ℛ𝐷 and 𝒴 ∈ ℛ𝐿 be a D-dimensional feature space and an L-dimensional label space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒟 = {(𝒙𝑖, 𝒚𝑖)}𝑖=1 𝑁 is the training set with N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝑿 = [𝒙1, 𝒙2, … , 𝒙𝑁] ∈ ℛ𝐷×𝑁 is the input matrix, and 𝒀 = [𝒚1, 𝒚2, … , 𝒚𝑁] ∈ ℛ𝐿×𝑁 is the output matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In multilabel learning, the label of an instance 𝒙𝑖 = [𝑥𝑖1, 𝑥𝑖2, … , 𝑥𝑖𝐷]T is given by a vector 𝒚𝑖 = [𝑦𝑖1, 𝑦𝑖2, … , 𝑦𝑖𝐿]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' If 𝒙𝑖 is related to the jth label, then 𝑦𝑖𝑗 = 1, otherwise, 𝑦𝑖𝑗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The aim of this study is to find a robust mapping function 𝑓: 𝒳 → 𝒴 that can reduce the influence of label noise and effectively predict the label vector for a new instance on the basis of transparent infer- ence rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' TSK Fuzzy System TSK FS is a classical inference model based on fuzzy rules with superior interpretability (transparency) and learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It has been successfully applied in different areas, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', transfer learning [25, 26], multiview learning [27], multitask learning [28] and others [29, 30, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For a classical TSK FS with K rules, the kth rule can be expressed as follows: IF: 𝑥1 𝑖𝑠 𝐴1 𝑘 ∧ 𝑥2 𝑖𝑠 𝐴2 𝑘 ∧ … ∧ 𝑥𝐷 𝑖𝑠 𝐴𝐷 𝑘, THEN: 𝑓𝑘(𝒙) = 𝑐0 𝑘 + 𝑐1 𝑘𝑥1 + ⋯ + 𝑐𝐷 𝑘𝑥𝐷, 𝑘 = 1, 2, … , 𝐾 (1) where D is the feature dimension, and 𝑓𝑘(𝒙) is the output of instance 𝒙 on the kth rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝐴𝑑 𝑘 (𝑑 = 1, 2, … , 𝐷) in IF-part represents the antecedent fuzzy set, which can be described by membership functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝑐𝑑 𝑘 in THEN-part is the consequent pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Depending on application scenarios, different membership functions can be chosen for the antecedent fuzzy sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Gaussian function, which is commonly used, is adopted in this paper and the corresponding membership function associated with 𝐴𝑑 𝑘 can be expressed as follows: 𝜇𝐴𝑑 𝑘(𝑥𝑑) = exp {− 1 2 ( 𝑥𝑑−𝑚𝑑 𝑘 𝛿𝑑 𝑘 )2} (2) where 𝑚𝑑 𝑘 and 𝛿𝑑 𝑘 can be obtained using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In the absence of domain knowledge, data-driven methods are usually utilized to estimate 𝑚𝑑 𝑘 and 𝛿𝑑 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, the Var- Part clustering has been used for this purpose [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It is insen- sitive to the parameters and is therefore beneficial in terms of stability and practicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hence, the Var-Part clustering is used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For TSK FS, the firing strength of instance 𝒙 on the kth rule can be computed as follows: 𝜇𝑘(𝒙) = ∏ 𝜇𝐴𝑑 𝑘(𝑥𝑑) 𝐷 𝑑=1 (3) 𝜇̃𝑘(𝒙) = 𝜇𝑘(𝒙) ∑ 𝜇𝑘′(𝒙) 𝐾 𝑘′=1 ⁄ (4) where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (4) is the normalized form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Finally, the output of TSK FS for instance 𝒙 can be ex- pressed as 𝑦 = 𝑓(𝒙) = ∑ 𝜇̃𝑘(𝒙)𝑓𝑘(𝒙) 𝐾 𝑘=1 (5) In fact, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (5) can also be expressed as a linear model in a new fuzzy feature space, that is, 𝑦 = 𝑓(𝒙) = 𝒄T𝒙𝑔 (6) where 𝒙𝑒 = [1, 𝒙T]T ∈ ℛ(𝐷+1)×1 (7) 𝒙̃𝑘 = 𝜇̃𝑘(𝒙)𝒙𝑒 ∈ ℛ(𝐷+1)×1 (8) 𝒙𝑔 = [(𝒙̃1)T, (𝒙̃2)T, … , (𝒙̃𝐾)T]T ∈ ℛ𝐾(𝐷+1)×1 (9) 𝒄𝑘 = [𝑐0 𝑘, 𝑐1 𝑘, … , 𝑐𝐷 𝑘]T ∈ ℛ(𝐷+1)×1 (10) 𝒄 = [(𝒄1)T, (𝒄2)T, … , (𝒄𝐾)T]T ∈ ℛ𝐾(𝐷+1)×1 (11) Here, 𝒙𝑔 is the fuzzy representation of instance 𝒙 in a new feature space generated by fuzzy rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒄 is the consequent pa- rameter vector of all the rules, which can be optimized by solv- ing the linear model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' PROPOSED METHOD: R-MLTSK-FS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' System Architecture The architecture of the R-MLTSK-FS proposed in this study is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It aims to provide a robust multilabel model with fuzzy inference ability, label correlation learning ability and resistance against noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' R-MLTSK-FS contains three mechanisms for soft label learning, soft multilabel loss function construction and correlation enhancement learning, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1 The architecture of the proposed R-MLTSK-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The first mechanism, soft label learning, maps the original labels to soft label space by linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Each soft la- bel in the soft label space is associated with all the original la- bels, which reduces the influence of label noise in the original label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It is the basis of the other two mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The second mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', soft multilabel loss function construc- tion, leverages the IF-part of the TSK FS to transform the orig- inal features into the fuzzy feature space, uses the THEN-part of the TSK FS to complete the inference between inputs and outputs, and then constructs the regression function between the fuzzy feature space and the soft label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Rule-based TSK FS makes R-MLTSK-FS transparent in modeling inference re- lationship between features and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The third mechanism, correlation enhancement learning, implements label correlation learning by establishing associations between any two soft la- bels and their corresponding fuzzy discriminative features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' This mechanism further enhances the learning ability of R-MLTSK- FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The details of R-MLTSK-FS are expanded in the following three sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The learning criteria of R-MLTSK-FS is intro- duced in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The optimization process and the algo- rithm description are given in Section III-C, and the computa- tional complexity is analyzed in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Learning Criteria of R-MLTSK-FS According to the analysis in Section III-A, the multilabel learning problem in this paper can be expressed as the following optimization objective criteria: min 𝜙1,𝜙2 𝛽 ∙ 𝑆𝑜𝑓_𝑙𝑎𝑏(𝒀|𝜙1) + 𝑆𝑜𝑓_𝑙𝑜𝑠(𝒀, 𝑿|𝜙1, 𝜙2) + 𝛾 ∙ 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) (12) The first term represents soft label learning, where 𝜙1 trans- forms the original labels to the soft labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The second term rep- resents soft multilabel loss function construction, where 𝜙2 is used to predict the labels from the original feature space to the soft label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The third term represents correlation enhance- ment learning, which is used to measure the association be- tween any two soft labels and their corresponding fuzzy dis- criminative features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The hyperparameters β and γ are used to balance the influences of different terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The solu- tions of 𝜙1 and 𝜙2 can be obtained by optimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The implementation of three terms is described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1) Soft Label Learning based on Original Label Space and Soft Label Space For the lth label 𝒀𝑙 ∈ ℛ1×𝑁 (1 ≤ 𝑙 ≤ 𝐿) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', the lth row in 𝒀), the interference of its label noise can be reduced by consid- ering the influence of all labels on 𝒀𝑙 comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Based on this, for soft label learning, we assume that each label is as- sociated with all the other original labels to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The learning process involves two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' First, we construct the label transformation 𝜙1 to effectively measure the interaction be- tween the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝜙1 maps the output matrix 𝒀 explicitly from the original label space to the soft label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In the soft label space, each soft label is associated with all the original labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The transformation function of 𝜙1 is defined as: 𝜙1(𝒀) = 𝑺𝒀 (13) where 𝑺 = [𝒔1, 𝒔2, … , 𝒔𝐿]T ∈ ℛ𝐿×𝐿 , and 𝒔𝑙 ∈ ℛ𝐿×1 (1 ≤ 𝑙 ≤ 𝐿) represents the influence weights of all the original labels on the lth soft label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Second, we preserve the expression consistency between the soft labels and original labels to ensure the classification per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Therefore, the overall soft label learning is defined as: min 𝜙1 𝑆𝑜𝑓_𝑙𝑎𝑏(𝒀|𝜙1) = min 𝑺 ‖(𝒀 − 𝑺𝒀)T‖2,1 (14) TSKFuzzy System Soft Label Space SoftLabel Soft Multilabel Loss Learning Function Construction Correlation EnhancementLearningAlthough different regularization norms can be used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (14), we choose the L2,1 norm for two reasons: (1) since L2,1 norm has the characteristic of row sparsity, we can screen out the original label subsets which have significant impact on the corresponding soft label, (2) L2,1 norm is well-known for its ability in robust group selection [34, 35, 36], which is helpful to reduce the impact of label noise on soft label learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2) Soft Multilabel Loss Function Construction based on TSK FS Multilabel loss function can be constructed by employing an evaluation metric as the multilabel objective function [37, 38], or by using linear regression to derive the multilabel loss func- tion [39, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Unlike these methods, we construct the loss function using soft label learning and TSK FS, which essen- tially constructs a rule-based transparent model that maps the original feature space to the soft label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The construction of the soft multilabel loss function is divided into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' First, the original feature matrix is transformed into the fuzzy feature space through the IF-part of the fuzzy rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Second, the inference between inputs and outputs is completed through the THEN-part of fuzzy rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Third, the regression loss function is constructed based on the fuzzy rules and soft labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' These de- tails are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' IF-part implementation of fuzzy rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In the multi-out- put TSK FS with K rules, the fuzzy feature matrix obtained by 𝑿 using fuzzy rules is given by 𝑿𝑔 = [𝒙𝑔,1, 𝒙𝑔,2, … , 𝒙𝑔,𝑁] ∈ ℛ𝐾(𝐷+1)×𝑁 (15) where 𝒙𝑔,𝑖 (1 ≤ 𝑖 ≤ 𝑁) is mapped by the instance 𝒙𝑖 through the IF-part of fuzzy rules, and it can be obtained by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (2)-(4) and (7)-(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Compared with the original features, the rule-based fuzzy features can empower R-MLTSK-FS to analyze the implicit inference relationship between features and labels [42], thereby strengthening the learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' THEN-part adaptation of fuzzy rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (6), the THEN-part of multi-output TSK FS is used to complete the inference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', 𝜙2(𝑿) = 𝑪𝑿𝑔 (16) where 𝑪 = [𝒄1, 𝒄2, … , 𝒄𝐿]T ∈ ℛ𝐿×𝐾(𝐷+1) (17) is composed of L consequent parameter vectors in THEN- part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' As defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (11), 𝒄𝑙 ∈ ℛ𝐾(𝐷+1)×1 (1 ≤ 𝑙 ≤ 𝐿) is the consequent parameter vector corresponding to the lth-output in multi-output TSK FS and the lth soft label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The main difference between multi-output TSK FS and sin- gle-output TSK FS is that the consequent parameters of sin- gle-output TSK FS are represented with a vector, whereas the consequent parameters of multi-output TSK FS are rep- resented by a matrix composed of multiple vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Construction of regression loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The loss function is a fundamental part of the optimization objective for multila- bel classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In this paper, it is built based on soft label learning and TSK FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (13) and (16), we construct the soft multilabel loss function as follows: min 𝜙1,𝜙2 𝑆𝑜𝑓_𝑙𝑜𝑠(𝒀, 𝑿|𝜙1, 𝜙2) = min 𝑺,𝑪 ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 2 (18) where α is a hyperparameter to balance the influence of the soft multilabel loss function and the regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Taking the Frobenius norm ‖∙‖𝐹 as the regularization term can not only reduce the risk of overfitting, but also facilitate the solution of consequent parameter matrix 𝑪.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3) Correlation Enhancement Learning based on Soft Label Space and Fuzzy Feature Space Section I has clarified that mining the correlation information between labels can effectively improve the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In this paper, we analyze the label correlation based on the fact that the correlation between two labels is consistent with the correlation between their discriminative features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, there is an intersection between the labels “Cat” and “Animal”, and then their discriminative features should par- tially overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Based on the above analysis, we utilize the correlation infor- mation on the basis of soft label learning and fuzzy features as follows: min 𝜙1,𝜙2 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) = min 𝑺,𝑪 ∑ ∑ ‖(𝒔𝑖 T𝒀 − 𝒔𝑗 T𝒀)T‖ 2𝒄𝑖 T𝒄𝑗 𝐿 𝑗=1 𝐿 𝑖=1 (19) where 𝒔𝑙 T𝒀 ∈ ℛ1×𝑁 (1 ≤ 𝑙 ≤ 𝐿) represents the lth soft label vector corresponding to N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒔𝑙 ∈ ℛ𝐿×1 represents the influence weights of all original labels on the lth soft label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒄𝑙 ∈ ℛ𝐾(𝐷+1)×1 (1 ≤ 𝑙 ≤ 𝐿) is used to learn the discriminative fea- tures from fuzzy feature space for the lth soft label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The larger the difference between the ith and jth soft labels, the more sig- nificant the difference between their fuzzy discriminative fea- tures, and further, the smaller the value of 𝒄𝑖 T𝒄𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Further, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (19) can be expressed as: min 𝜙1,𝜙2 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) = min 𝑺,𝑪 ∑ ∑ ‖(𝒔𝑖 T𝒀 − 𝒔𝑗 T𝒀)T‖ 2𝒄𝑖 T𝒄𝑗 𝐿 𝑗=1 𝐿 𝑖=1 = min 𝑺,𝑪 2Tr(𝒀T𝑺T𝑳𝑺𝒀) (20) where 𝑳 = 𝑫 − 𝑹, 𝑹 = 𝑪𝑪𝑇 ∈ ℛ𝐿×𝐿, 𝑫 ∈ ℛ𝐿×𝐿 is a diago- nal matrix, and 𝐷𝑖𝑖 = ∑ 𝑅𝑖𝑗 𝐿 𝑗=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Complete Objective Function and its Optimization By integrating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (14), (18) and (20), the multilabel learn- ing problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (12) is defined and the complete objective function of R-MLTSK-FS is expressed as: min 𝜙1,𝜙2 𝛽 ∙ 𝑆𝑜𝑓_𝑙𝑎𝑏(𝒀|𝜙1) + 𝑆𝑜𝑓_𝑙𝑜𝑠(𝒀, 𝑿|𝜙1, 𝜙2) + 𝛾 ∙ 𝐶𝑜𝑟_𝑒𝑛ℎ(𝒀, 𝑿|𝜙1, 𝜙2) = min 𝑺,𝑪 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + ‖(𝑺𝒀 − 𝑪𝑿𝑔) T‖ 2,1 + 𝛼‖𝑪‖𝐹 2 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) = min 𝑺,𝑪 ‖(𝑺𝒀 − 𝑪𝑿𝑔) T‖ 2,1 + 𝛼‖𝑪‖𝐹 2 + 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) (21) To optimize 𝑺 and 𝑪, we adopt the alternating direction minimization strategy, where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (21) is divided into two sub- problems, namely, the 𝑺-subproblem and the 𝑪-subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The optimization processes are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1) 𝑺-Subproblem By fixing 𝑪, the 𝑺-subproblem can be expressed as: 𝑺∗ = 𝑎𝑟𝑔𝑚𝑖𝑛𝑺 ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) (22) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (22), the Lagrange function for 𝑺 is 𝐿(𝑺) = ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛽‖(𝒀 − 𝑺𝒀)T‖2,1 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) (23) Set the derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (23) with respect to 𝑺 to 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', 𝜕𝐿(𝑺) 𝜕𝑺 ⁄ = 2𝑺𝒀𝑫𝑆1𝒀T − 2𝑪𝑿𝑔𝑫𝑆1𝒀T + 2𝛽𝑺𝒀𝑫𝑆2𝒀T −2𝛽𝒀𝑫𝑆2𝒀T + 4𝛾𝑳𝑺𝒀𝒀T = 0 (24) where 𝑫𝑆1 ∈ ℛ𝑁×𝑁 and 𝑫𝑆2 ∈ ℛ𝑁×𝑁 are diagonal matrices, and 𝐷𝑆1,𝑖𝑖 = 1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 T‖) ⁄ , 𝐷𝑆2,𝑖𝑖 = 1 (2‖(𝒀 − 𝑺𝒀)𝑖 T‖) ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (𝑨𝑖 T represents the ith row in 𝑨T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=') Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (24) can be re-expressed as (2𝛾𝑳)𝑺 + 𝑺(𝒀𝑫𝑆1𝒀T(𝒀𝒀T)−1 + 𝛽𝒀𝑫𝑆2𝒀T(𝒀𝒀T)−1) = 𝑪𝑿𝑔𝑫𝑆1𝒀T(𝒀𝒀T)−1 + 𝛽𝒀𝑫𝑆2𝒀T(𝒀𝒀T)−1 (25) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (25) is a classical optimization problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', the Sylvester equation, which has been thoroughly studied [43, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In general, for the Sylvester equation 𝑨𝑾 + 𝑾𝑩 = 𝒁 (𝑨 ∈ ℛ𝑚×𝑚, 𝑩 ∈ ℛ𝑛×𝑛, 𝒁 ∈ ℛ𝑚×𝑛, 𝑾 ∈ ℛ𝑚×𝑛), the matrix 𝑾 is the variable to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The specific solution formula of 𝑾 is as follows: 𝑾(: ) = (𝑰1⨂𝑨 + 𝑩T⨂𝑰2)−𝟏𝒁(: ) (26) where 𝑰1 ∈ ℛ𝑛×𝑛 and 𝑰2 ∈ ℛ𝑚×𝑚 are identity matrices, ⨂ is the Kronecker tensor product, 𝒁(: ) ∈ ℛ𝑚𝑛×1 and 𝑾(: ) ∈ ℛ𝑚𝑛×1 denote that the matrices 𝒁 and 𝑾 are single column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝑾(: ) can be reshaped to 𝑾∗ ∈ ℛ𝑚×𝑛, which is the solution of 𝑨𝑾 + 𝑾𝑩 = 𝒁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For simplicity, the solution 𝑾∗ is denoted as 𝑾∗ = 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝑨, 𝑩, 𝒁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Therefore, the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (25) is 𝑺∗ = 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(2𝛾𝑳, 𝒀(𝑫𝑆1 + 𝛽𝑫𝑆2)𝒀T(𝒀𝒀T)−1, (𝑪𝑿𝑔𝑫𝑆1 + 𝛽𝒀𝑫𝑆2)𝒀T(𝒀𝒀T)−1) (27) 2) 𝑪-Subproblem By fixing 𝑺, the 𝑪-subproblem can be expressed as: 𝑪∗ = 𝑎𝑟𝑔𝑚𝑖𝑛𝑪 ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 2 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) (28) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (28), the Lagrange function for 𝑪 is 𝐿(𝑪) = ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 2 + 2𝛾Tr(𝒀T𝑺T𝑳𝑺𝒀) = ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 2 + 2𝛾Tr(𝒀T𝑺T(𝑫 − 𝑹)𝑺𝒀) = ‖(𝑺𝒀 − 𝑪𝑿𝑔)T‖2,1 + 𝛼‖𝑪‖𝐹 2 + 2𝛾Tr(𝒀T𝑺T(𝑪𝑪T𝟏𝟏T ∘ 𝑰3 − 𝑪𝑪T)𝑺𝒀) (29) where 𝟏 ∈ ℛ𝐿×1 is a column vector with all elements equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The symbol (∘) represents the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝑰3 ∈ ℛ𝐿×𝐿 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Set the derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (29) with respect to 𝑪 to 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', 𝜕𝐿(𝑪) 𝜕𝑪 ⁄ = 2𝑪𝑿𝑔𝑫𝐶𝑿𝑔 T − 2𝑺𝒀𝑫𝐶𝑿𝑔 T + 2𝛼𝑪 + 2𝛾(((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3)T𝟏𝟏T𝑪 + 𝟏𝟏T((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3)𝑪 − 2𝑺𝒀𝒀T𝑺T𝑪) = 0 (30) where 𝑫𝐶 ∈ ℛ𝑁×𝑁 is a diagonal matrix, and 𝐷𝐶,𝑖𝑖 = 1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 𝑇‖) ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (𝑨𝑖 T is the ith row of 𝑨T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=') Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (30) is also a Sylvester equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Therefore, we can solve 𝑪 as follows: 𝑪∗ = 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝛼𝑰3 + 𝛾((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3) T𝟏𝟏T + 𝛾𝟏𝟏T((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3) − 2𝛾𝑺𝒀𝒀T𝑺T, 𝑿𝑔𝑫𝐶𝑿𝑔 T, 𝑺𝒀𝑫𝐶𝑿𝑔 T) (31) When the optimal 𝑺∗ and 𝑪∗ are obtained, the prediction output of the test instance 𝒙′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', 𝒚′ = [𝑦1 ′, … , 𝑦𝐿 ′]𝑇) can be formulated as follows: 𝒚′ = 𝜑𝜏(𝑪∗𝒙𝑔 ′ ) (32) where 𝒙𝑔 ′ is the fuzzy feature representation of 𝒙′ through fuzzy rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It can be obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (2)-(4) and (7)-(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝜑𝜏(∙) is a threshold function to convert the continuous output to the discrete output, and 𝜏 is the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Therefore, for the lth label 𝑦𝑙 ′ (1 ≤ 𝑙 ≤ 𝐿) in 𝒚′, its definition is 𝑦𝑙 ′ = {1, 𝑖𝑓 (𝑪∗𝒙𝑔 ′ )𝑙 > 𝜏 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (33) where (𝑪∗𝒙𝑔 ′ )𝑙 is the lth element in (𝑪∗𝒙𝑔 ′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The value of 𝜏 can be optimized by cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In this paper, we set it to the fixed value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Based on the above analysis, the procedure of the proposed R-MLTSK-FS is described in Algorithm I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Computational Complexity Analysis The computational complexity of R-MLTSK-FS is analyzed according to the steps in Algorithm I, which is expressed using the big-O notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 1, the complexity of initialization is 𝑂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 2, the computational complexity of trans- forming 𝑿 into 𝑿𝑔 is 𝑂(2𝑁𝐾𝐷 + 2𝑁𝐾) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The computa- tional complexity of step 4 is 𝑂(𝐿2𝑁 + 𝐿𝑁𝐾(𝐷 + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For the step 5, the computational complexity of 𝑻1 is 𝑂(2𝐿2𝑁 + 𝐿3 + 2𝐿2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 6, the computational complexity of 𝑻2 is 𝑂(𝑁2𝐾(𝐷 + 1) + 𝑁𝐾2(𝐷 + 1)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 7, the computa- tional complexity of calculating 𝑻3 is 𝑂(𝐿2𝑁 + 𝐿𝑁2 + Algorithm I R-MLTSK-FS Input: Input matrix 𝑿 ∈ ℛ𝐷×𝑁, output matrix 𝒀 ∈ ℛ𝐿×𝑁, rule number K, trade-off parameters α, β and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Procedure: 1: Initialize: 𝑺 = 𝟏𝐿×𝐿, 𝑪 = (1 𝐿 ⁄ )𝟏𝐿×𝐾(𝐷+1), 𝑫 = 𝟎𝐿×𝐿, 𝑫𝐶 = 𝟎𝑁×𝑁, 𝑫𝑆1 = 𝟎𝑁×𝑁, 𝑫𝑆2 = 𝟎𝑁×𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2: Transform 𝑿 into 𝑿𝑔 using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (2)-(4) and (7)-(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3: While not converged do 4: 𝐷𝐶,𝑖𝑖 = 1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 𝑇‖) ⁄ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 5: 𝑻1 ← 𝛼𝑰3 + 𝛾((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3)T𝟏𝟏T + 𝛾𝟏𝟏T((𝑺𝒀𝒀T𝑺T) ∘ 𝑰3) − 2𝛾𝑺𝒀𝒀T𝑺T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6: 𝑻2 ← 𝑿𝑔𝑫𝐶𝑿𝑔 T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 7: 𝑻3 ← 𝑺𝒀𝑫𝐶𝑿𝑔 T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 8: 𝑪 ← 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝑻1, 𝑻2, 𝑻3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 9: 𝐷𝑆1,𝑖𝑖 = 1 (2‖(𝑺𝒀 − 𝑪𝑿𝑔)𝑖 T‖) ⁄ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 10: 𝐷𝑆2,𝑖𝑖 = 1 (2‖(𝒀 − 𝑺𝒀)𝑖 T‖) ⁄ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 11: 𝑹 ← 𝑪𝑪T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 12: 𝐷𝑖𝑖 ← ∑ 𝑅𝑖𝑗 𝐿 𝑗=1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 13: 𝑳 = 𝑫 − 𝑹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 14: 𝑻4 ← 2γ𝑳;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 15: 𝑻5 ← 𝒀(𝑫𝑆1 + 𝛽𝑫𝑆2)𝒀T(𝒀𝒀T)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 16: 𝑻6 ← (𝑪𝑿𝑔𝑫𝑆1 + 𝛽𝒀𝑫𝑆2)𝒀T(𝒀𝒀T)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 17: 𝑺 ← 𝑠𝑦𝑙𝑣𝑒𝑠𝑡𝑒𝑟(𝑻4, 𝑻5, 𝑻6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 18: Check the convergence conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 19: End Output: 𝑺, 𝑪.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' TABLE I STATISTICS OF DATASETS Dataset #Instance #Feature #Label Arts 5000 462 26 Birds 645 260 19 CAL500 502 68 174 Corel5k 5000 499 374 Flags 194 19 7 Genbase 662 1185 27 Medical 978 1449 45 Mirflickr 25000 150 24 Recreation 5000 606 22 Science 5000 743 40 𝐿𝑁𝐾(𝐷 + 1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The computational complexity of step 8 is 𝑂(3𝐿4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 9, the complexity of calculating 𝑫𝑆1 is 𝑂(𝐿2𝑁 + 𝐿𝑁𝐾(𝐷 + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 10, the complexity of 𝑫𝑆2 is 𝑂(𝐿2𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The complexity of step 11 is 𝑂(𝐿2𝐾(𝐷 + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The complexity of steps 12-14 is 𝑂(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 15, the complexity of 𝑻5 is 𝑂(𝐿𝑁2 + 𝐿2𝑁 + 𝐿3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The complexity of step 16 is 𝑂(𝐿𝑁𝐾(𝐷 + 1) + 𝐿𝑁2 + 𝐿2𝑁 + 𝐿3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For step 17, the com- plexity is 𝑂(3𝐿2𝐾2(𝐷 + 1)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hence, the overall complexity of the whole algorithm is dominated by steps 6 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Let 𝑎 = max (𝐿, 𝐷, 𝐾) , 𝑏 = max (𝑁, 𝐾(𝐷 + 1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In general, 𝑎 ≪ 𝑏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Therefore, the maximum computational complexity of R- MLTSK-FS is 𝑂(𝑎3 + 𝑏(2𝑎𝑏 + 𝑎2 + 2𝑏2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' EXPERIMENTAL ANALYSIS Extensive experiments are conducted to fully assess the ef- fectiveness of R-MLTSK-FS, including classification perfor- mance evaluation, robustness analysis, effectiveness analysis of soft label learning and correlation enhancement learning, pa- rameter analysis, convergence analysis, and statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The datasets, evaluation metrics and the settings used in the ex- periments are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Datasets We adopt 10 benchmark multilabel datasets to evaluate the performance of R-MLTSK-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Table I shows the details of these datasets, where #Instance, #Feature, and #Label denote the instance number, the feature dimension, and the label space dimension respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' These datasets are available from the Github1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Evaluation Metrics Let {(𝒙̃𝑖, 𝒚̃𝑖)|1 ≤ 𝑖 ≤ 𝑁𝑡} be a test set with 𝑁𝑡 samples, 𝒚̂𝑖 be the predicted labels of 𝒙̃𝑖, 𝑓(𝒙̃𝑖, 𝑙) be the continuous output predicted by the multilabel method for the instance 𝒙̃𝑖 on the lth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The ranking function 𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙) is obtained according to 𝑓(𝒙̃𝑖, 𝑙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' If 𝑓(𝒙̃𝑖, 𝑙) > 𝑓(𝒙̃𝑖, 𝑙′) , then 𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙) < 𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Let 𝐿𝒙𝑖 be the label set related to 𝒙̃𝑖, and 𝐿𝒙𝑖 is the complement of 𝐿𝒙𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Based on the settings, the four metrics below, commonly used in multilabel learning, are employed in the experiments [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (1) Average Precision (AP): It is the average proportion of the related labels of an instance that are ranked lower than a given label l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The larger the value of AP, the better the classifi- cation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' AP = 1 𝑁𝑡 ∑ 1 |𝐿𝒙𝑖| ∑ |{𝑙′ ∈ 𝐿𝒙𝑖|𝑓(𝒙̃𝑖, 𝑙′) ≥ 𝑓(𝒙̃𝑖, 𝑙)}| 𝑟𝑎𝑛𝑘(𝒙̃𝑖,𝑙) 𝑙∈𝐿𝒙𝑖 𝑁𝑡 𝑖=1 (34) (2) Hamming Loss (HL): It is the average proportion of an instance that is predicted incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The smaller the value of HL, the better the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' HL = 1 𝑁𝑡 ∑ |𝒚̃𝑖⨁𝒚̂𝑖| 𝐿 𝑁𝑡 𝑖=1 (35) where ⨁ is the XOR operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (3) Ranking Loss (RL): It is the proportion of the related la- bels that are ranked higher than the unrelated labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The smaller the value of RL, the better the classification perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' RL = 1 𝑁𝑡 ∑ |{(𝑙, 𝑙′)|𝑓(𝒙̃𝑖, 𝑙) ≤ 𝑓(𝒙̃𝑖, 𝑙′), (𝑙, 𝑙′) ∈ 𝐿𝒙𝑖 × 𝐿𝒙𝑖}| |𝐿𝒙𝑖||𝐿𝒙𝑖| 𝑁𝑡 𝑖=1 (36) (4) Coverage (CV): It is the average number of times that all related labels of an instance are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The smaller the value of CV, the better the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' CV = 1 𝑁𝑡 ∑ max 𝑙∈𝐿𝒙𝑖 𝑟𝑎𝑛𝑘(𝒙̃𝑖, 𝑙) − 1 𝑁𝑡 𝑖=1 (37) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Experimental Settings In this paper, we employ eight methods for comparison, in- cluding binary relevance (BR) [47], multilabel k-nearest neigh- bor (MLkNN) [48], meta-label-specific features (MLSF) [49], ML-TSK FS [17], classifier chains (CC) [50], random k-label- sets (RAkEL) [51], correlated logistic models (CorrLog) [52] and hybrid noise-oriented multilabel learning (HNOML) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' These methods and the settings of the parameters for grid search are described in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' We adopt the 5-fold cross-validation strategy to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='com/ZesenChen/multi-label-dataset and https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='com/ KKimura360/MLC_toolbox/tree/master/dataset/matfile TABLE II DESCRIPTION OF METHODS Methods Description Parameter Setting BR This method is a first-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' To improve the robustness, it introduces \uf065-in- sensitive learning (a fuzzy method) by solving a system of linear inequalities (\uf065LSSLI) [54] as the binary classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝐶 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' ^(−5: 1: 5), 𝑀 = {2, 3, 4, 5, 6, 7, 8, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' MLkNN This method is a first-order method that predicts a new instance by maximizing the posterior probability of each label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The number of nearest neighbors affects the ro- bustness of the model to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝐾 = {1, 3, 5, 7, 9, 11, 13}, 𝑠 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='09}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' MLSF This method is a second-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It improves the performance through meta- label learning and specific feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝑘 = {2,4,6,8}, 𝜀 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,1,10}, 𝛼 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='9}, 𝛾 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,1,10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' ML-TSK FS This method is a second-order method that uses the correlation between any two la- bels to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' To realize the transparency, it uses fuzzy rules to model the inference relationship between features and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' This method does not consider the influence of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝐾 = {2,3,4,5}, 𝛼 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,1,10,100}, 𝛽 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,1,10,100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' CC This method is a high-order method which adds the prediction result of the previous label to the feature space to participate in the prediction of the next label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The \uf065-in- sensitive learning (a fuzzy method) by solving a system of linear inequalities (\uf065LSSLI) [54] is used as the binary classifier to improve the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝐶 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' ^(−5: 1: 5), 𝑀 = {2, 3, 4, 5, 6, 7, 8, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' RAkEL This method is a high-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In this method, the label space is randomly di- vided into multiple label subspaces, and the prediction result of a label is associated with other labels in the subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝑘 = 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='/(12: −2: 2) (N is the instance number), 𝛼 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' CorrLog This method is a high-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It achieves robustness by constructing the asso- ciation between a label and all other labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝑟ℎ𝑜1 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='003,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='005,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='007, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='009, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='07,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='09,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='9}, 𝑟ℎ𝑜2 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' HNOML This method is a high-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It designs a label enrichment matrix to improve the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝛼 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,1,10}, 𝛽 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,1,10,100}, 𝛾 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,1,10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' R-MLTSK-FS (ours) The method proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It is a second-order method and achieves the transparency and robustness against label noise through fuzzy rules, correlation en- hancement learning, soft multilabel loss function construction, and soft label learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝛼 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='001,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='005,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5,1,5,10,50,100}, 𝛽 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='001,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='005,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5,1,5,10,50,100}, 𝛾 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='001,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='005,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5,1,5,10,50,100}, 𝑘 = {2,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Performance Analysis 1) Classification Performance Evaluation To verify the effectiveness of R-MLTSK-FS, we compare the R-MLTSK-FS with eight methods on 10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The experi- mental results, expressed in terms of the mean and standard de- viation (inside brackets) of the four metrics, are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For each dataset, the best value of each metric is bold-faced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' We can see that compared to the eight methods, the overall per- formance of R-MLTSK-FS is the best on all the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' This is attributable to the three mechanisms introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2) Robustness Analysis In order to verify the robustness of R-MLTSK-FS against la- bel noise, we introduce label noise to the data and evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Specifically, we randomly select 0%, 10%, 20%, 30% and 40% samples from the training set, and then create noise by changing their related (unrelated) labels to unrelated (related) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The 5-fold cross-validation strategy is adopted in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2 shows the experimental results, from which the following findings are obtained: (1) Despite the increase in the amount of noise in the experi- ments, the proposed R-MLTSK-FS maintains outstanding clas- sification performance, indicating the effectiveness of the three mechanisms introduced in reducing the influence of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (2) Label noise has different effect on the comparison meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, the performance of MLkNN in the presence of label noise is unstable because the robustness of MLkNN against noisy labels is affected by the number of nearest neigh- bors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For RAkEL and CorrLog, their performance is unsatisfac- tory since they ignore label noise in modeling the correlation between labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For ML-TSK FS, its overall robustness is infe- rior to the proposed method as it also ignores the influence of label noise in model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3) Effectiveness Analysis of Soft Label Learning To evaluate the effectiveness of R-MLTSK-FS in soft label learning, we study the influence weights 𝑺 with three synthetic multilabel datasets, namely Independence dataset, Equality da- taset and Union dataset [55], each containing 1000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For each sample, the feature dimension is 20 and the label di- mension is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Each feature in the synthetic datasets is normal- ized in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Each synthetic dataset has five labels, 𝒴1, …, 𝒴5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For the first four labels, their logical relationships are designed as fol- lows: Independence dataset: The first four labels 𝒴1 , 𝒴2 , 𝒴3 and 𝒴4 are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Equality dataset: 𝒴1 = 𝒴2 and 𝒴3 = 𝒴4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' That is, for a sample (𝒙𝑖, 𝒚𝑖) (1 ≤ 𝑖 ≤ 1000), 𝑦𝑖1 = 𝑦𝑖2 and 𝑦𝑖3 = 𝑦𝑖4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Union dataset: 𝒴1 = 𝒴2 ∨ 𝒴3 ∨ 𝒴4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' That is, for a sample (𝒙𝑖, 𝒚𝑖) (1 ≤ 𝑖 ≤ 1000), if 𝑦𝑖2 = 1 or 𝑦𝑖3 = 1 or 𝑦𝑖4 = 1, then 𝑦𝑖1 = 1, otherwise, 𝑦𝑖1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' TABLE III MEAN (SD) OF THE METRICS OF THE MULTILABEL CLASSIFICATION METHODS Datasets Methods Met- rics BR MLkNN MLSF ML-TSK FS CC RAkEL CorrLog HNOML R-MLTSK-FS Arts AP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='6270 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0076) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='5454 (0.' metadata={'source': 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+page_content=' (Noise ratio is defined as the proportion of samples that are randomly selected from the training set and their related (unrelated) labels are changed to unrelated (related) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The larger the value of AP, the better the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=') TABLE IV INFLUENCE WEIGHTS (S) OF ORIGINAL LABELS ON A SOFT LABEL IN INDEPENDENCE DATASET original label 1 (𝒴1) original label 2 (𝒴2) original label 3 (𝒴3) original label 4 (𝒴4) original label 5 (𝒴5) soft label 1 (𝒴1 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2016 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2832 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒴1, 𝒴2, 𝒴3 and 𝒴4 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' TABLE V INFLUENCE WEIGHTS (S) OF ORIGINAL LABELS ON A SOFT LABEL IN EQUALITY DATASET original label 1 (𝒴1) original label 2 (𝒴2) original label 3 (𝒴3) original label 4 (𝒴4) original label 5 (𝒴5) soft label 1 (𝒴1 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='6172 soft label 2 (𝒴2 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='6172 soft label 3 (𝒴3 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='4350 soft label 4 (𝒴4 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='4350 soft label 5 (𝒴5 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='4480 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒴1 = 𝒴2 and 𝒴3 = 𝒴4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' TABLE VI INFLUENCE WEIGHTS (S) OF ORIGINAL LABELS ON A SOFT LABEL IN UNION DATASET original label 1 (𝒴1) original label 2 (𝒴2) original label 3 (𝒴3) original label 4 (𝒴4) original label 5 (𝒴5) soft label 1 (𝒴1 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2654 soft label 2 (𝒴2 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1327 soft label 3 (𝒴3 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2151 soft label 4 (𝒴4 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0106 soft label 5 (𝒴5 ′) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1057 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒴1 = 𝒴2 ∨ 𝒴3 ∨ 𝒴4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The fifth label is mutually exclusive with the first four labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Specifically, for a sample (𝒙𝑖, 𝒚𝑖) (1 ≤ 𝑖 ≤ 1000), if 𝑦𝑖1 = 0 and 𝑦𝑖2 = 0 and 𝑦𝑖3 = 0 and 𝑦𝑖4 = 0, then 𝑦𝑖5 = 1, otherwise, 𝑦𝑖5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The learned influence weights 𝑺 for each of the three syn- thetic datasets are shown in Tables IV-VI respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The fol- lowing findings can be obtained from the tables: (1) In Tables IV-VI, since the fifth label is mutually exclusive with the first four labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4)), reconstruction cannot be achieved with the first four labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' From the results of influence weights in Tables IV-VI, we can find that the influence of 𝒴5 on the soft label 𝒴5 ′ is most significant, whereas the influence of 𝒴1 ∼ 𝒴4 on 𝒴5 ′ is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (2) In Table IV, the first four labels 𝒴1, 𝒴2, 𝒴3 and 𝒴4 are independent of each other, and 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The results of influence weights in Table IV show that the effect of 𝒴1, 𝒴2, 𝒴3 and 𝒴4 on the soft labels 𝒴1 ′, 𝒴2 ′, 𝒴3 ′ and 𝒴4 ′, respectively, are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In addition, the contribution of 𝒴5 to 𝒴1 ′, 𝒴2 ′, 𝒴3 ′ and 𝒴4 ′ is also obvi- ous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (3) In Table V, 𝒴1 = 𝒴2 , 𝒴3 = 𝒴4, and 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The results of influence weights in Table V reveal that 𝒴5 has a greater influence on the soft la- bels 𝒴1 ′, 𝒴2 ′, 𝒴3 ′ and 𝒴4 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Meanwhile, it is obvious that 𝒴1 and 𝒴2 have the same influence on 𝒴1 ′ (𝒴2 ′ ), and 𝒴3 and 𝒴4 have the same influence on 𝒴3 ′ (𝒴4 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (4) In Table VI, 𝒴1 = 𝒴2 ∨ 𝒴3 ∨ 𝒴4 and 𝒴5 = (¬𝒴1) ∧ (¬𝒴2) ∧ (¬𝒴3) ∧ (¬𝒴4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' From the results of influence weights in Table VI, we can see that it is 𝒴1 and 𝒴5 that af- fect the soft label 𝒴1 ′ significantly, and that the effect of 𝒴2 ∼ 𝒴4 on the soft label 𝒴1 ′ are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The above findings are consistent with the logical relation- ship we designed for the labels, which validates that the soft label learning in R-MLTSK-FS is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4) Effectiveness Analysis of Correlation Enhancement Learn- ing In order to verify the effectiveness of the correlation en- hancement learning mechanism in guiding the consequent vec- tor optimization, we conduct correlation visualization experi- ment on the Science dataset, where the dimension of label space is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Specifically, the Pearson correlation coefficient is used to measure the correlation between two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The higher the value of Pearson correlation coefficient, the stronger the corre- lation between two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3, where Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3(a) visualizes the correlation between any two original labels, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3(b) visualizes the correlation be- tween any two optimized consequent vectors associated with the corresponding labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For an effective correlation enhance- ment learning mechanism, the correlation coefficient between two consequent vectors should be kept close to that between their corresponding labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3 Visualization of label correlation learning on the Science dataset: (a) visualization of the correlation coefficient between any two original label vectors, and (b) visualization of the correlation coefficient between any two consequent vectors associated with the corresponding labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The higher the value of correlation coefficient, the stronger the correlation between two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It is clear that there is little difference between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3(b), indicating that the correlation between the labels can closely guide the learning of the corresponding consequent vec- tors, and demonstrating the effectiveness of the correlation en- hancement learning mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 5) Parameter Analysis In this section, we analyze the influence of the hyperparam- eters α, β, γ and K on the classification performance of R- MLTSK-FS in terms of AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In the analysis, we study the sensi- tivity of the classification performance to a specific hyperpa- rameter by keeping the other three fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For example, we fix the values of β, γ and K, and adjust the value of α to analyze the effect of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The hyperparameters α, β and γ are varied within {10-3, 10-2, 10-1, 100, 101, 102} and K is varied within {2, 3, 4, 5, 6, 7, 8, 9, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The AP values of R-MLTSK-FS are obtained with the 5-fold cross-validation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (a) α (b) β (c) γ (d) K Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4 The influence of the hyperparameters (a) α, (b) β, (c) γ, and (d) K on AP of the R-MLTSK-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4, from which the following observations are obtained: (1) When α is in the range of (10-3, 100), the performance of R-MLTSK-FS in terms of AP is stable for most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In ad- dition, AP decreases with increasing α for most datasets when α is within (101, 102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For the CAL500 dataset, AP increases with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In general, R-MLTSK-FS is stable and can achieve op- timal performance when α is in the range of (10-2, 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (2) In general, R-MLTSK-FS is sensitive to β when it is in the range of (10-3, 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It is stable and can reach an optimal AP value for the 10 datasets when β is within (101, 102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (3) For the hyperparameter γ, AP fluctuates in a similar way for all the 10 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In general, the performance of R- MLTSK-FS is stable when γ is within (10-3, 10-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The AP value fluctuates significantly when γ is in the range of (10-1, 102), while exhibiting a decreasing trend with increasing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In general, optimal AP can be achieved for all the 10 datasets when γ is in the range of (10-3, 10-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (4) The AP value for the 10 datasets fluctuates slightly with increasing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Optimal values of AP can be obtained when K is within (4, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' According to the above analysis, it is necessary for R- MLTSK-FS to adopt the grid search strategy and the cross-val- idation strategy to get the optimal hyperparameters for different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6) Convergence Analysis The Birds and Flags datasets are adopted in this part to inves- tigate the convergence of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 5, where the vertical axis represents the absolute value of the difference between the previous and the current value of the objective function (denoted by df), and the hori- zontal axis represents the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 5 that for the Birds and Flags datasets, R-MLTSK-FS is convergent within 10 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 5 Convergence analysis for datasets (a) Birds and (b) Flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 7) Statistical Analysis We employ the Friedman test and the Bonferroni-Dunn test to evaluate the statistical significance of the difference observed between the proposed R-MLTSK-FS and the eight comparison methods [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The details are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' (1) Friedman Test: Based on the experimental results in Ta- ble III, we perform the Friedman test on the four metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', AP, HL, RL and CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The null hypothesis is that there is no sig- nificant difference between all the methods in terms of the four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For each metric, if the Friedman statistic FF is greater than a critical value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0698), the null hypothesis for that metric is rejected, which means the difference is statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The results of the Friedman test, corresponding to the results in Table III, are shown in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' It can be seen from Table VII that the null hypotheses on AP, HL, RL and CV are all rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' This means that the differences in classification performance of the nine methods are significant in terms of the four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Next, we conduct the post-hoc Bonferroni-Dunn test to evaluate whether the difference in performance between R-MLTSK-FS and the comparison methods is statistically sig- nificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' TABLE VII FRIEDMAN STATISTICS Evaluation met- ric FF Critical value (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05) AP 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='6045 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='0698 HL 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='6863 RL 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3718 CV 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='6201 (2) Bonferroni-Dunn Test: According to the results in Fried- man test, we conduct the post-hoc test based on the results of AP, HL, RL and CV respectively, where R-MLTSK-FS is set as the control method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' First, we calculate the average rank of the nine methods for each metric respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' We also calcu- late the critical difference (CD), which is a standard used for evaluating the difference in average rank between the methods, using the equation below: (a) AP (b) HL (c) RL (d) CV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6 Comparison of R-MLTSK-FS (as control) with the other meth- ods using the Bonferroni-Dunn test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The letter A refers to R-MLTSK- FS, B to BR, C to MLkNN, D to MLSF, E to ML-TSK FS, F to CC, G to RAkEL, H to CorrLog, and I to HNOML, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' CD = 𝑞𝛼√𝑛(𝑛 + 1) 6𝑀 ⁄ (38) where n and M are the number of methods (n = 9) and the num- ber of datasets (M = 10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' With confidence level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='05 and 𝑞𝛼 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='724, we have CD = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6 gives the average rank of the nine methods, which are shown on the horizontal line with ticks marking 1 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The smaller the average rank (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', closer to the right), the better the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' As R-MLTSK-FS is at the rightmost position on the horizontal line, for all the four metrics, it is the best among the nine methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' A red line of length one CD is drawn from R- MLTSK-FS to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' For a method located within the span of the red line, the difference in average rank between the method and R-MLTSK-FS is less than one CD, indicating that the per- formance difference between them is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Otherwise, the dif- ference is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' The following conclusions can be drawn from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Firstly, R-MLTSK-FS is superior to other methods on the four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Secondly, in general, the performance of ML-TSK FS is the second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Thirdly, the performance of MLkNN, CC, RAkEL and CorrLog are significantly lower than that of R-MLTSK-FS in terms of the four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Fourthly, for BR, MLSF and HNOML, their performance is mediocre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' CONCLUSION The robust multilabel learning method R-MLTSK-FS with strong fuzzy inference ability, label correlation learning ability and robustness against noisy labels is proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' From the aspect of soft label learning, R-MLTSK-FS constructs the soft label space to reduce the influence of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' From the aspect of soft multilabel loss function construction, R- MLTSK-FS utilizes the fuzzy rule-based TSK FS as a transpar- ent model to build the inference relationship between input fea- tures and soft labels, and then the loss function is constructed based on TSK FS and soft labels to enhance model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' From the aspect of correlation enhancement learning, R- MLTSK-FS utilizes the correlation information between soft la- bels to constrain the learning of model parameters and enhance the learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Experimental analyses on ten benchmark multilabel datasets and three synthetic multilabel datasets show the promising performance of R-MLTSK-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Further research on R-MLTSK-FS will proceed along two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' First, we will reduce the complexity of soft label learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Since R-MLTSK-FS considers all the original labels for a soft label, which is computationally intensive, research will be conducted to model with random label subsets for a soft label to reduce the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Second, we will simplify the rule base of TSK FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' In R-MLTSK-FS, the fuzzy system trans- forms all the original features into the fuzzy feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' If the dimension of the original feature space is large, the learning speed of R-MLTSK-FS will be slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hence, a screening mech- anism will be developed to identify representative subsets of the original features to improve the learning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' REFERENCE [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Shen, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Tsang, "The emerging trends of multi-label learning," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 7955-7974, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "A review on multi-label learning algorithms," IEEE Transactions on Knowledge and Data Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1819-1837, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Monfort, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Pan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ramakrishnan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Andonian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' McNamara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lascelles, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Fan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Gutfreund, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Feris, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Oliva, "Multi-moments in time: learning and interpreting models for multi-action video understanding," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 9434-9445, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Speth and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hand, "Automated label noise identification for facial attribute recognition," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 25-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [5] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhong, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, "Feature-induced labeling information enrichment for multi-label learning," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 32th AAAI Conference on Artificial Intelligence, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4446-4453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, "Incremental multi-label learning with active queries," Journal of Computer Science and Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 234-246, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [7] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Tan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Domeniconi, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, "Individuality-and commonality-based multiview multilabel learning," IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1716-1727, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, "Learning instance correlation functions for multilabel classification," IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 499-510, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Du and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Vong, "Robust online multilabel learning under dynamic changes in data distribution with labels," IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 50, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 374-385, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Kwok, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "Multi-label learning with global and local label correlation," IEEE Transactions on Knowledge and Data Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1081-1094, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Liu, "Towards class-imbalance aware multi-label learning," IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4459-4471, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chow, "Multilabel classification with label- specific features and classifiers: a coarse-and fine-tuned framework," IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1028-1042, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [13] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lughofer, "Evolving multi-label fuzzy classifier," Information Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 597, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1-23, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Cerri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Basgalupp, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Barros, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='F Carvalho, "Inducing hierarchical multi-label classification rules with genetic algorithms," Applied Soft Computing Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 77, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 584-604, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Shi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Gao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ma, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Qian, "A multi-label deep learning model with interpretable Grad- CAM for diabetic retinopathy classification," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' IEEE Engineering in Medicine and Biology Society, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1560-1563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chen, "Multi- label cause feature selection based on neighbourhood mutual information," International Journal of Machine Learning and Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3509-3522, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [17] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xiao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Choi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, "Multi-label Takagi- Sugeno-Kang fuzzy system," IEEE Transactions on Fuzzy Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3410-3425, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Cole, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Aodha, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lorieul, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Perona, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Morris, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Jojic, "Multi-label learning from single positive labels," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 933- 942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Han, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Shan, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chen, "Weakly supervised image classifiaction through noise regularization," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 11517-11525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xie and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, "CCMN: a general framework for learning with class-conditional multi-label noise," IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1109/TPAMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3141240 [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Aksoy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ravanbakhsh, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Demir, "Multi-label noise robust collaborative learning for remote sensing image classification," IEEE Transactions on Neural Networks and Learning Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='1109/TNNLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content='3209992 [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xie and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, "Partial multi-label learning with noisy label identification," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3676-3687, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Rajeswar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Rodriguez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Singhal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Vazquez, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Courville, "Multi-label iterated learning for image classification with label ambiguity," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4783-4793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [24] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Lyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Feng, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, "Noisy label tolerance: a new perspective of partial multi-label learning," Information Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 543, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 454-466, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [25] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xie, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Choi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, "Transductive joint- knowledge-transfer TSK FS for recognition of epileptic EEG signals," IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1481-1494, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Choi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, "Takagi-Sugeno-Kang transfer learning fuzzy logic system for the adaptive recognition of epileptic electroencephalogram signals," IEEE Transactions on Fuzzy Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1079-1094, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [27] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, "Multiview fuzzy logic system with the cooperation between visible and hidden views," IEEE Transactions on Fuzzy Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1162-1173, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Jiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Choi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chung, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, "A novel multi- task TSK fuzzy classifier and its enhanced version for labeling-risk-aware multi-task classification," Information Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 357, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' C, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 39-60, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Jiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chung, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Qian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Choi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=', "Recognition of epileptic EEG signals using a novel multiview TSK fuzzy system," IEEE Transactions on Fuzzy Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3-20, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [30] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Kong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' He, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Kaynak, "Fuzzy approximation- based finite-time control for a robot with actuator saturation under time- varying constraints of work space," IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4873-4884, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [31] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Liao and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Sun, "Sparse and decoupling control strategies based on Takagi–Sugeno fuzzy models," IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 947-960, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Tong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Feng, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, "Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems," IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1124-1135, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Choi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, "Transfer representation learning with TSK fuzzy system," IEEE Transactions on Fuzzy Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 649-663, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [34] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Guo and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xue, "Probabilistic multi-label classification with sparse feature learning," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 23rd International Joint Conference on Artificial Intelligence, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1373-1379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [35] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Shen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "L2,1-norm regularized discriminative feature selection for unsupervised learning," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 22nd International Joint Conference on Artificial Intelligence, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1589-1594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Nie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Cai, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ding, "Efficient and robust feature selection via joint ℓ2, 1-norms minimization," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 23rd International Conference on Neural Information Processing Systems, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1813-1821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "Multilabel neural networks with applications to functional genomics and text categorization," IEEE Transactions on Knowledge and Data Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1338-1351, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [38] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "Selective ensemble of classifier chains," in International Workshop on Multiple Classifier Systems, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 146-156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wu, "Learning label specific features for multi-label classification," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' IEEE International Conference on Data Mining, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 181-190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Gao, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "Fast multi-instance multi-label learning," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2614-2627, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ji, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Tang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Ye, "Extracting shared subspace for multi- label classification," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 381-389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [42] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Shen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hu, "Cascaded centralized TSK fuzzy system: universal approximator and high interpretation," Applied Soft Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 131-145, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [43] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hu and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Reichel, "Krylov-subspace methods for the sylvester equation," Linear Algebra and its Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 172, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 283- 313, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Song, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Wang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, "Semi-supervised multi-label learning by solving a sylvester equation," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the Siam International Conference on Data Mining, 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 410-419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [45] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Sorensen and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Antoulas, "The sylvester equation and approximate balanced reduction," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Linear Algebra and its Applications, 2002, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 351-352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Sun and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zong, "LCBM: a multi-view probabilistic model for multi- label classification," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2682-2696, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [47] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Luaces, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Díez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Barranquero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Coz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Bahamonde, "Binary relevance efficacy for multilabel classification," Progress in Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 303-313, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "ML-KNN: a lazy learning approach to multi-label learning," Pattern Recognition, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2038-2048, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [49] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Kudo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Kimura, "Multi-label classification with meta- label-specific features," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 23rd International Conference on Pattern Recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1613-1618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Read, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Pfahringer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Holmes, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Frank, "Classifier chains for multi-label classification," Machine Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 85, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 333-359, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [51] Tsoumakas, Grigorios, Katakis, Ioannis, and Vlahavas, "Random k- labelsets for multilabel classification," IEEE Transactions on Knowledge and Data Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1079-1089, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [52] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Bian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Xie, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Tao, "Corrlog: correlated logistic models for joint prediction of multiple labels," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 15th International Conference on Artificial Intelligence and Statistics, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 109-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [53] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Fu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Chen, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Hu, "Hybrid noise- oriented multilabel learning," IEEE Transactions on Cybermetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 50, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 2837-2850, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Łȩski, "Improving the generalization ability of neuro-fuzzy systems by ε-insensitive learning," International Journal of Applied Mathematics and Computer Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 437-447, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [55] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Yu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Zhou, "Multi-label hypothesis reuse," in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 525-533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' Demšar, "Statistical comparisons of classifiers over multiple data sets," Journal of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} +page_content=' 1-30, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE1T4oBgHgl3EQflQSM/content/2301.03283v1.pdf'} diff --git a/39FQT4oBgHgl3EQfHTWW/vector_store/index.pkl b/39FQT4oBgHgl3EQfHTWW/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3171b29e434e896a8477fca9df4eb176f623cb65 --- /dev/null +++ b/39FQT4oBgHgl3EQfHTWW/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fa4e8503d3e85b4b50b7f8952a19afa331aa89da598983d15467a8d2711ac495 +size 123336 diff --git a/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf b/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1358648c8d968e784fbc4040e5fffd6506f00ef0 --- /dev/null +++ b/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:17ee208b39c82caf31c6ec0f89f81088089808bab46f2d3796152cd1e77336f6 +size 2509023 diff --git 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The model of theta neurons (Theta model) captures, +in main, the bursting behavior of spiking cells in the brain of +biological beings, enduring periodic oscillations of the electric +potential in their membrane. +We study the following optimization problem: to design an +external stimulus (control), which steers all neurons of a given +population to their desired phases (i.e., excites/slows down its +spiking activity) with the highest probability. +This task is formulated as an optimal mean-field control +problem for the local continuity equation in the space of +probability measures. To solve this problem numerically, we +propose an indirect deterministic descent method based on an +exact representation of the increment (infinite-order variation) +of the objective functional. We discuss some aspects of practical +realization of the proposed method, and provide results of +numerical experiments. +I. INTRODUCTION +The phenomenon of synchronization of oscillatory pro- +cesses arise in many physical and natural systems involving +(relatively large) collections of structurally similar interacting +objects. This type of behavior — typically manifested in +practice by a formation of (desired or pathological) time- +periodic patterns — is demonstrated, e.g., by semiconductors +in laser physics [1], vibrating processes in mechanics [2], +biochemical reactions [3], [4], as well as in cardiac and +neural activity [5]–[7]. +In connection with oscillatory processes, there naturally +arise problems of designing artificial signals that can drive +open systems towards (or away from) synchronous oscil- +lations and frequency entrainment; important examples are +clinical treatment of neurological and cardiac deceases (such +*The authors acknowledge the financial support of the Foundation for +Science and Technology (FCT, Portugal) in the framework of the Associated +Laboratory “Advanced Production and Intelligent Systems” (AL ARISE, +ref. LA/P/0112/2020), R&D Unit SYSTEC (base UIDB/00147/2020 and +programmatic UIDP/00147/2020 funds), and projects SNAP (ref. NORTE- +01-0145-FEDER-000085) and MLDLCOV (ref. DSAIPA/CS/0086/2020). +1Roman Chertovskih, Maxim Staritsyn and Ant´onio Pedro Aguiar are +with Research Center for Systems and Technologies (SYSTEC), Faculty +of Engineering, University of Porto, Rua Dr. Roberto Frias, s/n 4200-465, +Porto, Portugal roman@fe.up.pt, staritsyn@fe.up.pt, +pedro.aguiar@fe.up.pt +2Nikolay Pogodaev is with Department of Mathematics “Tullio Levi- +Civita”, School of Sciences, University of Padova, Via Trieste, 63 - 35121 +Padova, Italy nickpogo@gmail.com +3 +Joaquim +Da +Silva +Sewane +is +with +Department +of +Mathe- +matics +and +Informatics, +Faculty +of +Sciences, +University +of +Ed- +uardo Mondlane, Av. Julius Nyerere, nr. 3453 Maputo, Mozambique +joaquimdasilvasewane@gmail.com +as Parkinson’s disease, epilepsy, and cardiac arrhythmias), +control of circadian rhythms [8], organization/destruction of +patterns in complex dynamic structures [9], and in neuro- +computing [10], [11]. +Starting from the pioneer works of Y. Kuramoto and +H. Araki, the mathematical imperative in the study of +oscillatory ensembles is the mean field dynamics, which +describes the behavior of an “averaged” representative of +the population instead of tracking all individuals in person. +This approach leads to a treatable (and elegant) mathematical +representation of the ensemble dynamics even in the case +when the cardinality of the population becomes very large, +and is naturally translated to the control-theoretical context: +in the most of applications, it is technically difficult (or even +impossible) to “isolate” the control influence for a particular +oscillatory unit; on the contrary, admissible signals usually +affect a significant part of the system, or the system as a +whole. The topic of control engineering which is focused on +designing “simultaneous” control signals for multi-agent sys- +tems is familiar under the name ensemble control. “Adaptive” +(distributed in the phase space) signals are called mean-field +type controls. +In this paper, we address a particular optimal control +problem of the type [12] based on a classical oscillatory +model [13] from the mathematical neuroscience. Namely, we +study the problem of in-phase synchronization of the mean +field of so-called theta neurons: to steer a given probability +distribution of harmonic phases towards a target one by a +simultaneous (ensemble) or individual (mean-field) control. +To solve our problem numerically, we propose a determin- +istic iterative method of sequential “control improvement”, +entailed by an an exact formula for the variation of the +objective functional. The proposed approach is based on the +optimal mean-field control theory (the dynamic optimization +in the space of probability measures) and is quite flexible: +it admits one to treat arbitrary statistical ensembles, and can +be applied to any problem of a “state-linear” structure, far +beyond the considered specific model. +II. PROBLEM STATEMENT. +MEAN-FIELD CONTROL SETUP +Consider a population of homotypic oscillatory systems +represented by the canonical Ermentrout-Kopell model [13], +[14]. This model describes the time-evolution of excitable +neurons (customary named “theta neurons”) which endure +periodic oscillations of their membrane potential. Each theta +arXiv:2301.11952v1 [math.OC] 27 Jan 2023 + +neuron in the population is characterized by its phase +θ(t) ∈ S1 .= R/2πZ +which satisfies the ODEs +d +dtθ .= ˙θ = vu(θ, η) .= (1 − cos θ) + (1 + cos θ) (u + η) . +Here, η is the baseline current in the neuron membrane, +which varies in a given interval I .= [a, b], and u is an external +stimulus. +Theta model provides a simple mathematical description +of the so-called spiking behavior. By convention, we say that +a neuron produces a spike at time t if θ(t) = π. If η > 0 (and +u ≡ 0) the neuron spikes periodically with the frequency +2√η. If η < 0, the neuron is excitable and can produce +spikes after a sufficiently intensive stimulus u. +In what follows, η is viewed as a parameter of the model +fluctuation. In the simplest case, this parameter runs through +a finite set {ηk, k = 1, N}, which corresponds to a finite +ensemble {θk, k = 1, N} of theta neurons, +˙θk = vu(θk, ηk), +k = 1, N. +(1) +In a more general setup to be discussed below, η can be +drawn from a given probability distribution. +Remark that (1) falls into the well-recognized Watanabe- +Strogatz class of phase oscillators driven by complex func- +tions t �→ Hk(t) ∈ C, +˙θk = ωk + Im +� +Hk(t) e−i θk� +, +k = 1, N, +where ωk is the natural (intrinsic) frequency of the kth +oscillator in the population, and Hk is the associated input, +modulated by a sinusoidal function (sometimes, this model is +called “sinusoidally coupled”); in general, both the natural +frequencies and the inputs can be effected by an external +driving parameter, furthermore, Hk can model interactions +between oscillators inside the population. Note that model +(1) fits the general statement with +ωk = ωk(u) .= u + ηk + 1, +Hk = Hk(u) .= i(u + ηk − 1), +which does not involve interaction terms (formally, equations +(1) are paired only by the common term u). In the context +of applications, this non-interacting model can be viewed +as a “first-order approximation” of a sufficiently sparsely +connected neural network (such are real biological ones), +especially, if the neurons’ activity is studied over relatively +short time periods. The case of interacting neurons will be +briefly discussed in section V. +A. Mean-Field Limit +We are interested in the behavior of system (1) for the case +when N → ∞. Introduce extra, “fictitious” states t �→ ηk(t) +as solutions to +˙ηk = 0, +(2) +accompanying (1), and consider the empirical probability +measure +µN +t = 1 +N +N +� +k=1 +δ(θk(t),ηk(t)), +(3) +(δx stands for the Dirac probability measure concentrated at +at a point x). +The measure-valued function t �→ µN +t +designates the +statistical behavior of the ensemble {(θk, ηk), k = 1, N}: +for any Borel set A ⊂ S1 × I, the value µN +t (A) shows the +number of neurons whose phase belongs to A. +It is well-known that the curve t �→ µN +t +satisfies, in the +weak sense, the local continuity equation [15] +∂tµt(θ, η) + ∂θ +� +vu(θ, η) µt(θ, η) +� += 0. +(4) +Recall that the map t �→ µt is said to be a weak (distribu- +tional) solution of (4) iff +0 = +� T +0 +dt +� +S1×I +� +∂tϕ + ∇xϕ · vu +� +dµt +∀ ϕ ∈ C1 +c ((0, T) × S1 × I). +(C1 +c ((0, T)×S1×I) denotes the space of continuously differ- +entiable functions (0, T)×S1 ×I �→ R with compact support +in (0, T) × S1 × I.) Under standard regularity assumptions, +the weak solution exists, it is unique, and it is absolutely +continuous as a function [0, T] �→ P(S1 ×I); here P(S1 ×I) +denotes the space of probability measures on S1×I endowed +with any Wasserstein distance Wp, p ≥ 1 [15]. +Equation (4) provides the macroscopic description of the +population of microscopic dynamical units (1) called the +mean field. This representation remains valid in the limit +N → ∞, when µN converges to some µ ∈ P(S1 × I) in +C([0, T]; P(S1 × I)). Moreover, (4) makes sense if phases +θ and currents η are drawn from an abstract probability +distribution on the cylinder S1 × I, +µ0 = ϑ ∈ P(S1 × I). +(5) +Indeed, one can immerse the system of ODEs (1) in a +deterministic (S1 × I)-valued random process +(t, ω) �→ Θt(ω), +defined on a probability space (Ω, F, P) of an arbitrary +nature (Ω is an abstract set, F is a sigma-algebra on Ω, +and P is a probability measure F �→ [0, 1]), and satisfying +the ODE +d +dtΘt(ω) = +� +vu +� +Θt(ω) +� +0 +� +. +It is a simple technical exercise to check that the function +t �→ µt .= (Θt)♯P +solves the Cauchy problem (4), (5) with ϑ .= (Θ0)♯P, where +the symbol ♯ denotes the operation of pushforward of a +measure by a (Borel) function Ω �→ S1 × I. Note that +empirical ensembles (3) fit this setup if Ω = {1, . . . , N} +and P is the normalized counting measure. + +Finally, observe that the variable η enters PDE (4) as a +parameter rather than state variable. This means that (4) can +be regarded as an η-parametric family of continuity equations +on the 1D space S1 rather than a PDE on the 2D space S1×I. +This observation is essential for the numerical treatment of +the problem (4) (see section IV). +B. Control Signals +Now, we shall fix the class of admissible control signal u. +Consider two options: +• u = u(t), i.e., the control effects all neurons of the +ensemble in the same way. We call this type of ex- +ternal influences the ensemble (simultaneous, common) +control. Such a control is statistical in its spirit as +it influences the whole ensemble “in average”. As a +natural space of such controls we choose +u ∈ U .= L2([0, T]; R). +(6) +• u = wt(θ, η), i.e., the stimulus is adopted to the +neuron’s individual characteristics and phase-dependent. +The use of such a distributed, mean-field type control +w ∈ W .= L2([0, T]; C(S1 × I; R)), +(7) +assumes some technical option to variate control signals +over the spatial domain. +It is natural to expect that the second-type control should +perform better. However, let us stress again that the practical +implementation of “personalized” control signals is hardly +realistic as soon as the number of driven objects is large +enough (for experiments that pretend to mimic the biological +neural tissue, this number should be astronomic!). In reality, +a meaningful class of control signals is U, or something “in +the middle” between the mentioned two options. +C. Performance Criterion +We study a generalization of the optimization problem +[12]: to steer the neural population to a target phase dis- +tribution at a prescribed (finite) time moment T > 0 with +care about the total energy of the control action. Assuming +that the target distribution is given by a (bounded continuous) +function η �→ ˇθ(η), our optimization problem reads: +(P1) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +min I[u] = +� +F +� +θ, ˇθ(η) +� +dµT (θ, η) ++α +2 +� T +0 +u2(t) dt, +α > 0, +subject to (4), (6), +where +F(θ, ω) = 1 +2(sin θ − sin ω)2 + 1 +2(cos θ − cos ω)2 +=1 − cos(θ − ω), +and +� +.= +� +S1×I +. +In this problem, the part of state variable is played by the +probability measure µt. +Note that the functional I and the dynamics (4) are linear +in µ (despite the non-linearity of the map (θ, η) �→ vu(θ, η)). +At the same time, (4) contains a product of µ and u, which +means that (P1) is, in fact, a bi-linear (non-convex) problem. +Standard arguments from the theory of transport equations +in the Wasserstein space [15] together with the classical +Weierstrass theorem ensure that problem (P1) is well posed, +i.e., it does have a minimizer within the admissible class U +of control signals (refer, e.g., to [16]). +An alternative version of problem (P1) is formulated in +terms of the mean-field type control: +(P2) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +min J[w] = +� +F +� +θ, ˇθ(η) +� +dµT ++α +2 +� T +0 +dt +� +w2 +t dµt, +subject to (4), (7). +In what follows, we shall focus on the “more realistic” +statement (P1), though all the forthcoming results can be +extended, at least formally, to problem (P2). +III. COST INCREMENT FORMULA. +NUMERICAL ALGORITHM +As it was remarked above, problem (P1) is linear in state- +measure. This fact allows us to represent the variation of +the cost functional I with respect to any variation of con- +trol u exactly (without any residual terms). The announced +representation follows from the duality with the co-state +from Pontryagin’s maximum principle [17], and generalizes +the classical exact increment formula for conventional state- +linear optimal control problems [18]. +Consider two arbitrary controls +¯u, u ∈ U, +u ̸= ¯u, +and let +t �→ ¯µt .= µt[¯u] and t �→ µt .= µt[u] +be the respective weak solutions to the continuity equation +(4). Let also +¯p .= p[¯u] : (t, θ, η) �→ ¯pt(θ, η) +be a classical solution to the following (non-conservative +transport) equation: +∂tpt(θ, η)+ ∂θpt(θ, η) · v¯u(t)(θ, η) = 0. +(8) +PDE (8) is known to be dual to the (conservative transport +equation) (4); the duality is formally established by the +observation that the map +t �→ +� +¯pt d¯µt +is constant on [0, T]. One can check that, under the common +regularity of the problem data, this map is an absolutely +continuous function [0, T] �→ R (refer to [15] for further +details). + +As soon as ¯p is chosen as a solution to (8) with the terminal +condition +pT (θ, η) = − F +� +θ, ˇθ(η) +� +, +(9) +the discussed duality makes it possible to represent the +increment (variation) +∆I .= I[u] − I[¯u] +of the functional I as follows: +−∆I = +� T +0 +� +H (µt, ∂θ ¯pt, u(t)) − H (µt, ∂θ ¯pt, ¯u(t)) +� +dt, +(10) +where +H(µ, ζ, u) .= u +� +ζ(θ, η) · (1 + cos θ) dµ(θ, η) − α +2 u2. +The derivation of this formula is dropped, since it is com- +pletely similar to [18]. +Based on representation (10), we can treat problem (P1) +in the following iterative way: given a reference control ¯u, +one looks for a new “target” signal u that “improves” the +functional value, i.e such that ∆I < 0. The best choice of +the target control is provided by the maximization of the +integrand of (10) in the variable u: +H (µt, ∂θ ¯pt, u) → max, +u ∈ R. +The unique solution of the latter problem is obtain in the +analytic form as +ut[µ] = 1 +α +� +∂θ ¯pt(θ, η) (1 + cos θ) dµ(θ, η). +(11) +Here, it is worthwhile to mention that the reference dual +state ¯p enters formula (11) only in the form of the partial +derivative +¯ξt(θ, η) .= ∂θ ¯pt(θ, η). +Differentiating (8) and (9) in θ one can easily check that +¯ξ solves the η-parametric family of the same continuity +equations (4) backward in time, starting from the terminal +condition +ξT = −∂θF +� +θ, ˇθ(η) +� .= sin +�ˇθ(η) − θ +� +. +(12) +Now, (11) can be reformulated in terms of the variable ¯ξ: +ut[µ] = 1 +α +� +¯ξt(θ, η) (1 + cos θ) dµ(θ, η). +(13) +Note that the map (t, µ) �→ ut[µ] can be used as a feedback +control +[0, T] × P(S1 × I) �→ R +of system (4) in the space of probability measures. Injecting +this control into (4), we obtain a nonlocal continuity equation +∂tµt + ∂θ +� +vu[µt] µt +� += 0, +µ0 = ϑ, +(14) +which is well-posed (thanks to the fact that function (θ, η) �→ +vu(θ, η) is smooth and bounded). Solving the last equation +Algorithm 1: Numerical algorithm for optimal en- +semble control +Data: ¯u ∈ U (initial guess), ε > 0 (tolerance) +Result: {uk}k≥0 ⊂ U such that I[uk+1] < I[uk] +k ← 0; +u0 ← ¯u; +repeat +µk ← ˆµ[uk]; +uk+1 ← u[µk]; +k ← k + 1; +until I[uk−1] − I[uk] < ε; +numerically, and substituting its solution t �→ ˆµt .= ˆµt[¯u] +into (11), we construct the “improved” signal: +u(t) = ut[ˆµt]. +This idea gives rise to the following Algorithm 1. +By construction, Algorithm 1 generates a sequence +{uk}k≥0 ⊂ U of controls with the property: +Ik+1 .= I[uk+1] < I[uk] .= Ik. +Since the sequence of numbers (Ik)k≥0 is bounded from +below by min(P) it converges. +Finally, remark that the same line of arguments can be +formally applied to problem (P2). The respective mean-field +type control takes the form +wt(θ, η) = 1 +α +¯ξt(θ, η) (1 + cos θ). +This construction gives rise to an iterative method, similar +to Algorithm 1. +IV. NUMERICAL RESULTS +Let us discuss several aspects of the numerical implemen- +tation of Algorithm 1. +First, note that the method proposed here does not involve +any intrinsic parametric optimization: the most of indirect +algorithms for optimal control require the dynamic adjust- +ment of some internal computational parameters; such are +standard methods based on Pontryagin’s maximum principle +[19], [20] that imply the internal such as line search for the +specification of the “depth” of the needle-shaped (or weak) +control variations. +Each iteration of Algorithm 1 requires numerical solution +of two problems: one is the linear problem (4), (12) (inte- +grated backward in time), and one for the nonlocal continuity +equation (14) (solved numerically forward in time). Since +both (4) and (14) have no terms involving partial derivatives +in η, one can think of η as a parameter and solve the corre- +sponding parametric families of one-dimensional continuity +equations. +Consider the problem (P) with initial distribution of +neurons µ0 given by the density function +ρ0(θ, η) = +� +2 + 3 cos(2θ) − 2 sin(2θ) +� +η, + +and with constant target function ˇθ(η) ≡ π. In other words, +our goal is to bring neurons’ states as close as possible to +the segment 0 × I by the time moment T with the aid of +sufficiently small controls. +Parameters for the computation: +T = 6, +I = [0.0, 1.0], +α = 1; +we used 512 Fourier harmonics in θ and grid steps +∆η = 0.002, +∆t = 0.002. +Equations (4) and (14) are integrated by the standard spectral +method [21] using the trigonometric Fourier expansion in θ +for each η from the grid. Parameters of the algorithm: ¯u ≡ 0, +ε = 0.01. +0 +1 +2 +3 +4 +5 +6 +t +−3 +−2 +−1 +0 +1 +u(t) +Fig. 1. +Control input computed by the Algorithm 1 +V. CONCLUSION +The goal of this paper is to present an approach based +on the mean-field control paradigm to solve problems of +optimization and synchronization of oscillatory processes +(here, the addressed Theta model is among the simplest +but prominent examples). The proposed technique can be +applied to any state-linear optimal control problem involving +(finite or infinite) non-interacting statistical ensembles of an +arbitrary nature. In particular, Algorithm 1 can be easily +adapted to some other neural model such as SNIPER model, +sinusoidal model etc. [12]. +We plan to continue this study in the way of natural +generalization of model (1) by admitting the interaction +between theta neurons, +˙θk = vu(θk, ηk) + 1 +N +N +� +j=1 +K(θk, θj), +k = 1, N, +where K is certain interaction potential formalizing the +spatial connectivity of neurons in the tissue. This will result +in control problems of the sort (P1,2) stated over the nonlocal +continuity equation +∂tµt + ∂θ +� +[vu + K ⋆ µt] µt +� += 0 +involving the term +(K ⋆ µ)(θ) .= +� +K(θ, ζ) dµ(ζ). +0 +π +2π +θ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +η +0 +π +2π +θ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +η +0 +π +2π +θ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +η +Fig. 2. +Trajectory µt(θ, µ) of (4) at time moments t = 0, 3 and 6 (from +top to bottom) computed for the optimal control input shown in Fig. 1. The +standard “rainbow” color table was used to code the isovalues: from black +(minimal values), violet, . . . , to red (maximal values). + +Such problems are not state-linear anymore, and the exact +formula (10) becomes inapplicable. For this case, a promis- +ing alternative could be an approach based on Pontryagin’s +maximum principle [16]. +REFERENCES +[1] I. +Fischer, +Y. +Liu, +and +P. +Davis, +“Synchronization +of +chaotic +semiconductor +laser +dynamics +on +subnanosecond +time +scales +and +its +potential +for +chaos +communication,” +Phys. +Rev. +A, +vol. +62, +p. +011801, +Jun +2000. +[Online]. +Available: +https://link.aps.org/doi/10.1103/PhysRevA.62.011801 +[2] I. +Blekhman, +I. +Blekhman, +and +E. +Rivin, +Synchro- +nization +in +Science +and +Technology, +ser. +ASME +press +translations. +ASME +Press, +1988. +[Online]. +Available: +https://books.google.ru/books?id=ao1QAAAAMAAJ +[3] T. Nishikawa, N. Gulbahce, and A. E. Motter, “Spontaneous +reaction silencing in metabolic optimization,” PLOS Computational +Biology, vol. 4, no. 12, pp. 1–12, 12 2008. [Online]. Available: +https://doi.org/10.1371/journal.pcbi.1000236 +[4] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, ser. +Dover books on chemistry. +Dover Publications, 2003. [Online]. +Available: https://books.google.ru/books?id=4ADt7smO5Q8C +[5] P. J. Uhlhaas and W. Singer, “Neural synchrony in brain disorders: +Relevance +for +cognitive +dysfunctions +and +pathophysiology,” +Neuron, vol. 52, no. 1, pp. 155–168, 2006. [Online]. Available: +https://www.sciencedirect.com/science/article/pii/S0896627306007276 +[6] S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, +and W. L. Ditto, “Controlling chaos in the brain,” Nature, vol. +370, +no. +6491, +pp. +615–620, +Aug +1994. +[Online]. +Available: +https://doi.org/10.1038/370615a0 +[7] L. +Glass, +“Cardiac +arrhythmias +and +circle +maps-a +classical +problem,” +Chaos: +An +Interdisciplinary +Journal +of +Nonlinear +Science, vol. 1, no. 1, pp. 13–19, 1991. [Online]. Available: +https://doi.org/10.1063/1.165810 +[8] A. Winfree, The Geometry of Biological Time, ser. Interdisciplinary +Applied Mathematics. Springer New York, 2013. [Online]. Available: +https://books.google.ru/books?id=7qjTBwAAQBAJ +[9] I. Z. Kiss, C. G. Rusin, H. Kori, and J. L. Hudson, “Engineering +complex dynamical structures: Sequential patterns and desynchro- +nization,” Science, vol. 316, no. 5833, pp. 1886–1889, 2007. [Online]. +Available: https://www.science.org/doi/abs/10.1126/science.1140858 +[10] F. C. Hoppensteadt and E. M. Izhikevich, “Synchronization of laser +oscillators, associative memory, and optical neurocomputing,” Phys. +Rev. E, vol. 62, pp. 4010–4013, Sep 2000. [Online]. Available: +https://link.aps.org/doi/10.1103/PhysRevE.62.4010 +[11] F. Hoppensteadt and E. Izhikevich, “Synchronization of mems res- +onators and mechanical neurocomputing,” IEEE Transactions on Cir- +cuits and Systems I: Fundamental Theory and Applications, vol. 48, +no. 2, pp. 133–138, 2001. +[12] J.-S. Li, I. Dasanayake, and J. Ruths, “Control and synchronization of +neuron ensembles,” IEEE Transactions on Automatic Control, vol. 58, +no. 8, pp. 1919–1930, 2013. +[13] G. B. Ermentrout and N. Kopell, “Parabolic bursting in an excitable +system coupled with a slow oscillation,” SIAM Journal on Applied +Mathematics, vol. 46, no. 2, pp. 233–253, 1986. [Online]. Available: +http://www.jstor.org/stable/2101582 +[14] S. H. Strogatz, Nonlinear dynamics and chaos : with applications to +physics, biology, chemistry, and engineering. Second edition. Boulder, +CO : Westview Press, a member of the Perseus Books Group, [2015], +[2015], includes bibliographical references and index. [Online]. +Available: https://search.library.wisc.edu/catalog/9910223127702121 +[15] L. Ambrosio and G. Savar´e, “Gradient flows of probability measures,” +in Handbook of differential equations: evolutionary equations. Vol. III, +ser. Handb. Differ. Equ. +Elsevier/North-Holland, Amsterdam, 2007, +pp. 1–136. +[16] N. Pogodaev and M. Staritsyn, “Impulsive control of nonlocal +transport +equations,” +Journal +of +Differential +Equations, +vol. +269, +no. +4, +pp. +3585–3623, +2020. +[Online]. +Available: +https://www.sciencedirect.com/science/article/pii/S002203962030108X +[17] N. Pogodaev, “Optimal control of continuity equations,” NoDEA +Nonlinear Differential Equations Appl., vol. 23, no. 2, pp. Art. 21, +24, 2016. +[18] M. Staritsyn, N. Pogodaev, R. Chertovskih, and F. L. Pereira, “Feed- +back maximum principle for ensemble control of local continuity +equations: An application to supervised machine learning,” IEEE +Control Systems Letters, vol. 6, pp. 1046–1051, 2022. +[19] A. +V. +Arguchintsev, +V. +A. +Dykhta, +and +V. +A. +Srochko, +“Optimal +control: +Nonlocal +conditions, +computational +methods, +and the variational principle of maximum,” Russian Mathematics, +vol. +53, +no. +1, +pp. +1–35, +Jan +2009. +[Online]. +Available: +https://doi.org/10.3103/S1066369X09010010 +[20] P. Drag, K. Styczen, M. Kwiatkowska, and A. Szczurek, “A review +on the direct and indirect methods for solving optimal control prob- +lems with differential-algebraic constraints,” Studies in Computational +Intelligence, vol. 610, pp. 91–105, 07 2016. +[21] J. P. Boyd, Chebyshev and Fourier spectral methods., 2nd ed. +Mi- +neola, NY: Dover Publications, 2001. + diff --git a/4NFKT4oBgHgl3EQf9C5Z/content/tmp_files/load_file.txt b/4NFKT4oBgHgl3EQf9C5Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c39f2209bcfa8b51d1d72963065882f032f91f4 --- /dev/null +++ b/4NFKT4oBgHgl3EQf9C5Z/content/tmp_files/load_file.txt @@ -0,0 +1,439 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf,len=438 +page_content='Optimization of External Stimuli for Populations of Theta Neurons via Mean-Field Feedback Control* Roman Chertovskih1, Nikolay Pogodaev2, Maxim Staritsyn1, Joaquim Da Silva Sewane3 and Ant´onio Pedro Aguiar1 Abstract— We study a problem of designing “robust” external excitations for control and synchronization of an assembly of homotypic harmonic oscillators representing so-called theta neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The model of theta neurons (Theta model) captures, in main, the bursting behavior of spiking cells in the brain of biological beings, enduring periodic oscillations of the electric potential in their membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' We study the following optimization problem: to design an external stimulus (control), which steers all neurons of a given population to their desired phases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', excites/slows down its spiking activity) with the highest probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This task is formulated as an optimal mean-field control problem for the local continuity equation in the space of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' To solve this problem numerically, we propose an indirect deterministic descent method based on an exact representation of the increment (infinite-order variation) of the objective functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' We discuss some aspects of practical realization of the proposed method, and provide results of numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' INTRODUCTION The phenomenon of synchronization of oscillatory pro- cesses arise in many physical and natural systems involving (relatively large) collections of structurally similar interacting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This type of behavior — typically manifested in practice by a formation of (desired or pathological) time- periodic patterns — is demonstrated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', by semiconductors in laser physics [1], vibrating processes in mechanics [2], biochemical reactions [3], [4], as well as in cardiac and neural activity [5]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In connection with oscillatory processes, there naturally arise problems of designing artificial signals that can drive open systems towards (or away from) synchronous oscil- lations and frequency entrainment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' important examples are clinical treatment of neurological and cardiac deceases (such The authors acknowledge the financial support of the Foundation for Science and Technology (FCT, Portugal) in the framework of the Associated Laboratory “Advanced Production and Intelligent Systems” (AL ARISE, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' LA/P/0112/2020), R&D Unit SYSTEC (base UIDB/00147/2020 and programmatic UIDP/00147/2020 funds), and projects SNAP (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' NORTE- 01-0145-FEDER-000085) and MLDLCOV (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' DSAIPA/CS/0086/2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1Roman Chertovskih, Maxim Staritsyn and Ant´onio Pedro Aguiar are with Research Center for Systems and Technologies (SYSTEC), Faculty of Engineering, University of Porto, Rua Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Roberto Frias, s/n 4200-465, Porto, Portugal roman@fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='pt, staritsyn@fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='pt, pedro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='aguiar@fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='pt 2Nikolay Pogodaev is with Department of Mathematics “Tullio Levi- Civita”, School of Sciences, University of Padova, Via Trieste, 63 - 35121 Padova, Italy nickpogo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='com 3 Joaquim Da Silva Sewane is with Department of Mathe- matics and Informatics, Faculty of Sciences, University of Ed- uardo Mondlane, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Julius Nyerere, nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 3453 Maputo, Mozambique joaquimdasilvasewane@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='com as Parkinson’s disease, epilepsy, and cardiac arrhythmias), control of circadian rhythms [8], organization/destruction of patterns in complex dynamic structures [9], and in neuro- computing [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Starting from the pioneer works of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Kuramoto and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Araki, the mathematical imperative in the study of oscillatory ensembles is the mean field dynamics, which describes the behavior of an “averaged” representative of the population instead of tracking all individuals in person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This approach leads to a treatable (and elegant) mathematical representation of the ensemble dynamics even in the case when the cardinality of the population becomes very large, and is naturally translated to the control-theoretical context: in the most of applications, it is technically difficult (or even impossible) to “isolate” the control influence for a particular oscillatory unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' on the contrary, admissible signals usually affect a significant part of the system, or the system as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The topic of control engineering which is focused on designing “simultaneous” control signals for multi-agent sys- tems is familiar under the name ensemble control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' “Adaptive” (distributed in the phase space) signals are called mean-field type controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In this paper, we address a particular optimal control problem of the type [12] based on a classical oscillatory model [13] from the mathematical neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Namely, we study the problem of in-phase synchronization of the mean field of so-called theta neurons: to steer a given probability distribution of harmonic phases towards a target one by a simultaneous (ensemble) or individual (mean-field) control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' To solve our problem numerically, we propose a determin- istic iterative method of sequential “control improvement”, entailed by an an exact formula for the variation of the objective functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The proposed approach is based on the optimal mean-field control theory (the dynamic optimization in the space of probability measures) and is quite flexible: it admits one to treat arbitrary statistical ensembles, and can be applied to any problem of a “state-linear” structure, far beyond the considered specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' PROBLEM STATEMENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' MEAN-FIELD CONTROL SETUP Consider a population of homotypic oscillatory systems represented by the canonical Ermentrout-Kopell model [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This model describes the time-evolution of excitable neurons (customary named “theta neurons”) which endure periodic oscillations of their membrane potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Each theta arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='11952v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='OC] 27 Jan 2023 neuron in the population is characterized by its phase θ(t) ∈ S1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= R/2πZ which satisfies the ODEs d dtθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= ˙θ = vu(θ, η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= (1 − cos θ) + (1 + cos θ) (u + η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Here, η is the baseline current in the neuron membrane, which varies in a given interval I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= [a, b], and u is an external stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Theta model provides a simple mathematical description of the so-called spiking behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' By convention, we say that a neuron produces a spike at time t if θ(t) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' If η > 0 (and u ≡ 0) the neuron spikes periodically with the frequency 2√η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' If η < 0, the neuron is excitable and can produce spikes after a sufficiently intensive stimulus u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In what follows, η is viewed as a parameter of the model fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In the simplest case, this parameter runs through a finite set {ηk, k = 1, N}, which corresponds to a finite ensemble {θk, k = 1, N} of theta neurons, ˙θk = vu(θk, ηk), k = 1, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (1) In a more general setup to be discussed below, η can be drawn from a given probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Remark that (1) falls into the well-recognized Watanabe- Strogatz class of phase oscillators driven by complex func- tions t �→ Hk(t) ∈ C, ˙θk = ωk + Im � Hk(t) e−i θk� , k = 1, N, where ωk is the natural (intrinsic) frequency of the kth oscillator in the population, and Hk is the associated input, modulated by a sinusoidal function (sometimes, this model is called “sinusoidally coupled”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' in general, both the natural frequencies and the inputs can be effected by an external driving parameter, furthermore, Hk can model interactions between oscillators inside the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Note that model (1) fits the general statement with ωk = ωk(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= u + ηk + 1, Hk = Hk(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= i(u + ηk − 1), which does not involve interaction terms (formally, equations (1) are paired only by the common term u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In the context of applications, this non-interacting model can be viewed as a “first-order approximation” of a sufficiently sparsely connected neural network (such are real biological ones), especially, if the neurons’ activity is studied over relatively short time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The case of interacting neurons will be briefly discussed in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Mean-Field Limit We are interested in the behavior of system (1) for the case when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Introduce extra, “fictitious” states t �→ ηk(t) as solutions to ˙ηk = 0, (2) accompanying (1), and consider the empirical probability measure µN t = 1 N N � k=1 δ(θk(t),ηk(t)), (3) (δx stands for the Dirac probability measure concentrated at at a point x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The measure-valued function t �→ µN t designates the statistical behavior of the ensemble {(θk, ηk), k = 1, N}: for any Borel set A ⊂ S1 × I, the value µN t (A) shows the number of neurons whose phase belongs to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' It is well-known that the curve t �→ µN t satisfies, in the weak sense, the local continuity equation [15] ∂tµt(θ, η) + ∂θ � vu(θ, η) µt(θ, η) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (4) Recall that the map t �→ µt is said to be a weak (distribu- tional) solution of (4) iff 0 = � T 0 dt � S1×I � ∂tϕ + ∇xϕ · vu � dµt ∀ ϕ ∈ C1 c ((0, T) × S1 × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (C1 c ((0, T)×S1×I) denotes the space of continuously differ- entiable functions (0, T)×S1 ×I �→ R with compact support in (0, T) × S1 × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=') Under standard regularity assumptions, the weak solution exists, it is unique, and it is absolutely continuous as a function [0, T] �→ P(S1 ×I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' here P(S1 ×I) denotes the space of probability measures on S1×I endowed with any Wasserstein distance Wp, p ≥ 1 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Equation (4) provides the macroscopic description of the population of microscopic dynamical units (1) called the mean field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This representation remains valid in the limit N → ∞, when µN converges to some µ ∈ P(S1 × I) in C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' P(S1 × I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Moreover, (4) makes sense if phases θ and currents η are drawn from an abstract probability distribution on the cylinder S1 × I, µ0 = ϑ ∈ P(S1 × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (5) Indeed, one can immerse the system of ODEs (1) in a deterministic (S1 × I)-valued random process (t, ω) �→ Θt(ω), defined on a probability space (Ω, F, P) of an arbitrary nature (Ω is an abstract set, F is a sigma-algebra on Ω, and P is a probability measure F �→ [0, 1]), and satisfying the ODE d dtΘt(ω) = � vu � Θt(ω) � 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' It is a simple technical exercise to check that the function t �→ µt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= (Θt)♯P solves the Cauchy problem (4), (5) with ϑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= (Θ0)♯P, where the symbol ♯ denotes the operation of pushforward of a measure by a (Borel) function Ω �→ S1 × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Note that empirical ensembles (3) fit this setup if Ω = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' , N} and P is the normalized counting measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Finally, observe that the variable η enters PDE (4) as a parameter rather than state variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This means that (4) can be regarded as an η-parametric family of continuity equations on the 1D space S1 rather than a PDE on the 2D space S1×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This observation is essential for the numerical treatment of the problem (4) (see section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Control Signals Now, we shall fix the class of admissible control signal u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Consider two options: u = u(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', the control effects all neurons of the ensemble in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' We call this type of ex- ternal influences the ensemble (simultaneous, common) control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Such a control is statistical in its spirit as it influences the whole ensemble “in average”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' As a natural space of such controls we choose u ∈ U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= L2([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (6) u = wt(θ, η), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', the stimulus is adopted to the neuron’s individual characteristics and phase-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The use of such a distributed, mean-field type control w ∈ W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= L2([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' C(S1 × I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' R)), (7) assumes some technical option to variate control signals over the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' It is natural to expect that the second-type control should perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' However, let us stress again that the practical implementation of “personalized” control signals is hardly realistic as soon as the number of driven objects is large enough (for experiments that pretend to mimic the biological neural tissue, this number should be astronomic!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In reality, a meaningful class of control signals is U, or something “in the middle” between the mentioned two options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Performance Criterion We study a generalization of the optimization problem [12]: to steer the neural population to a target phase dis- tribution at a prescribed (finite) time moment T > 0 with care about the total energy of the control action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Assuming that the target distribution is given by a (bounded continuous) function η �→ ˇθ(η), our optimization problem reads: (P1) � � � � � � � � � � � � � � � min I[u] = � F � θ, ˇθ(η) � dµT (θ, η) +α 2 � T 0 u2(t) dt, α > 0, subject to (4), (6), where F(θ, ω) = 1 2(sin θ − sin ω)2 + 1 2(cos θ − cos ω)2 =1 − cos(θ − ω), and � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= � S1×I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In this problem, the part of state variable is played by the probability measure µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Note that the functional I and the dynamics (4) are linear in µ (despite the non-linearity of the map (θ, η) �→ vu(θ, η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' At the same time, (4) contains a product of µ and u, which means that (P1) is, in fact, a bi-linear (non-convex) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Standard arguments from the theory of transport equations in the Wasserstein space [15] together with the classical Weierstrass theorem ensure that problem (P1) is well posed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', it does have a minimizer within the admissible class U of control signals (refer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', to [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' An alternative version of problem (P1) is formulated in terms of the mean-field type control: (P2) � � � � � � � � � � � � � � � min J[w] = � F � θ, ˇθ(η) � dµT +α 2 � T 0 dt � w2 t dµt, subject to (4), (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In what follows, we shall focus on the “more realistic” statement (P1), though all the forthcoming results can be extended, at least formally, to problem (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' COST INCREMENT FORMULA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' NUMERICAL ALGORITHM As it was remarked above, problem (P1) is linear in state- measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This fact allows us to represent the variation of the cost functional I with respect to any variation of con- trol u exactly (without any residual terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The announced representation follows from the duality with the co-state from Pontryagin’s maximum principle [17], and generalizes the classical exact increment formula for conventional state- linear optimal control problems [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Consider two arbitrary controls ¯u, u ∈ U, u ̸= ¯u, and let t �→ ¯µt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= µt[¯u] and t �→ µt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= µt[u] be the respective weak solutions to the continuity equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Let also ¯p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= p[¯u] : (t, θ, η) �→ ¯pt(θ, η) be a classical solution to the following (non-conservative transport) equation: ∂tpt(θ, η)+ ∂θpt(θ, η) · v¯u(t)(θ, η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (8) PDE (8) is known to be dual to the (conservative transport equation) (4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' the duality is formally established by the observation that the map t �→ � ¯pt d¯µt is constant on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' One can check that, under the common regularity of the problem data, this map is an absolutely continuous function [0, T] �→ R (refer to [15] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' As soon as ¯p is chosen as a solution to (8) with the terminal condition pT (θ, η) = − F � θ, ˇθ(η) � , (9) the discussed duality makes it possible to represent the increment (variation) ∆I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= I[u] − I[¯u] of the functional I as follows: −∆I = � T 0 � H (µt, ∂θ ¯pt, u(t)) − H (µt, ∂θ ¯pt, ¯u(t)) � dt, (10) where H(µ, ζ, u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= u � ζ(θ, η) · (1 + cos θ) dµ(θ, η) − α 2 u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The derivation of this formula is dropped, since it is com- pletely similar to [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Based on representation (10), we can treat problem (P1) in the following iterative way: given a reference control ¯u, one looks for a new “target” signal u that “improves” the functional value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='e such that ∆I < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The best choice of the target control is provided by the maximization of the integrand of (10) in the variable u: H (µt, ∂θ ¯pt, u) → max, u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The unique solution of the latter problem is obtain in the analytic form as ut[µ] = 1 α � ∂θ ¯pt(θ, η) (1 + cos θ) dµ(θ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (11) Here, it is worthwhile to mention that the reference dual state ¯p enters formula (11) only in the form of the partial derivative ¯ξt(θ, η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= ∂θ ¯pt(θ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Differentiating (8) and (9) in θ one can easily check that ¯ξ solves the η-parametric family of the same continuity equations (4) backward in time, starting from the terminal condition ξT = −∂θF � θ, ˇθ(η) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= sin �ˇθ(η) − θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (12) Now, (11) can be reformulated in terms of the variable ¯ξ: ut[µ] = 1 α � ¯ξt(θ, η) (1 + cos θ) dµ(θ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' (13) Note that the map (t, µ) �→ ut[µ] can be used as a feedback control [0, T] × P(S1 × I) �→ R of system (4) in the space of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Injecting this control into (4), we obtain a nonlocal continuity equation ∂tµt + ∂θ � vu[µt] µt � = 0, µ0 = ϑ, (14) which is well-posed (thanks to the fact that function (θ, η) �→ vu(θ, η) is smooth and bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Solving the last equation Algorithm 1: Numerical algorithm for optimal en- semble control Data: ¯u ∈ U (initial guess), ε > 0 (tolerance) Result: {uk}k≥0 ⊂ U such that I[uk+1] < I[uk] k ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' u0 ← ¯u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' repeat µk ← ˆµ[uk];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' uk+1 ← u[µk];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' k ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' until I[uk−1] − I[uk] < ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' numerically, and substituting its solution t �→ ˆµt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= ˆµt[¯u] into (11), we construct the “improved” signal: u(t) = ut[ˆµt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This idea gives rise to the following Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' By construction, Algorithm 1 generates a sequence {uk}k≥0 ⊂ U of controls with the property: Ik+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= I[uk+1] < I[uk] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Since the sequence of numbers (Ik)k≥0 is bounded from below by min(P) it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Finally, remark that the same line of arguments can be formally applied to problem (P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The respective mean-field type control takes the form wt(θ, η) = 1 α ¯ξt(θ, η) (1 + cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This construction gives rise to an iterative method, similar to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' NUMERICAL RESULTS Let us discuss several aspects of the numerical implemen- tation of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' First, note that the method proposed here does not involve any intrinsic parametric optimization: the most of indirect algorithms for optimal control require the dynamic adjust- ment of some internal computational parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' such are standard methods based on Pontryagin’s maximum principle [19], [20] that imply the internal such as line search for the specification of the “depth” of the needle-shaped (or weak) control variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Each iteration of Algorithm 1 requires numerical solution of two problems: one is the linear problem (4), (12) (inte- grated backward in time), and one for the nonlocal continuity equation (14) (solved numerically forward in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Since both (4) and (14) have no terms involving partial derivatives in η, one can think of η as a parameter and solve the corre- sponding parametric families of one-dimensional continuity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Consider the problem (P) with initial distribution of neurons µ0 given by the density function ρ0(θ, η) = � 2 + 3 cos(2θ) − 2 sin(2θ) � η, and with constant target function ˇθ(η) ≡ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In other words, our goal is to bring neurons’ states as close as possible to the segment 0 × I by the time moment T with the aid of sufficiently small controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Parameters for the computation: T = 6, I = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0], α = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' we used 512 Fourier harmonics in θ and grid steps ∆η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='002, ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Equations (4) and (14) are integrated by the standard spectral method [21] using the trigonometric Fourier expansion in θ for each η from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Parameters of the algorithm: ¯u ≡ 0, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 0 1 2 3 4 5 6 t −3 −2 −1 0 1 u(t) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Control input computed by the Algorithm 1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' CONCLUSION The goal of this paper is to present an approach based on the mean-field control paradigm to solve problems of optimization and synchronization of oscillatory processes (here, the addressed Theta model is among the simplest but prominent examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The proposed technique can be applied to any state-linear optimal control problem involving (finite or infinite) non-interacting statistical ensembles of an arbitrary nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' In particular, Algorithm 1 can be easily adapted to some other neural model such as SNIPER model, sinusoidal model etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' We plan to continue this study in the way of natural generalization of model (1) by admitting the interaction between theta neurons, ˙θk = vu(θk, ηk) + 1 N N � j=1 K(θk, θj), k = 1, N, where K is certain interaction potential formalizing the spatial connectivity of neurons in the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' This will result in control problems of the sort (P1,2) stated over the nonlocal continuity equation ∂tµt + ∂θ � [vu + K ⋆ µt] µt � = 0 involving the term (K ⋆ µ)(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='= � K(θ, ζ) dµ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 0 π 2π θ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0 η 0 π 2π θ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0 η 0 π 2π θ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='0 η Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Trajectory µt(θ, µ) of (4) at time moments t = 0, 3 and 6 (from top to bottom) computed for the optimal control input shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' The standard “rainbow” color table was used to code the isovalues: from black (minimal values), violet, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' , to red (maximal values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Such problems are not state-linear anymore, and the exact formula (10) becomes inapplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' For this case, a promis- ing alternative could be an approach based on Pontryagin’s maximum principle [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' REFERENCES [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Fischer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Liu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Davis, “Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 62, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 011801, Jun 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='011801 [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Blekhman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Blekhman, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Rivin, Synchro- nization in Science and Technology, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' ASME press translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' ASME Press, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='ru/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='id=ao1QAAAAMAAJ [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Nishikawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Gulbahce, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Motter, “Spontaneous reaction silencing in metabolic optimization,” PLOS Computational Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1–12, 12 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='pcbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1000236 [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Kuramoto, Chemical Oscillations, Waves, and Turbulence, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Dover books on chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Dover Publications, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='ru/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='id=4ADt7smO5Q8C [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Uhlhaas and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Singer, “Neural synchrony in brain disorders: Relevance for cognitive dysfunctions and pathophysiology,” Neuron, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 155–168, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='com/science/article/pii/S0896627306007276 [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Schiff, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Jerger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Duong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Chang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Spano, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Ditto, “Controlling chaos in the brain,” Nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 370, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 6491, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 615–620, Aug 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1038/370615a0 [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Glass, “Cardiac arrhythmias and circle maps-a classical problem,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 13–19, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='165810 [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Winfree, The Geometry of Biological Time, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Interdisciplinary Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Springer New York, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='ru/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='id=7qjTBwAAQBAJ [9] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Kiss, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Rusin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Kori, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Hudson, “Engineering complex dynamical structures: Sequential patterns and desynchro- nization,” Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 316, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 5833, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1886–1889, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/doi/abs/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1140858 [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Hoppensteadt and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Izhikevich, “Synchronization of laser oscillators, associative memory, and optical neurocomputing,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' E, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 62, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 4010–4013, Sep 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='4010 [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Hoppensteadt and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Izhikevich, “Synchronization of mems res- onators and mechanical neurocomputing,” IEEE Transactions on Cir- cuits and Systems I: Fundamental Theory and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 133–138, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Li, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Dasanayake, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Ruths, “Control and synchronization of neuron ensembles,” IEEE Transactions on Automatic Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1919–1930, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Ermentrout and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Kopell, “Parabolic bursting in an excitable system coupled with a slow oscillation,” SIAM Journal on Applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 233–253, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/stable/2101582 [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Strogatz, Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Boulder, CO : Westview Press, a member of the Perseus Books Group, [2015], [2015], includes bibliographical references and index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='edu/catalog/9910223127702121 [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Ambrosio and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Savar´e, “Gradient flows of probability measures,” in Handbook of differential equations: evolutionary equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' III, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Handb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Elsevier/North-Holland, Amsterdam, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [16] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Pogodaev and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Staritsyn, “Impulsive control of nonlocal transport equations,” Journal of Differential Equations, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 269, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 3585–3623, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='com/science/article/pii/S002203962030108X [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Pogodaev, “Optimal control of continuity equations,” NoDEA Nonlinear Differential Equations Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 21, 24, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Staritsyn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Pogodaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Chertovskih, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Pereira, “Feed- back maximum principle for ensemble control of local continuity equations: An application to supervised machine learning,” IEEE Control Systems Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1046–1051, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Arguchintsev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Dykhta, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Srochko, “Optimal control: Nonlocal conditions, computational methods, and the variational principle of maximum,” Russian Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 1–35, Jan 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content='3103/S1066369X09010010 [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Drag, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Styczen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Kwiatkowska, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Szczurek, “A review on the direct and indirect methods for solving optimal control prob- lems with differential-algebraic constraints,” Studies in Computational Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 610, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' 91–105, 07 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Boyd, Chebyshev and Fourier spectral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=', 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} +page_content=' Mi- neola, NY: Dover Publications, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFKT4oBgHgl3EQf9C5Z/content/2301.11952v1.pdf'} diff --git a/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf b/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9202cc09b0933593f14d8194f8c87bc174ca97b4 --- /dev/null +++ b/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:edc86a0dc45fa279c8c37480a4838925e0a80c8e0691d035ae2e121a7830243d +size 7678259 diff --git a/5tE1T4oBgHgl3EQfBAK1/content/tmp_files/2301.02847v1.pdf.txt b/5tE1T4oBgHgl3EQfBAK1/content/tmp_files/2301.02847v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..13b46615da4b524af82d49195f64dd288b1dd1c5 --- /dev/null +++ b/5tE1T4oBgHgl3EQfBAK1/content/tmp_files/2301.02847v1.pdf.txt @@ -0,0 +1,1976 @@ +USTC-ICTS/PCFT-22-27 +Irregular universe in the Nieh-Yan modified teleparallel gravity +Mingzhe Li +Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China and +Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China +Haomin Rao +School of Fundamental Physics and Mathematical Sciences, +Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China and +University of Chinese Academy of Sciences, 100190 Beijing, China +The Nieh-Yan modified teleparallel gravity is a model which modifies the general relativity equiv- +alent teleparallel gravity by a coupling between the Nieh-Yan density and an axion-like field. This +model predicts parity violations in the gravitational waves if the axion-like field has a non-trivial +background, and more importantly it is ghost free and avoids the pathologies presented in other +parity-violating gravity models. +The cosmological dynamics and perturbations of the Nieh-Yan +modified teleparallel gravity have been investigated in detail, but all these previous investigations +rely on the symmetry requirement that in the background universe both the metric and affine con- +nection are homogeneous and isotropic. In this paper we relax the symmetry constraint on the +connection and leave it arbitrary at the beginning, after all the cosmological principle only needs +the metric of the background spacetime to meet the symmetry requirement. We find a new flat +universe solution for the Nieh-Yan modified teleparallel gravity, for which the background dynamics +itself is unchanged but the perturbations around it present a new feature that the scalar and tensor +perturbations are coupled together at the linear level. The implications of this peculiar feature in +primordial perturbations from inflation are also discussed. +I. +INTRODUCTION +Stimulated by the experimental detections of gravitational waves (GWs) [1, 2] and the developments in the cosmic +microwave background radiation (CMB) experiments [3, 4], parity violating gravities attracted lots of interests in +recent years. A famous and frequently studied parity violating gravity model is the so-called Chern-Simons modified +gravity [5, 6] which within the framework of Riemannian geometry modifies general relativity (GR) by a gravitational +Chern-Simons term. The Chern-Simons modified gravity predicts the difference between the amplitudes of the left- +and right-handed polarized components of gravitational waves, i.e., the so-called amplitude birefringence phenomenon. +However, this model was found to suffer from the problem of vacuum instability because one of the circularly polarized +components of GWs becomes a ghost at high frequencies [7]. Further extensions [8–10] to this model did not circumvent +this difficulty because in these extended models the pathological behavior still appear at high energy scales, as shown +in Ref. [11]. It is very difficult to have a ghost-free parity violating gravity model within the framework of Riemannian +geometry. +Successful parity violating gravity models are available if we go beyond the Riemannian geometry. For example, +the Nieh-Yan modified teleparallel gravity (NYTG) [12, 13] is constructed within the framework of the teleparallel +gravity (TG) [14, 15], where the gravity is identified with the spacetime torsion in stead of the curvature. One may +have a GR equivalent TG model [16] (we may call it TGR). The NYTG model [12, 13] modifies TGR slightly by the +anomalous coupling θT �T between an axion-like field θ(x) and the Nieh-Yan density [17]: T �T ≡ (1/2)εµνρσT λ +µνTλρσ, +where T λ +µν is the torsion tensor, εµνρσ is Levi-Civita tensor which relates the totally antisymmetric symbol ϵµνρσ +and the determinant of the metric g through the equation εµνρσ = ϵµνρσ/√−g. The Nieh-Yan density is parity-odd, +so at the background with ∂µθ ̸= 0, the Nieh-Yan coupling term θT �T violates the parity symmetry spontaneously. +The NYTG model has been applied to cosmology in Refs. [12, 13], where it was found that this model predicts a +arXiv:2301.02847v1 [gr-qc] 7 Jan 2023 + +2 +difference between the propagating velocities of the left- and right-handed polarized components of GWs, i.e., the +so-called velocity birefringence phenomenon. More importantly, through detailed investigations on the cosmological +perturbations, it was shown in Refs. [12, 13] that the NYTG model is ghost-free. Recently, this model was found +to be compatible with the results of most local tests in the Solar System at the post-Newtonian order [18, 19], the +upper limit on its model parameters by the GWs data of LIGO/Virgo Collaboration was obtained in Ref. [20], and the +enhancement of primordial GWs during inflation due to the velocity birefringence of NYTG model and its implications +in the air-based GWs experiments were studied in Ref. [21]. Other recent studies on parity violating gravities can be +found in Refs. [22–33]. +In all the previous studies of the cosmological applications of the NYTG model, both the metric and the affine +connection of the background universe are required to be homogeneous and isotropic at the beginning. The spacetime +under this strong symmetry constraint is called the regular universe in this paper. The background solutions of the +regular universe have been well studied within the TG framework [34–36], and are universally applicable to almost +all TG models. In fact these solutions have been frequently adopted by different authors, e.g., [24, 37–39] 1. +However, the cosmological principle only needs the metric of the background universe to meet the high symmetry +requirement. In the Riemannian geometry, once we impose this symmetry requirement on the metric, the connection +(i.e., the Christoffel symbol) satisfies the same symmetry requirement automatically. In TG models, the symmetry +constraint on the affine connection is independent of the one on the metric. +If one drops this extra constraint +on the connection and leaves it arbitrary at the beginning, there will be final solutions for which the connection +is neither homogeneous nor isotropic. We call the universe which has a homogeneous and isotropic metric and a +non-homogeneous and non-isotropic affine connection the irregular universe. So far the irregular universe has rarely +aroused research interest, one example is the flat irregular universe solution found in Ref. [40] for the f(T) gravity. +The irregular universe does not violate the cosmological principle, but questions are in coming: What features and +new physical phenomena could exist in the irregular universe? Or might the irregular universe have properties that +are clearly contradictory to experiments so that only the regular universe is physically feasible? These questions +deserve detailed studies for any TG models. +In this paper, we will study the irregular universe in the NYTG model. Firstly, we will obtain a more general +flat universe solution than those in Refs. [12, 13] by solving the equations of motion of the NYTG model directly +under the condition that only the metric is required to be homogeneous and isotropic. By analyzing the symmetry +of the connection, we will show that the flat universe we obtain is generally an irregular flat universe, and in special +cases it reduces back to a regular universe. We will also show that even in the irregular flat universe, the background +equations in the NYTG model are exactly the same as those in GR. Secondly, we will study the linear cosmological +perturbations around the irregular flat universe. We will find that tensor perturbations and scalar perturbations are +coupled at the linear perturbation level. This is a peculiar feature that distinguishes the irregular universe from the +regular universe in the NYTG model. We speculate that this peculiar feature is caused by the fact that the interior +space does not satisfy the homogeneity and isotropy in the irregular universe. Finally, we will study the primordial +fluctuations generated by slow-roll inflation in the regular and irregular flat universes. We will show that the primordial +fluctuations of left- and right-handed GWs are different whether in the regular universe or in the irregular universe. +We will also show that there is a strong statistical correlation between primordial scalar fluctuations and primordial +tensor fluctuations generated by slow-roll inflation in the irregular universe. +This paper is organized as follows. In Sec. II, we briefly introduce the TG theory and the NYTG model. In Sec. III, +we study spatially flat cosmological background solutions that only requires the metric to be homogeneous and +isotropic in the NYTG model. In Sec. IV, through the quadratic actions for scalar, vector, and tensor perturbations, +we investigate linear perturbations around the regular and irregular flat universes. In Sec. V, we apply our result to +the early universe and discuss briefly the primordial perturbations generated by slow-roll inflation. +1 Actually, the cosmological background solution whose tetrad is eA +µ = diag(1, a, a, a) or eA +µ = diag(a, a, a, a) under the Weitzenb¨ock +gauge is the regular flat universe. However, most of the earlier literature did not clearly point out that the selection of such a tetrad +under the Weitzenb¨ock gauge actually requires the connection to satisfy the same symmetry of the metric. + +3 +In this paper, we adopt the unit 8πG = 1, and use the signature (+, −, −, −) for the metric. The tensor indices of +the interior space are denoted by A, B, C, ... = 0, 1, 2, 3 and by a, b, c, ... = 1, 2, 3 when limiting to spatial components. +They are lowered and raised by the Minkowski metric ηAB and its inverse ηAB. The spacetime tensor indices are +denoted by Greek µ, ν, ρ, ... = 0, 1, 2, 3 and by Latin i, j, k, ... = 1, 2, 3 when limiting to spatial components. They +are lowered and raised by the spacetime metric gµν and its inverse gµν. The antisymmetric symbol ϵµνρσ has the +properties: ϵ0ijk = ϵijk ≡ ϵijk, and ϵ123 = 1. In addition, we distinguish the spacetime affine connection ˆΓρ +µν and +its associated covariant derivative ˆ∇ from the Levi-Civita connection Γρ +µν and its associated covariant derivative ∇ +respectively. +II. +TG THEORY AND THE NYTG MODEL +The TG theory can be considered as a constrained metric-affine theory. It is formulated in a spacetime endowed +with a metric gµν and an affine connection ˆΓρ +µν, which is curvature free and metric compatible, +ˆRσ +ρµν = ∂µˆΓσ +νρ − ∂ν ˆΓσ +µρ + ˆΓσ +µλˆΓλ +νρ − ˆΓσ +νλˆΓλ +µρ = 0 , ˆ∇ρgµν = ∂ρgµν − ˆΓλ +ρµgλν − ˆΓλ +ρνgµλ = 0 . +(1) +Without curvature and nonmetricity, in the TG theory the gravity is identified with spacetime torsion T ρ +µν = 2ˆΓρ +[µν]. +One can also describe the TG theory using the language of the tetrad eA +µ and the spin connection ωA +Bµ. They relates +the metric gµν and the affine connection ˆΓρ +µν through the following relations +gµν = ηABeA +µeB +ν , +ˆΓρ +µν = e ρ +A (∂µeA +ν + ωA +BµeB +ν) . +(2) +The torsion tensor is written as +T ρ +µν = 2e ρ +A (∂[µeA +ν] + ωA +B[µeB +ν]) . +(3) +The teleparallel constraints (1) dictate that the spin connection can be in general expressed as +ωA +Bµ = (Λ−1)A +C∂µΛC +B , +(4) +where ΛA +B is arbitrary element of Lorentz transformation matrix which is position dependent and satisfies the relation +ηABΛA +CΛB +D = ηCD at any spacetime point. Therefore, the tetrad eA +µ and the Lorentz matrix ΛA +B can be regarded +as the basic variables of the TG theory. In this way, the teleparallel constraints (1) are automatically satisfied. +The TGR model, as the GR equivalent TG model, has the following action, +ST GR = 1 +2 +� +d4x |e| T ≡ +� +d4x |e| +� +−1 +2TµT µ + 1 +8TαβµT αβµ + 1 +4TαβµT βαµ +� +, +(5) +where |e| = √−g is the determinant of the tetrad, T is the torsion scalar, and Tµ = T α +µα is the torsion vector. +Since we have the identity −R(e) = T + 2∇µT µ, the action (5) is identical to the Einstein-Hilbert action up to a +surface term, where the curvature scalar R(e) is defined by the Levi-Civita connection and considered as being fully +constructed from the metric, and in turn from the tetrad. Since the surface term in the action does not affect the +equations of motion, we say that the TGR is equivalent to GR at the level of the equations of motion. +The NYTG model [12, 13] modifies the TGR model by introducing the coupling +SNY = c +4 +� +d4x |e| θ T �T , +(6) +between an axion-like field θ and the Nieh-Yan density T �T . The coupling constant c is dimensionless. Generally we +should also consider its own dynamics of the axion-like field and take other matter into account, so the full action of +the NYTG model is +SNY T G = +� +d4x |e| +�1 +2T + c +4 θ T �T + 1 +2∇µθ∇µθ − V (θ) +� ++ Sm . +(7) + +4 +Other matter with the action Sm is assumed to be coupled to spacetime minimally through the tetrad. +At the +background in which the axion-like field has non-zero spacetime derivatives, the Nieh-Yan coupling term breaks +parity spontaneously. Because only the first-order derivatives of the basic variables appears in the action, the NYTG +model can avoid the Ostrogradski ghost mode, which is expected to be originated from higher-order derivatives in the +action [41]. +As with most modified TG theories, the NYTG model apparently has two kinds of gauge symmetries: diffeomor- +phism invariance and local Lorentz invariance. The latter transformation makes the following change: +eA +µ → (L−1)A +BeB +µ , ΛA +B → ΛA +CLC +B , +(8) +where LA +B(x) are the element of Lorentz matrix. We would like to use different notations to distinguish two kinds +of Lorentz matrices: ΛA +B(x) is used to express the spin connection as in Eq. (4), but LA +B(x) represents the local +transformation that makes a shift from one local frame to another. Transformation (8) can be expressed in terms of +tetrad and spin connections as +eA +µ → (L−1)A +BeB +µ , ωA +Bµ → (L−1)A +CωC +DµLD +B + (L−1)A +C∂µLC +B . +(9) +It is easy to prove that the metric gµν and torsion tensor T ρ +µν are invariant under the local Lorentz transformation +(8), as is the action (7). Due to the local Lorentz invariance, one can choose the gauge ΛA +B = δA +B, i.e., ωA +Bµ = 0. +This is the Weitzenb¨ock connection, which has been frequently adopted in the literature. In addition, there is another +symmetry hidden in the NYTG model. The Nieh-Yan term (6) can be integrated by parts as +SNY = − c +2 +� +d4x ηABϵµνρσ(∂µθ)(ΛA +CeC +ν)∂ρ(ΛB +DeD +σ) . +(10) +It can be seen that the Nieh-Yan term (6) is invariant under the following transformation +(ΛA +CeC +µ) → LA +B(θ)(ΛB +CeC +µ) , +(11) +where LA +B(θ) is Lorentz matrix that depends only on axion-like field θ. Note that ΛA +CeC +µ is invariant under trans- +formation (8). Due to the Lorentz symmetry (8), the transformation (11) can always be attributed to the fact that +the tetrad eA +µ remains unchanged while the Lorentz matrix ΛA +B undergoes a Lorentz transformation. Obviously the +metric and the action of TGR model are invariant under such a transformation. So the total action of the NYTG +model is invariant under the transformation (11). +The equations of motion follow from the variation of the action (7) with respect to eA +µ and ΛA +B separately +Gµν + N µν = T µν + T µν +θ +, +(12) +N [µν] = 0 , +(13) +where N µν = (c/2)εµλρσ∂λθ T ν +ρσ, Gµν is the Einstein tensor, T µν = −(2/√−g)(δSm/δgµν) and T µν +θ += [V (θ) − +∇αθ∇αθ/2]gµν + ∇µθ∇νθ are the energy-momentum tensors for the matter and the axion-like field θ respectively. +Similar to most modified TG models, the equation of motion (13) from the variation of ΛA +B is not independent of +Eq. (12), it is just the antisymmetric part of the latter. As explained in Ref. [13], this is due to the local Lorentz +invariance of the action, any change caused by δΛA +B can always be equivalent to the change caused by δeA +µ, so +requiring the action to take the extremum under δeA +µ already includes the case where the action takes the extremum +under δΛA +B. There is another equation following from the variation of the action (7) with respect to θ, +□θ + V (1) − c +4T �T = 0 , +(14) +where □ = gµν∇µ∇ν and V (n) = dnV (θ)/dθn. All of these equations of motion are consistent with the Bianchi +identity ∇µGµν = 0 and the covariant conservation law ∇µT µν = 0. +Also in Refs. [12, 13], the cosmological perturbations of the NYTG model were analyzed in detail. It was found +that the NYTG model makes a difference between the propagating velocities of the left- and right-handed polarized + +5 +components of GWs, but makes no difference between their amplitudes. This phenomenon is called velocity birefrin- +gence, which is a clear physical signal of parity violation. More importantly, the NYTG model was confirmed to be +ghost free through the quadratic action of cosmological perturbations. +It is worth mentioning that the Nieh-Yan density T �T is not the only parity-odd term within the TG framework. +A more general model including all the parity-odd terms which are quadratic in the torsion tensor was considered in +Ref. [42]. But then it was found in Ref. [43] that this more general model suffers from the problem of ghost instability +again, unless it completely reduces to the NYTG model. Therefore, within the TG framework, for all parity-odd +terms which are quadratic in the torsion tensor, only the Nieh-Yan density T �T can avoid the ghost instability. This +means the NYTG model is robust to some extent. +III. +IRREGULAR FLAT UNIVERSE IN THE NYTG MODEL +So far all the studies on the cosmological applications of the NYTG model only considered the regular universe +as the background, that means both the metric and the affine connection are constrained to be homogeneous and +isotropic. +This constraint may be too strong, after all the cosmological principle which is supported by current +observations only needs the metric of the background spacetime to meet the high symmetry requirement. In this +paper, we will drop the symmetry requirement on the connection and leave it arbitrary at the beginning. After this +relaxation, it is expected that the NYTG model will have more interesting cosmological background solutions. We +are interested in the irregular universe solutions in which the metric homogeneous and isotropic but the connection +is neither homogeneous nor isotropic. For simplicity, we will only consider the spatially flat universe. +In flat universe, the metric can be expressed in rectangular coordinate as +ds2 = gµνdxµdxν = a2 � +dη2 − δijdxidxj� +, +(15) +where a = a(η) is the scale factor of the universe, η is the conformal time. This is the Friedmann-Robertson-Walker +(FRW) metric. There are 6 Killing vector fields {ξµ +I , I = 1, 2...6} in flat universe, which can be expressed as +ξµ +I = δ µ +I , ξµ +I+3 = ϵIijδµ +ixj , +I = 1, 2, 3 +(16) +where ξµ +1 , ξµ +2 , ξµ +3 are Killing vector fields representing the symmetry of spatial translation, and ξµ +4 , ξµ +5 , ξµ +6 are Killing +vector fields representing the symmetry of spatial rotation. One can prove that the FRW metric satisfies the condition: +LξIgµν = 0, where LξI is the Lie derivative along the Killing vector field ξµ +I . This reflects the result that the metric +is homogeneous and isotropic. One can also prove that LξIΓρ +µν = 0 for the Levi-Civita connection Γρ +µν, which is +automatically homogeneous and isotropic. This is why we do not need to pay extra attention to the symmetry of the +connection within the framework of Riemannian geometry. +A. +Regular flat universe +For TG models, even the metric is determined, the affine connection is still arbitrary to some extent. Usually, as +suggested in Refs [34–36], a further constraint was imposed that requires the connection is also homogeneous and +isotropic, that is, +LξI ˆΓρ +µν = ˆ∇µ ˆ∇ν ξρ +I − ˆ∇µ(T ρ +νσξσ +I ) = 0 . +(17) +Although ˆΓρ +µν is coordinate dependent, the Lie derivative of ˆΓρ +µν does not depend on the coordinate. Hence the +condition (17) is unambiguous. Combining Eqs. (15) and (17) selected the regular flat universe solution in which the +tetrad eA +µ and Lorentz matrix ΛA +B have the following forms: +eA +µ = aδA +µ , ΛA +B = ˚ΛA +B , +(18) + +6 +where ˚ΛA +B is a global Lorentz matrix, which does not depend on spacetime. All other solutions satisfying Eqs. (15) +and (17) differ from the solution (18) only by Lorentz transformation (8), so they are physically equivalent to the +solution (18). The above process does not depend on a specific TG theory, so the solution (18) is generally applicable +to most TG theories. +For the NYTG model, the solution (18) can automatically satisfy the constraint N [µν] = 0, so the solution (18) +is compatible with the NYTG model. Furthermore, solution (18) leads to N µν = 0 and T �T = 0, which means that +the Nieh-Yan term has no effect on the regular flat universe background. Therefore, the background equations of the +regular flat universe are exactly the same as those of GR [12, 13]. +B. +Irregular flat universe +To look for the irregular universe solution, we should give up the constraint (17) on the connection. After this +relaxation, the connection is left to be determined by the equation of motion. +In a flat universe, we can always simply find the non-zero components of Gµν, T µν and T µν +θ +as +G00 = 3H2 +a4 +, T 00 = ρ +a2 , T 00 +θ += ρθ +a2 , Gij = −2H′ + H2 +a4 +δij , T ij = p +a2 δij , T ij +θ = pθ +a2 δij , +(19) +where H = a′/a is the conformal Hubble rate, prime represents the derivative with respect to the conformal time η, +ρθ = θ′2/ +� +2a2� ++V and pθ = θ′2/ +� +2a2� +−V are the energy density and pressure of the θ field, and ρ and p denote the +energy density and pressure of other matter. Thanks to the Lorentz symmetry (8), we can always reduce the tetrad +to the simple form eA +µ = aδA +µ in flat universe. In order to facilitate further analysis, we decompose the independent +non-zero components of spin connections ωA +Bµ as follows +δa +iω0 +a0 = Ui , +δi +aδb +jωa +bk = Σϵijk + ϵijlΣkl + Σiδjk − Σjδik , +δi +aδb +jωa +b0 = ϵijkVk , +δa +iω0 +aj = σδij + σij + ϵijkσk , +(20) +where Σij and σij are symmetric and traceless spatial tensors. In the above decomposition we have exploited the +property ωABµ = −ωBAµ due to ˆ∇ρgµν = 0. Note that the variables σ, Σ, Ui, Vi, σi, Σi, σij, Σij are not completely +independent because we have not yet imposed ˆRσ +ρµν = 0 on the spin connection. Combining eA +µ = aδA +µ and Eq. (20), +N µν can be obtained as +N 00 = 0 , +N 0i = 0 , +N i0 = 2cθ′ +a4 σi , +N ij = cθ′ +a4 (2Σδij − Σij + ϵijkΣk) . +(21) +In order for Eqs. (12) and (13) to hold, there must be +σi = 0 , +Σi = 0 , +Σij = 0 , +Σ = Σ(η) . +(22) +Combining eA +µ = aδA +µ, Eqs. (20) and (22), Nieh-Yan density can be obtained as +T �T = 24Σ +a2 (H − σ) . +(23) +In order for Eq. (14) to hold, the Nieh-Yan density T �T can only be a function of time η, so σ = σ(η) when Σ ̸= 0. +Combining Eqs. (20) and (22), ˆRσ +ρµν = 0 gives +S′ +ij − Ui,j + ϵijkΣ Uk + ϵiklSjkVl = 0 , +(24) +Σ′δij − Vi,j + ϵijkΣ Uk − ϵiklSjkUl = 0 , +(25) +ϵiklSlj,k + Σ(Sij − Skkδij) = 0 , +(26) +ϵinmSjnSkm − Σ2ϵijk = 0 , +(27) +where Sij = σδij + σij and the subscript “, i” represents a derivative with respect to xi. The trace of Eq. (26) gives +σΣ = 0 . +(28) + +7 +This means that at least one of σ and Σ is zero. If σ = 0, the equation after the Hodge duality of the ”j, k” index in +Eq. (27) can be decomposed as follows according to the trace part and the traceless part: +6 Σ2 + σijσij = 0 , +σikσjk − 1 +3(σklσkl)δij = 0 . +(29) +The solution of Eq. (29) is Σ = 0, σij = 0. This means that Eqs. (27) and (28) must give +Σ = 0 . +(30) +Combining Eqs. (22) and (30) gives N µν = 0 and T �T = 0, which means that the Nieh-Yan term has no effect +even on the irregular flat universe background. Therefore, the background equations of the irregular flat universe are +exactly the same as those of GR. This is a somewhat unexpected result. But the fact that Nieh-Yan term has no effect +on the background does not mean that it has no effect on the perturbations. In order to analyze the perturbations, +we need to first find the background solution of the irregular flat universe. +Substituting Eq. (30) into Eqs. (24), (25), (26) and (27), we get +S′ +ij − Ui,j + ϵiklSjkVl = 0 , +(31) +Vi,j + ϵiklSjkUl = 0 , +(32) +ϵiklSlj,k = 0 , +(33) +ϵinmSjnSkm = 0 , +(34) +Although there are more equations than variables, this does not mean that Eqs. (31), (32), (33) and (34) have no +solution. It can be verified that the following are the solution of Eqs. (31), (32), (33) and (34) +Sij = vivjf(η)F (1)(⃗v · ⃗x) , +Vi = ga(η)αa +i (η, ⃗x) − ha(η)βa +i (η, ⃗x) , +Ui = ha(η)αa +i (η, ⃗x) + ga(η)βa +i (η, ⃗x) + vif (1)(η)F(⃗v · ⃗x) , +(35) +where +αa +i (η, ⃗x) = cosh [vf(η)F(⃗v · ⃗x)] δai + vavi +v2 +� +1 − cosh [vf(η)F(⃗v · ⃗x)] +� +, +βa +i (η, ⃗x) = ϵaij +vj +v sinh [vf(η)F(⃗v · ⃗x)] , +where v1, v2, v3 are constant parameters, v = +� +δijvivj, ⃗v · ⃗x = vixi, f(η), ga(η), ha(η) are arbitrary smooth function +of conformal time η, F(⃗v · ⃗x) is arbitrary smooth function of ⃗v · ⃗x, f (n)(η) is the n derivative of f(η) with respect to +conformal time η, and F (n)(⃗v · ⃗x) is the n derivative of F(⃗v · ⃗x) with respect to ⃗v · ⃗x. +Putting solutions (22), (30) and (35) into the decomposition (20), the spin connection ωA +Bµ when the tetrad is +eA +µ = aδA +µ can be obtained as +ωa +00 = ω0 +a0 = hc(η)αc +a(η, ⃗x) + gc(η)βc +a(η, ⃗x) + vaf (1)(η)F(⃗v · ⃗x) , +ωa +b0 = ϵabi [gc(η)αc +i(η, ⃗x) − hc(η)βc +i (η, ⃗x)] , +ω0 +ai = ωa +0i = vavif(η)F (1)(⃗v · ⃗x) , +ωa +bi = 0 . +(36) +It can be verified that the spin connection (36) does satisfy the teleparallel constraints (1). Due to the symmetry +(11), not every hI(η) and gI(η) represent a physically inequivalent solution. In order to see this better, we perform a +Lorentz transformation (9) on the above solution. The transformation matrix LA +B is +L0 +0 = cosh [vf(η)F(⃗v · ⃗x)] , L0 +a = La +0 = va +v sinh [vf(η)F(⃗v · ⃗x)] , +La +b = δab + vavb +v2 +� +cosh [vf(η)F(⃗v · ⃗x)] − 1 +� +, +(37) + +8 +Then, the tetrad ˜eA +µ = LA +BeB +µ and the corresponding spin connection ˜ωA +Bµ are +˜e0 +0 = a cosh [vf(η)F(⃗v · ⃗x)] , ˜ea +0 = δai˜e0 +i = ava +v sinh [vf(η)F(⃗v · ⃗x)] , +˜ea +i = a +� +δai + vavi +v2 +� +cosh [vf(η)F(⃗v · ⃗x)] − 1 +�� +, +˜ωa +00 = ˜ω0 +a0 = ha(η) , ˜ωa +b0 = ϵabcgb(η) , ˜ωA +Bi = 0 . +(38) +It can be verified that the metric gµν and connection ˆΓρ +µν given by solution (38) are the same as those given by +the tetrad eA +µ = aδA +µ and the spin connection (36). Since the solution (38) satisfies the teleparallel constraints (1), +the spin connection ˜ωA +Bµ in the solution (38) can be expressed by a Lorentz matrix ˜ΛA +B(η, ⃗x). And ˜ωA +Bi = 0 means +that ˜ΛA +B(η, ⃗x) = ˜ΛA +B(η). So taking different ha(η) and ga(η) is actually taking different ˜ΛA +B(η). Since θ = θ(η) in +the cosmological background, different ˜ΛA +B(η) can be converted to each other through the Lorentz transformation +˜ΛA +B(η) → LA +C(θ)˜ΛC +B(η). Therefore, the solutions with different ha(η) and ga(η) can be transformed into each other +by transformation (11), so they are physically equivalent. In this case, we only need to consider the simplest case +below, that is, the case where ha(η) = ga(η) = 0, so that the solution (36) can be simplified to +eA +µ = aδA +µ , +ωa +00 = ω0 +a0 = vaf (1)(η)F(⃗v · ⃗x) , ωa +b0 = 0 , +ωa +0i = ω0 +ai = vavif(η)F (1)(⃗v · ⃗x) , ωa +bi = 0 . +(39) +The solution (39) can be expressed by the tetrad eA +µ and the Lorentz matrix ΛA +B as +eA +µ = aδA +µ , +Λ = ˚Λ · exp +� +f(η)F(⃗v · ⃗x) vaKa� +, +(40) +where ˚Λ is a spacetime independent Lorentz matrix, and K1, K2, K3 are the boost matrices whose expression are +K1 = +� +� +� +� +� +0 1 0 0 +1 0 0 0 +0 0 0 0 +0 0 0 0 +� +� +� +� +� , +K2 = +� +� +� +� +� +0 0 1 0 +0 0 0 0 +1 0 0 0 +0 0 0 0 +� +� +� +� +� , +K3 = +� +� +� +� +� +0 0 0 1 +0 0 0 0 +0 0 0 0 +1 0 0 0 +� +� +� +� +� . +Regardless of the functional form of f(η) and F(⃗v · ⃗x), it can be verified that the solution (40) always satisfies the +teleparallel constraints (1) and makes Eqs. (12) and (14) self-consistent. Putting solution (40) into Eqs. (12) and (14), +we can get +3H2 = a2 (ρθ + ρ) , +2H′ + H2 = −a2 (pθ + p) , +θ′′ + 2Hθ′ + a2V (1) = 0 . +(41) +The background equations are exactly the same as those of GR. This means that the Nieh-Yan term has no effect +even on the irregular flat universe background. This is consistent with our analysis above. +Finally, let’s focus on the symmetry of the connection given by the solution (40). The non-zero components of +LξI ˆΓρ +µν given by the solution (40) are +LξI ˆΓ0 +0i = LξI ˆΓi +00 = vIvif (1)(η)F (1)(⃗v · ⃗x) , +LξI ˆΓ0 +ij = LξI ˆΓi +j0 = vIvivjf(η)F (2)(⃗v · ⃗x) , +LξI+3 ˆΓ0 +0i = LξI+3 ˆΓi +00 = −ϵIijvjf (1)(η)F(⃗v · ⃗x) + viϵIjkvjxkf (1)(η)F (1)(⃗v · ⃗x) , +LξI+3 ˆΓ0 +ij = LξI+3 ˆΓi +j0 = 2v(iϵj)Ikvkf(η)F (1)(⃗v · ⃗x) + vivjϵIklvkxlf(η)F (2)(⃗v · ⃗x) , +(42) +where I = 1, 2, 3 in Eq. (42), and the subscript parentheses denotes the symmetrization. The fact that LξI ˆΓρ +µν ̸= 0 +indicates that the spacetime connection given by the solution (40) is neither homogeneous nor isotropic. +So the +solution (40) does represent a irregular flat universe. When vi = 0 or f(η) = 0 or F(⃗v · ⃗x) = 0, there is LξI ˆΓρ +µν = 0, +and the solution (40) dose reduce to the regular flat universe solution (18). + +9 +IV. +PERTURBATIONS AROUND THE IRREGULAR FLAT UNIVERSE +In the previous section we studied the flat universe solution of the NYTG model that only requires the metric to +be homogeneous and isotropic. We found that the Nieh-Yan term has no effect even on the irregular flat universe +background. In order to explore the effect of the Nieh-Yan term on the irregular flat universe, we study the linear +cosmological perturbations around the irregular flat universe (40) in this section. For simplicity, we only consider the +case of F(⃗v · ⃗x) = ⃗v · ⃗x, which is equivalent to requiring that the coefficients of the equations of linear perturbations +do not depend on the spatial coordinates ⃗x (see below for details). And we also ignore other matter so that Sm = 0. +We use the following parametrization for perturbed tetrad [44]: +e0 +0 = a(1 + A) , e0 +i = a(β,i + βV +i ) , ec +0 = aδci(χ,i + χV +i ) , +ec +i = aδcj[(1 − ψ)δij + α,ij + αV +j,i − ϵijk(λ,k + λV +k ) + 1 +2hT +ij] , +(43) +So the perturbed metric components have the familiar forms: +g00 = a2(1 + 2A) , g0i = −a2(B,i + BV +i ) , +gij = −a2[(1 − 2ψ)δij + 2α,ij + αV +i,j + αV +j,i + hT +ij] , +(44) +where B = χ − β and BV +i += χV +i − βV +i . Besides the familiar scalar perturbations (A, B, ψ, α), vector perturbations +(BV +i , αV +i ), and tensor perturbations hT +ij in the metric, the parametrization of tetrad brings six extra variables, which +are scalar perturbation λ, χ + β and vector perturbation λV +i , χV +i + βV +i . All the vector perturbations are transverse +and denoted by the superscript V , both the tensor perturbations are transverse and traceless and denoted by the +superscript T. In addition, the scalar field θ is decomposed as θ(η, ⃗x) = ¯θ(η) + δθ(η, ⃗x). +Although we can perform a similar decomposition on the Lorentz matrix ΛA +B following the parametrization in +Ref. [13], we do not need to do so in this paper. Because we can always transform the perturbed Lorentz matrix into +the background Lorentz matrix in Eq. (40) through the infinitesimal Lorentz transformation (8). In other words, we +can always absorb the perturbations of the Lorentz matrix ΛA +B into the perturbations of the tetrad eA +µ through the +infinitesimal Lorentz transformation (8), so that we only need to deal with the perturbations of the the tetrad. +Due to the diffeomorphism invariance, it is safe to take the unitary gauge δθ = 0, α = 0, αV +i = 0. This simplifies +the calculations, for example, the gauge invariant scalar perturbation ζ = −(ψ + Hδθ/θ′) representing the curvature +perturbation of the hypersurfaces of constant θ reduces to −ψ under the unitary gauge. Since both α and αV +i +are +perturbations which enter the metric, the perturbations α, αV +i and δθ are invariant under the infinitesimal Lorentz +transformation (8). Therefore, the unitary gauge is compatible with the operation of absorbing the perturbations of +the Lorentz matrix into the perturbations of the tetrad. +The non-isotropic nature of the background connection may lead to coupling of scalar, vector and tensor perturba- +tions. Therefore, when studying linear perturbations around the irregular flat universe (40), we should not deal with +scalar, vector, or tensor perturbations individually, but should deal with all perturbation variables simultaneously. In +the following we choose A, ζ, B, BV +i , βi = β,i + βV +i , λi = λ,i + λV +i and hT +ij as independent variables, and we study +the linear perturbations around the irregular flat universe by means of quadratic action. +For the NYTG model (7) with Sm = 0, one can directly obtain the quadratic action as +S(2) = +� +d4x a2 +� +6Hζ′A − 3ζ′2 − (2A + ζ)ζ,ii − a2V A2 + 2(ζ′ − HA)B,ii + 1 +8 +� +hT ′ +ij hT ′ +ij − hT +ij,khT +ij,k +� +−1 +4BV +i BV +i,jj + cθ′� +2λiζ,i + 1 +2ϵijk(βiβj,k − λiλj,k) + ˆSijλiβj − 1 +2ϵijkSilhT +jlβk − 1 +8ϵijkhT +ilhT +jl,k +�� +. (45) +where Sij = vivjf(η)F (1)(⃗v · ⃗x) and ˆSij = (vivj − v2δij)f(η)F (1)(⃗v · ⃗x). In general, the coefficients Sij and ˆSij are +dependent on the spatial coordinate ⃗x. The coefficients of the equations for the linear perturbations are thus also +dependent on the spatial coordinate ⃗x. It means that the evolution equations for the linear perturbations are not + +10 +homogeneous. For simplicity, in the following we only consider the case of F(⃗v · ⃗x) = ⃗v · ⃗x 2. In this way, Sij and ˆSij +are constant coefficients. So the evolution equations for the linear perturbations are homogeneous. But it should be +noted that even in this case, the action (45) appears to be only homogeneous rather than homogeneous and isotropic, +because the constant coefficients Sij and ˆSij are not spatial rotation invariants. In addition, the terms ˆSijλiβj and +ϵijkSilhT +jlβk in the action (45) show that there is a coupling of scalar, vector and tensor perturbations. But such +coupling may be eliminated by the constraints imposed by the action (45) itself. Therefore, only after the constraints +are lifted can we know whether there is really a coupling of scalar, vector and tensor perturbations. +To further simplify the quadratic action, we change to the momentum space in terms of Fourier transformations, +ζ(η, ⃗x) = +� +d3k +(2π) +3 +2 ζ(η,⃗k) ei⃗k·⃗x , +(46) +and we also expand the variables A, B, λi, βi and hT +ij in the same way. The tensor perturbation hT +ij can be further +expanded as +hT +ij(η,⃗k) = +� +A +hA(η,⃗k) ˆeA +ij(⃗k) , +(47) +where {ˆeA +ij(⃗k), A = L, R} are circular polarization bases 3 satisfying ˆklϵlikˆeA +jk(⃗k) = ipAˆeA +ij(⃗k), where ˆk is the unit +vector of ⃗k, pL = −1 and pR = 1. Note that we use the normal letter A for the left- and right- hand indices to +distinguish it from the italic letter A used to represent the tetrad indices. The quadratic action in the momentum +space can be expressed as +S(2) = +� +dη +� +d3k a2 +� +6Hζ′A∗ − 3ζ∗′ζ′ + k2(2A + ζ)ζ∗ + 2k2(HA − ζ′)B∗ +−a2V A∗A + 1 +4k2BV ∗ +i +BV +i + 1 +4 +� +A +� +h∗′ +Ah′ +A − (k2 − cθ′pAk)h∗ +AhA +� ++cθ′� +2ikiλ∗ +i ζ + i +2ϵijkki(β∗ +j βk − λ∗ +jλk) + ˆSijλ∗ +i βj − 1 +2β∗ +i +� � +A +SA +i hA +��� +, +(48) +where SA +i (⃗k) = ϵijkSjlˆeA +kl(⃗k). It can be seen that A, B, BV +i , λi and βi are all non-dynamical fields and the variations +of the action (48) with them lead to the following constraints: +BV +i = 0 , +(49) +HA − ζ′ = 0 , +(50) +3Hζ′ + k2ζ − a2V A + Hk2B = 0 , +(51) +ϵijkkjλk − i ˆSijβj + 2kiζ = 0 , +(52) +− ˆSijλj + iϵijkkjβk + 1 +2 +� +A +SA +i hA = 0 . +(53) +For the regular flat universe case with vi = 0 or f(η) = 0, there are ˆSij = 0 and SA +i = 0, so the solution of Eqs. +(49), (50), (51), (52) and (53) is +ζ = 0 , A = 0 , B = 0 , BV +i = 0 , λi = ikiλ , βi = ikiβ , +(54) +2 The expression of F(⃗v · ⃗x) can differ by a constant term, which does not change the coefficients Sij and ˆSij. And a constant factor of +the difference of F(⃗v · ⃗x) can be absorbed into f(η). +3 Note that the choice of circular polarization bases is not unique, ˆeA +ij(⃗k) can be rotated along the ⃗k-axis while maintaining all the properties +of the circular polarization bases. For the case where there is a constant vector ⃗v ̸= 0 on the background, we can always choose the +circular polarization bases to satisfy vivjˆeA +ij(⃗k) = (v2/ +√ +2) sin2 ϑ, where ϑ is the angle between ⃗k and ⃗v. This choice maximally simplifies +the quadratic action (57), so we adopt this choice in this paper. + +11 +where λ and β are arbitrary scalar perturbations. Substituting the Eq. (54) back into the action (48), the action (48) +can be simplified as +S(2) = +� +dη +� +d3k a2 +4 +� +A +� +|h′ +A|2 − ω2 +A|hA|2� +, +(55) +where ω2 +A = k2 −cθ′pAk. It can be seen that there is no scalar dynamical degree of freedom at the linear perturbation +level. This is a bit strange because the action (7) clearly shows that there is a scalar dynamical degree of freedom. +Further research in Ref. [13] shows that the missing scalar dynamical degree of freedom reappears in the regular curved +universe. The phenomenon of degrees of freedom being hidden under special background also appears in f(T) gravity +[45] and massive gravity [46]. This implies that such a special background is likely to suffer from strong coupling +issue [47]. It can also be seen that the modified dispersion relation ω2 +A is helicity dependent. This means that GWs +with different helicities will have different propagation velocities. This phenomenon is called velocity birefringence, +which is a direct reflection of the parity violation in the NYTG model. These results are consistent with the results +in Refs. [12, 13] 4. +For the irregular flat universe case with vi ̸= 0 and f(η) ̸= 0, the solution of Eqs. (49), (50), (51), (52) and (53) is +A = ζ′/H , B = − +� +θ′2ζ′ + 2k2Hζ +� +/2k2H2 , BV +i = 0 , +λi = +� 2 cos ϑ +kv sin2 ϑϵijkkjvk +� +ζ − +i +2 +√ +2k ki� � +A +pAhA +� +, +βi = +� +2i +v2f(η) sin2 ϑki + 2ivf(η) cos ϑ +k sin2 ϑ +vi +� +ζ + ivf(η) cos ϑ +2 +√ +2k +vi +� � +A +hA +� +, +(56) +where ϑ is the angle between ⃗k and ⃗v. Substituting the above results back into the action (48), the action (48) can +be simplified as +S(2) = +� +dη +� +d3k +�z2 +2 +� +|ζ′|2 − k2|ζ|2� ++ a2 +4 +� +A +� +|h′ +A|2 − ω2 +A|hA|2� +− ca2θ′k +√ +2 +ζ∗� � +A +pAhA +�� +, +(57) +where z2 = a2θ′2/H2. For the action (57), the following points need to be emphasized. Firstly, it can be seen that there +is indeed a scalar dynamical degree of freedom, which again verifies that there is a scalar dynamical degree of freedom +hidden under the regular flat universe at the linear perturbation level. Secondly, there are two tensor dynamics degrees +of freedom and the dispersion relation ω2 +A is helicity dependent, as is the case for the regular universe. This means +that the velocity birefringence phenomenon of GWs also exists in the irregular universe. Thirdly, it is surprising that +vi and f(η) are completely cancelled in the step of lifting the constraints, so that the action (57) no longer depends on +vi and f(η). This makes the case of vi = 0, f(η) = 0 not the limit of the case of vi → 0, f(η) → 0. This is somewhat +analogous to the case where a massless photon is not the limit of a photon with mass tends to zero. Fourth, it can be +seen that the coefficients in the action (57) are homogeneous and isotropic. This means that the evolution equations of +the scalar perturbation ζ and the tensor perturbations hA are homogeneous and isotropic. Finally, it can be seen that +even after the constraints are lifted, there is still a coupling of scalar and tensor degrees of freedom. This is a feature +that neither in the regular flat universe nor in the regular curved universe. This means that scalar perturbations and +tensor perturbations can influence each other at the linear perturbation level. This can be seen more clearly from the +perspective of the equations of motion. From the action (57), the linear equations of ζ and hA can be obtained as +ζ′′ + 2z′ +z ζ′ + k2ζ + ca2θ′k +√ +2z2 +� � +A +pAhA +� += 0 , +(58) +h′′ +A + 2Hh′ +A + ω2 +AhA + +√ +2cθ′pAkζ = 0 . +(59) +4 The subtle difference in the dispersion relation ω2 +A is due to the difference between expanding by ei⃗k·⃗x and expanding by e−i⃗k·⃗x in the +Fourier transformation. + +12 +Eq. (58) shows that the tensor perturbations hA can be used as a source of the scalar perturbation ζ. The scalar +perturbation ζ can be excited when left- and right- handed GWs have different amplitudes or phases. And Eq. (59) +shows that the scalar perturbation ζ can be used as a source of the tensor perturbations hA. It is worth noting that +the source of the tensor perturbations hA caused by ζ is helicity-dependent, that is, the excitation effects caused by +ζ on the left- and right-handed GWs are different. +V. +PRIMORDIAL FLUCTUATIONS GENERATED BY INFLATION +In the previous section, we preliminarily studied the the linear perturbations around the regular and irregular flat +universe, and obtained the quadratic action after the constraints was lifted. In this section, we will preliminarily +study the primordial fluctuations generated by slow-roll inflation in the regular and irregular flat universe. +A. +The case of the regular universe +For the case of regular universe, the quadratic action (55) can be expressed as +S(2) = +� +dη +� +d3k a2 +2 +� +A +���� 1 +√ +2h′ +A +��� +2 +− +� +k2 − cθ′pAk +� ��� 1 +√ +2hA +��� +2� +. +(60) +Note that since there are only tensor degrees of freedom in the regular flat universe at the linear perturbation +level, a scalar field other than θ needs to be introduced to generate the primordial scalar perturbation [12, 21]. In +this subsection we do not consider the case of introducing additional scalar fields, and we only focus on the tensor +perturbations. +Next we consider the case of slow-roll inflation dominated by the axion-like field θ. Since the background equations +of the regular flat universe are exactly the same as those in GR, the background evolution during inflation will be +exactly the same as the case of slow-roll inflation in GR [48, 49]. So we don’t need to repeat the analysis of the details +of single scalar field inflation. We introduce two commonly used slow-roll parameters +ε ≡ − +˙H +H2 , δ ≡ +¨θ +H ˙θ +, +(61) +where H = ˙a/a = H/a is the Hubble rate, the upper dot represents the derivative with respect to the physical time +t. We assume ε ∼ |δ| ≪ 1, | ˙ε/H| ≪ |ε| and | ˙δ/H| ≪ |δ| during inflation. Under the slow-roll approximation, +H ≈ −1 + ε +η +, +θ′ ≈ +√ +2ε +η +. +(62) +Without loss of generality, in Eq. (62) we have assumed that the value of θ decreases during inflation. +Next, by combining Eqs (60) and (62), the correlation function of hA can be obtained through the process in +Appendix C: +⟨h† +AhA⟩ ≈ H2e−pA√ +ε/2cπk−(3+2ε) , +(63) +and ⟨h† +LhR⟩ = 0. Through the correlation functions (63), the power spectrum of the left- and right-handed GWs can +be obtained as +PA(k) = k3 +π2 ⟨h† +AhA⟩ ≈ H2 +π2 e−pA√ +ε/2cπk−2ε . +(64) +The power spectrum of the tensor perturbations can be obtained as +PT (k) = PL(k) + PR(k) ≈ H2 +π2 +� +1 + cosh +��ε +2cπ +�� +k−2ε . +(65) + +13 +The relative different between the power spectrum of the left- and right-handed GWs can be obtained as +Π ≡ PR − PL +PR + PL +≈ − tanh +��ε +2cπ +� +≈ − +�ε +2cπ . +(66) +Π ̸= 0 means that the magnitudes of the primordial fluctuations of left- and right-handed GWs are different. This is +a clear physical signal of parity violation. But this seems to contradict the conclusion in Refs. [12, 13] that there is +only velocity birefringence of GWs but no amplitude birefringence of GWs in the NYTG model. The reason for this +contradiction is that θ′ is approximated as a constant in the analysis of the evolution of GWs in Refs. [12, 13]. Of +course, this approximation is valid when studying the propagation of GWs in a slowly expanding universe. However, +θ′ = a ˙θ ∝ 1/η cannot be approximated as a constant during the slow-roll inflation dominated by θ. We know that for +a harmonic oscillator (the equation of motion is ¨x+ω2x = 0), the amplitude of the harmonic oscillator can be changed +when the frequency ω is time-dependent. And when the time dependence of θ′ is not negligible, the time dependence +of ωL and ωR will be different, resulting in different effects on the amplitudes of left- and right-hand GWs. This is +why the magnitudes of the primordial fluctuations of left- and right-handed GWs generated by slow-roll inflation in +the regular flat universe are different. If ε → 0, it can be seen from Eq. (62) that θ′ ≈ 0 can be approximated as a +constant, and from Eq. (66), it can be seen that Π → 0 too, that is, the magnitudes of the primordial fluctuation of +the left- and right-handed GWs are the same. +Finally, let’s look at the case when the coupling constant c → 0, then +PT (k) ≈ 2H2 +π2 k−2ε , +Π ≈ 0 . +(67) +This is exactly the result of the slow-roll inflation of single scalar field in GR. +B. +The case of the irregular universe +For the case of irregular universe, since the coupling of ζ and hA in the action (57) makes it difficult to analyze the +quantum fluctuations, we first diagonalize the variables ζ and hA below. Firstly, for the convenience of analysis, we +introduce new variables ξ1 = (z/a)ζ, ξ2 = (1/ +√ +2)hL and ξ3 = (1/ +√ +2)hR, so that the action (57) can be simplified as +S(2) = +� +dη +� +d3k a2 +2 +� +3 +� +s=1 +ξ∗′ +s ξs − +3 +� +s1=1 +3 +� +s2=1 +Ms1s2ξ∗ +s1ξs2 +� +, +with M = +� +� +� +k2 − Ω +−κ +κ +−κ +k2 − σ +0 +κ +0 +k2 + σ +� +� +� , +(68) +where Ω = z′′/z − a′′/a, σ = −cθ′k and κ = cHk are background quantities. Secondly, we introduce an orthogonal +matrix T that can diagonalize the matrix M, and its expression is +T = +� +� +� +tT +1 +tT +2 +tT +3 +� +� +� , with ts = +−s2 + 5s − 5 +� +1 + (τs−σ)2 +κ2 ++ +� +1 − (τs−σ)(τs+Ω) +κ2 +�2 +� +� +� +(τs − σ)/κ +1 − (τs − σ)(τs + Ω)/κ2 +1 +� +� +� , +(69) +where the superscript T means transpose, and {τs, s = 1, 2, 3} are the solutions of the cubic equation +τ 3 + Ωτ 2 − (2κ2 + σ2)τ − σ2Ω = 0 . +(70) +The specific expressions of {τs, s = 1, 2, 3} are in Appendix A. Finally, we introduce new variables {qs, s = 1, 2, 3}, +which are defined as +� +� +� +q1 +q2 +q3 +� +� +� = T +� +� +� +ξ1 +ξ2 +ξ3 +� +� +� . +(71) + +14 +Thus, the action (68) can be further simplified as +S(2) = +3 +� +s=1 +� +dη +� +d3k a2 +2 +� +|q′ +s|2 − (k2 + τs)|qs|2� +. +(72) +So far, we have simplified the action (57) with coupling between variables to the action (72) without coupling between +variables. The latter form makes it easier to calculate the primordial fluctuations generated by inflation. +Next we consider the case of slow-roll inflation dominated by the axion-like field θ. Since in Sec. III we proved that +the background equations of the irregular flat universe are exactly the same as those in GR, the background evolution +during inflation will be exactly the same as the case of slow-roll inflation in GR. Under the slow-roll approximation, +the background quantities Ω, σ and κ can be approximately expressed as +Ω ≈ 3(ε + δ) +2η2 +, σ ≈ − +√ +2εck +η +, κ ≈ −(1 + ε)ck +η +. +(73) +In this section, we also assume that the coupling constant c ∼ 1 (it can also be seen as a requirement of naturalness), +so that c ≫ √ε. Ignoring high-order small quantities such as ε2, {τs, s = 1, 2, 3} in Eq. (A3) can be approximated as +τ1 ≈ (2 + 3ε)ck +√ +2η +− 3(ε + δ) +2η2 +, τ2 ≈ 0 , τ3 ≈ −(2 + 3ε)ck +√ +2η +− 3(ε + δ) +2η2 +. +(74) +If only up to the order of √ε is retained, the orthogonal matrix T can be approximated as +T ≈ +� +� +� +� +1 +√ +2 +1+√ε +2 +− 1−√ε +2 +−√ε +1 +√ +2 +1 +√ +2 +1 +√ +2 +− 1−√ε +2 +1+√ε +2 +� +� +� +� +(75) +Regarding the approximate expression (75), there are two points that need additional explanation. First, the order √ε +is the lowest order approximation required to preserve the difference in the power spectrum of left- and right-handed +GWs. If we further ignore the contribution of √ε in T, the difference in the power spectrum of left- and right-handed +GWs disappears. And if we keep the higher-order terms, it brings only more complex but less important corrections +in the power spectrum. Second, it can be seen that the matrix T does not tend to the identity matrix as c → 0 +in the approximate expression (75). This is confusing because the three variables are all decoupled as c → 0 in the +action (68). The reason for this confusing phenomenon is that we have used the approximation c ≫ √ε in Eqs. (74) +and (75). If c is too small, neither the Eq. (74) nor Eq. (75) hold. See Appendix B for the approximate behavior of +orthogonal matrix T when c → 0. +Next, by combining Eqs (72) and (74), the correlation function between variables qs can be obtained through the +process in Appendix C: +⟨q† +1q1⟩ ≈ H2 +2 e +cπ +√ +2 k−(3+3ε+δ) , ⟨q† +2q2⟩ ≈ H2 +2 k−(3+2ε) , ⟨q† +3q3⟩ ≈ H2 +2 e− cπ +√ +2 k−(3+3ε+δ) , +(76) +and ⟨q† +s1qs2⟩ = 0 when s1 ̸= s2. Then, using the approximation techniques in Appendix D and combining Eqs (71), +(75) and (76), the correlation functions for the variables ζ and hA can be obtained as +⟨ζ†ζ⟩ ≈ 1 +2ε cosh +� cπ +√ +2 +� +H2knS−4 , +⟨h† +AhA⟩ ≈ +�1 +2 + 1 +2 cosh +� cπ +√ +2 +� +− pA +√ε sinh +� cπ +√ +2 +�� +H2knT −3 , +⟨ζ†hA⟩ ≈ − pA +2 +√ +2ε sinh +� cπ +√ +2 +� +H2k−(3+3ε+δ) , +⟨h† +LhR⟩ ≈ 1 +2 +� +1 − cosh +� cπ +√ +2 +�� +H2k +−(3+3ε+δ)− 1 +2 csch2� +cπ +2 +√ +2 +� +(ε+δ) , +(77) + +15 +where +nS ≈ 1 − (δ + 3ε) , +nT ≈ −(3ε + δ) + 1 +2 sech2 +� cπ +2 +√ +2 +� +(ε + δ) . +(78) +It should be noted that since Eqs. (74) and (75) are approximately true only when c ≫ √ε, Eqs. (77) and (78) are +also approximately true only when c ≫ √ε. +Through the correlation functions (77), the power spectrum of the scalar perturbation ζ can be obtained as +PS(k) = k3 +2π2 ⟨ζ†ζ⟩ ≈ H2 +8π2ε cosh +� cπ +√ +2 +� +knS−1 . +(79) +The power spectrum of the left- and right-handed GWs can be obtained as +PA(k) = k3 +π2 ⟨h† +AhA⟩ ≈ H2 +2π2 +� +1 + cosh +� cπ +√ +2 +� +− 2pA +√ε sinh +� cπ +√ +2 +�� +knT . +(80) +The power spectrum of the tensor perturbations can be obtained as +PT (k) = PL(k) + PR(k) ≈ H2 +π2 +� +1 + cosh +� cπ +√ +2 +�� +knT . +(81) +The tensor-to-scalar ratio r can be obtained as +r ≡ PT +PS += 8 +� +1 + sech +� cπ +√ +2 +�� +ε . +(82) +The relative different between the power spectrum of the left- and right-handed GWs can be obtained as +Π ≡ PR − PL +PR + PL +≈ −2√ε tanh +� cπ +√ +2 +� +. +(83) +Strictly speaking, since Eqs. (77) and (78) are only approximately true when c ≫ √ε, Eqs. (79)-(83) are also approx- +imately true only when c ≫ √ε. But If we ignore this fact and force c → 0, then +PS ≈ H2 +8π2εknS−1 , PT ≈ 2H2 +π2 knT , r ≈ 16ε , Π ≈ 0 . +(84) +It can be seen that except for the spectral indices nS and nT , Eq. (84) is the result of the slow-roll inflation in GR. +From the Planck 2018 [50], we know that the scalar spectral index nS ≈ 0.966 and the tensor-to-scalar ratio +r < 0.101. This means that the allowable value range of the slow-roll parameters ε and δ is +0 < ε < +0.101 +8 +� +1 + sech +� +cπ/ +√ +2 +�� < 0.012625 , +δ ≈ 0.034 − 3ε . +(85) +It can be seen that the maximum value of ε depends on the coupling constant c, but will not exceed 0.012625 (the +upper limit of ε when c → ∞). The allowable value of δ is determined by ε. FIG. 1 shows the allowable value range +of slow-roll parameters ε and δ when c = 1. +Although by comparing the results in subsections V A and V B, we can find that the power spectrum of the left- +and right-handed GWs given by the irregular universe is different from that of the regular universe. But this is not +the main difference between irregular and regular universes for primordial fluctuations. For primordial fluctuations, +the most important feature of the irregular universe compared to the regular universe is that the correlation function +of scalar perturbation and tensor perturbations ⟨ζ†hA⟩ ̸= 0 at the linear perturbation level. This means that there +is a strong statistical correlation between primordial scalar fluctuations and primordial tensor fluctuations generated +by slow-roll inflation in the irregular universe. The apparent reason for this phenomenon is that the quadratic action +contains the coupling of scalar perturbations and tensor perturbations in the irregular universe, as exhibited by the +action (57). The deeper reason may be that the condition LξI ˆΓρ +µν ̸= 0 destroys the homogeneity and isotropy of the +interior space, so that the scalar fluctuations and the tensor fluctuations can interact with each other in the irregular +universe. + +16 +0.002 +0.004 +0.006 +0.008 +0.010 ε +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +δ +FIG. 1: In the ε-δ plane, the blue line is the allowable value range when c = 1. +VI. +CONCLUSION +As a step towards exploring the irregular universe within the TG framework, in this paper, we studied the irregular +flat universe of the NYTG model. Firstly, we obtained the irregular flat universe solution of the NYTG model under +the condition that only the symmetry of the metric is required. We found that the cosmological background equations +of the NYTG model are exactly the same as those of GR in both the regular flat universe and the irregular flat +universe. Secondly, we studied the linear cosmological perturbations around the irregular flat universes. We found a +peculiar feature of the irregular flat universe: the tensor and scalar perturbations are coupled together at the linear +perturbation level. We speculate that this peculiar feature is caused by the fact that the interior space does not satisfy +the homogeneity and isotropy in the irregular universe. Finally, we applied the NYTG model to the early universe +and studied the primordial perturbations generated by slow-roll inflation in the regular and irregular flat universes. +We found that the left- and right-handed primordial GWs are different in both the regular flat universe and the +irregular flat universe. We also found that there is a strong statistical correlation between the primordial scalar and +tensor perturbations generated by slow-roll inflation in the case of irregular universe, this is a direct consequence of +the direct coupling between the scalar and tensor perturbations at linear order. +Acknowledgement: +This work is supported in part by National Key R&D Program of China Grant No. +2021YFC2203102, and by NSFC under Grant No. 12075231 and 12047502. +Appendix A: Solutions of the cubic equation +Consider a cubic equation with respect to the variable τ as +aτ 3 + bτ 2 + cτ + d = 0 , +(A1) +where a, b, c and d are real coefficients. In order to express the solution of Eq. (A1) conveniently, we introduce the +following parameters +A = b2 − 3ac , B = bc − 9ad , C = c2 − 3bd , ∆ = B2 − 4AC , Θ = 1 +3 arccos +�2Ab − 3Ba +2A3/2 +� +. +(A2) + +17 +When ∆ < 0, Eq. (A1) has three real solutions, which are +τ1 = − 1 +3a +� +b + 2 +√ +A cos Θ +� +, +τ2 = 1 +3a +� +−b + +√ +A +� +cos Θ − +√ +3 sin Θ +�� +, +τ3 = 1 +3a +� +−b + +√ +A +� +cos Θ + +√ +3 sin Θ +�� +, +(A3) +The Eq. (70) in the main text is the result of taking a = 1, b = Ω, c = −(2κ2 + σ2) and d = −σ2Ω in Eq. (A1). +In this case, there are always A ≥ 0 and ∆ ≤ 0, where the equal sign holds if and only if κ = σ = Ω = 0. And when +κ = σ = Ω = 0, obviously the three solutions of Eq. (70) are τ1 = τ2 = τ3 = 0, and the orthogonal matrix T is the +identity matrix. +Appendix B: The orthogonal matrix T when c → 0 +In this appendix, we discuss the approximate behavior of the orthogonal matrix T in Eq. (69) as c → 0 in a more +general background (not only during inflation). Since σ ∝ c and κ ∝ c, then +κ +Ω ∝ c , σ +Ω ∝ c , +κ2 +2σΩ ∝ c . +(B1) +When c is much smaller than any other background quantities such as √ε, ˙θ and H−1, ignoring the quadratic and +higher terms of c, the solutions of Eq. (70) can be approximately expressed as +τ1 ≈ Ω , τ2 ≈ σ , τ3 ≈ σ . +(B2) +So the orthogonal matrix T in Eq. (69) can be approximately expressed as +T = +� +� +� +1 +κ +Ω +− κ +Ω +− κ +Ω +1 +κ2 +2σΩ +κ +Ω +− κ2 +2σΩ +1 +� +� +� +when c→0 +−−−−−−−→ +� +� +� +1 0 0 +0 1 0 +0 0 1 +� +� +� +(B3) +It can be easily seen from Eqs. (B1) and (B3) that when c → 0, the orthogonal matrix T does tend to the identity +matrix. This is consistent with the fact that all variables in the action (68) tend to be decoupled when c → 0. +Appendix C: Correlation function generated by inflation +The purpose of this appendix is to show how to calculate the correlation function generated by inflation. Consider +a univariate system whose effective action during inflation is +S = 1 +2 +� +dη d3k a2 +� +|q′ +⃗k|2 − +� +k2 − 2ak +η +− 3b +η2 +� +|q⃗k|2 +� +, +(C1) +where a and b are real parameters, and b has the same order of magnitude as the slow-roll parameter ε. Here q(η, ⃗x) is +the variable and we have changed to the Fourier space q⃗k(η). After quantization, the variable q⃗k(η) can be expanded +as +q⃗k(η) = +1 +a(η) +� +vk(η)ˆa⃗k + v∗ +k(η)ˆa† +⃗k +� +, +(C2) +where ˆa† +⃗k and ˆa⃗k are the generation and annihilation operators that satisfy the following commutation relations +[ˆa⃗k ˆa† +⃗k′] = δ(3)(⃗k − ⃗k′) , +[ˆa⃗k ˆa⃗k′] = [ˆa† +⃗k ˆa† +⃗k′] = 0 , +(C3) + +18 +and vk(η) satisfies the following equation +v′′ +k + +� +k2 − 2ak +η +− µ2 − 1/4 +η2 +� +vk = 0 , +(C4) +where µ ≈ 3/2+ε+b. Note that in Eq. (C4), we used the approximation a′′/a ≈ [(3/2+ε)2 −1/4]/η, and we ignored +the higher-order terms of ε and b. Next we choose the Bunch-Davies vacuum at η → −∞, that is, +lim +η→−∞ vk = +1 +√ +2k +e−ikη . +(C5) +Under this condition, the solution for Eq. (C4) is (for more detail, see [51]) +vk(η) = e−ikη(−2kη)µ(−η) +1 +2 e−iπ( 1 +4 + µ +2 )U (1/2 + µ − ia, 1 + 2µ; 2ikη) e− aπ +2 , +(C6) +where U(c1, c2; z) is the confluent hypergeometric function. The |vk| has the following asymptotic form when kη → 0− +(super-horizon scale) +|vk| ≈ 2µ−1π− 1 +2 Γ(µ)k−µ(−η) +1 +2 −µe− aπ +2 ≈ 2− 1 +2 e− aπ +2 aHk−µ +(C7) +where Γ(z) is the Gamma function. In the last approximately equal sign in Eq. (C7), we used the approximations +µ ≈ 3/2 and (−η)−1 ≈ aH. Combining Eqs. (C2), (C3) and (C7), we can obtain the correlation function on the +super-horizon scale as +⟨0|q† +⃗kq⃗k′|0⟩ ≈ H2 +2 e−aπk−(3+2ε+2b)δ(3)(⃗k + ⃗k′) . +(C8) +where |0⟩ is the vacuum state, which satisfies ˆa⃗k|0⟩ = 0. For the sake of convenience, we can omit the subscript ⃗k and +throw away the annoying delta function δ(3)(⃗k + ⃗k′), so that the correlation function (C8) can be abbreviated as +⟨q†q⟩ ≈ H2 +2 e−aπk−(3+2ε+2b) . +(C9) +Appendix D: Summation of nearly scale-invariant functions +Consider there are N nearly scale-invariant functions {fi(k) = Cikni, i = 1, 2, ..., N}, where |ni| ≪ 1. Then the +sum of these functions should also be a nearly scale-invariant function, so it can be approximated as +f(k) = +N +� +i=1 +fi(k) = +N +� +i=1 +Cikni ≈ Ckn , with |n| ≪ 1 . +(D1) +Next we need to find the coefficient C and the exponent n in Eq. (D1). Since ni ≈ 0 and n ≈ 0, we can approximately +let ni = n = 0 in Eq. (D1), so that Eq. (D1) becomes +C ≈ +N +� +i=1 +Ci . +(D2) +Next, let Eq. (D1) take the derivative of k and then let ni = n = 0 on the exponent of k. Then the approximate +expression of n can be obtained as +n ≈ 1 +C +N +� +i=1 +Cini . +(D3) +[1] B. +P. +Abbott +et +al. +[LIGO +Scientific +and +Virgo], +Phys. +Rev. +Lett. +116, +no.6, +061102 +(2016) +doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]]. + +19 +[2] B. +P. +Abbott +et +al. +[LIGO +Scientific +and +Virgo], +Phys. +Rev. +Lett. +119, +no.16, +161101 +(2017) +doi:10.1103/PhysRevLett.119.161101 [arXiv:1710.05832 [gr-qc]]. +[3] H. Li, S. Y. Li, Y. Liu, Y. P. Li, Y. Cai, M. Li, G. B. Zhao, C. Z. Liu, Z. W. Li and H. Xu, et al. “Probing Primordial +Gravitational Waves: Ali CMB Polarization Telescope,” Natl. Sci. Rev. 6, no.1, 145-154 (2019) doi:10.1093/nsr/nwy019 +[arXiv:1710.03047 [astro-ph.CO]]. +[4] K. Abazajian et al. [CMB-S4], Astrophys. J. 926, no.1, 54 (2022) doi:10.3847/1538-4357/ac1596 [arXiv:2008.12619 [astro- +ph.CO]]. +[5] R. Jackiw and S. Y. Pi, Phys. Rev. D 68, 104012 (2003) doi:10.1103/PhysRevD.68.104012 [arXiv:gr-qc/0308071 [gr-qc]]. +[6] S. Alexander and N. Yunes, Phys. Rept. 480, 1-55 (2009) doi:10.1016/j.physrep.2009.07.002 [arXiv:0907.2562 [hep-th]]. +[7] S. Dyda, E. E. Flanagan and M. Kamionkowski, Phys. Rev. D 86, 124031 (2012) doi:10.1103/PhysRevD.86.124031 +[arXiv:1208.4871 [gr-qc]]. +[8] M. +Crisostomi, +K. +Noui, +C. +Charmousis +and +D. +Langlois, +Phys. +Rev. +D +97, +no.4, +044034 +(2018) +doi:10.1103/PhysRevD.97.044034 [arXiv:1710.04531 [hep-th]]. +[9] X. Gao and X. Y. Hong, Phys. Rev. D 101, no.6, 064057 (2020) doi:10.1103/PhysRevD.101.064057 [arXiv:1906.07131 +[gr-qc]]. +[10] W. Zhao, T. Zhu, J. Qiao and A. Wang, “Waveform of gravitational waves in the general parity-violating gravities,” Phys. +Rev. D 101, no.2, 024002 (2020) doi:10.1103/PhysRevD.101.024002 [arXiv:1909.10887 [gr-qc]]. +[11] N. Bartolo, L. Caloni, G. Orlando and A. Ricciardone, JCAP 03, 073 (2021) doi:10.1088/1475-7516/2021/03/073 +[arXiv:2008.01715 [astro-ph.CO]]. +[12] M. Li, H. Rao and D. Zhao, JCAP 11, 023 (2020) doi:10.1088/1475-7516/2020/11/023 [arXiv:2007.08038 [gr-qc]]. +[13] M. Li, H. Rao and Y. Tong, Phys. Rev. D 104, no.8, 084077 (2021) doi:10.1103/PhysRevD.104.084077 [arXiv:2104.05917 +[gr-qc]]. +[14] R. Aldrovandi and J. G. Pereira, Teleparallel Gravity, Vol. 173. Springer, 23 Dordrecht, (2013). +[15] S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. L. Said, +J. Mifsud and E. Di Valentino, [arXiv:2106.13793 [gr-qc]]. +[16] J. W. Maluf, Annalen Phys. 525, 339-357 (2013) doi:10.1002/andp.201200272 [arXiv:1303.3897 [gr-qc]]. +[17] H. T. Nieh and M. L. Yan, J. Math. Phys. 23, 373 (1982) doi:10.1063/1.525379 +[18] H. Rao, Phys. Rev. D 104, no.12, 124084 (2021) doi:10.1103/PhysRevD.104.124084 [arXiv:2107.08597 [gr-qc]]. +[19] J. Qiao, T. Zhu, G. Li and W. Zhao, JCAP 04, no.04, 054 (2022) doi:10.1088/1475-7516/2022/04/054 [arXiv:2110.09033 +[gr-qc]]. +[20] Q. Wu, T. Zhu, R. Niu, W. Zhao and A. Wang, Phys. Rev. D 105, no.2, 024035 (2022) doi:10.1103/PhysRevD.105.024035 +[arXiv:2110.13870 [gr-qc]]. +[21] R. G. Cai, C. Fu and W. W. Yu, Phys. Rev. D 105, no.10, 103520 (2022) doi:10.1103/PhysRevD.105.103520 +[arXiv:2112.04794 [astro-ph.CO]]. +[22] M. Li and D. Zhao, Phys. Lett. B 827 (2022), 136968 doi:10.1016/j.physletb.2022.136968 [arXiv:2108.01337 [gr-qc]]. +[23] C. Gong, T. Zhu, R. Niu, Q. Wu, J. L. Cui, X. Zhang, W. Zhao and A. Wang, Phys. Rev. D 105 (2022) no.4, 044034 +doi:10.1103/PhysRevD.105.044034 [arXiv:2112.06446 [gr-qc]]. +[24] M. Hohmann and C. Pfeifer, Phys. Lett. B 834 (2022), 137437 doi:10.1016/j.physletb.2022.137437 [arXiv:2203.01856 [gr- +qc]]. +[25] X. Tong and Z. Z. Xianyu, JHEP 10 (2022), 194 doi:10.1007/JHEP10(2022)194 [arXiv:2203.06349 [hep-ph]]. +[26] M. Li, Y. Tong and D. Zhao, Phys. Rev. D 105 (2022) no.10, 104002 doi:10.1103/PhysRevD.105.104002 [arXiv:2203.06912 +[gr-qc]]. +[27] F. Zhang, J. X. Feng and X. Gao, JCAP 10 (2022), 054 doi:10.1088/1475-7516/2022/10/054 [arXiv:2205.12045 [gr-qc]]. +[28] T. Zhu, W. Zhao and A. Wang, [arXiv:2210.05259 [gr-qc]]. +[29] T. Zhu, W. Zhao and A. Wang, [arXiv:2211.04711 [gr-qc]]. +[30] A. A. A. Filho, J. R. Nascimento, A. Y. Petrov and P. J. Porf´ırio, [arXiv:2211.11821 [gr-qc]]. +[31] J. Qiao, Z. Li, T. Zhu, R. Ji, G. Li and W. Zhao, [arXiv:2211.16825 [gr-qc]]. +[32] Y. Cai, [arXiv:2212.10893 [gr-qc]]. +[33] Z. Chen, Y. Yu and X. Gao, [arXiv:2212.14362 [gr-qc]]. +[34] M. Hohmann, L. J¨arv, M. Krˇsˇs´ak and C. Pfeifer, Phys. Rev. D 100, no.8, 084002 (2019) doi:10.1103/PhysRevD.100.084002 +[arXiv:1901.05472 [gr-qc]]. +[35] M. Hohmann, +Int. J. Geom. Meth. Mod. Phys. 18, +no.supp01, +2140005 (2021) doi:10.1142/S0219887821400053 +[arXiv:2008.12186 [gr-qc]]. + +20 +[36] A. A. Coley, R. J. v. Hoogen and D. D. McNutt, [arXiv:2205.10719 [gr-qc]]. +[37] R. Myrzakulov, Eur. Phys. J. C 71, 1752 (2011) doi:10.1140/epjc/s10052-011-1752-9 [arXiv:1006.1120 [gr-qc]]. +[38] Y. F. Cai, S. H. Chen, J. B. Dent, S. Dutta and E. N. Saridakis, Class. Quant. Grav. 28, 215011 (2011) doi:10.1088/0264- +9381/28/21/215011 [arXiv:1104.4349 [astro-ph.CO]]. +[39] Y. F. Cai, S. Capozziello, M. De Laurentis and E. N. Saridakis, Rept. Prog. Phys. 79, no.10, 106901 (2016) +doi:10.1088/0034-4885/79/10/106901 [arXiv:1511.07586 [gr-qc]]. +[40] C. Bejarano, R. Ferraro and M. J. Guzm´an, Eur. Phys. J. C 77, no.12, 825 (2017) doi:10.1140/epjc/s10052-017-5394-4 +[arXiv:1707.06637 [gr-qc]]. +[41] R. P. Woodard, Lect. Notes Phys. 720, 403-433 (2007) doi:10.1007/978-3-540-71013-4 14 [arXiv:astro-ph/0601672 [astro- +ph]]. +[42] M. Hohmann and C. Pfeifer, Eur. Phys. J. C 81, no.4, 376 (2021) doi:10.1140/epjc/s10052-021-09165-x [arXiv:2012.14423 +[gr-qc]]. +[43] M. Li, Z. Li and H. Rao, Phys. Lett. B 834, 137395 (2022) doi:10.1016/j.physletb.2022.137395 [arXiv:2201.02357 [gr-qc]]. +[44] K. Izumi and Y. C. Ong, JCAP 06, 029 (2013) doi:10.1088/1475-7516/2013/06/029 [arXiv:1212.5774 [gr-qc]]. +[45] Y. C. Ong, K. Izumi, J. M. Nester and P. Chen, Phys. Rev. D 88, 024019 (2013) doi:10.1103/PhysRevD.88.024019 +[arXiv:1303.0993 [gr-qc]]. +[46] A. +De +Felice, +A. +E. +Gumrukcuoglu +and +S. +Mukohyama, +Phys. +Rev. +Lett. +109, +171101 +(2012) +doi:10.1103/PhysRevLett.109.171101 [arXiv:1206.2080 [hep-th]]. +[47] A. Delhom, A. Jim´enez-Cano and F. J. Maldonado Torralba, [arXiv:2207.13431 [gr-qc]]. +[48] D. Baumann, doi:10.1142/9789814327183 0010 [arXiv:0907.5424 [hep-th]]. +[49] Y. Wang, Commun. Theor. Phys. 62, 109-166 (2014) doi:10.1088/0253-6102/62/1/19 [arXiv:1303.1523 [hep-th]]. +[50] N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A1 doi:10.1051/0004-6361/201833880 [arXiv:1807.06205 [astro- +ph.CO]]. +[51] M. Satoh, JCAP 11, 024 (2010) doi:10.1088/1475-7516/2010/11/024 [arXiv:1008.2724 [astro-ph.CO]]. + diff --git a/5tE1T4oBgHgl3EQfBAK1/content/tmp_files/load_file.txt b/5tE1T4oBgHgl3EQfBAK1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c5037dccdf4831832b2f94708a70fd842fd3b98 --- /dev/null +++ b/5tE1T4oBgHgl3EQfBAK1/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf,len=1041 +page_content='USTC-ICTS/PCFT-22-27 Irregular universe in the Nieh-Yan modified teleparallel gravity Mingzhe Li Interdisciplinary Center for Theoretical Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' China and Peng Huanwu Center for Fundamental Theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' China Haomin Rao School of Fundamental Physics and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hangzhou Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' UCAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hangzhou 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' China and University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 100190 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' China The Nieh-Yan modified teleparallel gravity is a model which modifies the general relativity equiv- alent teleparallel gravity by a coupling between the Nieh-Yan density and an axion-like field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This model predicts parity violations in the gravitational waves if the axion-like field has a non-trivial background, and more importantly it is ghost free and avoids the pathologies presented in other parity-violating gravity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The cosmological dynamics and perturbations of the Nieh-Yan modified teleparallel gravity have been investigated in detail, but all these previous investigations rely on the symmetry requirement that in the background universe both the metric and affine con- nection are homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this paper we relax the symmetry constraint on the connection and leave it arbitrary at the beginning, after all the cosmological principle only needs the metric of the background spacetime to meet the symmetry requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We find a new flat universe solution for the Nieh-Yan modified teleparallel gravity, for which the background dynamics itself is unchanged but the perturbations around it present a new feature that the scalar and tensor perturbations are coupled together at the linear level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The implications of this peculiar feature in primordial perturbations from inflation are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' INTRODUCTION Stimulated by the experimental detections of gravitational waves (GWs) [1, 2] and the developments in the cosmic microwave background radiation (CMB) experiments [3, 4], parity violating gravities attracted lots of interests in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' A famous and frequently studied parity violating gravity model is the so-called Chern-Simons modified gravity [5, 6] which within the framework of Riemannian geometry modifies general relativity (GR) by a gravitational Chern-Simons term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The Chern-Simons modified gravity predicts the difference between the amplitudes of the left- and right-handed polarized components of gravitational waves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=', the so-called amplitude birefringence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' However, this model was found to suffer from the problem of vacuum instability because one of the circularly polarized components of GWs becomes a ghost at high frequencies [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Further extensions [8–10] to this model did not circumvent this difficulty because in these extended models the pathological behavior still appear at high energy scales, as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It is very difficult to have a ghost-free parity violating gravity model within the framework of Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Successful parity violating gravity models are available if we go beyond the Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For example, the Nieh-Yan modified teleparallel gravity (NYTG) [12, 13] is constructed within the framework of the teleparallel gravity (TG) [14, 15], where the gravity is identified with the spacetime torsion in stead of the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' One may have a GR equivalent TG model [16] (we may call it TGR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The NYTG model [12, 13] modifies TGR slightly by the anomalous coupling θT �T between an axion-like field θ(x) and the Nieh-Yan density [17]: T �T ≡ (1/2)εµνρσT λ µνTλρσ, where T λ µν is the torsion tensor, εµνρσ is Levi-Civita tensor which relates the totally antisymmetric symbol ϵµνρσ and the determinant of the metric g through the equation εµνρσ = ϵµνρσ/√−g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The Nieh-Yan density is parity-odd, so at the background with ∂µθ ̸= 0, the Nieh-Yan coupling term θT �T violates the parity symmetry spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The NYTG model has been applied to cosmology in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12, 13], where it was found that this model predicts a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='02847v1 [gr-qc] 7 Jan 2023 2 difference between the propagating velocities of the left- and right-handed polarized components of GWs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=', the so-called velocity birefringence phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' More importantly, through detailed investigations on the cosmological perturbations, it was shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12, 13] that the NYTG model is ghost-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Recently, this model was found to be compatible with the results of most local tests in the Solar System at the post-Newtonian order [18, 19], the upper limit on its model parameters by the GWs data of LIGO/Virgo Collaboration was obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [20], and the enhancement of primordial GWs during inflation due to the velocity birefringence of NYTG model and its implications in the air-based GWs experiments were studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Other recent studies on parity violating gravities can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [22–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In all the previous studies of the cosmological applications of the NYTG model, both the metric and the affine connection of the background universe are required to be homogeneous and isotropic at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The spacetime under this strong symmetry constraint is called the regular universe in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The background solutions of the regular universe have been well studied within the TG framework [34–36], and are universally applicable to almost all TG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In fact these solutions have been frequently adopted by different authors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=', [24, 37–39] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' However, the cosmological principle only needs the metric of the background universe to meet the high symmetry requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In the Riemannian geometry, once we impose this symmetry requirement on the metric, the connection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=', the Christoffel symbol) satisfies the same symmetry requirement automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In TG models, the symmetry constraint on the affine connection is independent of the one on the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' If one drops this extra constraint on the connection and leaves it arbitrary at the beginning, there will be final solutions for which the connection is neither homogeneous nor isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We call the universe which has a homogeneous and isotropic metric and a non-homogeneous and non-isotropic affine connection the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' So far the irregular universe has rarely aroused research interest, one example is the flat irregular universe solution found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [40] for the f(T) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The irregular universe does not violate the cosmological principle, but questions are in coming: What features and new physical phenomena could exist in the irregular universe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Or might the irregular universe have properties that are clearly contradictory to experiments so that only the regular universe is physically feasible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' These questions deserve detailed studies for any TG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this paper, we will study the irregular universe in the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Firstly, we will obtain a more general flat universe solution than those in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12, 13] by solving the equations of motion of the NYTG model directly under the condition that only the metric is required to be homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' By analyzing the symmetry of the connection, we will show that the flat universe we obtain is generally an irregular flat universe, and in special cases it reduces back to a regular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We will also show that even in the irregular flat universe, the background equations in the NYTG model are exactly the same as those in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Secondly, we will study the linear cosmological perturbations around the irregular flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We will find that tensor perturbations and scalar perturbations are coupled at the linear perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is a peculiar feature that distinguishes the irregular universe from the regular universe in the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We speculate that this peculiar feature is caused by the fact that the interior space does not satisfy the homogeneity and isotropy in the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Finally, we will study the primordial fluctuations generated by slow-roll inflation in the regular and irregular flat universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We will show that the primordial fluctuations of left- and right-handed GWs are different whether in the regular universe or in the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We will also show that there is a strong statistical correlation between primordial scalar fluctuations and primordial tensor fluctuations generated by slow-roll inflation in the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' II, we briefly introduce the TG theory and the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' III, we study spatially flat cosmological background solutions that only requires the metric to be homogeneous and isotropic in the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' IV, through the quadratic actions for scalar, vector, and tensor perturbations, we investigate linear perturbations around the regular and irregular flat universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' V, we apply our result to the early universe and discuss briefly the primordial perturbations generated by slow-roll inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 1 Actually, the cosmological background solution whose tetrad is eA µ = diag(1, a, a, a) or eA µ = diag(a, a, a, a) under the Weitzenb¨ock gauge is the regular flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' However, most of the earlier literature did not clearly point out that the selection of such a tetrad under the Weitzenb¨ock gauge actually requires the connection to satisfy the same symmetry of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 3 In this paper, we adopt the unit 8πG = 1, and use the signature (+, −, −, −) for the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The tensor indices of the interior space are denoted by A, B, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' = 0, 1, 2, 3 and by a, b, c, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' = 1, 2, 3 when limiting to spatial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' They are lowered and raised by the Minkowski metric ηAB and its inverse ηAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The spacetime tensor indices are denoted by Greek µ, ν, ρ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' = 0, 1, 2, 3 and by Latin i, j, k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' = 1, 2, 3 when limiting to spatial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' They are lowered and raised by the spacetime metric gµν and its inverse gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The antisymmetric symbol ϵµνρσ has the properties: ϵ0ijk = ϵijk ≡ ϵijk, and ϵ123 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In addition, we distinguish the spacetime affine connection ˆΓρ µν and its associated covariant derivative ˆ∇ from the Levi-Civita connection Γρ µν and its associated covariant derivative ∇ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' TG THEORY AND THE NYTG MODEL The TG theory can be considered as a constrained metric-affine theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It is formulated in a spacetime endowed with a metric gµν and an affine connection ˆΓρ µν, which is curvature free and metric compatible, ˆRσ ρµν = ∂µˆΓσ νρ − ∂ν ˆΓσ µρ + ˆΓσ µλˆΓλ νρ − ˆΓσ νλˆΓλ µρ = 0 , ˆ∇ρgµν = ∂ρgµν − ˆΓλ ρµgλν − ˆΓλ ρνgµλ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (1) Without curvature and nonmetricity, in the TG theory the gravity is identified with spacetime torsion T ρ µν = 2ˆΓρ [µν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' One can also describe the TG theory using the language of the tetrad eA µ and the spin connection ωA Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' They relates the metric gµν and the affine connection ˆΓρ µν through the following relations gµν = ηABeA µeB ν , ˆΓρ µν = e ρ A (∂µeA ν + ωA BµeB ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (2) The torsion tensor is written as T ρ µν = 2e ρ A (∂[µeA ν] + ωA B[µeB ν]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (3) The teleparallel constraints (1) dictate that the spin connection can be in general expressed as ωA Bµ = (Λ−1)A C∂µΛC B , (4) where ΛA B is arbitrary element of Lorentz transformation matrix which is position dependent and satisfies the relation ηABΛA CΛB D = ηCD at any spacetime point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, the tetrad eA µ and the Lorentz matrix ΛA B can be regarded as the basic variables of the TG theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this way, the teleparallel constraints (1) are automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The TGR model, as the GR equivalent TG model, has the following action, ST GR = 1 2 � d4x |e| T ≡ � d4x |e| � −1 2TµT µ + 1 8TαβµT αβµ + 1 4TαβµT βαµ � , (5) where |e| = √−g is the determinant of the tetrad, T is the torsion scalar, and Tµ = T α µα is the torsion vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since we have the identity −R(e) = T + 2∇µT µ, the action (5) is identical to the Einstein-Hilbert action up to a surface term, where the curvature scalar R(e) is defined by the Levi-Civita connection and considered as being fully constructed from the metric, and in turn from the tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since the surface term in the action does not affect the equations of motion, we say that the TGR is equivalent to GR at the level of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The NYTG model [12, 13] modifies the TGR model by introducing the coupling SNY = c 4 � d4x |e| θ T �T , (6) between an axion-like field θ and the Nieh-Yan density T �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The coupling constant c is dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Generally we should also consider its own dynamics of the axion-like field and take other matter into account, so the full action of the NYTG model is SNY T G = � d4x |e| �1 2T + c 4 θ T �T + 1 2∇µθ∇µθ − V (θ) � + Sm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (7) 4 Other matter with the action Sm is assumed to be coupled to spacetime minimally through the tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' At the background in which the axion-like field has non-zero spacetime derivatives, the Nieh-Yan coupling term breaks parity spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Because only the first-order derivatives of the basic variables appears in the action, the NYTG model can avoid the Ostrogradski ghost mode, which is expected to be originated from higher-order derivatives in the action [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' As with most modified TG theories, the NYTG model apparently has two kinds of gauge symmetries: diffeomor- phism invariance and local Lorentz invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The latter transformation makes the following change: eA µ → (L−1)A BeB µ , ΛA B → ΛA CLC B , (8) where LA B(x) are the element of Lorentz matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We would like to use different notations to distinguish two kinds of Lorentz matrices: ΛA B(x) is used to express the spin connection as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (4), but LA B(x) represents the local transformation that makes a shift from one local frame to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Transformation (8) can be expressed in terms of tetrad and spin connections as eA µ → (L−1)A BeB µ , ωA Bµ → (L−1)A CωC DµLD B + (L−1)A C∂µLC B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (9) It is easy to prove that the metric gµν and torsion tensor T ρ µν are invariant under the local Lorentz transformation (8), as is the action (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Due to the local Lorentz invariance, one can choose the gauge ΛA B = δA B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=', ωA Bµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is the Weitzenb¨ock connection, which has been frequently adopted in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In addition, there is another symmetry hidden in the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The Nieh-Yan term (6) can be integrated by parts as SNY = − c 2 � d4x ηABϵµνρσ(∂µθ)(ΛA CeC ν)∂ρ(ΛB DeD σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (10) It can be seen that the Nieh-Yan term (6) is invariant under the following transformation (ΛA CeC µ) → LA B(θ)(ΛB CeC µ) , (11) where LA B(θ) is Lorentz matrix that depends only on axion-like field θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Note that ΛA CeC µ is invariant under trans- formation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Due to the Lorentz symmetry (8), the transformation (11) can always be attributed to the fact that the tetrad eA µ remains unchanged while the Lorentz matrix ΛA B undergoes a Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Obviously the metric and the action of TGR model are invariant under such a transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' So the total action of the NYTG model is invariant under the transformation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The equations of motion follow from the variation of the action (7) with respect to eA µ and ΛA B separately Gµν + N µν = T µν + T µν θ , (12) N [µν] = 0 , (13) where N µν = (c/2)εµλρσ∂λθ T ν ρσ, Gµν is the Einstein tensor, T µν = −(2/√−g)(δSm/δgµν) and T µν θ = [V (θ) − ∇αθ∇αθ/2]gµν + ∇µθ∇νθ are the energy-momentum tensors for the matter and the axion-like field θ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Similar to most modified TG models, the equation of motion (13) from the variation of ΛA B is not independent of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (12), it is just the antisymmetric part of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' As explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [13], this is due to the local Lorentz invariance of the action, any change caused by δΛA B can always be equivalent to the change caused by δeA µ, so requiring the action to take the extremum under δeA µ already includes the case where the action takes the extremum under δΛA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' There is another equation following from the variation of the action (7) with respect to θ, □θ + V (1) − c 4T �T = 0 , (14) where □ = gµν∇µ∇ν and V (n) = dnV (θ)/dθn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' All of these equations of motion are consistent with the Bianchi identity ∇µGµν = 0 and the covariant conservation law ∇µT µν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Also in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12, 13], the cosmological perturbations of the NYTG model were analyzed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It was found that the NYTG model makes a difference between the propagating velocities of the left- and right-handed polarized 5 components of GWs, but makes no difference between their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This phenomenon is called velocity birefrin- gence, which is a clear physical signal of parity violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' More importantly, the NYTG model was confirmed to be ghost free through the quadratic action of cosmological perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It is worth mentioning that the Nieh-Yan density T �T is not the only parity-odd term within the TG framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' A more general model including all the parity-odd terms which are quadratic in the torsion tensor was considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' But then it was found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [43] that this more general model suffers from the problem of ghost instability again, unless it completely reduces to the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, within the TG framework, for all parity-odd terms which are quadratic in the torsion tensor, only the Nieh-Yan density T �T can avoid the ghost instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means the NYTG model is robust to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' IRREGULAR FLAT UNIVERSE IN THE NYTG MODEL So far all the studies on the cosmological applications of the NYTG model only considered the regular universe as the background, that means both the metric and the affine connection are constrained to be homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This constraint may be too strong, after all the cosmological principle which is supported by current observations only needs the metric of the background spacetime to meet the high symmetry requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this paper, we will drop the symmetry requirement on the connection and leave it arbitrary at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' After this relaxation, it is expected that the NYTG model will have more interesting cosmological background solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We are interested in the irregular universe solutions in which the metric homogeneous and isotropic but the connection is neither homogeneous nor isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For simplicity, we will only consider the spatially flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In flat universe, the metric can be expressed in rectangular coordinate as ds2 = gµνdxµdxν = a2 � dη2 − δijdxidxj� , (15) where a = a(η) is the scale factor of the universe, η is the conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is the Friedmann-Robertson-Walker (FRW) metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' There are 6 Killing vector fields {ξµ I , I = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='6} in flat universe, which can be expressed as ξµ I = δ µ I , ξµ I+3 = ϵIijδµ ixj , I = 1, 2, 3 (16) where ξµ 1 , ξµ 2 , ξµ 3 are Killing vector fields representing the symmetry of spatial translation, and ξµ 4 , ξµ 5 , ξµ 6 are Killing vector fields representing the symmetry of spatial rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' One can prove that the FRW metric satisfies the condition: LξIgµν = 0, where LξI is the Lie derivative along the Killing vector field ξµ I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This reflects the result that the metric is homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' One can also prove that LξIΓρ µν = 0 for the Levi-Civita connection Γρ µν, which is automatically homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is why we do not need to pay extra attention to the symmetry of the connection within the framework of Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Regular flat universe For TG models, even the metric is determined, the affine connection is still arbitrary to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Usually, as suggested in Refs [34–36], a further constraint was imposed that requires the connection is also homogeneous and isotropic, that is, LξI ˆΓρ µν = ˆ∇µ ˆ∇ν ξρ I − ˆ∇µ(T ρ νσξσ I ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (17) Although ˆΓρ µν is coordinate dependent, the Lie derivative of ˆΓρ µν does not depend on the coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hence the condition (17) is unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (15) and (17) selected the regular flat universe solution in which the tetrad eA µ and Lorentz matrix ΛA B have the following forms: eA µ = aδA µ , ΛA B = ˚ΛA B , (18) 6 where ˚ΛA B is a global Lorentz matrix, which does not depend on spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' All other solutions satisfying Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (15) and (17) differ from the solution (18) only by Lorentz transformation (8), so they are physically equivalent to the solution (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The above process does not depend on a specific TG theory, so the solution (18) is generally applicable to most TG theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For the NYTG model, the solution (18) can automatically satisfy the constraint N [µν] = 0, so the solution (18) is compatible with the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Furthermore, solution (18) leads to N µν = 0 and T �T = 0, which means that the Nieh-Yan term has no effect on the regular flat universe background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, the background equations of the regular flat universe are exactly the same as those of GR [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Irregular flat universe To look for the irregular universe solution, we should give up the constraint (17) on the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' After this relaxation, the connection is left to be determined by the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In a flat universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' we can always simply find the non-zero components of Gµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' T µν and T µν θ as G00 = 3H2 a4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' T 00 = ρ a2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' T 00 θ = ρθ a2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Gij = −2H′ + H2 a4 δij ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' T ij = p a2 δij ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' T ij θ = pθ a2 δij ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (19) where H = a′/a is the conformal Hubble rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' prime represents the derivative with respect to the conformal time η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ρθ = θ′2/ � 2a2� +V and pθ = θ′2/ � 2a2� −V are the energy density and pressure of the θ field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' and ρ and p denote the energy density and pressure of other matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Thanks to the Lorentz symmetry (8), we can always reduce the tetrad to the simple form eA µ = aδA µ in flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In order to facilitate further analysis, we decompose the independent non-zero components of spin connections ωA Bµ as follows δa iω0 a0 = Ui , δi aδb jωa bk = Σϵijk + ϵijlΣkl + Σiδjk − Σjδik , δi aδb jωa b0 = ϵijkVk , δa iω0 aj = σδij + σij + ϵijkσk , (20) where Σij and σij are symmetric and traceless spatial tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In the above decomposition we have exploited the property ωABµ = −ωBAµ due to ˆ∇ρgµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Note that the variables σ, Σ, Ui, Vi, σi, Σi, σij, Σij are not completely independent because we have not yet imposed ˆRσ ρµν = 0 on the spin connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Combining eA µ = aδA µ and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (20), N µν can be obtained as N 00 = 0 , N 0i = 0 , N i0 = 2cθ′ a4 σi , N ij = cθ′ a4 (2Σδij − Σij + ϵijkΣk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (21) In order for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (12) and (13) to hold, there must be σi = 0 , Σi = 0 , Σij = 0 , Σ = Σ(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (22) Combining eA µ = aδA µ, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (20) and (22), Nieh-Yan density can be obtained as T �T = 24Σ a2 (H − σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (23) In order for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (14) to hold, the Nieh-Yan density T �T can only be a function of time η, so σ = σ(η) when Σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (20) and (22), ˆRσ ρµν = 0 gives S′ ij − Ui,j + ϵijkΣ Uk + ϵiklSjkVl = 0 , (24) Σ′δij − Vi,j + ϵijkΣ Uk − ϵiklSjkUl = 0 , (25) ϵiklSlj,k + Σ(Sij − Skkδij) = 0 , (26) ϵinmSjnSkm − Σ2ϵijk = 0 , (27) where Sij = σδij + σij and the subscript “, i” represents a derivative with respect to xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The trace of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (26) gives σΣ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (28) 7 This means that at least one of σ and Σ is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' If σ = 0, the equation after the Hodge duality of the ”j, k” index in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (27) can be decomposed as follows according to the trace part and the traceless part: 6 Σ2 + σijσij = 0 , σikσjk − 1 3(σklσkl)δij = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (29) The solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (29) is Σ = 0, σij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (27) and (28) must give Σ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (30) Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (22) and (30) gives N µν = 0 and T �T = 0, which means that the Nieh-Yan term has no effect even on the irregular flat universe background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, the background equations of the irregular flat universe are exactly the same as those of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is a somewhat unexpected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' But the fact that Nieh-Yan term has no effect on the background does not mean that it has no effect on the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In order to analyze the perturbations, we need to first find the background solution of the irregular flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (30) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (24), (25), (26) and (27), we get S′ ij − Ui,j + ϵiklSjkVl = 0 , (31) Vi,j + ϵiklSjkUl = 0 , (32) ϵiklSlj,k = 0 , (33) ϵinmSjnSkm = 0 , (34) Although there are more equations than variables, this does not mean that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (31), (32), (33) and (34) have no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It can be verified that the following are the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (33) and (34) Sij = vivjf(η)F (1)(⃗v · ⃗x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Vi = ga(η)αa i (η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⃗x) − ha(η)βa i (η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⃗x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Ui = ha(η)αa i (η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⃗x) + ga(η)βa i (η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⃗x) + vif (1)(η)F(⃗v · ⃗x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (35) where αa i (η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⃗x) = cosh [vf(η)F(⃗v · ⃗x)] δai + vavi v2 � 1 − cosh [vf(η)F(⃗v · ⃗x)] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' βa i (η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⃗x) = ϵaij vj v sinh [vf(η)F(⃗v · ⃗x)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' where v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' v3 are constant parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' v = � δijvivj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⃗v · ⃗x = vixi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' f(η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ga(η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ha(η) are arbitrary smooth function of conformal time η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' F(⃗v · ⃗x) is arbitrary smooth function of ⃗v · ⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' f (n)(η) is the n derivative of f(η) with respect to conformal time η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' and F (n)(⃗v · ⃗x) is the n derivative of F(⃗v · ⃗x) with respect to ⃗v · ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Putting solutions (22), (30) and (35) into the decomposition (20), the spin connection ωA Bµ when the tetrad is eA µ = aδA µ can be obtained as ωa 00 = ω0 a0 = hc(η)αc a(η, ⃗x) + gc(η)βc a(η, ⃗x) + vaf (1)(η)F(⃗v · ⃗x) , ωa b0 = ϵabi [gc(η)αc i(η, ⃗x) − hc(η)βc i (η, ⃗x)] , ω0 ai = ωa 0i = vavif(η)F (1)(⃗v · ⃗x) , ωa bi = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (36) It can be verified that the spin connection (36) does satisfy the teleparallel constraints (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Due to the symmetry (11), not every hI(η) and gI(η) represent a physically inequivalent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In order to see this better, we perform a Lorentz transformation (9) on the above solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The transformation matrix LA B is L0 0 = cosh [vf(η)F(⃗v · ⃗x)] , L0 a = La 0 = va v sinh [vf(η)F(⃗v · ⃗x)] , La b = δab + vavb v2 � cosh [vf(η)F(⃗v · ⃗x)] − 1 � , (37) 8 Then, the tetrad ˜eA µ = LA BeB µ and the corresponding spin connection ˜ωA Bµ are ˜e0 0 = a cosh [vf(η)F(⃗v · ⃗x)] , ˜ea 0 = δai˜e0 i = ava v sinh [vf(η)F(⃗v · ⃗x)] , ˜ea i = a � δai + vavi v2 � cosh [vf(η)F(⃗v · ⃗x)] − 1 �� , ˜ωa 00 = ˜ω0 a0 = ha(η) , ˜ωa b0 = ϵabcgb(η) , ˜ωA Bi = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (38) It can be verified that the metric gµν and connection ˆΓρ µν given by solution (38) are the same as those given by the tetrad eA µ = aδA µ and the spin connection (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since the solution (38) satisfies the teleparallel constraints (1), the spin connection ˜ωA Bµ in the solution (38) can be expressed by a Lorentz matrix ˜ΛA B(η, ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' And ˜ωA Bi = 0 means that ˜ΛA B(η, ⃗x) = ˜ΛA B(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' So taking different ha(η) and ga(η) is actually taking different ˜ΛA B(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since θ = θ(η) in the cosmological background, different ˜ΛA B(η) can be converted to each other through the Lorentz transformation ˜ΛA B(η) → LA C(θ)˜ΛC B(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, the solutions with different ha(η) and ga(η) can be transformed into each other by transformation (11), so they are physically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this case, we only need to consider the simplest case below, that is, the case where ha(η) = ga(η) = 0, so that the solution (36) can be simplified to eA µ = aδA µ , ωa 00 = ω0 a0 = vaf (1)(η)F(⃗v · ⃗x) , ωa b0 = 0 , ωa 0i = ω0 ai = vavif(η)F (1)(⃗v · ⃗x) , ωa bi = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (39) The solution (39) can be expressed by the tetrad eA µ and the Lorentz matrix ΛA B as eA µ = aδA µ , Λ = ˚Λ · exp � f(η)F(⃗v · ⃗x) vaKa� , (40) where ˚Λ is a spacetime independent Lorentz matrix, and K1, K2, K3 are the boost matrices whose expression are K1 = � � � � � 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 � � � � � , K2 = � � � � � 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 � � � � � , K3 = � � � � � 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Regardless of the functional form of f(η) and F(⃗v · ⃗x), it can be verified that the solution (40) always satisfies the teleparallel constraints (1) and makes Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (12) and (14) self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Putting solution (40) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (12) and (14), we can get 3H2 = a2 (ρθ + ρ) , 2H′ + H2 = −a2 (pθ + p) , θ′′ + 2Hθ′ + a2V (1) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (41) The background equations are exactly the same as those of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that the Nieh-Yan term has no effect even on the irregular flat universe background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is consistent with our analysis above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Finally, let’s focus on the symmetry of the connection given by the solution (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The non-zero components of LξI ˆΓρ µν given by the solution (40) are LξI ˆΓ0 0i = LξI ˆΓi 00 = vIvif (1)(η)F (1)(⃗v · ⃗x) , LξI ˆΓ0 ij = LξI ˆΓi j0 = vIvivjf(η)F (2)(⃗v · ⃗x) , LξI+3 ˆΓ0 0i = LξI+3 ˆΓi 00 = −ϵIijvjf (1)(η)F(⃗v · ⃗x) + viϵIjkvjxkf (1)(η)F (1)(⃗v · ⃗x) , LξI+3 ˆΓ0 ij = LξI+3 ˆΓi j0 = 2v(iϵj)Ikvkf(η)F (1)(⃗v · ⃗x) + vivjϵIklvkxlf(η)F (2)(⃗v · ⃗x) , (42) where I = 1, 2, 3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (42), and the subscript parentheses denotes the symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The fact that LξI ˆΓρ µν ̸= 0 indicates that the spacetime connection given by the solution (40) is neither homogeneous nor isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' So the solution (40) does represent a irregular flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' When vi = 0 or f(η) = 0 or F(⃗v · ⃗x) = 0, there is LξI ˆΓρ µν = 0, and the solution (40) dose reduce to the regular flat universe solution (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 9 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' PERTURBATIONS AROUND THE IRREGULAR FLAT UNIVERSE In the previous section we studied the flat universe solution of the NYTG model that only requires the metric to be homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We found that the Nieh-Yan term has no effect even on the irregular flat universe background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In order to explore the effect of the Nieh-Yan term on the irregular flat universe, we study the linear cosmological perturbations around the irregular flat universe (40) in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For simplicity, we only consider the case of F(⃗v · ⃗x) = ⃗v · ⃗x, which is equivalent to requiring that the coefficients of the equations of linear perturbations do not depend on the spatial coordinates ⃗x (see below for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' And we also ignore other matter so that Sm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We use the following parametrization for perturbed tetrad [44]: e0 0 = a(1 + A) , e0 i = a(β,i + βV i ) , ec 0 = aδci(χ,i + χV i ) , ec i = aδcj[(1 − ψ)δij + α,ij + αV j,i − ϵijk(λ,k + λV k ) + 1 2hT ij] , (43) So the perturbed metric components have the familiar forms: g00 = a2(1 + 2A) , g0i = −a2(B,i + BV i ) , gij = −a2[(1 − 2ψ)δij + 2α,ij + αV i,j + αV j,i + hT ij] , (44) where B = χ − β and BV i = χV i − βV i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Besides the familiar scalar perturbations (A, B, ψ, α), vector perturbations (BV i , αV i ), and tensor perturbations hT ij in the metric, the parametrization of tetrad brings six extra variables, which are scalar perturbation λ, χ + β and vector perturbation λV i , χV i + βV i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' All the vector perturbations are transverse and denoted by the superscript V , both the tensor perturbations are transverse and traceless and denoted by the superscript T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In addition, the scalar field θ is decomposed as θ(η, ⃗x) = ¯θ(η) + δθ(η, ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Although we can perform a similar decomposition on the Lorentz matrix ΛA B following the parametrization in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [13], we do not need to do so in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Because we can always transform the perturbed Lorentz matrix into the background Lorentz matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (40) through the infinitesimal Lorentz transformation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In other words, we can always absorb the perturbations of the Lorentz matrix ΛA B into the perturbations of the tetrad eA µ through the infinitesimal Lorentz transformation (8), so that we only need to deal with the perturbations of the the tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Due to the diffeomorphism invariance, it is safe to take the unitary gauge δθ = 0, α = 0, αV i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This simplifies the calculations, for example, the gauge invariant scalar perturbation ζ = −(ψ + Hδθ/θ′) representing the curvature perturbation of the hypersurfaces of constant θ reduces to −ψ under the unitary gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since both α and αV i are perturbations which enter the metric, the perturbations α, αV i and δθ are invariant under the infinitesimal Lorentz transformation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, the unitary gauge is compatible with the operation of absorbing the perturbations of the Lorentz matrix into the perturbations of the tetrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The non-isotropic nature of the background connection may lead to coupling of scalar, vector and tensor perturba- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, when studying linear perturbations around the irregular flat universe (40), we should not deal with scalar, vector, or tensor perturbations individually, but should deal with all perturbation variables simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In the following we choose A, ζ, B, BV i , βi = β,i + βV i , λi = λ,i + λV i and hT ij as independent variables, and we study the linear perturbations around the irregular flat universe by means of quadratic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For the NYTG model (7) with Sm = 0, one can directly obtain the quadratic action as S(2) = � d4x a2 � 6Hζ′A − 3ζ′2 − (2A + ζ)ζ,ii − a2V A2 + 2(ζ′ − HA)B,ii + 1 8 � hT ′ ij hT ′ ij − hT ij,khT ij,k � −1 4BV i BV i,jj + cθ′� 2λiζ,i + 1 2ϵijk(βiβj,k − λiλj,k) + ˆSijλiβj − 1 2ϵijkSilhT jlβk − 1 8ϵijkhT ilhT jl,k �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (45) where Sij = vivjf(η)F (1)(⃗v · ⃗x) and ˆSij = (vivj − v2δij)f(η)F (1)(⃗v · ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In general, the coefficients Sij and ˆSij are dependent on the spatial coordinate ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The coefficients of the equations for the linear perturbations are thus also dependent on the spatial coordinate ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It means that the evolution equations for the linear perturbations are not 10 homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For simplicity, in the following we only consider the case of F(⃗v · ⃗x) = ⃗v · ⃗x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this way, Sij and ˆSij are constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' So the evolution equations for the linear perturbations are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' But it should be noted that even in this case, the action (45) appears to be only homogeneous rather than homogeneous and isotropic, because the constant coefficients Sij and ˆSij are not spatial rotation invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In addition, the terms ˆSijλiβj and ϵijkSilhT jlβk in the action (45) show that there is a coupling of scalar, vector and tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' But such coupling may be eliminated by the constraints imposed by the action (45) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Therefore, only after the constraints are lifted can we know whether there is really a coupling of scalar, vector and tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' To further simplify the quadratic action, we change to the momentum space in terms of Fourier transformations, ζ(η, ⃗x) = � d3k (2π) 3 2 ζ(η,⃗k) ei⃗k·⃗x , (46) and we also expand the variables A, B, λi, βi and hT ij in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The tensor perturbation hT ij can be further expanded as hT ij(η,⃗k) = � A hA(η,⃗k) ˆeA ij(⃗k) , (47) where {ˆeA ij(⃗k), A = L, R} are circular polarization bases 3 satisfying ˆklϵlikˆeA jk(⃗k) = ipAˆeA ij(⃗k), where ˆk is the unit vector of ⃗k, pL = −1 and pR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Note that we use the normal letter A for the left- and right- hand indices to distinguish it from the italic letter A used to represent the tetrad indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The quadratic action in the momentum space can be expressed as S(2) = � dη � d3k a2 � 6Hζ′A∗ − 3ζ∗′ζ′ + k2(2A + ζ)ζ∗ + 2k2(HA − ζ′)B∗ −a2V A∗A + 1 4k2BV ∗ i BV i + 1 4 � A � h∗′ Ah′ A − (k2 − cθ′pAk)h∗ AhA � +cθ′� 2ikiλ∗ i ζ + i 2ϵijkki(β∗ j βk − λ∗ jλk) + ˆSijλ∗ i βj − 1 2β∗ i � � A SA i hA ��� , (48) where SA i (⃗k) = ϵijkSjlˆeA kl(⃗k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It can be seen that A, B, BV i , λi and βi are all non-dynamical fields and the variations of the action (48) with them lead to the following constraints: BV i = 0 , (49) HA − ζ′ = 0 , (50) 3Hζ′ + k2ζ − a2V A + Hk2B = 0 , (51) ϵijkkjλk − i ˆSijβj + 2kiζ = 0 , (52) − ˆSijλj + iϵijkkjβk + 1 2 � A SA i hA = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (53) For the regular flat universe case with vi = 0 or f(η) = 0, there are ˆSij = 0 and SA i = 0, so the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (49), (50), (51), (52) and (53) is ζ = 0 , A = 0 , B = 0 , BV i = 0 , λi = ikiλ , βi = ikiβ , (54) 2 The expression of F(⃗v · ⃗x) can differ by a constant term, which does not change the coefficients Sij and ˆSij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' And a constant factor of the difference of F(⃗v · ⃗x) can be absorbed into f(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 3 Note that the choice of circular polarization bases is not unique, ˆeA ij(⃗k) can be rotated along the ⃗k-axis while maintaining all the properties of the circular polarization bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For the case where there is a constant vector ⃗v ̸= 0 on the background, we can always choose the circular polarization bases to satisfy vivjˆeA ij(⃗k) = (v2/ √ 2) sin2 ϑ, where ϑ is the angle between ⃗k and ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This choice maximally simplifies the quadratic action (57), so we adopt this choice in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 11 where λ and β are arbitrary scalar perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Substituting the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (54) back into the action (48), the action (48) can be simplified as S(2) = � dη � d3k a2 4 � A � |h′ A|2 − ω2 A|hA|2� , (55) where ω2 A = k2 −cθ′pAk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It can be seen that there is no scalar dynamical degree of freedom at the linear perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is a bit strange because the action (7) clearly shows that there is a scalar dynamical degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Further research in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [13] shows that the missing scalar dynamical degree of freedom reappears in the regular curved universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The phenomenon of degrees of freedom being hidden under special background also appears in f(T) gravity [45] and massive gravity [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This implies that such a special background is likely to suffer from strong coupling issue [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It can also be seen that the modified dispersion relation ω2 A is helicity dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that GWs with different helicities will have different propagation velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This phenomenon is called velocity birefringence, which is a direct reflection of the parity violation in the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' These results are consistent with the results in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12, 13] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For the irregular flat universe case with vi ̸= 0 and f(η) ̸= 0, the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (49), (50), (51), (52) and (53) is A = ζ′/H , B = − � θ′2ζ′ + 2k2Hζ � /2k2H2 , BV i = 0 , λi = � 2 cos ϑ kv sin2 ϑϵijkkjvk � ζ − i 2 √ 2k ki� � A pAhA � , βi = � 2i v2f(η) sin2 ϑki + 2ivf(η) cos ϑ k sin2 ϑ vi � ζ + ivf(η) cos ϑ 2 √ 2k vi � � A hA � , (56) where ϑ is the angle between ⃗k and ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Substituting the above results back into the action (48), the action (48) can be simplified as S(2) = � dη � d3k �z2 2 � |ζ′|2 − k2|ζ|2� + a2 4 � A � |h′ A|2 − ω2 A|hA|2� − ca2θ′k √ 2 ζ∗� � A pAhA �� , (57) where z2 = a2θ′2/H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For the action (57), the following points need to be emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Firstly, it can be seen that there is indeed a scalar dynamical degree of freedom, which again verifies that there is a scalar dynamical degree of freedom hidden under the regular flat universe at the linear perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Secondly, there are two tensor dynamics degrees of freedom and the dispersion relation ω2 A is helicity dependent, as is the case for the regular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that the velocity birefringence phenomenon of GWs also exists in the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Thirdly, it is surprising that vi and f(η) are completely cancelled in the step of lifting the constraints, so that the action (57) no longer depends on vi and f(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This makes the case of vi = 0, f(η) = 0 not the limit of the case of vi → 0, f(η) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is somewhat analogous to the case where a massless photon is not the limit of a photon with mass tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Fourth, it can be seen that the coefficients in the action (57) are homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that the evolution equations of the scalar perturbation ζ and the tensor perturbations hA are homogeneous and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Finally, it can be seen that even after the constraints are lifted, there is still a coupling of scalar and tensor degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is a feature that neither in the regular flat universe nor in the regular curved universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that scalar perturbations and tensor perturbations can influence each other at the linear perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This can be seen more clearly from the perspective of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' From the action (57), the linear equations of ζ and hA can be obtained as ζ′′ + 2z′ z ζ′ + k2ζ + ca2θ′k √ 2z2 � � A pAhA � = 0 , (58) h′′ A + 2Hh′ A + ω2 AhA + √ 2cθ′pAkζ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (59) 4 The subtle difference in the dispersion relation ω2 A is due to the difference between expanding by ei⃗k·⃗x and expanding by e−i⃗k·⃗x in the Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 12 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (58) shows that the tensor perturbations hA can be used as a source of the scalar perturbation ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The scalar perturbation ζ can be excited when left- and right- handed GWs have different amplitudes or phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' And Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (59) shows that the scalar perturbation ζ can be used as a source of the tensor perturbations hA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' It is worth noting that the source of the tensor perturbations hA caused by ζ is helicity-dependent, that is, the excitation effects caused by ζ on the left- and right-handed GWs are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' PRIMORDIAL FLUCTUATIONS GENERATED BY INFLATION In the previous section, we preliminarily studied the the linear perturbations around the regular and irregular flat universe, and obtained the quadratic action after the constraints was lifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this section, we will preliminarily study the primordial fluctuations generated by slow-roll inflation in the regular and irregular flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The case of the regular universe For the case of regular universe, the quadratic action (55) can be expressed as S(2) = � dη � d3k a2 2 � A ���� 1 √ 2h′ A ��� 2 − � k2 − cθ′pAk � ��� 1 √ 2hA ��� 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (60) Note that since there are only tensor degrees of freedom in the regular flat universe at the linear perturbation level, a scalar field other than θ needs to be introduced to generate the primordial scalar perturbation [12, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this subsection we do not consider the case of introducing additional scalar fields, and we only focus on the tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Next we consider the case of slow-roll inflation dominated by the axion-like field θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since the background equations of the regular flat universe are exactly the same as those in GR, the background evolution during inflation will be exactly the same as the case of slow-roll inflation in GR [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' So we don’t need to repeat the analysis of the details of single scalar field inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We introduce two commonly used slow-roll parameters ε ≡ − ˙H H2 , δ ≡ ¨θ H ˙θ , (61) where H = ˙a/a = H/a is the Hubble rate, the upper dot represents the derivative with respect to the physical time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We assume ε ∼ |δ| ≪ 1, | ˙ε/H| ≪ |ε| and | ˙δ/H| ≪ |δ| during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Under the slow-roll approximation, H ≈ −1 + ε η , θ′ ≈ √ 2ε η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (62) Without loss of generality, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (62) we have assumed that the value of θ decreases during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Next, by combining Eqs (60) and (62), the correlation function of hA can be obtained through the process in Appendix C: ⟨h† AhA⟩ ≈ H2e−pA√ ε/2cπk−(3+2ε) , (63) and ⟨h† LhR⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Through the correlation functions (63), the power spectrum of the left- and right-handed GWs can be obtained as PA(k) = k3 π2 ⟨h† AhA⟩ ≈ H2 π2 e−pA√ ε/2cπk−2ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (64) The power spectrum of the tensor perturbations can be obtained as PT (k) = PL(k) + PR(k) ≈ H2 π2 � 1 + cosh ��ε 2cπ �� k−2ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (65) 13 The relative different between the power spectrum of the left- and right-handed GWs can be obtained as Π ≡ PR − PL PR + PL ≈ − tanh ��ε 2cπ � ≈ − �ε 2cπ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (66) Π ̸= 0 means that the magnitudes of the primordial fluctuations of left- and right-handed GWs are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is a clear physical signal of parity violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' But this seems to contradict the conclusion in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12, 13] that there is only velocity birefringence of GWs but no amplitude birefringence of GWs in the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The reason for this contradiction is that θ′ is approximated as a constant in the analysis of the evolution of GWs in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Of course, this approximation is valid when studying the propagation of GWs in a slowly expanding universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' However, θ′ = a ˙θ ∝ 1/η cannot be approximated as a constant during the slow-roll inflation dominated by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We know that for a harmonic oscillator (the equation of motion is ¨x+ω2x = 0), the amplitude of the harmonic oscillator can be changed when the frequency ω is time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' And when the time dependence of θ′ is not negligible, the time dependence of ωL and ωR will be different, resulting in different effects on the amplitudes of left- and right-hand GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is why the magnitudes of the primordial fluctuations of left- and right-handed GWs generated by slow-roll inflation in the regular flat universe are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' If ε → 0, it can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (62) that θ′ ≈ 0 can be approximated as a constant, and from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (66), it can be seen that Π → 0 too, that is, the magnitudes of the primordial fluctuation of the left- and right-handed GWs are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Finally, let’s look at the case when the coupling constant c → 0, then PT (k) ≈ 2H2 π2 k−2ε , Π ≈ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (67) This is exactly the result of the slow-roll inflation of single scalar field in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The case of the irregular universe For the case of irregular universe, since the coupling of ζ and hA in the action (57) makes it difficult to analyze the quantum fluctuations, we first diagonalize the variables ζ and hA below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Firstly, for the convenience of analysis, we introduce new variables ξ1 = (z/a)ζ, ξ2 = (1/ √ 2)hL and ξ3 = (1/ √ 2)hR, so that the action (57) can be simplified as S(2) = � dη � d3k a2 2 � 3 � s=1 ξ∗′ s ξs − 3 � s1=1 3 � s2=1 Ms1s2ξ∗ s1ξs2 � , with M = � � � k2 − Ω −κ κ −κ k2 − σ 0 κ 0 k2 + σ � � � , (68) where Ω = z′′/z − a′′/a, σ = −cθ′k and κ = cHk are background quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Secondly, we introduce an orthogonal matrix T that can diagonalize the matrix M, and its expression is T = � � � tT 1 tT 2 tT 3 � � � , with ts = −s2 + 5s − 5 � 1 + (τs−σ)2 κ2 + � 1 − (τs−σ)(τs+Ω) κ2 �2 � � � (τs − σ)/κ 1 − (τs − σ)(τs + Ω)/κ2 1 � � � , (69) where the superscript T means transpose, and {τs, s = 1, 2, 3} are the solutions of the cubic equation τ 3 + Ωτ 2 − (2κ2 + σ2)τ − σ2Ω = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (70) The specific expressions of {τs, s = 1, 2, 3} are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Finally, we introduce new variables {qs, s = 1, 2, 3}, which are defined as � � � q1 q2 q3 � � � = T � � � ξ1 ξ2 ξ3 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (71) 14 Thus, the action (68) can be further simplified as S(2) = 3 � s=1 � dη � d3k a2 2 � |q′ s|2 − (k2 + τs)|qs|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (72) So far, we have simplified the action (57) with coupling between variables to the action (72) without coupling between variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The latter form makes it easier to calculate the primordial fluctuations generated by inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Next we consider the case of slow-roll inflation dominated by the axion-like field θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' III we proved that the background equations of the irregular flat universe are exactly the same as those in GR, the background evolution during inflation will be exactly the same as the case of slow-roll inflation in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Under the slow-roll approximation, the background quantities Ω, σ and κ can be approximately expressed as Ω ≈ 3(ε + δ) 2η2 , σ ≈ − √ 2εck η , κ ≈ −(1 + ε)ck η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (73) In this section, we also assume that the coupling constant c ∼ 1 (it can also be seen as a requirement of naturalness), so that c ≫ √ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Ignoring high-order small quantities such as ε2, {τs, s = 1, 2, 3} in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (A3) can be approximated as τ1 ≈ (2 + 3ε)ck √ 2η − 3(ε + δ) 2η2 , τ2 ≈ 0 , τ3 ≈ −(2 + 3ε)ck √ 2η − 3(ε + δ) 2η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (74) If only up to the order of √ε is retained, the orthogonal matrix T can be approximated as T ≈ � � � � 1 √ 2 1+√ε 2 − 1−√ε 2 −√ε 1 √ 2 1 √ 2 1 √ 2 − 1−√ε 2 1+√ε 2 � � � � (75) Regarding the approximate expression (75), there are two points that need additional explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' First, the order √ε is the lowest order approximation required to preserve the difference in the power spectrum of left- and right-handed GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' If we further ignore the contribution of √ε in T, the difference in the power spectrum of left- and right-handed GWs disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' And if we keep the higher-order terms, it brings only more complex but less important corrections in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Second, it can be seen that the matrix T does not tend to the identity matrix as c → 0 in the approximate expression (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is confusing because the three variables are all decoupled as c → 0 in the action (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The reason for this confusing phenomenon is that we have used the approximation c ≫ √ε in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (74) and (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' If c is too small, neither the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (74) nor Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (75) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' See Appendix B for the approximate behavior of orthogonal matrix T when c → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Next, by combining Eqs (72) and (74), the correlation function between variables qs can be obtained through the process in Appendix C: ⟨q† 1q1⟩ ≈ H2 2 e cπ √ 2 k−(3+3ε+δ) , ⟨q† 2q2⟩ ≈ H2 2 k−(3+2ε) , ⟨q† 3q3⟩ ≈ H2 2 e− cπ √ 2 k−(3+3ε+δ) , (76) and ⟨q† s1qs2⟩ = 0 when s1 ̸= s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' using the approximation techniques in Appendix D and combining Eqs (71),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (75) and (76),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' the correlation functions for the variables ζ and hA can be obtained as ⟨ζ†ζ⟩ ≈ 1 2ε cosh � cπ √ 2 � H2knS−4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⟨h† AhA⟩ ≈ �1 2 + 1 2 cosh � cπ √ 2 � − pA √ε sinh � cπ √ 2 �� H2knT −3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⟨ζ†hA⟩ ≈ − pA 2 √ 2ε sinh � cπ √ 2 � H2k−(3+3ε+δ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' ⟨h† LhR⟩ ≈ 1 2 � 1 − cosh � cπ √ 2 �� H2k −(3+3ε+δ)− 1 2 csch2� cπ 2 √ 2 � (ε+δ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (77) 15 where nS ≈ 1 − (δ + 3ε) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' nT ≈ −(3ε + δ) + 1 2 sech2 � cπ 2 √ 2 � (ε + δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (78) It should be noted that since Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (74) and (75) are approximately true only when c ≫ √ε, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (77) and (78) are also approximately true only when c ≫ √ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Through the correlation functions (77), the power spectrum of the scalar perturbation ζ can be obtained as PS(k) = k3 2π2 ⟨ζ†ζ⟩ ≈ H2 8π2ε cosh � cπ √ 2 � knS−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (79) The power spectrum of the left- and right-handed GWs can be obtained as PA(k) = k3 π2 ⟨h† AhA⟩ ≈ H2 2π2 � 1 + cosh � cπ √ 2 � − 2pA √ε sinh � cπ √ 2 �� knT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (80) The power spectrum of the tensor perturbations can be obtained as PT (k) = PL(k) + PR(k) ≈ H2 π2 � 1 + cosh � cπ √ 2 �� knT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (81) The tensor-to-scalar ratio r can be obtained as r ≡ PT PS = 8 � 1 + sech � cπ √ 2 �� ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (82) The relative different between the power spectrum of the left- and right-handed GWs can be obtained as Π ≡ PR − PL PR + PL ≈ −2√ε tanh � cπ √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (83) Strictly speaking, since Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (77) and (78) are only approximately true when c ≫ √ε, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (79)-(83) are also approx- imately true only when c ≫ √ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' But If we ignore this fact and force c → 0, then PS ≈ H2 8π2εknS−1 , PT ≈ 2H2 π2 knT , r ≈ 16ε , Π ≈ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (84) It can be seen that except for the spectral indices nS and nT , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (84) is the result of the slow-roll inflation in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' From the Planck 2018 [50], we know that the scalar spectral index nS ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='966 and the tensor-to-scalar ratio r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that the allowable value range of the slow-roll parameters ε and δ is 0 < ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='101 8 � 1 + sech � cπ/ √ 2 �� < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='012625 , δ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='034 − 3ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (85) It can be seen that the maximum value of ε depends on the coupling constant c, but will not exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='012625 (the upper limit of ε when c → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The allowable value of δ is determined by ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 1 shows the allowable value range of slow-roll parameters ε and δ when c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Although by comparing the results in subsections V A and V B, we can find that the power spectrum of the left- and right-handed GWs given by the irregular universe is different from that of the regular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' But this is not the main difference between irregular and regular universes for primordial fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For primordial fluctuations, the most important feature of the irregular universe compared to the regular universe is that the correlation function of scalar perturbation and tensor perturbations ⟨ζ†hA⟩ ̸= 0 at the linear perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This means that there is a strong statistical correlation between primordial scalar fluctuations and primordial tensor fluctuations generated by slow-roll inflation in the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The apparent reason for this phenomenon is that the quadratic action contains the coupling of scalar perturbations and tensor perturbations in the irregular universe, as exhibited by the action (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The deeper reason may be that the condition LξI ˆΓρ µν ̸= 0 destroys the homogeneity and isotropy of the interior space, so that the scalar fluctuations and the tensor fluctuations can interact with each other in the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='010 ε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='035 δ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 1: In the ε-δ plane, the blue line is the allowable value range when c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' CONCLUSION As a step towards exploring the irregular universe within the TG framework, in this paper, we studied the irregular flat universe of the NYTG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Firstly, we obtained the irregular flat universe solution of the NYTG model under the condition that only the symmetry of the metric is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We found that the cosmological background equations of the NYTG model are exactly the same as those of GR in both the regular flat universe and the irregular flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Secondly, we studied the linear cosmological perturbations around the irregular flat universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We found a peculiar feature of the irregular flat universe: the tensor and scalar perturbations are coupled together at the linear perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We speculate that this peculiar feature is caused by the fact that the interior space does not satisfy the homogeneity and isotropy in the irregular universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Finally, we applied the NYTG model to the early universe and studied the primordial perturbations generated by slow-roll inflation in the regular and irregular flat universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We found that the left- and right-handed primordial GWs are different in both the regular flat universe and the irregular flat universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' We also found that there is a strong statistical correlation between the primordial scalar and tensor perturbations generated by slow-roll inflation in the case of irregular universe, this is a direct consequence of the direct coupling between the scalar and tensor perturbations at linear order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Acknowledgement: This work is supported in part by National Key R&D Program of China Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 2021YFC2203102, and by NSFC under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 12075231 and 12047502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Appendix A: Solutions of the cubic equation Consider a cubic equation with respect to the variable τ as aτ 3 + bτ 2 + cτ + d = 0 , (A1) where a, b, c and d are real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In order to express the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (A1) conveniently, we introduce the following parameters A = b2 − 3ac , B = bc − 9ad , C = c2 − 3bd , ∆ = B2 − 4AC , Θ = 1 3 arccos �2Ab − 3Ba 2A3/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (A2) 17 When ∆ < 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (A1) has three real solutions, which are τ1 = − 1 3a � b + 2 √ A cos Θ � , τ2 = 1 3a � −b + √ A � cos Θ − √ 3 sin Θ �� , τ3 = 1 3a � −b + √ A � cos Θ + √ 3 sin Θ �� , (A3) The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (70) in the main text is the result of taking a = 1, b = Ω, c = −(2κ2 + σ2) and d = −σ2Ω in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In this case, there are always A ≥ 0 and ∆ ≤ 0, where the equal sign holds if and only if κ = σ = Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' And when κ = σ = Ω = 0, obviously the three solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (70) are τ1 = τ2 = τ3 = 0, and the orthogonal matrix T is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Appendix B: The orthogonal matrix T when c → 0 In this appendix, we discuss the approximate behavior of the orthogonal matrix T in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (69) as c → 0 in a more general background (not only during inflation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since σ ∝ c and κ ∝ c, then κ Ω ∝ c , σ Ω ∝ c , κ2 2σΩ ∝ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (B1) When c is much smaller than any other background quantities such as √ε, ˙θ and H−1, ignoring the quadratic and higher terms of c, the solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (70) can be approximately expressed as τ1 ≈ Ω , τ2 ≈ σ , τ3 ≈ σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (B2) So the orthogonal matrix T in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (69) can be approximately expressed as T = � � � 1 κ Ω − κ Ω − κ Ω 1 κ2 2σΩ κ Ω − κ2 2σΩ 1 � � � when c→0 −−−−−−−→ � � � 1 0 0 0 1 0 0 0 1 � � � (B3) It can be easily seen from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (B1) and (B3) that when c → 0, the orthogonal matrix T does tend to the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' This is consistent with the fact that all variables in the action (68) tend to be decoupled when c → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Appendix C: Correlation function generated by inflation The purpose of this appendix is to show how to calculate the correlation function generated by inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Consider a univariate system whose effective action during inflation is S = 1 2 � dη d3k a2 � |q′ ⃗k|2 − � k2 − 2ak η − 3b η2 � |q⃗k|2 � , (C1) where a and b are real parameters, and b has the same order of magnitude as the slow-roll parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Here q(η, ⃗x) is the variable and we have changed to the Fourier space q⃗k(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' After quantization, the variable q⃗k(η) can be expanded as q⃗k(η) = 1 a(η) � vk(η)ˆa⃗k + v∗ k(η)ˆa† ⃗k � , (C2) where ˆa† ⃗k and ˆa⃗k are the generation and annihilation operators that satisfy the following commutation relations [ˆa⃗k ˆa† ⃗k′] = δ(3)(⃗k − ⃗k′) , [ˆa⃗k ˆa⃗k′] = [ˆa† ⃗k ˆa† ⃗k′] = 0 , (C3) 18 and vk(η) satisfies the following equation v′′ k + � k2 − 2ak η − µ2 − 1/4 η2 � vk = 0 , (C4) where µ ≈ 3/2+ε+b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (C4), we used the approximation a′′/a ≈ [(3/2+ε)2 −1/4]/η, and we ignored the higher-order terms of ε and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Next we choose the Bunch-Davies vacuum at η → −∞, that is, lim η→−∞ vk = 1 √ 2k e−ikη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (C5) Under this condition, the solution for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (C4) is (for more detail, see [51]) vk(η) = e−ikη(−2kη)µ(−η) 1 2 e−iπ( 1 4 + µ 2 )U (1/2 + µ − ia, 1 + 2µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 2ikη) e− aπ 2 , (C6) where U(c1, c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' z) is the confluent hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' The |vk| has the following asymptotic form when kη → 0− (super-horizon scale) |vk| ≈ 2µ−1π− 1 2 Γ(µ)k−µ(−η) 1 2 −µe− aπ 2 ≈ 2− 1 2 e− aπ 2 aHk−µ (C7) where Γ(z) is the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' In the last approximately equal sign in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (C7), we used the approximations µ ≈ 3/2 and (−η)−1 ≈ aH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (C2), (C3) and (C7), we can obtain the correlation function on the super-horizon scale as ⟨0|q† ⃗kq⃗k′|0⟩ ≈ H2 2 e−aπk−(3+2ε+2b)δ(3)(⃗k + ⃗k′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (C8) where |0⟩ is the vacuum state, which satisfies ˆa⃗k|0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' For the sake of convenience, we can omit the subscript ⃗k and throw away the annoying delta function δ(3)(⃗k + ⃗k′), so that the correlation function (C8) can be abbreviated as ⟨q†q⟩ ≈ H2 2 e−aπk−(3+2ε+2b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (C9) Appendix D: Summation of nearly scale-invariant functions Consider there are N nearly scale-invariant functions {fi(k) = Cikni, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=', N}, where |ni| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Then the sum of these functions should also be a nearly scale-invariant function, so it can be approximated as f(k) = N � i=1 fi(k) = N � i=1 Cikni ≈ Ckn , with |n| ≪ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (D1) Next we need to find the coefficient C and the exponent n in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Since ni ≈ 0 and n ≈ 0, we can approximately let ni = n = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (D1), so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (D1) becomes C ≈ N � i=1 Ci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (D2) Next, let Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (D1) take the derivative of k and then let ni = n = 0 on the exponent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Then the approximate expression of n can be obtained as n ≈ 1 C N � i=1 Cini .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' (D3) [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [LIGO Scientific and Virgo], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 116, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='6, 061102 (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='061102 [arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='03837 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 19 [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [LIGO Scientific and Virgo], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 119, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='16, 161101 (2017) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='161101 [arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='05832 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Cai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Xu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' “Probing Primordial Gravitational Waves: Ali CMB Polarization Telescope,” Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1, 145-154 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1093/nsr/nwy019 [arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='03047 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [CMB-S4], Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 926, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1, 54 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='3847/1538-4357/ac1596 [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='12619 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Jackiw and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Pi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 68, 104012 (2003) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='104012 [arXiv:gr-qc/0308071 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Alexander and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Yunes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 480, 1-55 (2009) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='physrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='002 [arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2562 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Dyda, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Flanagan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Kamionkowski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 86, 124031 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='124031 [arXiv:1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='4871 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Crisostomi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Noui, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Charmousis and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Langlois, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 97, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='4, 044034 (2018) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='044034 [arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='04531 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [9] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Gao and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hong, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 101, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='6, 064057 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='064057 [arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='07131 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Qiao and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Wang, “Waveform of gravitational waves in the general parity-violating gravities,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 101, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2, 024002 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='024002 [arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='10887 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [11] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Bartolo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Caloni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Orlando and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Ricciardone, JCAP 03, 073 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/1475-7516/2021/03/073 [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='01715 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rao and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao, JCAP 11, 023 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/1475-7516/2020/11/023 [arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='08038 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rao and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Tong, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 104, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='8, 084077 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='084077 [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='05917 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Aldrovandi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Pereira, Teleparallel Gravity, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Springer, 23 Dordrecht, (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Bahamonde, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Dialektopoulos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Escamilla-Rivera, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Farrugia, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Gakis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hendry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hohmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Said, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Mifsud and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Di Valentino, [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='13793 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Maluf, Annalen Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 525, 339-357 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1002/andp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='201200272 [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='3897 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Nieh and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Yan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 23, 373 (1982) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='525379 [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 104, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='12, 124084 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='124084 [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='08597 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Qiao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao, JCAP 04, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='04, 054 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 105, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2, 024035 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='024035 [arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='13870 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Cai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Fu and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Yu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 105, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='10, 103520 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='103520 [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='04794 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' B 827 (2022), 136968 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='136968 [arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='01337 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [23] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Gong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Niu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Cui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 105 (2022) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='4, 044034 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='044034 [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='06446 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hohmann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Pfeifer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' B 834 (2022), 137437 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='137437 [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='01856 [gr- qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [25] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Tong and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Xianyu, JHEP 10 (2022), 194 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1007/JHEP10(2022)194 [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='06349 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Tong and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 105 (2022) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='10, 104002 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='104002 [arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='06912 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Feng and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Gao, JCAP 10 (2022), 054 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/1475-7516/2022/10/054 [arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='12045 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Wang, [arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='05259 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Wang, [arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='04711 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Filho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Nascimento, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Petrov and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Porf´ırio, [arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='11821 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Qiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Ji, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Zhao, [arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='16825 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Cai, [arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='10893 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [33] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Yu and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Gao, [arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='14362 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hohmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J¨arv, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Krˇsˇs´ak and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Pfeifer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 100, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='8, 084002 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='084002 [arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='05472 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hohmann, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='supp01, 2140005 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1142/S0219887821400053 [arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='12186 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 20 [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Coley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hoogen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' McNutt, [arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='10719 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [37] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Myrzakulov, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' C 71, 1752 (2011) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1140/epjc/s10052-011-1752-9 [arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1120 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [38] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Cai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Dent, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Dutta and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Saridakis, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 28, 215011 (2011) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/0264- 9381/28/21/215011 [arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='4349 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [39] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Cai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Capozziello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' De Laurentis and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Saridakis, Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 79, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='10, 106901 (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/0034-4885/79/10/106901 [arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='07586 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Bejarano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Ferraro and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Guzm´an, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' C 77, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='12, 825 (2017) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1140/epjc/s10052-017-5394-4 [arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='06637 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Woodard, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 720, 403-433 (2007) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1007/978-3-540-71013-4 14 [arXiv:astro-ph/0601672 [astro- ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Hohmann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Pfeifer, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' C 81, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='4, 376 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1140/epjc/s10052-021-09165-x [arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='14423 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Li and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' B 834, 137395 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='137395 [arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='02357 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [44] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Izumi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Ong, JCAP 06, 029 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/1475-7516/2013/06/029 [arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='5774 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [45] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Ong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Izumi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Nester and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' D 88, 024019 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='024019 [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='0993 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' De Felice, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Gumrukcuoglu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Mukohyama, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 109, 171101 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='171101 [arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2080 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Delhom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Jim´enez-Cano and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Maldonado Torralba, [arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='13431 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [48] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Baumann, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1142/9789814327183 0010 [arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='5424 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [49] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Wang, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 62, 109-166 (2014) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/0253-6102/62/1/19 [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1523 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [50] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [Planck], Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' 641 (2020), A1 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1051/0004-6361/201833880 [arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='06205 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' [51] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content=' Satoh, JCAP 11, 024 (2010) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='1088/1475-7516/2010/11/024 [arXiv:1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='2724 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfBAK1/content/2301.02847v1.pdf'} diff --git a/6tFAT4oBgHgl3EQfnx0W/content/tmp_files/2301.08630v1.pdf.txt b/6tFAT4oBgHgl3EQfnx0W/content/tmp_files/2301.08630v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..10fb491dd5c8d12e3d634f75794b177ccf92e5ce --- /dev/null +++ b/6tFAT4oBgHgl3EQfnx0W/content/tmp_files/2301.08630v1.pdf.txt @@ -0,0 +1,1508 @@ +Evaluating approaches for on-the-fly machine learning interatomic potential for +activated mechanisms sampling with the activation-relaxation technique nouveau +Eugène Sanscartier,1 Félix Saint-Denis,1 Karl-Étienne Bolduc,1 and Normand Mousseau1 +1Département de physique and Regroupement québécois sur les matériaux de pointe, +Université de Montréal, Case Postale 6128, Succursale Centre-ville, Montréal, Québec H3C 3J7, Canada +(Dated: January 23, 2023) +In the last few years, much efforts have gone into developing universal machine-learning potentials +able to describe interactions for a wide range of structures and phases. Yet, as attention turns to +more complex materials including alloys, disordered and heterogeneous systems, the challenge of +providing reliable description for all possible environment become ever more costly. In this work, we +evaluate the benefits of using specific versus general potentials for the study of activated mechanisms +in solid-state materials. More specifically, we tests three machine-learning fitting approaches using +the moment-tensor potential to reproduce a reference potential when exploring the energy landscape +around a vacancy in Stillinger-Weber silicon crystal and silicon-germanium zincblende structure +using the activation-relaxation technique nouveau (ARTn). We find that a a targeted on-the-fly +approach specific and integrated to ARTn generates the highest precision on the energetic and +geometry of activated barriers, while remaining cost-effective. This approach expands the type of +problems that can be addressed with high-accuracy ML potentials. +I. +INTRODUCTION +As computational materials scientists turn to atten- +tion to ever more complex systems, they are faced with +two major challenges : (i) how to describe correctly their +physics and (ii) how to reach the appropriate size and +time scale to capture the properties of interest. +The +first challenge is generally solved by turning to ab ini- +tio methods,1 that allow the solution Heisenberg’s equa- +tion with reasonably controlled approximations. Theses +approaches, however, suffer from N 4 scaling which lim- +its their application to small system sizes and short time +scales. The second challenge is met by a variety of meth- +ods that cover different scales. Molecular dynamics2, for +example, which directly solves Newton’s equation, ac- +cesses typical time scales between picoseconds and mi- +croseconds, at the very best. +Other approaches, such +as lattice3,4 and off-lattices kinetic Monte-Carlo5,6, by +focusing on physically relevant mechanisms, can extend +this time scale to seconds and more, as long the diffusion +takes place through activated processes. +Even though +these methods are efficient, each trajectory can require +hundreds of thousands to millions of forces evaluations, +which becomes too costly with ab initio approaches, forc- +ing modellers to use empirical potentials in spite of their +incapacity at describing correctly complex environments. +Building on ab initio energy and forces, machine- +learned potentials7–10 open the door to lifting some of +this difficulties, by offering much more reliable physics as +a small fraction of the cost of ab initio evaluations. +Since their introduction, ML potentials have been +largely coupled with MD and focusing on the search for +universal potentials, able to describe a full range of struc- +tures and phases for a given material11–13. As we turn +to more complex systems such as alloys and disordered +and heterogeneous systems, it becomes more and more +difficult to generate such universal potentials, since the +number of possible environments grows rapidly with this +complexity. In this context, the development of specific +potentials, with on-the-fly learning that makes it possible +to adapt to new environments, becomes a strategy worth +exploring. +In this work, we focus on the construction of machine- +learned potentials adapted to the sampling of energy +landscape dominated by activated mechanisms, +i.e., +solid-state systems with local activated diffusion and evo- +lution. A correct computational sampling, using methods +such as the activation-relaxation technique (ART)14 and +its revised version (ART nouveau or ARTn)15,16, requires +a precise description of local minima and of the land- +scape surrounding the first-order saddle points that char- +acterize diffusion according to the transition-state theory +(TST)17. These barriers can be high — reaching many +electron-volts — and involve strained configurations that +can be visited only very rarely with standard molecular +dynamics. +More specifically, we compare three machine learning +procedures in which we change the context where lean- +ing on-the-fly occur to train a Moment Tensor Poten- +tial (MTP)10,18 that describes the diffusion of vacancy +in Stillinger-Weber silicon19 and silicon-germanium20 as +sampled with ARTn. The first one uses a pure MD learn- +ing procedure, fitted at various temperatures, in a proce- +dure that echoes the work of Novoselov et al.21, a second- +one adds an on-the-fly adjustment during an ARTn run +and the third one focuses on purely OTF-ARTn potential +adjustment. +Results underline the efficiency gain in developing tar- +geted ML potentials for specific applications, comparing +the cost of fitting Si with SiGe, it also shows the rapid +increase in computation complexity associated with mov- +ing from element to alloy systems, which emphasizes the +usefulness of a specific approach such as the one applied +here to activated processes. +arXiv:2301.08630v1 [cond-mat.mtrl-sci] 20 Jan 2023 + +2 +II. +METHODOLOGY +A. +ML Potential +The Moment Tensor Potential (MTP)10,18 is a linear +model of functions Bα(ri) built from contractions of mo- +ment tensor descriptors defined by the local neighbor- +hood relative position ri of atom i within a sphere of +influence of radius rc respecting a set invariances. This +model has been shown to be fast while giving accuracy +on the order of ∼meV/atom and requiring few hundreds +to thousands of reference potential calls22 on-the-fly. +MTP have been used on a wide variety of problems in- +cluding on-the-fly MD simulation18,21,23, search and min- +imization of new alloys24,25 and diffusion processes21 on +systems counting one or multiple species. +MTP approximates atomic configuration energy as +sum of local contributions. A local contribution is ob- +tained through a sum over the included basis {Bα(ri)} +as a linear combination of B(ri) and ξα, +V (ri) = +m +� +α=1 +ξαBα(ri) +(1) +The “level” of a potential gives the number of different +possible tensor Mµ,ν (ri) descriptors. The {Bα(ri)} func- +tions of Eq. 1 are constructed by a tensorial contraction +of different Mµ,ν (ri) and the number of different tenso- +rial contraction sets m in Eq. 1. More information on +MTP is available in Ref. 18. +The total energy of a N-atom configuration (R) is then +given by the sum of N local contributions +E(R) = +N +� +i=1 +V (ri) = +N +� +i=1 +m +� +α=1 +ξαBα(ri) +(2) +and the forces are obtained by taking the gradient of this +quantity +F(R) = −∇ +N +� +i=1 +m +� +α=1 +ξαBα(ri) +(3) +The parameters ξα are obtained by minimizing the loss +function: +� +R∈A +� +we +���E(R) − ˆE(R) +��� +2 +2 + wf +N +� +i +���fi(R) −ˆfi(R) +��� +2 +2 +� +→ min +ξ +(4) +Here A is the training set made of configurations with +known energy and forces. The goal is to minimize the +difference between E(R), fi(R)(real value) and ˆE(R), +ˆfi(R)(predicted by model), respectively, for all element +in A. Weights on contribution from energy and forces +(we and wf) are set to one. +B. +Learning On-The-Fly Tools +On-the-fly atomic machine learning potential (OTF) +involves the repeated training of the model potential as +new atomic environments are generated through various +procedures. +Following the work of Shapeev and collaborators18, the +reliability of the potential to describe a given configura- +tion is evaluated using the D-optimality criterion to grade +to which extend a configuration extrapolate. This grade +is used along with a selection algorithm (MaxVol) to as- +sess whether the new configuration should be added to +the training set or replace a configuration already in it. +While a detailed description can be found in Ref.23, we +provide here a brief summary of the retained approach. +The selection and extrapolation-grade algorithm can +be applied using either a local-energy or a global-energy +descriptor. +The local-energy descriptor is presented as a rectangu- +lar matrix Gm×N formed by the basis elements {Bα(ri)} +associated with the neighborhood ri of all N atoms: +G = +� +� +� +B1(r1) . . . Bm(r1) +... +... +... +B1(rN) . . . Bm(rN) +� +� +� +T +For a given configuration, the global-energy description +reduces this information to a vector g +g = +� b1(R) . . . bm(R) � +where each term, {bα(R)} is a sum over all neighborhoods +for a specific basis element {Bα(ri)}: +{bα(R)} = +N +� +i=0 +{Bα(ri)} +For the global-energy descriptor, evaluating the over- +lap of a new configuration with the training set A is done +by solving for cj, in +A +� c1 . . . cm +� += g, +(5) +The coefficients {cj} can be understood as expressing g +through A. The extrapolation grade, γ, is then defined +as the largest component of {cj}, +γ(R) = max |cj| . +(6) +The same approach is used for the local-energy descrip- +tion, applying Eq. 5 with the rows of matrix G rather +than the vector g and solve for a matrix of cj,k and Eq. 6 +becomes γ(R) = max |cj,k|. +For γ(R) below a certain threshold γ0, the new con- +figuration is considered to overlap sufficiently with the +training set to allow the model to interpolate with confi- +dence. For γ0 < γ(R) < γmax, the model cannot be ap- +plied with confidence, but can be adapted by adding this + +3 +configuration to the training set. When γ(R) > γmax, +the configuration is too far from the training set and it +is rejected as the model cannot be adapted with confi- +dence. In this work, we set γ0 = 1.1 and γmax = 2.2, +unless specified otherwise. +C. +On-The-Fly Learning Cycle Workflow +Our workflow is similar to that of Ref.18, with main +differences discussed in Section II F. We follow the same +general machine-learning on-the-fly workflow for all sam- +pling approaches tested here. +We split each simulation in one or multiple sequences +of atomic configurations generated using either MD or +ARTn. Each run unrolls as follows (see fig. 1): +1. Launch a sequence during which configurations are +generated according to a sampling algorithm (MD +or ARTn). +At each iteration step the extrapolation-grade γ is +evaluated. +(a) If 0 < γ < γmax, the energy and forces of the +configuration are evaluated with MTP; +(b) if γ0 < γ < γmax, the configuration is set aside +for an update of MTP parameters; +(c) else if γ > γmax, energy and forces of the con- +figuration are not evaluated with MTP and +the configuration is not kept for update. The +sequence is stopped and we go directly to the +update step (step 3). +2. Move on next to the iteration in the sequence (step +1). +3. The model is updated, if at at least one configura- +tion as been set aside for an update of MPT (i) at +the end of a sequence or (ii) at any moment during +the sequence if γ > γmax. +4. If there is an update, restart a new sequence (go to +step 1), else stop if no configurations with γ > γ0 +have been set aside during the predefined maximum +length of the sequence. +The moment tensor potential model update is defined +as follows (see Fig. 1, right-hand side): +1. A selection is made from the set aside configura- +tions (with γ > γ0) using MaxVol23. +2. Each selected configuration is evaluated by the ref- +erence model +3. The training set is updated with the new evaluated +configurations +4. The moment tensor potential is fitted on the new +training set accordingly to Eq. 4 +More details of this procedure can be found in Ref.23. +Simulation +Configuration + < < +Evaluate + < < +it+1 +Evaluate +Configuration +Configuration +Set for MTP +Update + > +itmax +it=0 +Update MTP +No +Next Sequence +Yes +it=0 +Update MTP +Update MTP +Selection +Selected + New Configuration +Evaluated by +Reference Model +Update New TS +Retrain MTP +Figure 1. +On-the-fly machine learning workflow used with +MD and ARTn (on the left). +A potential update can take +place at two points: when the sequence ends or when γ > +γmax. The updating procedures are given in the box on the +right. +D. +MD and ARTn +Two sampling approaches are used to generate a +sequence of configurations: +(1) molecular dynamics +(MD) as implemented within LAMMPS26 and (2) the +activation-relaxation technique nouveau (ARTn) algo- +rithm developed by Mousseau and collaborators14,15,27. +Since MD is well known, we only give below a brief sum- +mary of ARTn. +ARTn is designed to explore the potential energy land- +scape of atomic systems through the identification of lo- +cal transition states connecting nearby local minima. Its +workflow can be summarized in three main steps (see, for +a recent in depth discussion of the ARTn version used in +this work, see Ref.27): +1. Leaving the harmonic well: starting from an energy +minimum, an atom and its neighbours are moved +iteratively in a direction selected at random un- +til a direction of negative curvature on the poten- +tial energy surfaces, d(λmin) with λmin, the lowest +eigenvalue of the Hessian matrix, smaller than zero, +emerges; this indicates the presence of a nearby +first-order saddle point; +2. Converging to a first-order saddle point: the system +is then pushed in the direction of negative curvature +d(λmin) while the force is minimized in the perpen- +dicular plane, until the total force F passes below a +threshold near F0, which indicates the saddle point +have been reached; +3. Relaxing into a new minimum: the system is then +pushed over the saddle point and relaxed into a +connected new minimum. + +4 +At each step λmin and d(λmin) are found using an it- +erative Lanczos method16,28,29. Perpendicular relaxation +during activation and global minimization are done using +the Fast Inertial Relaxation Engine (FIRE) algorithm30. +Generated events are accepted or rejected according to +the Metropolis algorithm, where the acceptation proba- +bility p is given by +p = min +� +1, e−β∆E� +(7) +with ∆E = Esaddle − Eminimum, the energy difference +between the saddle and a connected minima and β = +1/kBT where kB is the Boltzmann factor and T is a fic- +titious temperature, since thermal deformations are not +taken into account. +Potential energy landscape explo- +ration consist of generating a number of event. +E. +Systems studied +The fitting approaches are tested on two physical sys- +tems: (i) a Si diamond structure with Stillinger-Weber as +a reference potential19; and (ii) a SiGe zincblende struc- +ture using the Stillinger-Weber potential with parame- +ters from Ref.20. Both models count 215 atoms and a +vacancy. +The Si system is fitted with a ML potential set at level +16, with 92 moment tensor functions (B(R), Eq. +1). +For SiGe, a potential at this level (16) generates errors +on the barrier of the order of 0.5 eV, which indicates +that a richer set of parameters is needed to describe the +chemical diversity and a level 20 is chosen for this system, +with 288 moment tensor functions. The relation between +the number of moment tensor functions for Si and energy +error is presented in Supplemental Fig. 1. +F. +Fitting approaches +To evaluate the reliability of the various on-the-fly ap- +proaches to reproduce the reference potential on config- +urations of interest for complex materials, the training +set is limited to structures visited during MD or ARTn +simulations within the conditions described below. No +additional information regarding alternative crystalline +structures, defects, surfaces, pressure, etc. is provided. +For each of these two systems, we compare the follow- +ing approaches: +1. ML-MD: The MTP potential is train OTF on MD +simulations. The potential is then evaluated, with- +out further update, in ARTn simulation. +2. OTF-MDART: Starting from the ML-MD gener- +ated potential, the MTP is re-trained following the +OTF procedure during ARTn simulations. +3. OTF-ART: Training of the potential is done +uniquely during ARTn runs with OTF. +The ML-MD approach is in line with21 where a po- +tential is trained OTF during MD. However, while the +potential is trained with MD, its accuracy is evaluated +during ARTn activated process search. +1. +ML-MD: simulations details +Nine sets of MTP ML-MD potentials are developed +and trained independently during NVT MD simulations. +Each set is trained at one specific simulation temperature +ranging from 300 K to 2700 K by step of 300 K and +starting from the same 215 atom crystalline structure +with a vacancy. Each set consists of ten independently +constructed MTP potentials for statistical purpose. +Training takes place on a series of sequences, each run +for a maximum of 100 ps, with steps of 1 fs, with an +average of 75 ps per cycle. MTP potentials require about +34 ± 14 and 93 ± 43 learning cycles for Si and SiGe to +be converged: the MTP potential is considered having +learned the potential when no configuration generated +during a 100 ps second is found in the extrapolating zone +of the potential (with γ > γmax). +As long as this is not the case, the sequence is restarted +from the same initial structure with different initial ve- +locities. To facilitate convergence, ML-MD potentials are +fitted over three sets of progressively more restricted re- +liability extrapolation parameter γ0. Moreover because +MD leads to global deformation, the extrapolation is +computed using global descriptors (see tab. I). +The final potential is then evaluated, in a fixed form, +in ARTn simulations. +Table I. Extrapolation and selection hyper-parameter values +used for the three on-the-fly approaches used in this work. +approach: +γ0 +γmax +grade- +mode +ML-MD +5.5/3.3/1.1 60/10/2.2 global +OTF-MDART +1.1 +2.2 +local +OTF-ART +1.1 +2.2 +local +2. +OTF ARTn simulations details +Each ARTn simulation is launched for 1500 events, +with 24 parallel independent searches, for a total of +36 000 generated events. +For ARTn, a sequence is ei- +ther a search for a saddle point (successful or failed) or +a minimization from the saddle to minimum. +At each point, 24 sequences are generated in parallel, +and the configuration selected for an update of the po- +tential is made on the combined set of configurations to +generate one training set. Sequence are restarted from +the last accepted position or, in the case of the vacancy +in Si, the ground state. When an activation step gener- +ates a configuration with γ(R) > γmax, it is relaunched + +5 +with the same initial deformation. As with MD, ten in- +dependent ARTn runs are launched for statistics. +In the bulk, diffusion of the vacancy in Si takes place +through a symmetric mechanism bringing the vacancy +from one state to an identical one so all ARTn event +searches are effectively started from the same state. +Starting from a zincblende structure, SiGe evolves ac- +cording to an accept-reject Metropolis with a fictitious +temperature of 0.5 eV31. +Since the configurations ex- +plored by ARTn are locally deformed; the extrapolation +grade for ARTn generated configurations used for the +OTF-MDART and OTF-ART approaches are evaluated +with the local descriptors. +G. +Analysis +Following the standard approach, the error is com- +puted on the energy and force differences between the +MLP and reference potentials computed on the same +structures. Here, however, this error is only measured +on configurations generated during the ARTn procedure. +For the energy: +∆E = |EMLP (XMLP ) − Eref(XMLP )|, +(8) +For the forces: +∆F = 1 +N +N +� +i=0 +� +∥f (i) +MLP (XMLP ) − f (i) +ref(Xref)∥2, +(9) +where the positions XMLP are obtained from a simu- +lation run with the machine-learned potential and the +energy on this exact configuration is computed with the +reference and the machine-learned potentials. The same +is done for the error on forces. +Since this work is focused on the correct description of +first-order transition states, we also compute the mini- +mum and saddle barrier positions and energy convergence +errors(∆Xconv, ∆Econv) as +∆Xconv += +��N +i=0 ∥x(i) +MLP − x(i) +ref∥2, +(10) +∆Econv = |EMLP (XMLP ) − Eref(Xref)|, +(11) +where XMLP and Xref are the positions corresponding +to minimum or saddle point as defined by the MLP and +the reference potentials respectively, with EMLP (XMLP ) +and Eref(Xref) the corresponding energies; by definition, +forces are zero at these points defined by the respective +potentials. +While XMLP and EMLP (XMLP ) are obtained on the +ARTn trajectories, Xref and Eref(Xref) are obtained af- +ter reconverging the minima or the saddle point using the +reference potential starting from XMLP and following the +ARTn procedure. +From an energy barrier δE(X), the energy barrier error +∆δEbarrier is given by, +∆δEbarrier = |δEMLP (XMLP ) − δEref(Xref)| +(12) +If no trend is observed between the different temper- +atures where potentials are trained, we calculate their +average and deviation in order to to effectively compare +them with other approach. +III. +RESULTS +0 +500 +1000 +1500 +2000 +2500 +T(K) +200 +400 +600 +800 +1000 +1200 +1400 +Number of reference potential calls +253±60 +369±85 +1232±177 +505±109 +628±283 +ml-md +new: otf-mdart +total: otf-mdart+md +otf-art +Figure 2. +Number of calls to the reference potential for +each of the machine-learned potentials developed for Si as +function of the temperature referring to the one used during +MD training. Since configurations are relaxed to zero K in +ARTn simulations, there is no associated temperature for this +procedure. +In this section, we first examine results for a vacancy +in c-Si to establish the methods then consider the same +approaches on the more complex SiGe alloy. +A. +ML-MD +The ML-MD approach serves as a benchmark to assess +the efficiency of the various approaches in sampling en- +ergy barriers and diffusion mechanisms. Here, ten inde- +pendent ML potentials are generated through on-the-fly +MD simulations at 9 different target temperatures rang- +ing from 300 to 2700 K by step of 300 K and require +between 253 ± 60, at 300 K, and 369 ± 85 evaluations of +the reference potential, at 2700 K, to complete learning +cycles (see Fig. 2). +For the purpose of this work, the quality of the ML-MD +potential is evaluated on configurations generated with +ARTn as local activated events associated with vacancy +in a crystalline environment are generated. To avoid non- +physical results, when a ARTn-generated configuration +shows a γ > 200, the configuration is rejected, the event +search is stopped and a new event search is launched from +the same initial minimum. + +6 +0 +1 +2 +3 +4 +5 +6 +7 +Energy error per atom (meV/atom) +1500 +2000 +0.2 +0.3 +0.4 +0.5 +0.6 +ml-md +otf-mdart +otf-art +0 +500 +1000 +1500 +2000 +2500 +T(K) +0.0100 +0.0125 +0.0150 +0.0175 +0.0200 +0.0225 +0.0250 +0.0275 +0.0300 +Force error (eV/Å) +Figure 3. +Average energy (top) and mean absolute forces +(bottom) errors per atom for Si measured over all configu- +rations generated along pathways in ARTn for the three ap- +proaches. +Temperature refers to the one used during MD +training. +Fig. 3 shows the standard validation error on en- +ergy and forces calculated over all configurations gen- +erated along pathways for the 36 000 successful events +and 10 080 failed saddle searches (a success rate of +78 %). +The error on energy increases almost expo- +nentially with the sampling temperature, ranging from +0.44 ± 0.36 meV/atom at 300 K to 5.1 ± 1.7 meV/atom +at 2700K. The error on forces is essentially constant +at 0.0123 eV/Å, on average, between 300 and 1800 K, +and increases rapidly at high temperature, to reach +0.0256 eV/Å at 2700 K. +Since, the focus of this work is on transition states, +Fig. 4 displays the error on the energy barriers as a func- +tion of MD-fitting temperature, computed with Eq. 10 +and averaged over all generated barriers. This error is +relatively uncorrelated of the MD temperature simula- +tion with an average of 0.056 ± 0.022 eV, with minimum +error of 0.024 ± 0.01 eV at 2400 K and maximum of +0 +500 +1000 +1500 +2000 +2500 +T(K) +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Energy barrier error (eV) +ml-md +otf-mdart +otf-art +Figure 4. +Average energy barrier error for Si as defined +by Eq. 12 for all events generated in ARTn for the three ap- +proaches. +Temperature refers to the one used during MD +training. +0 +500 +1000 +1500 +2000 +2500 +T(K) +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Saddle position error (Å) +ml-md +otf-mdart +otf-art +Figure 5. +Mean position error on all saddle point for Si. +Temperature refers to the one used during MD training. +0.08±0.03 eV at 1200 K. This error is lower than that for +a general point on the energy landscape (Fig. 3) in part +because it is computed as a difference between saddle and +initial minimum. +Errors on the position of the saddle point, associated +with the capacity to reproduce correctly their geome- +try, are given in Fig. 5. +The top panel indicates the +average distance between saddle points converged with +the reference and the ML potentials: it decreases from +0.16 ± 0.05 Å at 300 K to a minimum of 0.09 ± 0.02 Å +between 1500 and 2100 K, going up at the two highest +temperatures (2400 and 2700 K). +Overall, this straightforward fitting approach based on +constant-temperature MD runs provides accurate diffu- +sion barriers, ranging from 0.51 to more than 4 eV, for + +7 +a vacancy in crystalline silicon at a low computational +costs (263 to 369 evaluations of the reference potential). +B. +Revisiting ML-MD potential in ARTn: the +OTF-MDART adjusting approach +To evaluate the possibility of improving on ML-MD +potentials for activated events, potentials are on-the-fly +re-trained during ARTn learning cycles (OTF-MDART). +Fig. 2 gives the number of calls to the reference poten- +tial for this procedure during the ARTn runs (dashed +orange line) as well as the total number of calls, includ- +ing those made during ML-MD fitting (solid orange line). +The number of calls during ARTn learning cycles ranges +from 979±153 at 300 K to to 136±38 at 2700 K for a to- +tal of 1232±177 to 505±109 respectively, when including +ML-MD calls. +The error on energy and forces remains correlated with +the ML-MD temperature: it is higher when the error is +higher at ML-MD trained temperature. This correlation +is particularly strong when retraining MD potentials fit- +ted between 1500 and 2700 K (Fig. 3, solid orange line). +Error on energy for OTF-MDART is almost constant +between 300 and 2400 K, at 0.22 meV/atom, rising to +1.9 meV/atom at 2700 K, lower by 50 to 63 % than ML- +MD. As similar improvement is observed on the forces, +which range from 0.0103 eV/Å, on average, between 300 +and 1800 K, increasing to 0.0173 eV/Å at 2700 K, repre- +senting a 16 % to 32 % decrease in error. +Table II. Average energy barrier error and mean position error +on all saddle point for Si. The average error for ML-MD and +OTF-MDART training is taken over all temperature sets. +Errors +ML-MD +OTF-MDART +OTF-ART +Energy (eV) +0.056±0.022 +0.039±0.008 +0.040±0.012 +Position (Å) +0.114±0.029 +0.072±0.006 +0.072±0.010 +Between 300 and 1500 K, retrained potentials with +OTF-MDART show more constant energy barrier errors +than pure ML-MD models (Fig. 4), with an error of about +0.036 eV (OTF-MDART) vs average of 0.064 eV (ML- +MD) a 44 % improvement. At the highest temperature — +1800 to 2700 K, however, as OTF-MDART calls for less +learning cycles, errors and fluctuations are not reduced +with respect to ML-MD. Interestingly, though, improve- +ments on the saddle position is observed at all tempera- +tures for OTF-MDART (Fig. 5) with an average error of +0.072 ± 0.010 Å. +Overall, by retraining ML-MD potential in ARTn, er- +rors are reduced and results are more consistent, i.e., er- +ror distributions are narrower, irrespective of the tem- +perature used in the initial MD training. This additional +retraining leads to a 50 % to 96 % decrease in energy +error (Fig. 3), a 29 % improvement for average energy +barrier errors (Tab. II) and a 37 % reduction on mean +saddle positions errors but with an additional number of +calls to the reference potential increasing between 37 to +490 %. +500 +1000 +1500 +2000 +2500 +T (K) +0 +10 +20 +30 +40 +50 +Proportion of MD configuration in training set (%) + +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +Kept MD configuration in training set (%) + +Figure 6. +Fraction of original MD configurations (left scale) +and total number of MD configurations (right scale) remain- +ing in the final training set (TS) for Si. Temperature refers +to the one used during MD training. +These results can be understood by looking at the frac- +tion of MD-generated configurations that remain in the +training set at the end of the simulation (Fig. 6): at +temperatures between 300 and 1200 K, none of the ML- +MD configurations remain in the final training set for +training temperatures between 300 and 1200 K; this pro- +portions goes from from 1.3 to 38 % between 1500 and +2700 K (left-hand axis, blue line). +At these tempera- +tures, the system melts and generates a wider range of +configurations. Since these configurations are far from +ARTn-generated configurations, the selection algorithm +keeps them in the set even though they do not help re- +duce errors for the configurational space of interest with +ARTn. +C. +The OTF-ART adjusting approach +Given the results for OTF-MDART, we now turn to +an OTF approach entirely integrated in ARTn, in an at- +tempt to increase accuracy, and reduce the cost and waste +of evaluations of the reference potential. +Ten independents on-the-fly ML potential are gener- +ated entirely in ARTn for a total of 36 000 events start- +ing from the same initial minimum. +Each potential is +trained initially from the same one configuration (the ini- +tial minimum), in the training set. Each parallel event +search goes trow a learning cycle if needed and as the +simulation progresses learning cycle become rarer. The +values are averaged over the ten simulation and as the +simulation go through learning. +With an average total of 628 ± 283 reference po- +tential evaluations, the cost of the OTF-ART is be- + +8 +tween that of ML-MD and OTF-MDART. Along path- +ways, the average energy error for these potentials is of +0.22 ± 0.03 meV/atom, on par with OTF-MDART po- +tential based on low-temperature ML-MD fitting, and +49 % lower than the 300 K ML-MD potential. Errors +on forces, at 0.011±0.001 eV/Å, are in between ML- +MD (0.012 eV/Å) and OTF-MDART (0.010 eV/Å) at +low training temperature. Comparing with the 2700 K +potential fitting in MD, OFT-ART error is 57 % lower +than ML-MD (0.026 eV/Å) and 36 % lower than OTF- +MDART (0.017 eV/Å). +Focusing on barrier energy, the average error is 0.039± +0.008 eV (see Fig. 4), about 2.5 % lower than OTF- +MDART and 30.3 % better than ML-MD. The error of +0.072 ± 0.006 Å on the converged saddle position is sim- +ilar to the 0.072 ± 0.010 Å obtained with OTF-MDART +and 37 % lower than with ML-MD (0.114 Å). +D. +Reproducing the dominant diffusion mechanism +0 +1 +2 +3 +4 +5 +Energy barrier (eV) +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +Probability +MTP +Refined +Figure 7. +ARTn-generated energy barrier distributions for +vacancy-diffusion events in Si, including direct barrier (from +ground state) and inverse barriers (from excited states), as +generated with the MTP model (orange) and re-converged +using the reference model (blue) from saddles and minima +position originally found with the MTP model. +The exploration of the energy landscape around the +vacancy leads to a generation of wide range of activated +mechanisms and associated barriers (both forward, as- +sociated with the diffusion of the vacancy, and back- +ward, from the final minima back to the saddle point). +Fig. 7 presents the complete distribution of generated di- +rect and inverse barriers connected to the ground state. +The peak near 0 eV (around 10−2 to 10−1 eV) is associ- +ated with the inverse barrier to to direct saddle at 2.38, +2.70 eV and higher (up to 5.5 eV), except for the in- +verse 0.45 eV barrier which is linked to the 2.87 eV direct +barrier. Direct barriers at 0.51 eV represent symmetric +first neighbor vacancy diffusion while barriers at 2.38 and +2.70 eV are associated with more complex vacancy diffu- +sion mechanism32. Events with barriers at 2.38, 2.70 eV, +for example, involve a vacancy diffusion through com- +plex bond-exchanges. Spectator events33 where the dia- +mond network around the vacancy is transformed by a +bond switching are also generated. This mechanism was +proposed by Wooten, Winer, and Weaire (WWW) to de- +scribe the amorphization of silicon34. The main spectator +event occurs as two neighbors of the vacancy are pushed +together allowing the creation of a bound associated with +the 2.87 eV barrier. Other mechanisms involve strong lat- +tice distortion and bond formation not involving direct +neighbors of the vacancy with very high energy barriers32 +of in between 3.2 and 4.0 eV. +Table III. +Average energy barrier errors and mean saddle +position error on the 0.51 eV vacancy diffusion for Si. The +average error for ML-MD and OTF-MDART training is taken +over all temperature sets. +Errors +ML-MD +OTF-MDART +OTF-ART +Energy (eV) +0.026±0.015 +0.022±0.011 +0.019±0.005 +Position (Å) +0.088±0.036 +0.040±0.017 +0.047±0.018 +Since vacancy diffusion for this system is dominated +by a 0.51 eV single barrier mechanism, with the next +barrier at 2.35 eV, an accurate description of the dom- +inant mechanism is essential to correctly capture defect +kinetics in Si. Tab. III presents the error on this barrier +for the three approaches described above. With an error +of 0.019±0.005 eV, a relative error of 3.7 %, OTF-ART +offers the closest reproduction of the reference barrier, +followed by OTF-MDART and ML-MD, with a respec- +tive error of 0.022±0.011 (relative error of 4.3 %) and +0.026±0.015 (5.1 %). Overall, the error on energy bar- +rier is lower than that on the total energy presented above +(0.046±0.006 eV for OTF-ART, for example), due to a +partial error cancellation associated with energy differ- +ence taken to measure the barrier. +The validity of the barrier is also measured by the +precision on the saddle geometry. +For the e 0.51 eV +barrier, ML-MD converges with an error on the posi- +tion of 0.088±0.036Å, with OTF-MDART and OTF- +ART giving error almost 50 % lower, at 0.040±0.017Å +and 0.047±0.018Å respectively. +E. +SiGe system +Having shown the interest of developing a specific po- +tential by applying on-the-fly learning directly to acti- +vated events on a simple system such as c-Si with a va- +cancy, we test this approach with a more complex al- +loy with the same overall reference potential to facilitate +comparison. Starting from a ordered zincblende struc- +ture, the diffusion of a vacancy creates chemical disorder +that complexifies the landscape visited as shown by the + +9 +0 +1 +2 +3 +4 +5 +Energy barrier (eV) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Probability +MTP +Refined +Figure 8. +SiGe barrier histogram, including direct bar- +rier (from ground state) and inverse barriers (from excited +states), as found on-the-fly by the MTP model(orange) and +re-converge by the reference model(blue) from saddles and +minimums position originally given by MTP. +0 +500 +1000 +1500 +2000 +2500 +T(K) +1000 +2000 +3000 +4000 +Number of reference potential calls +380±125 +1549±705 +3465±844 +3329±265 +ml-md +new: otf-mdart +total: otf-mdart+md +mean: otf-mdart+md +otf-art +Figure 9. +Number of calls to the reference potential for each +of the OTF machine-learned potentials developed for SiGe +as a function of the temperature referring to the one used +during MD training. Since configurations are relaxed to zero +K in ARTn simulations, there is no associated temperature +for this procedure. +continuous distribution of activated barriers, including +both direct and inverse barriers, found as the vacancy +diffuses (Fig. 8); we note that the lowest barrier for a +vacancy diffusing is around 0.6 eV, with lower barriers +associated, as for Si, with reverse jumps from metastable +states. The energy barrier distribution for a vacancy dif- +fusing in SiGe (Fig. 8) is much more complex than for Si +due to the chemical disorder that builds as the vacancy +diffuses. +As stated in the methodology, the additional complex- +ity of the system imposes a richer machined-learning po- +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Energy error per atom (meV/atom) +ml-md +otf-mdart +otf-art +0 +500 +1000 +1500 +2000 +2500 +T(K) +0.015 +0.020 +0.025 +0.030 +Force error (eV/Å) +Figure 10. +Average energy (top) and mean absolute forces +(bottom) errors for SiGe measured over all configurations gen- +erated along pathways in ARTn for the three approaches. +Temperature refers to the one used during MD training. +Table IV. Average energy barrier errors and mean saddle po- +sition error on all barriers for SiGe. +The average error for +ML-MD and OTF-MDART training is taken over all temper- +ature sets. +Errors +ML-MD +OTF-MDART +OTF-ART +Energy (eV) +0.082±0.024 +0.072±0.014 +0.066±0.015 +Position (Å) +0.091±0.020 +0.076±0.013 +0.070±0.014 +tential, with a larger set of parameters to encompass the +greater diversity in the components and the configura- +tions, due to chemical disorder. +Combined, these two +levels of complexity (set of parameters and configura- +tional) result in an overall higher numbers of calls to the +reference potential as compared to Si, irrespective of the +approach used (see Fig. 9 (SiGe) vs. Fig. 2(Si)): while +ML-MD requires between 380 evaluations of the reference +potential at 300 K and 1549 at 2700 K, OTF-MDART +needs a total of around 3465 calculations of the reference + +10 +potential, irrespective of the temperature as original ML- +MD configurations are progressively removed from the +training set. +This efforts results in a number of calls +to the reference potential for OTF-MDART 4 % higher +than with OTF-ART (3329 on average). To reduce com- +putational costs, we omit the 1500 K run, as statistical +behavior is smooth in this temperature region. +500 +1000 +1500 +2000 +2500 +T(K) +0 +20 +40 +60 +80 +Percent interruption ( > 200) per event search (%) +Si +SiGe +Figure 11. +Percentage of search interruptions during ML- +MD potential evaluation in ARTn (γ > 200) for Si and SiGe +as function of ML-MD training temperature. +To disentangle the two contributions, we compare with +the cost of fitting a Si potential with the same level 20 +potential as used for SiGe. Following the full OTF-ART +procedure, creating a Si MLP requires 2926 calls to the +reference potential. The intrinsic complexity of the land- +scape contributes therefore to about a 14 % increase of +the Si baseline calls count. +In terms of accuracy, the +Si MLP level 20 leads to an average error on energy of +0.1 meV/atom, about 50 % lower than with the level 16 +potential described above (0.22 meV/atom). For SiGe, +this error is (0.42 meV/atom), two times higher than for +Si MLP level 16 and four times that of Si MLP level 20. +This can be understood by the number of different con- +figurations visited: as opposed to the Si system where +each initial minimum is identical (as the vacancy moves +in an otherwise perfect elemental crystal), the binary sys- +tem is transformed as the vacancy diffuses, as the chemi- +cal order is slowly destroyed: each of the 24 ARTn parallel +trajectories used to define the potential over 1500 events +evolves independently according to a probability given +by the Metropolis algorithm with a fictitious tempera- +ture (since network itself is structurally at 0K) of 0.5 eV +(Eq. 7), providing a rich range of local environments. +Fitting a potential is clearly harder: with the param- +eters used — when a configuration graded at γ > 200 is +encountered, the ARTn event search is stopped —, not +significantly enough event could be generated using the +ML-MD potential at 300 K and 600 K, which explains the +absence of data for this temperatures in Fig. 10 and IV. +For SiGe, the error on energy (see Fig. 10) with the ML- +MD at 900 K and above ranges from 0.5 meV/atom to +1.4 meV/atom, as a function of temperature. On aver- +age, these errors are between 14 % and 69 % lower with +OTF-MDART or OTF-ART at around 0.43 meV/atom. +The OTF-ART approach gives an error in energy bar- +rier of 0.066 ± 0.015 eV which represent a 19.5 % and +8.3 % lower error from the ML-MD (0.082±0.024 eV) and +OTF-MDART (0.072 ± 0.014 eV) respectively (Tab. IV). +The errors on the converged saddle position for OTF- +ART and OTF-MDART are similar at 0.070 ± 0.014 Å +and 0.076 ± 0.013 Å, respectively, and represent a 23 % +lower error than with ML-MD (0.092 Å). This accuracy +is similar to that obtained with Si, in contrast to total +energy and energy barrier errors. +We note that the advantage of ML-MD for SiGe is +overstated as shown by the proportion of events gener- +ated with ML-MD potential that are interrupted due to +a too large extrapolation grade, γ > 200 for both SiGe +and Si (Fig. 11): for SiGe between 85 % and 30 % of +events are aborted between 300 K and 1200 K, respec- +tively. +This proportion falls to zero percent failure at +1800 K. +IV. +DISCUSSION +We compare three approaches aimed at the construc- +tion of potentials with machine learning on-the-fly for +the exploration of activated mechanism of the potential +energy landscape. We evaluate these by computing their +efficiency at reproducing the energy landscape around a +vacancy in two systems, a relatively simple Si diamond +system (Fig. 7) and a more complex SiGe zincblende sys- +tem that disorders under vacancy diffusion (Fig. 8). +The first approach, which sets the comparison level, +constructs a more general machine learning potential +with molecular dynamics (ML-MD), the second on-the- +fly adjusts this this generated potential, during the search +for activated events using ARTn, while the third ap- +proaches constructs a specifically on-the-fly trained a po- +tential during search of activated events (OTF-ART). +The efficiency of these three procedures is measured on +the quality of the reproduction of the reference potential +during the search for activated event. +The baseline, defined by the ML-MD, is competitive +with previously published work. Energy errors for the +more standard ML MD approach with a level 16 po- +tential range from 0.44 ± 0.36 meV/atom at 300 K to +5.1 ± 1.7 meV/atom at 2700 K (Fig. 3), an order of mag- +nitude lower or similar than the 4 meV/atom on an MTP +potential of level 24 for Si obtained by Zuo et al.22, with +the difference explained by the fact that activated events +involve local deformations from a zero-temperature crys- +tal with a vacancy and that DFT potentials are more +difficult to fit than empirical ones18. +Similarly, the relative energy error on the dominant +0.51 eV diffusion barrier for SW Si is of 5.1 % (0.026 eV) + +11 +with the ML-MD approach and 3.7 % (0.019 eV) with +the OTF-ART. Using the same MTP potential trained +using an OTF MD with an ab initio reference potential, +Novoselov et al. find a 0.20 eV barrier for vacancy dif- +fusion in Si as compared with 0.18 eV with the reference +potential, an error of 0.02 eV or a 10.0 % relative error. +Overall, the ML-MD approach, especially when run at +temperatures between 900 and 1800 K, can generate a +generic ML potential with reasonable precision for de- +scribing activated mechanisms in Si and SiGe. +Devel- +oping a more specific OTF potential, generated directly +with ARTn on activated trajectories, however, offers a +more accurate description of both the energy and geom- +etry at the barriers. +It is possible to recover this precision by adjusting the +original MD potential during ARTn runs, however, this +increases the number of calls to the reference potential, +raising the total costs beyond that of OTF-ART while +largely erasing work made during ML-MD training phase: +for Si, between 300 and 1200 K, none of the ML-MD +configurations are retained while around 1.3 to 12.5 % +are retained for the potential trained in range of 1500 +and 2700 K (Fig. 6, right-hand axis, orange line), but at +the cost of lowering the precision on barriers. +Moving to a more complex system, such a binary al- +loy, increases the overall cost of the procedure in terms of +calls to the reference potential, as more parameters need +to be fit. Here also, the gain on using a specific poten- +tial constructed with from ARTn trajectories is notable, +both in the average errors and their fluctuations. Indeed, +the ML-MD potential presents considerable instabilities +when generated activated trajectories as can be seen by +the number of configurations considered out-side of the +potential’s scope (γ > 200), see Fig. 11. +CONCLUSION +We compare the advantage of using a more general vs +specific machine-learned potential (MLP) to describe ac- +tivated mechanisms in solid. To do so, we generate first +an MLP constructed with the Moment Tensor Potential +formalism10,18 to replicate Stillinger-Weber potential for +Si and SiGe crystals with a single vacancy using a stan- +dard molecular dynamics procedure (MD-ML). +Comparing the quality of the reproduction of activated +mechanisms with a ML potential further refined during +an activation-relaxation technique nouveau sampling of +the energy landscape and a potential unique constructed +on-the-fly within ARTn, we show that while a general +potential can deliver high accuracy for both the barrier +geometries and their related energies, error and fluctua- +tions around the average value are significantly lowered +by constructing a specific potential, with a number of +calls to the reference potential that is lower than a com- +bined approach (MD + ARTn) for a similar precision. +The advantage of using a specific potential remains +when looking at more complex materials, such the SiGe +alloys considered here, even though the advantage in +terms of calls to the reference is strongly reduced. +Having demonstrated that a specific machine-learned +potential developed with methods such as MTP and +ARTn can reproduce with high precision the activated +mechanisms at the origin of kinetic for complex mate- +rials, the next steps will involve applying this strategy +to attack problems that have long been out of reach of +computational materials sciences, allowing a much closer +connection between modeling and experience. +V. +CODE AND DATA AVAILABILITY +The ARTn packages as well as the data reported here +are distributed freely. Please contact Normand Mousseau +(normand.mousseau@umontreal.ca). +ACKNOWLEDGMENTS +This project is supported through a Discovery grant +from the Natural Science and Engineering Research +Council of Canada (NSERC). Karl-Étienne Bolduc is +grateful to NSERC and IVADO for summer scholarchips. +We are grateful to Calcul Québec and Compute Canada +for generous allocation of computational resources. +1 W. Kohn and L. J. Sham. +Self-consistent equations in- +cluding exchange and correlation effects. +Phys. Rev., +140:A1133–A1138, Nov 1965. +2 Erik R Lindahl. Molecular dynamics simulations. In Molec- +ular modeling of proteins, pages 3–23. Springer, 2008. +3 Arthur F Voter and Jimmie D Doll. Transition state theory +description of surface self-diffusion: Comparison with clas- +sical trajectory results. The Journal of chemical physics, +80(11):5832–5838, 1984. +4 Arthur F Voter. Introduction to the kinetic monte carlo +method. +In Radiation effects in solids, pages 1–23. +Springer, 2007. +5 Graeme Henkelman and Hannes Jónsson. Long time scale +kinetic monte carlo simulations without lattice approxima- +tion and predefined event table. The Journal of Chemical +Physics, 115(21):9657–9666, 2001. +6 Fedwa El-Mellouhi, Normand Mousseau, and Laurent J. +Lewis. +Kinetic activation-relaxation technique: +An off- +lattice self-learning kinetic Monte Carlo algorithm. Physi- +cal Review B, 78(15):153202, October 2008. +7 Jörg Behler and Michele Parrinello. Generalized Neural- +Network Representation of High-Dimensional Potential- +Energy Surfaces. +Phys. Rev. Lett., 98(14):146401, April +2007. + +12 +8 Albert P Bartók, Risi Kondor, and Gábor Csányi. +On +representing chemical environments. Physical Review B, +87(18):184115, 2013. +9 Aidan P Thompson, Laura P Swiler, Christian R Trott, +Stephen M Foiles, and Garritt J Tucker. Spectral neigh- +bor analysis method for automated generation of quantum- +accurate interatomic potentials. Journal of Computational +Physics, 285:316–330, 2015. +10 Alexander V Shapeev. Moment tensor potentials: A class +of systematically improvable interatomic potentials. Mul- +tiscale Modeling & Simulation, 14(3):1153–1173, 2016. +11 Ganesh Sivaraman, Jicheng Guo, Logan Ward, Nathaniel +Hoyt, Mark Williamson, Ian Foster, Chris Benmore, and +Nicholas Jackson. Automated development of molten salt +machine learning potentials: application to licl. The Jour- +nal of Physical Chemistry Letters, 12(17):4278–4285, 2021. +12 Pei-Lin Kang, Cheng Shang, and Zhi-Pan Liu. +Large- +scale atomic simulation via machine learning potentials +constructed by global potential energy surface exploration. +Accounts of Chemical Research, 53(10):2119–2129, 2020. +13 Ganesh Sivaraman, Anand Narayanan Krishnamoorthy, +Matthias Baur, Christian Holm, Marius Stan, Gábor +Csányi, Chris Benmore, and Álvaro Vázquez-Mayagoitia. +Machine-learned interatomic potentials by active learning: +amorphous and liquid hafnium dioxide. npj Computational +Materials, 6(1):1–8, 2020. +14 G. T. Barkema and Normand Mousseau. Event-Based Re- +laxation of Continuous Disordered Systems. Physical Re- +view Letters, 77(21):4358–4361, November 1996. +15 Rachid +Malek +and +Normand +Mousseau. +Dynamics +of Lennard-Jones clusters: +A characterization of the +activation-relaxation +technique. +Physical +Review +E, +62(6):7723–7728, December 2000. +16 Antoine Jay, Miha Gunde, Nicolas Salles, Matic Poberžnik, +Layla Martin-Samos, Nicolas Richard, Stefano de Giron- +coli, Normand Mousseau, and Anne Hémeryck. +Activa- +tion–Relaxation Technique: An efficient way to find min- +ima and saddle points of potential energy surfaces. Com- +putational Materials Science, 209:111363, June 2022. +17 Donald G Truhlar, Bruce C Garrett, and Stephen J Klip- +penstein. Current status of transition-state theory. The +Journal of physical chemistry, 100(31):12771–12800, 1996. +18 Ivan +S +Novikov, +Konstantin +Gubaev, +Evgeny +V +Podryabinkin, and Alexander V Shapeev. +The mlip +package: moment tensor potentials with mpi and active +learning. +Machine Learning: +Science and Technology, +2(2):025002, 2020. +19 Frank H Stillinger and Thomas A Weber. Computer simu- +lation of local order in condensed phases of silicon. Physical +review B, 31(8):5262, 1985. +20 Stéphane Ethier and Laurent J Lewis. Epitaxial growth of +si1- xgex on si (100) 2× 1: A molecular-dynamics study. +Journal of materials research, 7(10):2817–2827, 1992. +21 II +Novoselov, +AV +Yanilkin, +AV +Shapeev, +and +EV +Podryabinkin. +Moment +tensor +potentials +as +a +promising tool to study diffusion processes. +Computa- +tional Materials Science, 164:46–56, 2019. +22 Yunxing Zuo, Chi Chen, Xiangguo Li, Zhi Deng, Yiming +Chen, Jorg Behler, Gábor Csányi, Alexander V Shapeev, +Aidan P Thompson, Mitchell A Wood, et al. Performance +and cost assessment of machine learning interatomic po- +tentials. The Journal of Physical Chemistry A, 124(4):731– +745, 2020. +23 Evgeny V Podryabinkin and Alexander V Shapeev. Active +learning of linearly parametrized interatomic potentials. +Computational Materials Science, 140:171–180, 2017. +24 Evgeny V Podryabinkin, Evgeny V Tikhonov, Alexander V +Shapeev, and Artem R Oganov. Accelerating crystal struc- +ture prediction by machine-learning interatomic potentials +with active learning. +Physical Review B, 99(6):064114, +2019. +25 Konstantin Gubaev, Evgeny V Podryabinkin, Gus LW +Hart, and Alexander V Shapeev. +Accelerating high- +throughput searches for new alloys with active learning of +interatomic potentials. Computational Materials Science, +156:148–156, 2019. +26 A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolin- +tineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld, +A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, +M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton. +LAMMPS - a flexible simulation tool for particle-based +materials modeling at the atomic, meso, and continuum +scales. Comp. Phys. Comm., 271:108171, 2022. +27 Antoine +Jay, +Miha +Gunde, +Nicolas +Salles, +Matic +Poberžnik, Layla Martin-Samos, Nicolas Richard, Stefano +de Gironcoli, Normand Mousseau, and Anne Hémeryck. +Activation–relaxation technique: An efficient way to find +minima and saddle points of potential energy surfaces. +Computational Materials Science, 209:111363, 2022. +28 Cornelius Lanczos. An iteration method for the solution +of the eigenvalue problem of linear differential and integral +operators. 1950. +29 Rachid +Malek +and +Normand +Mousseau. +Dynamics +of lennard-jones clusters: +A characterization of the +activation-relaxation +technique. +Physical +Review +E, +62(6):7723, 2000. +30 Erik Bitzek, Pekka Koskinen, Franz Gähler, Michael +Moseler, and Peter Gumbsch. Structural relaxation made +simple. Physical review letters, 97(17):170201, 2006. +31 Normand Mousseau and Gerard T. Barkema. Exploring +High-Dimensional Energy Landscapes. Computing in Sci- +ence & Engineering, 1(2):74–82, March 1999. +32 Fedwa El-Mellouhi, Normand Mousseau, and Pablo Orde- +jón. Sampling the diffusion paths of a neutral vacancy in +silicon with quantum mechanical calculations. Phys. Rev. +B, 70:205202, Nov 2004. +33 Yuko Kumeda, David J Wales, and Lindsey J Munro. +Transition states and rearrangement mechanisms from hy- +brid eigenvector-following and density functional theory.: +application to c10h10 and defect migration in crystalline +silicon. Chemical physics letters, 341(1-2):185–194, 2001. +34 F. Wooten, K. Winer, and D. Weaire. Computer genera- +tion of structural models of amorphous si and ge. Phys. +Rev. Lett., 54:1392–1395, Apr 1985. + diff --git a/6tFAT4oBgHgl3EQfnx0W/content/tmp_files/load_file.txt b/6tFAT4oBgHgl3EQfnx0W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f90b5f89ee8c300bd12163e7024de631b1ff7ff0 --- /dev/null +++ b/6tFAT4oBgHgl3EQfnx0W/content/tmp_files/load_file.txt @@ -0,0 +1,813 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf,len=812 +page_content='Evaluating approaches for on-the-fly machine learning interatomic potential for activated mechanisms sampling with the activation-relaxation technique nouveau Eugène Sanscartier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 Félix Saint-Denis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 Karl-Étienne Bolduc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 and Normand Mousseau1 1Département de physique and Regroupement québécois sur les matériaux de pointe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Université de Montréal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Case Postale 6128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Succursale Centre-ville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Montréal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Québec H3C 3J7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Canada (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2023) In the last few years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' much efforts have gone into developing universal machine-learning potentials able to describe interactions for a wide range of structures and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Yet, as attention turns to more complex materials including alloys, disordered and heterogeneous systems, the challenge of providing reliable description for all possible environment become ever more costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In this work, we evaluate the benefits of using specific versus general potentials for the study of activated mechanisms in solid-state materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' More specifically, we tests three machine-learning fitting approaches using the moment-tensor potential to reproduce a reference potential when exploring the energy landscape around a vacancy in Stillinger-Weber silicon crystal and silicon-germanium zincblende structure using the activation-relaxation technique nouveau (ARTn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' We find that a a targeted on-the-fly approach specific and integrated to ARTn generates the highest precision on the energetic and geometry of activated barriers, while remaining cost-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This approach expands the type of problems that can be addressed with high-accuracy ML potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' INTRODUCTION As computational materials scientists turn to atten- tion to ever more complex systems, they are faced with two major challenges : (i) how to describe correctly their physics and (ii) how to reach the appropriate size and time scale to capture the properties of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The first challenge is generally solved by turning to ab ini- tio methods,1 that allow the solution Heisenberg’s equa- tion with reasonably controlled approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Theses approaches, however, suffer from N 4 scaling which lim- its their application to small system sizes and short time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The second challenge is met by a variety of meth- ods that cover different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Molecular dynamics2, for example, which directly solves Newton’s equation, ac- cesses typical time scales between picoseconds and mi- croseconds, at the very best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Other approaches, such as lattice3,4 and off-lattices kinetic Monte-Carlo5,6, by focusing on physically relevant mechanisms, can extend this time scale to seconds and more, as long the diffusion takes place through activated processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Even though these methods are efficient, each trajectory can require hundreds of thousands to millions of forces evaluations, which becomes too costly with ab initio approaches, forc- ing modellers to use empirical potentials in spite of their incapacity at describing correctly complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Building on ab initio energy and forces, machine- learned potentials7–10 open the door to lifting some of this difficulties, by offering much more reliable physics as a small fraction of the cost of ab initio evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since their introduction, ML potentials have been largely coupled with MD and focusing on the search for universal potentials, able to describe a full range of struc- tures and phases for a given material11–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' As we turn to more complex systems such as alloys and disordered and heterogeneous systems, it becomes more and more difficult to generate such universal potentials, since the number of possible environments grows rapidly with this complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In this context, the development of specific potentials, with on-the-fly learning that makes it possible to adapt to new environments, becomes a strategy worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In this work, we focus on the construction of machine- learned potentials adapted to the sampling of energy landscape dominated by activated mechanisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=', solid-state systems with local activated diffusion and evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' A correct computational sampling, using methods such as the activation-relaxation technique (ART)14 and its revised version (ART nouveau or ARTn)15,16, requires a precise description of local minima and of the land- scape surrounding the first-order saddle points that char- acterize diffusion according to the transition-state theory (TST)17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' These barriers can be high — reaching many electron-volts — and involve strained configurations that can be visited only very rarely with standard molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' More specifically, we compare three machine learning procedures in which we change the context where lean- ing on-the-fly occur to train a Moment Tensor Poten- tial (MTP)10,18 that describes the diffusion of vacancy in Stillinger-Weber silicon19 and silicon-germanium20 as sampled with ARTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The first one uses a pure MD learn- ing procedure, fitted at various temperatures, in a proce- dure that echoes the work of Novoselov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='21, a second- one adds an on-the-fly adjustment during an ARTn run and the third one focuses on purely OTF-ARTn potential adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Results underline the efficiency gain in developing tar- geted ML potentials for specific applications, comparing the cost of fitting Si with SiGe, it also shows the rapid increase in computation complexity associated with mov- ing from element to alloy systems, which emphasizes the usefulness of a specific approach such as the one applied here to activated processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='08630v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='mtrl-sci] 20 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ML Potential The Moment Tensor Potential (MTP)10,18 is a linear model of functions Bα(ri) built from contractions of mo- ment tensor descriptors defined by the local neighbor- hood relative position ri of atom i within a sphere of influence of radius rc respecting a set invariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This model has been shown to be fast while giving accuracy on the order of ∼meV/atom and requiring few hundreds to thousands of reference potential calls22 on-the-fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' MTP have been used on a wide variety of problems in- cluding on-the-fly MD simulation18,21,23, search and min- imization of new alloys24,25 and diffusion processes21 on systems counting one or multiple species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' MTP approximates atomic configuration energy as sum of local contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' A local contribution is ob- tained through a sum over the included basis {Bα(ri)} as a linear combination of B(ri) and ξα, V (ri) = m � α=1 ξαBα(ri) (1) The “level” of a potential gives the number of different possible tensor Mµ,ν (ri) descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The {Bα(ri)} func- tions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1 are constructed by a tensorial contraction of different Mµ,ν (ri) and the number of different tenso- rial contraction sets m in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' More information on MTP is available in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='The total energy of a N-atom configuration (R) is then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='given by the sum of N local contributions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='E(R) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='V (ri) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='α=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='ξαBα(ri) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='and the forces are obtained by taking the gradient of this ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='quantity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='F(R) = −∇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='α=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='ξαBα(ri) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='The parameters ξα are obtained by minimizing the loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='function: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='R∈A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='we ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='���E(R) − ˆE(R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 + wf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='���fi(R) −ˆfi(R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='→ min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='Here A is the training set made of configurations with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='known energy and forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The goal is to minimize the difference between E(R), fi(R)(real value) and ˆE(R), ˆfi(R)(predicted by model), respectively, for all element in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Weights on contribution from energy and forces (we and wf) are set to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Learning On-The-Fly Tools On-the-fly atomic machine learning potential (OTF) involves the repeated training of the model potential as new atomic environments are generated through various procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Following the work of Shapeev and collaborators18, the reliability of the potential to describe a given configura- tion is evaluated using the D-optimality criterion to grade to which extend a configuration extrapolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This grade is used along with a selection algorithm (MaxVol) to as- sess whether the new configuration should be added to the training set or replace a configuration already in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' While a detailed description can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='23, we provide here a brief summary of the retained approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The selection and extrapolation-grade algorithm can be applied using either a local-energy or a global-energy descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The local-energy descriptor is presented as a rectangu- lar matrix Gm×N formed by the basis elements {Bα(ri)} associated with the neighborhood ri of all N atoms: G = � � � B1(r1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Bm(r1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' B1(rN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Bm(rN) � � � T For a given configuration, the global-energy description reduces this information to a vector g g = � b1(R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' bm(R) � where each term, {bα(R)} is a sum over all neighborhoods for a specific basis element {Bα(ri)}: {bα(R)} = N � i=0 {Bα(ri)} For the global-energy descriptor, evaluating the over- lap of a new configuration with the training set A is done by solving for cj, in A � c1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' cm � = g, (5) The coefficients {cj} can be understood as expressing g through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The extrapolation grade, γ, is then defined as the largest component of {cj}, γ(R) = max |cj| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' (6) The same approach is used for the local-energy descrip- tion, applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 5 with the rows of matrix G rather than the vector g and solve for a matrix of cj,k and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 6 becomes γ(R) = max |cj,k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For γ(R) below a certain threshold γ0, the new con- figuration is considered to overlap sufficiently with the training set to allow the model to interpolate with confi- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For γ0 < γ(R) < γmax, the model cannot be ap- plied with confidence, but can be adapted by adding this 3 configuration to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' When γ(R) > γmax, the configuration is too far from the training set and it is rejected as the model cannot be adapted with confi- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In this work, we set γ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 and γmax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2, unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' On-The-Fly Learning Cycle Workflow Our workflow is similar to that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='18, with main differences discussed in Section II F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' We follow the same general machine-learning on-the-fly workflow for all sam- pling approaches tested here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' We split each simulation in one or multiple sequences of atomic configurations generated using either MD or ARTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Each run unrolls as follows (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Launch a sequence during which configurations are generated according to a sampling algorithm (MD or ARTn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' At each iteration step the extrapolation-grade γ is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' (a) If 0 < γ < γmax, the energy and forces of the configuration are evaluated with MTP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' (b) if γ0 < γ < γmax, the configuration is set aside for an update of MTP parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' (c) else if γ > γmax, energy and forces of the con- figuration are not evaluated with MTP and the configuration is not kept for update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The sequence is stopped and we go directly to the update step (step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Move on next to the iteration in the sequence (step 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The model is updated, if at at least one configura- tion as been set aside for an update of MPT (i) at the end of a sequence or (ii) at any moment during the sequence if γ > γmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' If there is an update, restart a new sequence (go to step 1), else stop if no configurations with γ > γ0 have been set aside during the predefined maximum length of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The moment tensor potential model update is defined as follows (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1, right-hand side): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' A selection is made from the set aside configura- tions (with γ > γ0) using MaxVol23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Each selected configuration is evaluated by the ref- erence model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The training set is updated with the new evaluated configurations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The moment tensor potential is fitted on the new training set accordingly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 4 More details of this procedure can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Simulation Configuration < < Evaluate < < it+1 Evaluate Configuration Configuration Set for MTP Update > itmax it=0 Update MTP No Next Sequence Yes it=0 Update MTP Update MTP Selection Selected New Configuration Evaluated by Reference Model Update New TS Retrain MTP Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' On-the-fly machine learning workflow used with MD and ARTn (on the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' A potential update can take place at two points: when the sequence ends or when γ > γmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The updating procedures are given in the box on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' MD and ARTn Two sampling approaches are used to generate a sequence of configurations: (1) molecular dynamics (MD) as implemented within LAMMPS26 and (2) the activation-relaxation technique nouveau (ARTn) algo- rithm developed by Mousseau and collaborators14,15,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since MD is well known, we only give below a brief sum- mary of ARTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ARTn is designed to explore the potential energy land- scape of atomic systems through the identification of lo- cal transition states connecting nearby local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Its workflow can be summarized in three main steps (see, for a recent in depth discussion of the ARTn version used in this work, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='27): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Leaving the harmonic well: starting from an energy minimum, an atom and its neighbours are moved iteratively in a direction selected at random un- til a direction of negative curvature on the poten- tial energy surfaces, d(λmin) with λmin, the lowest eigenvalue of the Hessian matrix, smaller than zero, emerges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' this indicates the presence of a nearby first-order saddle point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Converging to a first-order saddle point: the system is then pushed in the direction of negative curvature d(λmin) while the force is minimized in the perpen- dicular plane, until the total force F passes below a threshold near F0, which indicates the saddle point have been reached;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Relaxing into a new minimum: the system is then pushed over the saddle point and relaxed into a connected new minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 4 At each step λmin and d(λmin) are found using an it- erative Lanczos method16,28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Perpendicular relaxation during activation and global minimization are done using the Fast Inertial Relaxation Engine (FIRE) algorithm30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Generated events are accepted or rejected according to the Metropolis algorithm, where the acceptation proba- bility p is given by p = min � 1, e−β∆E� (7) with ∆E = Esaddle − Eminimum, the energy difference between the saddle and a connected minima and β = 1/kBT where kB is the Boltzmann factor and T is a fic- titious temperature, since thermal deformations are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Potential energy landscape explo- ration consist of generating a number of event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Systems studied The fitting approaches are tested on two physical sys- tems: (i) a Si diamond structure with Stillinger-Weber as a reference potential19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' and (ii) a SiGe zincblende struc- ture using the Stillinger-Weber potential with parame- ters from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Both models count 215 atoms and a vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The Si system is fitted with a ML potential set at level 16, with 92 moment tensor functions (B(R), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For SiGe, a potential at this level (16) generates errors on the barrier of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 eV, which indicates that a richer set of parameters is needed to describe the chemical diversity and a level 20 is chosen for this system, with 288 moment tensor functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The relation between the number of moment tensor functions for Si and energy error is presented in Supplemental Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Fitting approaches To evaluate the reliability of the various on-the-fly ap- proaches to reproduce the reference potential on config- urations of interest for complex materials, the training set is limited to structures visited during MD or ARTn simulations within the conditions described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' No additional information regarding alternative crystalline structures, defects, surfaces, pressure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For each of these two systems, we compare the follow- ing approaches: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ML-MD: The MTP potential is train OTF on MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The potential is then evaluated, with- out further update, in ARTn simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' OTF-MDART: Starting from the ML-MD gener- ated potential, the MTP is re-trained following the OTF procedure during ARTn simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' OTF-ART: Training of the potential is done uniquely during ARTn runs with OTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The ML-MD approach is in line with21 where a po- tential is trained OTF during MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' However, while the potential is trained with MD, its accuracy is evaluated during ARTn activated process search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ML-MD: simulations details Nine sets of MTP ML-MD potentials are developed and trained independently during NVT MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Each set is trained at one specific simulation temperature ranging from 300 K to 2700 K by step of 300 K and starting from the same 215 atom crystalline structure with a vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Each set consists of ten independently constructed MTP potentials for statistical purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Training takes place on a series of sequences, each run for a maximum of 100 ps, with steps of 1 fs, with an average of 75 ps per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' MTP potentials require about 34 ± 14 and 93 ± 43 learning cycles for Si and SiGe to be converged: the MTP potential is considered having learned the potential when no configuration generated during a 100 ps second is found in the extrapolating zone of the potential (with γ > γmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' As long as this is not the case, the sequence is restarted from the same initial structure with different initial ve- locities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' To facilitate convergence, ML-MD potentials are fitted over three sets of progressively more restricted re- liability extrapolation parameter γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Moreover because MD leads to global deformation, the extrapolation is computed using global descriptors (see tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The final potential is then evaluated, in a fixed form, in ARTn simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Extrapolation and selection hyper-parameter values used for the three on-the-fly approaches used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' approach: γ0 γmax grade- mode ML-MD 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='3/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 60/10/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 global OTF-MDART 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 local OTF-ART 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 local 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' OTF ARTn simulations details Each ARTn simulation is launched for 1500 events, with 24 parallel independent searches, for a total of 36 000 generated events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For ARTn, a sequence is ei- ther a search for a saddle point (successful or failed) or a minimization from the saddle to minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' At each point, 24 sequences are generated in parallel, and the configuration selected for an update of the po- tential is made on the combined set of configurations to generate one training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Sequence are restarted from the last accepted position or, in the case of the vacancy in Si, the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' When an activation step gener- ates a configuration with γ(R) > γmax, it is relaunched 5 with the same initial deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' As with MD, ten in- dependent ARTn runs are launched for statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In the bulk, diffusion of the vacancy in Si takes place through a symmetric mechanism bringing the vacancy from one state to an identical one so all ARTn event searches are effectively started from the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Starting from a zincblende structure, SiGe evolves ac- cording to an accept-reject Metropolis with a fictitious temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 eV31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since the configurations ex- plored by ARTn are locally deformed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' the extrapolation grade for ARTn generated configurations used for the OTF-MDART and OTF-ART approaches are evaluated with the local descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Analysis Following the standard approach, the error is com- puted on the energy and force differences between the MLP and reference potentials computed on the same structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Here, however, this error is only measured on configurations generated during the ARTn procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For the energy: ∆E = |EMLP (XMLP ) − Eref(XMLP )|, (8) For the forces: ∆F = 1 N N � i=0 � ∥f (i) MLP (XMLP ) − f (i) ref(Xref)∥2, (9) where the positions XMLP are obtained from a simu- lation run with the machine-learned potential and the energy on this exact configuration is computed with the reference and the machine-learned potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The same is done for the error on forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since this work is focused on the correct description of first-order transition states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' we also compute the mini- mum and saddle barrier positions and energy convergence errors(∆Xconv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ∆Econv) as ∆Xconv = ��N i=0 ∥x(i) MLP − x(i) ref∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' (10) ∆Econv = |EMLP (XMLP ) − Eref(Xref)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' (11) where XMLP and Xref are the positions corresponding to minimum or saddle point as defined by the MLP and the reference potentials respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' with EMLP (XMLP ) and Eref(Xref) the corresponding energies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' by definition, forces are zero at these points defined by the respective potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' While XMLP and EMLP (XMLP ) are obtained on the ARTn trajectories, Xref and Eref(Xref) are obtained af- ter reconverging the minima or the saddle point using the reference potential starting from XMLP and following the ARTn procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' From an energy barrier δE(X), the energy barrier error ∆δEbarrier is given by, ∆δEbarrier = |δEMLP (XMLP ) − δEref(Xref)| (12) If no trend is observed between the different temper- atures where potentials are trained, we calculate their average and deviation in order to to effectively compare them with other approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' RESULTS 0 500 1000 1500 2000 2500 T(K) 200 400 600 800 1000 1200 1400 Number of reference potential calls 253±60 369±85 1232±177 505±109 628±283 ml-md new: otf-mdart total: otf-mdart+md otf-art Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Number of calls to the reference potential for each of the machine-learned potentials developed for Si as function of the temperature referring to the one used during MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since configurations are relaxed to zero K in ARTn simulations, there is no associated temperature for this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In this section, we first examine results for a vacancy in c-Si to establish the methods then consider the same approaches on the more complex SiGe alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ML-MD The ML-MD approach serves as a benchmark to assess the efficiency of the various approaches in sampling en- ergy barriers and diffusion mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Here, ten inde- pendent ML potentials are generated through on-the-fly MD simulations at 9 different target temperatures rang- ing from 300 to 2700 K by step of 300 K and require between 253 ± 60, at 300 K, and 369 ± 85 evaluations of the reference potential, at 2700 K, to complete learning cycles (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For the purpose of this work, the quality of the ML-MD potential is evaluated on configurations generated with ARTn as local activated events associated with vacancy in a crystalline environment are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' To avoid non- physical results, when a ARTn-generated configuration shows a γ > 200, the configuration is rejected, the event search is stopped and a new event search is launched from the same initial minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 6 0 1 2 3 4 5 6 7 Energy error per atom (meV/atom) 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='6 ml-md otf-mdart otf-art 0 500 1000 1500 2000 2500 T(K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0300 Force error (eV/Å) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Average energy (top) and mean absolute forces (bottom) errors per atom for Si measured over all configu- rations generated along pathways in ARTn for the three ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Temperature refers to the one used during MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3 shows the standard validation error on en- ergy and forces calculated over all configurations gen- erated along pathways for the 36 000 successful events and 10 080 failed saddle searches (a success rate of 78 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The error on energy increases almost expo- nentially with the sampling temperature, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='36 meV/atom at 300 K to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='7 meV/atom at 2700K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The error on forces is essentially constant at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0123 eV/Å, on average, between 300 and 1800 K, and increases rapidly at high temperature, to reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0256 eV/Å at 2700 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since, the focus of this work is on transition states, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 4 displays the error on the energy barriers as a func- tion of MD-fitting temperature, computed with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 10 and averaged over all generated barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This error is relatively uncorrelated of the MD temperature simula- tion with an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='056 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='022 eV, with minimum error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='024 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='01 eV at 2400 K and maximum of 0 500 1000 1500 2000 2500 T(K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='12 Energy barrier error (eV) ml-md otf-mdart otf-art Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Average energy barrier error for Si as defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 12 for all events generated in ARTn for the three ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Temperature refers to the one used during MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 T(K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='200 Saddle position error (Å) ml-md otf-mdart otf-art Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Mean position error on all saddle point for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Temperature refers to the one used during MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='03 eV at 1200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This error is lower than that for a general point on the energy landscape (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3) in part because it is computed as a difference between saddle and initial minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Errors on the position of the saddle point, associated with the capacity to reproduce correctly their geome- try, are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The top panel indicates the average distance between saddle points converged with the reference and the ML potentials: it decreases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='05 Å at 300 K to a minimum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='02 Å between 1500 and 2100 K, going up at the two highest temperatures (2400 and 2700 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Overall, this straightforward fitting approach based on constant-temperature MD runs provides accurate diffu- sion barriers, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='51 to more than 4 eV, for 7 a vacancy in crystalline silicon at a low computational costs (263 to 369 evaluations of the reference potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Revisiting ML-MD potential in ARTn: the OTF-MDART adjusting approach To evaluate the possibility of improving on ML-MD potentials for activated events, potentials are on-the-fly re-trained during ARTn learning cycles (OTF-MDART).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2 gives the number of calls to the reference poten- tial for this procedure during the ARTn runs (dashed orange line) as well as the total number of calls, includ- ing those made during ML-MD fitting (solid orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The number of calls during ARTn learning cycles ranges from 979±153 at 300 K to to 136±38 at 2700 K for a to- tal of 1232±177 to 505±109 respectively, when including ML-MD calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The error on energy and forces remains correlated with the ML-MD temperature: it is higher when the error is higher at ML-MD trained temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This correlation is particularly strong when retraining MD potentials fit- ted between 1500 and 2700 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3, solid orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Error on energy for OTF-MDART is almost constant between 300 and 2400 K, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='22 meV/atom, rising to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='9 meV/atom at 2700 K, lower by 50 to 63 % than ML- MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' As similar improvement is observed on the forces, which range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0103 eV/Å, on average, between 300 and 1800 K, increasing to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0173 eV/Å at 2700 K, repre- senting a 16 % to 32 % decrease in error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Average energy barrier error and mean position error on all saddle point for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The average error for ML-MD and OTF-MDART training is taken over all temperature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Errors ML-MD OTF-MDART OTF-ART Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='056±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='039±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='040±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='012 Position (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='114±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='072±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='072±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='010 Between 300 and 1500 K, retrained potentials with OTF-MDART show more constant energy barrier errors than pure ML-MD models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 4), with an error of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='036 eV (OTF-MDART) vs average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='064 eV (ML- MD) a 44 % improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' At the highest temperature — 1800 to 2700 K, however, as OTF-MDART calls for less learning cycles, errors and fluctuations are not reduced with respect to ML-MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Interestingly, though, improve- ments on the saddle position is observed at all tempera- tures for OTF-MDART (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 5) with an average error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='072 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='010 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Overall, by retraining ML-MD potential in ARTn, er- rors are reduced and results are more consistent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=', er- ror distributions are narrower, irrespective of the tem- perature used in the initial MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This additional retraining leads to a 50 % to 96 % decrease in energy error (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3), a 29 % improvement for average energy barrier errors (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' II) and a 37 % reduction on mean saddle positions errors but with an additional number of calls to the reference potential increasing between 37 to 490 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 500 1000 1500 2000 2500 T (K) 0 10 20 30 40 50 Proportion of MD configuration in training set (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 Kept MD configuration in training set (%) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Fraction of original MD configurations (left scale) and total number of MD configurations (right scale) remain- ing in the final training set (TS) for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Temperature refers to the one used during MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' These results can be understood by looking at the frac- tion of MD-generated configurations that remain in the training set at the end of the simulation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 6): at temperatures between 300 and 1200 K, none of the ML- MD configurations remain in the final training set for training temperatures between 300 and 1200 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' this pro- portions goes from from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='3 to 38 % between 1500 and 2700 K (left-hand axis, blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' At these tempera- tures, the system melts and generates a wider range of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since these configurations are far from ARTn-generated configurations, the selection algorithm keeps them in the set even though they do not help re- duce errors for the configurational space of interest with ARTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The OTF-ART adjusting approach Given the results for OTF-MDART, we now turn to an OTF approach entirely integrated in ARTn, in an at- tempt to increase accuracy, and reduce the cost and waste of evaluations of the reference potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Ten independents on-the-fly ML potential are gener- ated entirely in ARTn for a total of 36 000 events start- ing from the same initial minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Each potential is trained initially from the same one configuration (the ini- tial minimum), in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Each parallel event search goes trow a learning cycle if needed and as the simulation progresses learning cycle become rarer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The values are averaged over the ten simulation and as the simulation go through learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' With an average total of 628 ± 283 reference po- tential evaluations, the cost of the OTF-ART is be- 8 tween that of ML-MD and OTF-MDART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Along path- ways, the average energy error for these potentials is of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='03 meV/atom, on par with OTF-MDART po- tential based on low-temperature ML-MD fitting, and 49 % lower than the 300 K ML-MD potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Errors on forces, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='011±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='001 eV/Å, are in between ML- MD (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='012 eV/Å) and OTF-MDART (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='010 eV/Å) at low training temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Comparing with the 2700 K potential fitting in MD, OFT-ART error is 57 % lower than ML-MD (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='026 eV/Å) and 36 % lower than OTF- MDART (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='017 eV/Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Focusing on barrier energy, the average error is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='039± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='008 eV (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 4), about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 % lower than OTF- MDART and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='3 % better than ML-MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='072 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='006 Å on the converged saddle position is sim- ilar to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='072 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='010 Å obtained with OTF-MDART and 37 % lower than with ML-MD (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='114 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Reproducing the dominant diffusion mechanism 0 1 2 3 4 5 Energy barrier (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 Probability MTP Refined Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ARTn-generated energy barrier distributions for vacancy-diffusion events in Si, including direct barrier (from ground state) and inverse barriers (from excited states), as generated with the MTP model (orange) and re-converged using the reference model (blue) from saddles and minima position originally found with the MTP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The exploration of the energy landscape around the vacancy leads to a generation of wide range of activated mechanisms and associated barriers (both forward, as- sociated with the diffusion of the vacancy, and back- ward, from the final minima back to the saddle point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 7 presents the complete distribution of generated di- rect and inverse barriers connected to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The peak near 0 eV (around 10−2 to 10−1 eV) is associ- ated with the inverse barrier to to direct saddle at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='38, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='70 eV and higher (up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 eV), except for the in- verse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='45 eV barrier which is linked to the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='87 eV direct barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Direct barriers at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='51 eV represent symmetric first neighbor vacancy diffusion while barriers at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='38 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='70 eV are associated with more complex vacancy diffu- sion mechanism32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Events with barriers at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='38, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='70 eV, for example, involve a vacancy diffusion through com- plex bond-exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Spectator events33 where the dia- mond network around the vacancy is transformed by a bond switching are also generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This mechanism was proposed by Wooten, Winer, and Weaire (WWW) to de- scribe the amorphization of silicon34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The main spectator event occurs as two neighbors of the vacancy are pushed together allowing the creation of a bound associated with the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='87 eV barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Other mechanisms involve strong lat- tice distortion and bond formation not involving direct neighbors of the vacancy with very high energy barriers32 of in between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Average energy barrier errors and mean saddle position error on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='51 eV vacancy diffusion for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The average error for ML-MD and OTF-MDART training is taken over all temperature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Errors ML-MD OTF-MDART OTF-ART Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='026±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='022±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='019±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='005 Position (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='088±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='040±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='047±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='018 Since vacancy diffusion for this system is dominated by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='51 eV single barrier mechanism, with the next barrier at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='35 eV, an accurate description of the dom- inant mechanism is essential to correctly capture defect kinetics in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' III presents the error on this barrier for the three approaches described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' With an error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='019±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='005 eV, a relative error of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='7 %, OTF-ART offers the closest reproduction of the reference barrier, followed by OTF-MDART and ML-MD, with a respec- tive error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='022±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='011 (relative error of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='3 %) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='026±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='015 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Overall, the error on energy bar- rier is lower than that on the total energy presented above (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='046±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='006 eV for OTF-ART, for example), due to a partial error cancellation associated with energy differ- ence taken to measure the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The validity of the barrier is also measured by the precision on the saddle geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For the e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='51 eV barrier, ML-MD converges with an error on the posi- tion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='088±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='036Å, with OTF-MDART and OTF- ART giving error almost 50 % lower, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='040±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='017Å and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='047±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='018Å respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' SiGe system Having shown the interest of developing a specific po- tential by applying on-the-fly learning directly to acti- vated events on a simple system such as c-Si with a va- cancy, we test this approach with a more complex al- loy with the same overall reference potential to facilitate comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Starting from a ordered zincblende struc- ture, the diffusion of a vacancy creates chemical disorder that complexifies the landscape visited as shown by the 9 0 1 2 3 4 5 Energy barrier (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 Probability MTP Refined Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' SiGe barrier histogram, including direct bar- rier (from ground state) and inverse barriers (from excited states), as found on-the-fly by the MTP model(orange) and re-converge by the reference model(blue) from saddles and minimums position originally given by MTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 T(K) 1000 2000 3000 4000 Number of reference potential calls 380±125 1549±705 3465±844 3329±265 ml-md new: otf-mdart total: otf-mdart+md mean: otf-mdart+md otf-art Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Number of calls to the reference potential for each of the OTF machine-learned potentials developed for SiGe as a function of the temperature referring to the one used during MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Since configurations are relaxed to zero K in ARTn simulations, there is no associated temperature for this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' continuous distribution of activated barriers, including both direct and inverse barriers, found as the vacancy diffuses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' we note that the lowest barrier for a vacancy diffusing is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='6 eV, with lower barriers associated, as for Si, with reverse jumps from metastable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The energy barrier distribution for a vacancy dif- fusing in SiGe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 8) is much more complex than for Si due to the chemical disorder that builds as the vacancy diffuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' As stated in the methodology, the additional complex- ity of the system imposes a richer machined-learning po- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='75 Energy error per atom (meV/atom) ml-md otf-mdart otf-art 0 500 1000 1500 2000 2500 T(K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='030 Force error (eV/Å) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Average energy (top) and mean absolute forces (bottom) errors for SiGe measured over all configurations gen- erated along pathways in ARTn for the three approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Temperature refers to the one used during MD training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Average energy barrier errors and mean saddle po- sition error on all barriers for SiGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The average error for ML-MD and OTF-MDART training is taken over all temper- ature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Errors ML-MD OTF-MDART OTF-ART Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='082±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='072±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='066±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='015 Position (Å) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='091±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='076±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='070±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='014 tential, with a larger set of parameters to encompass the greater diversity in the components and the configura- tions, due to chemical disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Combined, these two levels of complexity (set of parameters and configura- tional) result in an overall higher numbers of calls to the reference potential as compared to Si, irrespective of the approach used (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 9 (SiGe) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2(Si)): while ML-MD requires between 380 evaluations of the reference potential at 300 K and 1549 at 2700 K, OTF-MDART needs a total of around 3465 calculations of the reference 10 potential, irrespective of the temperature as original ML- MD configurations are progressively removed from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This efforts results in a number of calls to the reference potential for OTF-MDART 4 % higher than with OTF-ART (3329 on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' To reduce com- putational costs, we omit the 1500 K run, as statistical behavior is smooth in this temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 500 1000 1500 2000 2500 T(K) 0 20 40 60 80 Percent interruption ( > 200) per event search (%) Si SiGe Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Percentage of search interruptions during ML- MD potential evaluation in ARTn (γ > 200) for Si and SiGe as function of ML-MD training temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' To disentangle the two contributions, we compare with the cost of fitting a Si potential with the same level 20 potential as used for SiGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Following the full OTF-ART procedure, creating a Si MLP requires 2926 calls to the reference potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The intrinsic complexity of the land- scape contributes therefore to about a 14 % increase of the Si baseline calls count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In terms of accuracy, the Si MLP level 20 leads to an average error on energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 meV/atom, about 50 % lower than with the level 16 potential described above (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='22 meV/atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For SiGe, this error is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='42 meV/atom), two times higher than for Si MLP level 16 and four times that of Si MLP level 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This can be understood by the number of different con- figurations visited: as opposed to the Si system where each initial minimum is identical (as the vacancy moves in an otherwise perfect elemental crystal),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' the binary sys- tem is transformed as the vacancy diffuses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' as the chemi- cal order is slowly destroyed: each of the 24 ARTn parallel trajectories used to define the potential over 1500 events evolves independently according to a probability given by the Metropolis algorithm with a fictitious tempera- ture (since network itself is structurally at 0K) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 eV (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 7), providing a rich range of local environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Fitting a potential is clearly harder: with the param- eters used — when a configuration graded at γ > 200 is encountered, the ARTn event search is stopped —, not significantly enough event could be generated using the ML-MD potential at 300 K and 600 K, which explains the absence of data for this temperatures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 10 and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' For SiGe, the error on energy (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 10) with the ML- MD at 900 K and above ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 meV/atom to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='4 meV/atom, as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' On aver- age, these errors are between 14 % and 69 % lower with OTF-MDART or OTF-ART at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='43 meV/atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The OTF-ART approach gives an error in energy bar- rier of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='066 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='015 eV which represent a 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 % and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='3 % lower error from the ML-MD (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='082±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='024 eV) and OTF-MDART (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='072 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='014 eV) respectively (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The errors on the converged saddle position for OTF- ART and OTF-MDART are similar at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='070 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='014 Å and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='013 Å, respectively, and represent a 23 % lower error than with ML-MD (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='092 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This accuracy is similar to that obtained with Si, in contrast to total energy and energy barrier errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' We note that the advantage of ML-MD for SiGe is overstated as shown by the proportion of events gener- ated with ML-MD potential that are interrupted due to a too large extrapolation grade, γ > 200 for both SiGe and Si (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 11): for SiGe between 85 % and 30 % of events are aborted between 300 K and 1200 K, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' This proportion falls to zero percent failure at 1800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' DISCUSSION We compare three approaches aimed at the construc- tion of potentials with machine learning on-the-fly for the exploration of activated mechanism of the potential energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' We evaluate these by computing their efficiency at reproducing the energy landscape around a vacancy in two systems, a relatively simple Si diamond system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 7) and a more complex SiGe zincblende sys- tem that disorders under vacancy diffusion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The first approach, which sets the comparison level, constructs a more general machine learning potential with molecular dynamics (ML-MD), the second on-the- fly adjusts this this generated potential, during the search for activated events using ARTn, while the third ap- proaches constructs a specifically on-the-fly trained a po- tential during search of activated events (OTF-ART).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The efficiency of these three procedures is measured on the quality of the reproduction of the reference potential during the search for activated event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The baseline, defined by the ML-MD, is competitive with previously published work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Energy errors for the more standard ML MD approach with a level 16 po- tential range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='36 meV/atom at 300 K to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='7 meV/atom at 2700 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3), an order of mag- nitude lower or similar than the 4 meV/atom on an MTP potential of level 24 for Si obtained by Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='22, with the difference explained by the fact that activated events involve local deformations from a zero-temperature crys- tal with a vacancy and that DFT potentials are more difficult to fit than empirical ones18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Similarly, the relative energy error on the dominant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='51 eV diffusion barrier for SW Si is of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='1 % (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='026 eV) 11 with the ML-MD approach and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='7 % (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='019 eV) with the OTF-ART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Using the same MTP potential trained using an OTF MD with an ab initio reference potential, Novoselov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' find a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='20 eV barrier for vacancy dif- fusion in Si as compared with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='18 eV with the reference potential, an error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='02 eV or a 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='0 % relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Overall, the ML-MD approach, especially when run at temperatures between 900 and 1800 K, can generate a generic ML potential with reasonable precision for de- scribing activated mechanisms in Si and SiGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Devel- oping a more specific OTF potential, generated directly with ARTn on activated trajectories, however, offers a more accurate description of both the energy and geom- etry at the barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' It is possible to recover this precision by adjusting the original MD potential during ARTn runs, however, this increases the number of calls to the reference potential, raising the total costs beyond that of OTF-ART while largely erasing work made during ML-MD training phase: for Si, between 300 and 1200 K, none of the ML-MD configurations are retained while around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='3 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='5 % are retained for the potential trained in range of 1500 and 2700 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 6, right-hand axis, orange line), but at the cost of lowering the precision on barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Moving to a more complex system, such a binary al- loy, increases the overall cost of the procedure in terms of calls to the reference potential, as more parameters need to be fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Here also, the gain on using a specific poten- tial constructed with from ARTn trajectories is notable, both in the average errors and their fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Indeed, the ML-MD potential presents considerable instabilities when generated activated trajectories as can be seen by the number of configurations considered out-side of the potential’s scope (γ > 200), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' CONCLUSION We compare the advantage of using a more general vs specific machine-learned potential (MLP) to describe ac- tivated mechanisms in solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' To do so, we generate first an MLP constructed with the Moment Tensor Potential formalism10,18 to replicate Stillinger-Weber potential for Si and SiGe crystals with a single vacancy using a stan- dard molecular dynamics procedure (MD-ML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Comparing the quality of the reproduction of activated mechanisms with a ML potential further refined during an activation-relaxation technique nouveau sampling of the energy landscape and a potential unique constructed on-the-fly within ARTn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' we show that while a general potential can deliver high accuracy for both the barrier geometries and their related energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' error and fluctua- tions around the average value are significantly lowered by constructing a specific potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' with a number of calls to the reference potential that is lower than a com- bined approach (MD + ARTn) for a similar precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The advantage of using a specific potential remains when looking at more complex materials, such the SiGe alloys considered here, even though the advantage in terms of calls to the reference is strongly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Having demonstrated that a specific machine-learned potential developed with methods such as MTP and ARTn can reproduce with high precision the activated mechanisms at the origin of kinetic for complex mate- rials, the next steps will involve applying this strategy to attack problems that have long been out of reach of computational materials sciences, allowing a much closer connection between modeling and experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' CODE AND DATA AVAILABILITY The ARTn packages as well as the data reported here are distributed freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Please contact Normand Mousseau (normand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='mousseau@umontreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' ACKNOWLEDGMENTS This project is supported through a Discovery grant from the Natural Science and Engineering Research Council of Canada (NSERC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Karl-Étienne Bolduc is grateful to NSERC and IVADO for summer scholarchips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' We are grateful to Calcul Québec and Compute Canada for generous allocation of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Kohn and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Sham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Self-consistent equations in- cluding exchange and correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=', 140:A1133–A1138, Nov 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 2 Erik R Lindahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In Molec- ular modeling of proteins, pages 3–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 3 Arthur F Voter and Jimmie D Doll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Transition state theory description of surface self-diffusion: Comparison with clas- sical trajectory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The Journal of chemical physics, 80(11):5832–5838, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 4 Arthur F Voter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Introduction to the kinetic monte carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' In Radiation effects in solids, pages 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Springer, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 5 Graeme Henkelman and Hannes Jónsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Long time scale kinetic monte carlo simulations without lattice approxima- tion and predefined event table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The Journal of Chemical Physics, 115(21):9657–9666, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 6 Fedwa El-Mellouhi, Normand Mousseau, and Laurent J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Lewis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Kinetic activation-relaxation technique: An off- lattice self-learning kinetic Monte Carlo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physi- cal Review B, 78(15):153202, October 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 7 Jörg Behler and Michele Parrinello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Generalized Neural- Network Representation of High-Dimensional Potential- Energy Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=', 98(14):146401, April 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 12 8 Albert P Bartók, Risi Kondor, and Gábor Csányi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' On representing chemical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physical Review B, 87(18):184115, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 9 Aidan P Thompson, Laura P Swiler, Christian R Trott, Stephen M Foiles, and Garritt J Tucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Spectral neigh- bor analysis method for automated generation of quantum- accurate interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Journal of Computational Physics, 285:316–330, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 10 Alexander V Shapeev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Moment tensor potentials: A class of systematically improvable interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Mul- tiscale Modeling & Simulation, 14(3):1153–1173, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 11 Ganesh Sivaraman, Jicheng Guo, Logan Ward, Nathaniel Hoyt, Mark Williamson, Ian Foster, Chris Benmore, and Nicholas Jackson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Automated development of molten salt machine learning potentials: application to licl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The Jour- nal of Physical Chemistry Letters, 12(17):4278–4285, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 12 Pei-Lin Kang, Cheng Shang, and Zhi-Pan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Large- scale atomic simulation via machine learning potentials constructed by global potential energy surface exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Accounts of Chemical Research, 53(10):2119–2129, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 13 Ganesh Sivaraman, Anand Narayanan Krishnamoorthy, Matthias Baur, Christian Holm, Marius Stan, Gábor Csányi, Chris Benmore, and Álvaro Vázquez-Mayagoitia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Machine-learned interatomic potentials by active learning: amorphous and liquid hafnium dioxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' npj Computational Materials, 6(1):1–8, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 14 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Barkema and Normand Mousseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Event-Based Re- laxation of Continuous Disordered Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physical Re- view Letters, 77(21):4358–4361, November 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 15 Rachid Malek and Normand Mousseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Dynamics of Lennard-Jones clusters: A characterization of the activation-relaxation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physical Review E, 62(6):7723–7728, December 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 16 Antoine Jay, Miha Gunde, Nicolas Salles, Matic Poberžnik, Layla Martin-Samos, Nicolas Richard, Stefano de Giron- coli, Normand Mousseau, and Anne Hémeryck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Activa- tion–Relaxation Technique: An efficient way to find min- ima and saddle points of potential energy surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Com- putational Materials Science, 209:111363, June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 17 Donald G Truhlar, Bruce C Garrett, and Stephen J Klip- penstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Current status of transition-state theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The Journal of physical chemistry, 100(31):12771–12800, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 18 Ivan S Novikov, Konstantin Gubaev, Evgeny V Podryabinkin, and Alexander V Shapeev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The mlip package: moment tensor potentials with mpi and active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Machine Learning: Science and Technology, 2(2):025002, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 19 Frank H Stillinger and Thomas A Weber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Computer simu- lation of local order in condensed phases of silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physical review B, 31(8):5262, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 20 Stéphane Ethier and Laurent J Lewis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Epitaxial growth of si1- xgex on si (100) 2× 1: A molecular-dynamics study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Journal of materials research, 7(10):2817–2827, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 21 II Novoselov, AV Yanilkin, AV Shapeev, and EV Podryabinkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Moment tensor potentials as a promising tool to study diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Computa- tional Materials Science, 164:46–56, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 22 Yunxing Zuo, Chi Chen, Xiangguo Li, Zhi Deng, Yiming Chen, Jorg Behler, Gábor Csányi, Alexander V Shapeev, Aidan P Thompson, Mitchell A Wood, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Performance and cost assessment of machine learning interatomic po- tentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' The Journal of Physical Chemistry A, 124(4):731– 745, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 23 Evgeny V Podryabinkin and Alexander V Shapeev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Active learning of linearly parametrized interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Computational Materials Science, 140:171–180, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 24 Evgeny V Podryabinkin, Evgeny V Tikhonov, Alexander V Shapeev, and Artem R Oganov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Accelerating crystal struc- ture prediction by machine-learning interatomic potentials with active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physical Review B, 99(6):064114, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 25 Konstantin Gubaev, Evgeny V Podryabinkin, Gus LW Hart, and Alexander V Shapeev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Accelerating high- throughput searches for new alloys with active learning of interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Computational Materials Science, 156:148–156, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Thompson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Aktulga, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Berger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Bolin- tineanu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Brown, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Crozier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' in ’t Veld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Kohlmeyer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Moore, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Nguyen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Shan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Stevens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Tranchida, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Trott, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Plimpton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=', 271:108171, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 27 Antoine Jay, Miha Gunde, Nicolas Salles, Matic Poberžnik, Layla Martin-Samos, Nicolas Richard, Stefano de Gironcoli, Normand Mousseau, and Anne Hémeryck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Activation–relaxation technique: An efficient way to find minima and saddle points of potential energy surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Computational Materials Science, 209:111363, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 28 Cornelius Lanczos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' An iteration method for the solution of the eigenvalue problem of linear differential and integral operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 29 Rachid Malek and Normand Mousseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Dynamics of lennard-jones clusters: A characterization of the activation-relaxation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physical Review E, 62(6):7723, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 30 Erik Bitzek, Pekka Koskinen, Franz Gähler, Michael Moseler, and Peter Gumbsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Structural relaxation made simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Physical review letters, 97(17):170201, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 31 Normand Mousseau and Gerard T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Barkema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Exploring High-Dimensional Energy Landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Computing in Sci- ence & Engineering, 1(2):74–82, March 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 32 Fedwa El-Mellouhi, Normand Mousseau, and Pablo Orde- jón.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Sampling the diffusion paths of a neutral vacancy in silicon with quantum mechanical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' B, 70:205202, Nov 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 33 Yuko Kumeda, David J Wales, and Lindsey J Munro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Transition states and rearrangement mechanisms from hy- brid eigenvector-following and density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' : application to c10h10 and defect migration in crystalline silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Chemical physics letters, 341(1-2):185–194, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' 34 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Wooten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Winer, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Weaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Computer genera- tion of structural models of amorphous si and ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} +page_content=', 54:1392–1395, Apr 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tFAT4oBgHgl3EQfnx0W/content/2301.08630v1.pdf'} diff --git a/8NAzT4oBgHgl3EQfgfyv/vector_store/index.pkl b/8NAzT4oBgHgl3EQfgfyv/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..12b55d7ab7ad1e86cbbd96732ddd8a0e51a738d0 --- /dev/null +++ b/8NAzT4oBgHgl3EQfgfyv/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0590c851e63d05837b8558354c9ecaf46ba41cfc875d209bae9adf119dcf934a +size 72483 diff --git a/8NFLT4oBgHgl3EQfsy_c/content/tmp_files/load_file.txt b/8NFLT4oBgHgl3EQfsy_c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6694ee53eff2b83f1806c58b905a9f800a2557d --- /dev/null +++ b/8NFLT4oBgHgl3EQfsy_c/content/tmp_files/load_file.txt @@ -0,0 +1,1020 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf,len=1019 +page_content='POSTER V2: A simpler and stronger facial expression recognition network Jiawei Mao† Rui Xu† Xuesong Yin* Yuanqi Chang Binling Nie Aibin Huang∗ School of Media and Design, Hangzhou Dianzi University, Hangzhou, China {jiaweima0,211330017,yinxs,yuanqichang,binlingnie,huangaibin}@hdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='cn Abstract Facial expression recognition (FER) plays an impor- tant role in a variety of real-world applications such as human-computer interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 achieves the state-of-the-art (SOTA) performance in FER by effectively combining facial landmark and image features through two-stream pyramid cross-fusion design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' However, the architecture of POSTER V1 is undoubtedly complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It causes expensive computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In order to relieve the computational pressure of POSTER V1, in this pa- per, we propose POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It improves POSTER V1 in three directions: cross-fusion, two-stream, and multi- scale feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In cross-fusion, we use window- based cross-attention mechanism replacing vanilla cross- attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We remove the image-to-landmark branch in the two-stream design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For multi-scale feature extraction, POSTER V2 combines images with landmark’s multi-scale features to replace POSTER V1’s pyramid de- sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Extensive experiments on several standard datasets show that our POSTER V2 achieves the SOTA FER perfor- mance with the minimum computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For exam- ple, POSTER V2 reached 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='21% on RAF-DB, 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='49% on AffectNet (7 cls) and 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='77% on AffectNet (8 cls), respec- tively, using only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4G floating point operations (FLOPs) and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7M parameters (Param).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This demonstrates the ef- fectiveness of our improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The code and models are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='com/Talented-Q/ POSTER_V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Introduction With the continuous development of technology and the continuous improvement of automation, the need for human-computer interaction is becoming increasingly strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Facial expression recognition (FER) helps ma- chines to understand human emotions from facial expres- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This makes it as a core task for human-computer in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Besides, with its powerful expression understand- Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='†Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V2 results on RAF-DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We compare POSTER V2 with three variants of POSTER V1 and other FER algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The results indicate that POSTER V2 weighs the number of pa- rameters and accuracy better than other FER methods on RAF- DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' ing ability, FER has great potential applications in psychol- ogy, intelligent robotics, intelligent surveillance, virtual re- ality and synthetic animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Therefore, research on FER is very necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Due to the increasing attention of FER, it has been able to develop rapidly in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Early FER works [55, 59, 33, 20] used manual features [6, 34, 23] for the anal- ysis of facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' However, FER algorithms based on manual features are often only applicable to specific FER tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' When applied to real world scenarios, it is difficult for these algorithms to achieve the same results as in the experi- mental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' With the development of deep learning, con- volutional neural networks (CNNs) are introduced to FER for improving the robustness of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Savchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [38] first verified the effectiveness of CNNs such as Mo- bileNet [19], EfficientNet [41] and RexNet [15] for FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' proposed an efficient and robust FER network EfficientFace [57] for the analysis of facial expressions in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Nevertheless, convolution-based FER algorithms cannot consider the global information of the image due to the limitation of convolutional local receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Influ- enced by the vision transformer, Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [51] designed the first transformer-based FER network to model long-range arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='12149v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='CV] 28 Jan 2023 93 POSTER V2 POSTER V1 92 POSTER V1-S POSTER V1-T ★ Acc TransFER 91 RAF-DB Top-1 90 DMUE 89 VTFF 88 87 40 45 50 55 60 65 70 75 80 Param(M)dependencies for FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [24] improved the vision transformer (ViT) to combine both global and local features so that ViT can be adapted to FER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Among many excellent FER works, POSTER V1 [58] stands out with state-of-the-art (SOTA) performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 mainly solves three key issues of FER at the same time: inter-class similarity, intra-class discrepancy and scale sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 cleverly combines facial landmark with image features through a network design of two-stream pyramidal cross-fusion transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' With the difference and sparsity of landmark, POSTER V1 success- fully solves the issue of inter-class similarity and intra-class discrepancy in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The network design of pyramid archi- tecture introduces multi-scale features for POSTER V1 to solve the scale sensitivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Along with the solution of the three main issues of FER, POSTER V1 shows the amazing expression analysis ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Although POSTER V1 works so well on FER, the huge number of parameters and expensive computational cost brought by its network architecture affects the efficiency of FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' To address this issue, we revisit the network de- sign of POSTER V1 and improve it to obtain POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We mainly improve POSTER V1 in three directions: two-stream, cross-fusion and multi-scale feature extrac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 contains two main branches: image- to-landmark and landmark-to-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Landmark-to-image branch is essential as the core of POSTER V1 to solve inter- class similarity and intra-class discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The image- to-landmark branch is only used to provide information to landmark that it fails to take into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This does not contribute to solving the three main issues of FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' There- fore, in POSTER V2, we remove the image-to-landmark branch from the two-stream design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This greatly reduces the computational cost on POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For cross-fusion, we use a window-based cross-attention mechanism instead of the vanilla cross-attention mechanism in POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The window-based cross-attention mechanism not only pro- vides linear computational complexity for POSTER V2 but also enhances the local modeling capability of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In addition, POSTER V2 no longer uses an additional pyra- mid architecture for multi-scale feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We per- form multi-scale feature extraction directly from the image backbone as well as from the facial landmark detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the extracted multi-scale features, we use a vision trans- former network consisting of only two layers of transformer modules for integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Based on the above designs, our POSTER V2 becomes a simpler and more powerful facial expression recognition network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It achieves SOTA perfor- mance on several standard FER datasets with only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4G floating point operations (FLOPs) and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7M parameters (Param).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 1 demonstrates the superiority of POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Specially, POSTER V2 reached 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='21% on RAF-DB [29], 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='49% on AffectNet [32] (7 cls) and 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='77% on Af- fecNet (8 cls), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This is better than POSTER V1 (RAF-DB with 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05%, AffectNet (7 cls) with 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='31% and AffectNet (8 cls) with 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='34%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' And POSTER V2 of- fers a smaller Param (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7M vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='8M) and FLOPs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We hope that our work could contribute to the design of future FER models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In general, we summarize the contributions of this paper as follows: 1) We design POSTER V2 by modifying POSTER V1 from three perspectives: two-stream, cross-fusion and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Compared with POSTER V1, POSTER V2 is simpler and stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2) POSTER V2 shows state-of-the-art performance on several standard FER datasets such as RAF-DB, Affec- Net and CAER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This shows the powerful expression analysis capability of POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3) POSTER V2 greatly reduces the FLOPs and Param of POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Specifically, POSTER V2 reduces 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='1M of Param and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3G of FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This greatly im- proves the computational efficiency of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Facial Expression Recognition The study of FER has become very popular in re- cent years as more and more researchers focus on human- computer interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [55] used the manual fea- ture LBP [34] for the research of FER with good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [59] proposed a two-stage multitask sparse learning framework (MTSL) for the FER task by explor- ing some common and specific information among differ- ent expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Savchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [38] studied lightweight convolutional neural networks for FER task learning and verified the effectiveness of CNNs for FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Sang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [37] focused on reducing intra-class variation in facial expres- sion depth features and introduced a dense convolutional network [21] for the FER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' PSR [45] solves the prac- tical issues associated with individual wild images in FER in terms of pose, orientation and input resolution with its super-resolution pyramidal network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [54] proposed an erasing attention consistency method to handle the noise-labeled facial expression recognition task that is more challenging than the conventional FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' With the rise of transformer in the field of computer vi- sion, many FER methods combined with transformer have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The vision transformer was first used for the study of FER by Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [51] and achieved state-of-the-art per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' VTFF [31] excels in dealing with facial expres- sion recognition tasks in the wild by virtue of its feature fu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' designed the teacher-student model PID- ViT [22] for modeling the probability distribution of frontal Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Pipeline of POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 mainly contains facial landmark detector, image backbone, cross-fusion transformer encoders and pyramid network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' and multi-pose facial expressions, and solved the problem of pose change and occlusion in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [9] com- bined global and local attention in order to address the two key issues of occlusion and pose change in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 [58] solves the intra-class discrepancy, inter-class sim- ilarity and scale sensitivity issues of FER in the same time by integrating image features with facial landmark features through two-stream, cross-fusion and pyramid design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' However, the huge computational cost of POSTER V1 has prevented researchers from investigating further im- provements in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' To solve this issue, we improved the architecture of POSTER V1 and proposed POSTER V2, which is simpler and more powerful for FER tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Vision Transformer Recently vision transformer has been widely used for computer vision tasks on large scale datasets with its ex- cellent ability to model long distance dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [8] pioneered the introduction of trans- former from the field of natural language processing to com- puter vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [42] used a teacher-student strategy to accelerate the training of transformer by distill- ing tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [60] found that the reason why the transformer quickly saturates at deeper levels is that the attention map becomes increasingly similar as the trans- former goes deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Based on this observation, they pro- posed the Re-attention model to regenerate the attention map in order to enhance the diversity among layers at a small computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Touvron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' also designed CaiT [43], a deep vision transformer for optimal image classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' To solve the issue that ViT is inferior to traditional ResNet [17] on datasets without huge data size, Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' proposed T2T-ViT [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Besides, Hassani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' proposed CCT [16] which uses convolution rather than patch em- bedding layer for self-attention processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This introduces convolutional inductive bias for the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' proposed CrossViT [4], which combines image patches of different sizes by dual branches to produce stronger im- age features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Heo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [18] also verified whether pooling layers bring advantages to ViT as they do in convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [30] reduced the atten- tion mechanism from quadratic computational complexity to linear by window attention and the design of a shift win- dow scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' grafted CNN with Transformer to obtain LeViT [13] with higher accuracy and faster speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' have designed a new architecture called convo- lutional visual transformer CVT [50], which improves the performance and efficiency of ViT by introducing convolu- tion into vision transformer to produce the better results of both designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' proposed a new architecture with a pyramidal structure and a novel region-to-local-attention vision transformer, RegionViT [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [48] intro- duced ViT into a CNN-like pyramid structure for intensive prediction tasks such as object detection and semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The architectural design of these vision transformer ef- forts inspires our improvements for POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This leads to a better trade-off between accuracy and computa- tional complexity in FER with our POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Method In this section, we first review the POSTER V1 process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We then describe the overall architecture of POSTER V2 and discuss the specific details of POSTER V2 in three di- rections: two-stream, cross-fusion, and multi-scale feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' A brief review of POSTER V1 POSTER V1 contains four main core designs: facial landmark detector, image backbone, cross-fusion trans- former encoders and pyramid network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Given the input im- age X ∈ RH×W ×3, POSTER V1 obtain the image features Ximg and landmark features Xlm by facial landmark detec- tor and image backbone, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The image features Ximg ∈ RN×D as well as the land- mark features Xlm ∈ RN×D are mapped into three ma- trices respectively: image query matrix Qimg, image key matrix Kimg, image value matrix Vimg and landmark query Cross-fusion Transform Landmark Feature Encoders Input Image Landmark Dector Cross-fusion Transform head Encoders Concat Image Backbone Cross-fusion Transform Image Feature EncodersFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The overview of POSTER V2 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' LMFi and IMFi denotes facial landmark features and image features at the i-th level of POSTER V2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' matrix Qlm, landmark key matrix Klm, landmark value ma- trix Vlm in the cross-fusion transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Specifi- cally expressed as: Qimg = XimgWq1, Qlm = XlmWq2, Kimg = XimgWk1, Klm = XlmWk2, Vimg = XimgWv1, Vlm = XlmWv2, (1) where Wq1, Wq2, Wk1, Wk2, Wv1 and Wv2 ∈ RD×D are the mapping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The cross-fusion transformer encoder uses the vanilla cross-attention mechanism to interact image features and landmark features respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It is defined as follows: Attention(img) = softmax(QlmKT img/ √ d)Vimg, Attention(lm) = softmax(QimgKT lm/ √ d)Vlm, (2) where softmax(·) is softmax [1] activation function and 1 √ d is an appropriately normalized scaling factor used to prevent the gradient from being too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In summary cross-fusion transformer encoder can be de- noted as: X’img = Attention(img) + Ximg, Ximg o = MLP(Norm(X’img)) + X’img, X’lm = Attention(lm) + Xlm, Xlm o = MLP(Norm(X’lm)) + X’lm, (3) where MLP (·) is multi-layer perceptron and Norm (·) denotes the normalization operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Finally, POSTER V1 extracts and integrates multi-scale features of images and landmarks by the pyramid network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The specific details are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Architecture Figure 3 shows the pipeline for POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The POSTER V2 keeps the facial landmark detector and im- age backbone in POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In difference, we remove the POSTER V1 pyramid architecture and the image-to- landmark branch of the two-stream design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Meanwhile, we perform multi-scale feature extraction directly from the fa- cial landmark detector and image backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' And we in- troduce a small vision transformer consisting of only two layers of vanilla tranformer blocks in POSTER V2 to in- tegrate multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Moreover, we design the new cross-fusion transformer encoder with window-based cross- attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Next, we discuss the detailed modifi- cations to POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Two-stream Methods RAF-DB AffectNet Baseline 91 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='06 POSTER V1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='31 POSTER w/o image to landmark branch 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='82 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='96 POSTER w/o landmark to image branch 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='62 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='28 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Ablation study of two branches in cross-fusion of POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The baseline in the table keeps the baseline setting in POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Although two-stream is central to the design of POSTER V1, POSTER V1 does not explore which branch of two- stream actually plays a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Thus, in this section, we first perform an ablation study of the two-stream to learn the contribution of the two branches to the FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Table 1 shows the ablation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We see that on the RAF-DB dataset, the accuracy of POSTER V1 slips by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='23 after missing the im- age to landmark branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' If the landmark-to-image branch is missing, the accuracy of POSTER V1 on RAF-DB is re- duced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Meanwhile, we observe a similar situation on the AffectNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This indicates that although the image-to-landmark branch contributes to the POSTER V1 FER performance, it is the landmark-to-image branch that plays a decisive role in POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Next, we analyze the above results at the theoretical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The two-stream design in POSTER V1 is mainly used to solve the issues of intra-class discrepancy and inter-class similarity in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It includes landmark-to- 2nd POSTER-V2 level 1st POSTER-V2 level 3rd POSTER-V2 level Landmark Stage 2 Stage LMFi LMF2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='. Landmark Input Image LMFs LMF2 ViT Model LMFi head IMFi IMF2 IMFs Image Stage 2 Image Stage 3 IMFi IMF2 Low-Level Feature Extraction (LFE) High-Level Feature Extraction (HFE) Multi-Level Feature Integration (MFI)Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Input images (row 1), facial landmark images (row 2), landmark-to-image branching attention visualization results (row 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We visualize the attention map belonging to the last layer of the landmarks to image branching for high-level features in POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We can observe that with the help of landmark features, the attention map focuses more on the outstanding areas of face and less on the areas common to face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' image and image-to-landmark branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We revisit the influence of the two branches on POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In the landmark-to-image branch, the landmark features inter- act with the image features as queries Qlm in the cross- attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Image features are guided by land- mark features to more easily represent salient regions of fa- cial expressions when dealing with intra-class discrepancy issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Also benefiting from the sparsity of landmark fea- tures, image features guided by landmark features reduce the focus on regions where faces are prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This helps to reduce the impact of inter-class similarity in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The results of the visualization of landmark-to-image branch attention in Figure 4 also validate the above statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Therefore, the landmark-to-image branch in the two-stream is essential and needs to be retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the image-to- landmark branch, the image features interact with the land- mark features as query Qimg to compensate for the lack of landmark features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Although this also benefits the FER task to some extent, it does not contribute to solving the issues of inter-class similarity and intra-class discrepancy as well as comes with a huge computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This is consistent with the results we observed in the ablation ex- periments of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Thus, by making a trade-off between computational cost and accuracy, we eventually remove the image-to-landmark branch in the two-stream design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Cross-fusion In POSTER V2 we use window-based cross-attention mechanism instead of vanilla cross-attention mechanism in POSTER V1 for the purpose of linear computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Fig- ure 5 illustrates the detailed differences between the two cross-attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For image features Ximg ∈ RN×D, we first divide them into several non-overlapping windows zimg ∈ RM×D, where zimg contains M to- Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Window-based cross attention mechanism and vanilla cross attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' kens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the landmark feature Xlm ∈ RC×H×W , we first down-sample it to the window size zlm ∈ Rc×h×w, where c = D, M = h × w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Then we reshape it according to the shape of Zimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' At this point, the cross-attention with I heads in a local window can be formulated as: q = zlmwq, k = zimgwk, v = zimgwv, o(i) = θ(q(i)k(i)T/ √ d + b)v(i), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=',I, o = [o(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' , o(I)]wo, (4) where wq, wk, wv, wo are the mapping matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' θ (·) is the softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' [·] denotes the merge operation and b ∈ RI×I is the relative position bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We perform the above cross-attention calculation for all windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We refer to this cross-attention mechanism as window-based multi-head cross-attention (W-MCSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Thus the cross-fusion transformer encoder in POSTER V2 can be expressed as follows: X’img = W-MCSA(img) + Ximg, Ximg o = MLP(Norm(X’img)) + X’img, (5) Computational Complexity Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Since the query in the two types of cross-attention computation keeps the same shape as the key, value, we can use the multi-head self- attention and the window-based multi-head self-attention complexity to represent their computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This can be indicated as follows: Ω(MCSA) = 4ND2 + 2N2D, Ω(W-MCSA) = 4ND2 + 2M2ND, (6) Attention Query Window-based Cross Attention Mechanism Vanilla Cross Attention MechanismAccording to Eqn 6, we can find that the window-based cross-attention mechanism we use successfully reduces the computational complexity of cross-fusion in POSTER V1 from square level to linear level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This further improves the computational efficiency of POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Multi-scale feature extraction From Figure 3, we can observe that POSTER V2 re- moves the pyramid design from POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Moreover, in POSTER V2, we extract multi-scale features directly from facial landmark detector and image backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' And we also add a small vision transformer network to POSTER V2 for the integration of multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the obtained multi-scale features o1, o2, o3, we directly merge in the to- ken dimension and using the vanilla transformer blocks for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This process is specifically described as: o = [o1, o2, o3], o’ = MSA(o) + o, oout = MLP(Norm(o’)) + o’, (7) where MSA (·) represents multi-head self-attention mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For above design we discuss as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 adopts the pyramid structure to solve the scale sensitivity problem in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' However, we consider that the pyramid structure design is only an up- sampling and down-sampling operation on the basis of the same scale feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Although it provides multi-scale information to some extent, we believe that it is not as good as multi-scale feature extraction directly from the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The method analysis in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3 also proves our point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the integration of multi-scale features, we believe that the vanilla transformer block is sufficient for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We combine the tokens of all scale feature maps together, and the attention mechanism can model long-range dependen- cies for all scale tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Thus, different scales of token in- formation are delivered in the transformer block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Experiments We verify the effectiveness of POSTER V2 on several standard FER datasets such as RAF-DB [29], AffectNet [32] and CAER-S [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In the following, we first compare POSTER V2 with SOTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We then conduct a se- ries of method analysis and ablation studies on POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' More detailed experimental setup, more experimen- tal results and visualization results are detailed in the Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Experiment Setup Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We evaluat the FER performance of POSTER V2 on the widely used RAF-DB, AffectNet and CAER-S Dataset Train size Test size Classes RAF-DB 12271 3068 7 AffectNet (7 cls) 280401 3500 7 AffectNet (8 cls) 283501 4000 8 CAER-S 44996 20987 7 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Detailed size of the experimental dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The Real-world Affective Faces Database (RAF- DB) is a large-scale database of facial expressions, anno- tated by 315 staff members (students and faculty members of the University).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the selection of expressions, RAF- DB selected six basic emotions as well as neutral emotions from a range of expressions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=', smile, cackle, cry, anger, fear, dread, fear, shock, surprise, disgust, and no expres- sion), for a total of seven expressions for expression anno- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It mainly contains 12,271 training images as well as 3,068 test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' AffectNet is currently the largest pub- licly available dataset in the FER field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It contains about 1M images of faces associated with emotional words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It mainly contains 8 categories of primary emotions (neutral, happy, angry, sad, fear, surprise, disgust,contempt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We mainly use AffectNet settings based on class 7 (excluding contempt) as well as class 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' AffectNet (7 cls) consists of 280K training images and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='500 validation images (500 images per cat- egory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' AffectNet (8 cls) consists of 283K training images and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='000 validation images (500 images per category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The CAER-S dataset was obtained from the CAER dataset con- taining 65,983 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It is mainly divided into 7 types of expressions: neutral, happy, sad, surprised, fear, disgust and anger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In the FER task we used 44996 images for training and 20987 images for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The specific dataset configu- ration is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Methods Year RAF-DB AffectNet (7 cls) AffectNet (8 cls) SCN [46] CVPR 2020 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='03 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='23 PSR [45] CVPR 2020 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='77 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='68 LDL-ALSG [5] CVPR 2020 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='53 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='35 RAN [47] TIP 2020 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='9 DACL [11] WACV 2020 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='78 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 KTN [28] TIP 2021 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='07 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='97 DMUE [39] CVPR 2021 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='42 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='11 FDRL [36] CVPR 2021 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='47 VTFF [31] TAC 2021 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='14 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='85 ARM [40] 2021 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='42 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='33 TransFER [51] ICCV 2021 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='91 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='23 DAN [49] 2021 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='69 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='09 EfficientFace [57] AAAI 2021 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='36 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='23 MA-Net [56] TIP 2021 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='42 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='53 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='29 Meta-Face2Exp [53] CVPR 2022 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='54 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='23 EAC [54] ECCV 2022 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='35 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='32 POSTER V1 [58] 2022 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='31 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='34 POSTER V2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='21 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='49 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='77 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Comparison results with SOTA FER algorithm on RAF- DB and AffectNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Similar to POSTER V1 [58], we also use the ir50 [7] network pre-trained on the Ms-Celeb-1M [14] dataset as the image backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' And MobileFaceNet [2] with frozen weights is used as our facial landmark detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We employ Dataset Method Neutral Happy Sad Surprise Fear Disgust Anger Contempt mean Acc RAF-DB POSTER V1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='35 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='96 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='21 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='27 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='57 75 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='89 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='04 RAF-DB POSTER V2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='06 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='22 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='89 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='58 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='92 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='88 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='27 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='97 AffectNet (7 cls) POSTER V1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 89 67 64 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='8 56 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='23 AffectNet (7 cls) POSTER V2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4 68 66 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4 65 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='45 AffectNet (8 cls) POSTER V1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='8 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='71 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='34 AffectNet (8 cls) POSTER V2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6 63 58 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='52 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='76 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Class-wise accuracy of POSTER V1 and POSTER V2 on RAF-DB, AffectNet (7 cls), and AffectNet (8 cls) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Green, blue and red mark the highest value of single category in RAF-DB, AffectNet (7 cls) and AffectNet (8 cls) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' the Adam [25] optimizer for 200 epochs training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' A train- ing scheme with a batch size of 144, a learning rate of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5e-4 and a weight decay of 1e-4 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We use random hor- izontal flipping and random erasing as our data augmenta- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the loss function, we choose the standard cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We eventually realized POSTER V2 on a single NVIDIA RTX 3090 via Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Comparison with SOTA FER Methods Results on RAF-DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We compare POSTER V2 with the SOTA FER algorithms in recent years on the RAF-DB datasets in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The experimental results show that POSTER V2 exhibits SOTA performance on RAF-DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Compared with POSTER V1 (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05), POSTER V2 im- proved by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='86 for POSTER V2 over EAC (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='35), and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3 for POSTER V2 over TransFER (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This shows the superiority of PSTER V2 on RAF-DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Table 4 shows the comparison of POSTER V2 with POSTER V1 for RAF-DB individual classes and average accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Al- though POSTER V2 outperformed POSTER V1 in sev- eral categories, the average accuracy was slightly inferior to POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Results on AffectNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Table 3, we also conduct FER ex- periments on AffectNet (7 cls) as well as AffectNet (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We observe that POSTER V2 exhibits SOTA FER effect in both AffectNet (7 cls) and AffectNet (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Compared with POSTER V1 (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='31, 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='34), POSTER V2 increases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='43 on AffectNet (7 cls) and AffectNet (8 cls), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' On AffectNet (8 cls), POSTER V2 is higher than DAN (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='09) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' On AffectNet (7 cls), POSTER V2 is greater than TransFER (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='23) with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This demon- strates that POSTER V2 can maintain excellent FER perfor- mance even on larger datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Table 4 shows that POSTER V2 exceeds POSTER V1 for the majority of individual class accuracies in both AffectNet (7 cls) and AffectNet (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' As a result, POSTER V2 achieves better average accuracy than POSTER V1 on AffectNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Results on CAER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We compare POSTER V2 with SOTA FER methods of recent years on the CAER-S dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Our POSTER V2 in Table 5 performs extremely well on the CAER-S dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Specifically, POSTER V2 scored 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98 on CAER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='27 for POSTER V2 over POSTER Methods Year CAER-S DSN [10] ICML 2018 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='19 CAER-Net-S [27] ICCV 2019 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='51 GRERN [12] IEEE Access 2020 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='31 EfficientFace [57] AAAI 2021 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='87 MA-Net [56] TIP 2021 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='42 GLAMOR-Net [26] NCA 2021 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='88 POSTER V1 [58] 2022 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='73 POSTER V2 93 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Comparison results with SOTA FER algorithm on CAER- S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' V1 (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='12 for POSTER V2 over GLAMOR-Net (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='88), and +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='58 for POSTER V2 over MA-Net (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='13 for POSTER V2 over EfficientFace (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The excellent results on CAER-S prove that the success of POSTER V2 is no accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' It shows the powerful gen- eralization ability of POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' FLOPs and Param Comparison Methods #Param #FLOPs RAF-DB AffectNet POSTER V1-T 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2M 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6G 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='36 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='87 POSTER V1-S 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='0M 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7G 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='54 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='13 POSTER V1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='8M 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7G 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='31 POSTER V2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='7M 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4G 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='21 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='49 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Comparison of Param and FLOPs with POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' From Table 6, we can see that POSTER V2 achieves better FER results with smaller Param and FLOPs than POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Compared to POSTER V1-T, POSTER V2 reduces 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5M Param and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2G FLOPs, while increasing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='85% on RAF-DB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='62% on AffectNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Compared to POSTER V1-S, POSTER V2 reduces 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3M Param and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3G FLOPs, while increasing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='67% on RAF-DB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='36% on AffectNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Compared to POSTER V1, POSTER V2 reduces 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='1M Param and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='3G FLOPs, while increas- ing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='16% on RAF-DB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='18% on AffectNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Therefore, POSTER V2 would be a better choice for the FER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Method Analysis In this sub-section, we present a method analysis for the small ViT model we used in POSTER V2 on RAF-DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Influence of different depth ViT models on POSTER V2 for RAF-DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Vit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Here, we investigate the impact of different depths for ViT on the FER performance of POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Figure 6, we show the influence of the ViT model with depth {2,4,6,8} on POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We observe that for multi- scale integration we do not need to increase the depth of the ViT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The ViT model with a depth of 2 is sufficient to handle the FER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' A deeper ViT model hurts the perfor- mance of POSTER V2 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' ViT w/ pre-trained weights RAF-DB AffectNet \x13 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='21 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='49 \x17 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='49 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Impact of pre-trained ViT models for POSTER V2 on FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Pre-trained Vit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We study the influence of the pre-trained ViT model on POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We use the ViT pre-trained weights on ImagenNet-21K [35] for POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Table 7 shows that the performance of POSTER V2 on FER drops after using the pre-trained ViT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We argue that this is mainly due to the fact that the pre-trained ViT model acts mainly on the feature extraction of the image-level inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' However, in POSTER V2, ViT performs the multi-scale fea- ture integration task of feature-level inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The difference in input and task resulted in the pre-trained ViT not working on POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Ablation Study Methods RAF-DB AffectNet POSTER V2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='21 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='49 w/o multi-scale feature extraction 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='47 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='51 w/o ViT 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='86 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='92 w/o W-MCSA 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='56 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='24 w/o cross-fusion 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='39 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='35 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Results of ablation experiments of key components of POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We validate the effectiveness of our POSTER V1 im- provement component on the RAF-DB as well as on the AffectNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Multi-scale feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We first verify the effec- tiveness of extracting multi-scale features directly in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In this ablation experiment, we only use the im- age backbone as well as the last layer of feature maps from the facial landmark detector for cross-fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' From Table 8 we observe that POSTER V2 degrades significantly on the RAF-DB and AffectNet datasets when multi-scale feature extraction is not performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This shows that our method of directly extracting multi-scale features can also solve the scale sensitivity issue of FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Also, this indicates the im- portance of multi-scale features for FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Vit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For the ViT used for multi-scale feature integration, we ablate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We directly sum several different scale features for FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' According to the experimental results in Table 8, POSTER V2 decreases by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='35 on RAF-DB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='57 on AffectNet when multi-scale feature integration is not per- formed by ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This suggests that ViT facilitates multi-scale feature integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' W-MCSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We validate the effectiveness of W-MCSA for cross-fusion by ablation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In this experiment, we use the vanilla cross-attention mechanism to replace our window-based cross-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We observed that POSTER V2 degraded on both RAF-DB and Affect- Net datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This shows that the W-MCSA we use both improves the FER accuracy and reduces the computational complexity of POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Thus, W-MCSA is essential for POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Cross-fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This experiment mainly verifies the role of landmark-to-image branch for POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In the abla- tion experiments on cross-fusion, we merge the extracted image multi-scale features and landmark multi-scale fea- tures directly and integrate them by ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Table 8 shows that the effectiveness of POSTER V2 on RAF-DB as well as AffectNet drops sharply when cross-fusion is not applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This shows that cross-fusion is the key for POSTER V2 to achieve SOTA FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Also, this indicates that addressing inter-class similarity and intra-class discrepancy are partic- ularly important for FER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Conclusion In this paper, we improve POSTER V1 from three direc- tions: two-stream, cross-fusion, and multi-scale feature ex- traction to obtain a simpler and stronger vision transformer for FER, POSTER V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Extensive FER experimental results show that POSTER V2 achieves the state-of-the-art FER performance while greatly reducing the Param and FLOPs of POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' This suggests that POSTER V2 achieves a better trade-off between accuracy and computational com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Therefore, POSTER V2 is a better choice for the FER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 RAF-DB Top-1 Accuracy (%) 92 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='2 2 4 6 8 ViT DepthAcknowledge This work was supported by Public-welfare Technology Application Research of Zhejiang Province in China un- der Grant LGG22F020032, and Key Research and Devel- opment Project of Zhejiang Province in China under Grant 2021C03137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' References [1] Peter F Brown, Vincent J Della Pietra, Peter V Desouza, Jennifer C Lai, and Robert L Mercer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Class-based n-gram models of natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Computational linguistics, 18(4):467–480, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 4 [2] Cunjian Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' PyTorch Face Landmark: A fast and accurate facial landmark detector, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [3] Chun-Fu Chen, Rameswar Panda, and Quanfu Fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Re- gionvit: Regional-to-local attention for vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='02689, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [4] Chun-Fu Richard Chen, Quanfu Fan, and Rameswar Panda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Crossvit: Cross-attention multi-scale vision transformer for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF in- ternational conference on computer vision, pages 357–366, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [5] Shikai Chen, Jianfeng Wang, Yuedong Chen, Zhongchao Shi, Xin Geng, and Yong Rui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Label distribution learning on auxiliary label space graphs for facial expression recog- nition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on com- puter vision and pattern recognition, pages 13984–13993, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [6] Navneet Dalal and Bill Triggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Histograms of oriented gra- dients for human detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In 2005 IEEE computer soci- ety conference on computer vision and pattern recognition (CVPR’05), volume 1, pages 886–893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Ieee, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1 [7] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Arcface: Additive angular margin loss for deep face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF con- ference on computer vision and pattern recognition, pages 4690–4699, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [8] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl- vain Gelly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' An image is worth 16x16 words: Trans- formers for image recognition at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='11929, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [9] Tongle Fan, Guanglei Wang, Yan Li, and Hongrui Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Ma-net: A multi-scale attention network for liver and tumor segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Access, 8:179656–179665, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [10] Yingruo Fan, Jacqueline CK Lam, and Victor OK Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Video- based emotion recognition using deeply-supervised neu- ral networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the 20th ACM Interna- tional Conference on Multimodal Interaction, pages 584– 588, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 7 [11] Amir Hossein Farzaneh and Xiaojun Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Facial expression recognition in the wild via deep attentive center loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Pro- ceedings of the IEEE/CVF winter conference on applications of computer vision, pages 2402–2411, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [12] Qinquan Gao, Hanxin Zeng, Gen Li, and Tong Tong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Graph reasoning-based emotion recognition network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Access, 9:6488–6497, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 7 [13] Benjamin Graham, Alaaeldin El-Nouby, Hugo Touvron, Pierre Stock, Armand Joulin, Herv´e J´egou, and Matthijs Douze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Levit: a vision transformer in convnet’s clothing for faster inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF interna- tional conference on computer vision, pages 12259–12269, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [14] Yandong Guo, Lei Zhang, Yuxiao Hu, Xiaodong He, and Jianfeng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Ms-celeb-1m: A dataset and benchmark for large-scale face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In European conference on computer vision, pages 87–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [15] Dongyoon Han, Sangdoo Yun, Byeongho Heo, and YoungJoon Yoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Rexnet: Diminishing representational bot- tleneck on convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00992, 6, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1 [16] Ali Hassani, Steven Walton, Nikhil Shah, Abulikemu Abuduweili, Jiachen Li, and Humphrey Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Escaping the big data paradigm with compact transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05704, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [17] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceed- ings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [18] Byeongho Heo, Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Junsuk Choe, and Seong Joon Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Rethinking spa- tial dimensions of vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vi- sion, pages 11936–11945, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [19] Andrew G Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco An- dreetto, and Hartwig Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Mobilenets: Efficient convolu- tional neural networks for mobile vision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='04861, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1 [20] Yuxiao Hu, Zhihong Zeng, Lijun Yin, Xiaozhou Wei, Xi Zhou, and Thomas S Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Multi-view facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In 2008 8th IEEE International Conference on Automatic Face & Gesture Recognition, pages 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1 [21] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kil- ian Q Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Densely connected convolutional net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2 [22] Yin-Fu Huang and Chia-Hsin Tsai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Pidvit: Pose-invariant distilled vision transformer for facial expression recognition in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Transactions on Affective Computing, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2 [23] Taskeed Jabid, Md Hasanul Kabir, and Oksam Chae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Lo- cal directional pattern (ldp) for face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In 2010 digest of technical papers international conference on con- sumer electronics (ICCE), pages 329–330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1 [24] Sangwon Kim, Jaeyeal Nam, and Byoung Chul Ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Facial expression recognition based on squeeze vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Sensors, 22(10):3729, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2 [25] Diederik P Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 7 [26] Nhat Le, Khanh Nguyen, Anh Nguyen, and Bac Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Global- local attention for emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Neural Computing and Applications, 34(24):21625–21639, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 7 [27] Jiyoung Lee, Seungryong Kim, Sunok Kim, Jungin Park, and Kwanghoon Sohn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Context-aware emotion recognition net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF international con- ference on computer vision, pages 10143–10152, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6, 7 [28] Hangyu Li, Nannan Wang, Xinpeng Ding, Xi Yang, and Xinbo Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Adaptively learning facial expression represen- tation via cf labels and distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Transactions on Image Processing, 30:2016–2028, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [29] Shan Li, Weihong Deng, and JunPing Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Reliable crowd- sourcing and deep locality-preserving learning for expres- sion recognition in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 2584–2593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2, 6 [30] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Swin transformer: Hierarchical vision transformer using shifted windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 10012–10022, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [31] Fuyan Ma, Bin Sun, and Shutao Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Facial expression recog- nition with visual transformers and attentional selective fu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Transactions on Affective Computing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2, 6 [32] Ali Mollahosseini, Behzad Hasani, and Mohammad H Ma- hoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Affectnet: A database for facial expression, valence, and arousal computing in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Transactions on Affective Computing, 10(1):18–31, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2, 6 [33] Stephen Moore and Richard Bowden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Local binary patterns for multi-view facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Computer vi- sion and image understanding, 115(4):541–558, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1 [34] Timo Ojala, Matti Pietikainen, and Topi Maenpaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Multires- olution gray-scale and rotation invariant texture classification with local binary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Transactions on pattern analysis and machine intelligence, 24(7):971–987, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1, 2 [35] Tal Ridnik, Emanuel Ben-Baruch, Asaf Noy, and Lihi Zelnik-Manor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Imagenet-21k pretraining for the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='10972, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 8 [36] Delian Ruan, Yan Yan, Shenqi Lai, Zhenhua Chai, Chunhua Shen, and Hanzi Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Feature decomposition and recon- struction learning for effective facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7660–7669, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [37] Dinh Viet Sang, Pham Thai Ha, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Discriminative deep feature learning for facial emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In 2018 1st International Conference on Multimedia Analysis and Pat- tern Recognition (MAPR), pages 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2 [38] Andrey V Savchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Facial expression and attributes recog- nition based on multi-task learning of lightweight neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In 2021 IEEE 19th International Symposium on Intel- ligent Systems and Informatics (SISY), pages 119–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1, 2 [39] Jiahui She, Yibo Hu, Hailin Shi, Jun Wang, Qiu Shen, and Tao Mei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Dive into ambiguity: Latent distribution min- ing and pairwise uncertainty estimation for facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6248– 6257, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [40] Jiawei Shi, Songhao Zhu, and Zhiwei Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Learning to amend facial expression representation via de-albino and affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='10189, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [41] Mingxing Tan and Quoc Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Efficientnet: Rethinking model scaling for convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In International conference on machine learning, pages 6105–6114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1 [42] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Herv´e J´egou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Training data-efficient image transformers & distillation through at- tention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 10347–10357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [43] Hugo Touvron, Matthieu Cord, Alexandre Sablayrolles, Gabriel Synnaeve, and Herv´e J´egou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Going deeper with im- age transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Interna- tional Conference on Computer Vision, pages 32–42, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [44] Laurens Van der Maaten and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Visualiz- ing data using t-sne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Journal of machine learning research, 9(11), 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 14 [45] Thanh-Hung Vo, Guee-Sang Lee, Hyung-Jeong Yang, and Soo-Hyung Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Pyramid with super resolution for in-the- wild facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Access, 8:131988– 132001, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2, 6 [46] Kai Wang, Xiaojiang Peng, Jianfei Yang, Shijian Lu, and Yu Qiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Suppressing uncertainties for large-scale facial ex- pression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF con- ference on computer vision and pattern recognition, pages 6897–6906, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [47] Kai Wang, Xiaojiang Peng, Jianfei Yang, Debin Meng, and Yu Qiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Region attention networks for pose and occlusion robust facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Transactions on Image Processing, 29:4057–4069, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [48] Wenhai Wang, Enze Xie, Xiang Li, Deng-Ping Fan, Kaitao Song, Ding Liang, Tong Lu, Ping Luo, and Ling Shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Pyramid vision transformer: A versatile backbone for dense prediction without convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 568–578, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [49] Zhengyao Wen, Wenzhong Lin, Tao Wang, and Ge Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Distract your attention: Multi-head cross attention net- work for facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='07270, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [50] Haiping Wu, Bin Xiao, Noel Codella, Mengchen Liu, Xiyang Dai, Lu Yuan, and Lei Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Cvt: Introducing convolutions to vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 22–31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [51] Fanglei Xue, Qiangchang Wang, and Guodong Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Trans- fer: Learning relation-aware facial expression representa- tions with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 3601– 3610, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1, 2, 6 [52] Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zi-Hang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Tokens-to-token vit: Training vision transformers from scratch on imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF In- ternational Conference on Computer Vision, pages 558–567, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 [53] Dan Zeng, Zhiyuan Lin, Xiao Yan, Yuting Liu, Fei Wang, and Bo Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Face2exp: Combating data biases for facial ex- pression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Con- ference on Computer Vision and Pattern Recognition, pages 20291–20300, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6 [54] Yuhang Zhang, Chengrui Wang, Xu Ling, and Weihong Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Learn from all: Erasing attention consistency for noisy label facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In European Con- ference on Computer Vision, pages 418–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2, 6 [55] Guoying Zhao and Matti Pietikainen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Dynamic texture recognition using local binary patterns with an application to facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 29(6):915–928, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1, 2 [56] Zengqun Zhao, Qingshan Liu, and Shanmin Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Learning deep global multi-scale and local attention features for facial expression recognition in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE Transactions on Image Processing, 30:6544–6556, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 6, 7 [57] Zengqun Zhao, Qingshan Liu, and Feng Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Robust lightweight facial expression recognition network with la- bel distribution training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In Proceedings of the AAAI confer- ence on artificial intelligence, volume 35, pages 3510–3519, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1, 6, 7 [58] Ce Zheng, Matias Mendieta, and Chen Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Poster: A pyra- mid cross-fusion transformer network for facial expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='04083, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 2, 3, 6, 7 [59] Lin Zhong, Qingshan Liu, Peng Yang, Bo Liu, Junzhou Huang, and Dimitris N Metaxas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Learning active facial patches for expression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In 2012 IEEE Conference on Computer Vision and Pattern Recognition, pages 2562– 2569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 1, 2 [60] Daquan Zhou, Bingyi Kang, Xiaojie Jin, Linjie Yang, Xi- aochen Lian, Zihang Jiang, Qibin Hou, and Jiashi Feng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Deepvit: Towards deeper vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='11886, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 3 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Implementation Details For POSTER V2, we conduct FER experiments on the RAF-DB, AffectNet, and CAER-S datasets, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For different datasets, we adopt different detail set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Specifically, for different datasets, we exploit differ- ent learning rates for training according to the settings of POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Moreover, for AffectNet (8 cls), POSTER V2 uses a classification head with a category number of 8 for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The rest of the settings are consistent with the experimental sections in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' config value optimizer Adam base learning rate 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='50E-05 weight decay 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00E-04 batch size 144 training epochs 200 learning rate schedule ExponentialLR (gamma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98) augmentation RandomHorizontalFlip(), RandomErasing(scale=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' drop path linspace(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5, 5) num classes 7 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Supervised training POSTER V2 from scratch on RAF- DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' RAF-DB Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We use the Adam optimizer with a learning rate of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5e-5 for 200 epochs training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The batch size is maintained at 144 and the weight decay remains at 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The learning rate schedule uses an exponential decay with a gamma of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Data augmentation includes random horizontal flipping and random erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The specific set- tings are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' config value optimizer Adam base learning rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00E-06 weight decay 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00E-04 batch size 144 training epochs 200 learning rate schedule ExponentialLR (gamma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98) augmentation RandomHorizontalFlip(), RandomErasing(p=1, scale=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' drop path linspace(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5, 5) num classes 7 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Supervised training POSTER V2 from scratch on Af- fectNet (7 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' AffectNet (7 cls) Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' On the AffcetNet (7 cls) dataset, we adjust the learning rate to 1e-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The training epochs re- mains at 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The batch size is maintained at 144 and the weight decay remains at 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The learning rate schedule uses an exponential decay with a gamma of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Data aug- mentation includes random horizontal flipping and random erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The detailed settings are shown in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' config value optimizer Adam base learning rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00E-06 weight decay 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00E-04 batch size 144 training epochs 200 learning rate schedule ExponentialLR (gamma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98) augmentation RandomHorizontalFlip(), RandomErasing(p=1, scale=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' drop path linspace(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5, 5) num classes 8 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Supervised training POSTER V2 from scratch on Af- fectNet (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' AffectNet (8 cls) Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We use the Adam optimizer with a learning rate of 1e-6 for 200 epochs training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The batch size is maintained at 144 and the weight decay re- mains at 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The learning rate schedule uses an expo- nential decay with a gamma of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Data augmentation includes random horizontal flipping and random erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' In addition, we set the number of categories to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Table 11 shows the specific experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' config value optimizer Adam base learning rate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00E-05 weight decay 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='00E-04 batch size 144 training epochs 200 learning rate schedule ExponentialLR (gamma=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98) augmentation RandomHorizontalFlip(), RandomErasing(p=1, scale=(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='05)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' drop path linspace(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='5, 5) num classes 7 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Supervised training POSTER V2 from scratch on CAER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' CAER-S Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' On the CAER-S dataset, we employ the Adam optimizer with a learning rate of 4e-5 for 200 epochs of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The batch size is maintained at 144 and the weight decay remains at 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The learning rate schedule uses an exponential decay with a gamma of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Data aug- mentation includes random horizontal flipping and random erasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The specific settings are shown in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Detailed Experimental Results In this section, we show more detailed experimental re- sults of POSTER V2 on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' And we also show the confusion matrix of POSTER V2 in each dataset in Fig- ure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' RAF-DB Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 8 shows the specific training pro- cess of POSTER V2 on RAF-DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We observe that the train- ing loss and validation loss of POSTER V2 decrease un- til saturation during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Furthermore, the training accuracy and validation accuracy of POSTER V2 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The confusion matrix of POSTER V2 on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The specific training process of POSTER V2 on RAF- DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' continue to increase until a small fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The detailed training process of POSTER V2 on Affect- Net (7 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' AffectNet (7 cls) Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We show in Figure 9 the detailed training of POSTER V2 on AffectNet (7 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V2 achieves the best training results on AffectNet (7 cls) at an early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' At this point, POSTER V2 achieves the highest accuracy on AffectNet (7 cls) for both the training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Therefore, we stop training in time to save training costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' AffectNet (8 cls) Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 10 shows the exact per- formance of POSTER V2 on AffectNet (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We observe a similar phenomenon on AffectNet (8 cls) as POSTER V2 did on AffectNet (7 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V2 also reach saturation in the early stages of AffectNet (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V2 train- ing loss continues to show a decreasing trend, yet there is a small increase in validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Nevertheless, the training Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The detailed training process of POSTER V2 on Af- fectNet (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' accuracy of POSTER V2 on AffectNet (8 cls) continues to increase, and the validation accuracy has largely been op- timal and remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Therefore, we take the same early end operation for POSTER V2 on AffectNet (8 cls) as we do for AffectNet (7 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' The detailed training process of POSTER V2 on CAER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' CAER-S Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We show the specific training perfor- mance of POSTER V2 on CAER-S in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Compared with other datasets, POSTER V2 has a relatively long sat- uration time on the CAER-S dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' During the training process, the loss on the POSTER V2 training and validation sets decreases and saturates at a late stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Meanwhile, the accuracy of POSTER V2 on both the training and validation sets has been increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' the accuracy/loss curve of train/val 100 95 90 85 80 75 70 65 60 accuracy 55 45 40 35 30 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='. 25 20 15 10 train-accuracy valid-accuracy 5 valid-loss-x30 0 10 15 20 25 OE 35 40 55 90 95 100 105 110 115 180 185 195 200 the training epochthe accuracy/loss curve of train/val 100 95 90 85 80 75 70 65 60 45 35 OE 25 20 15 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='++,+++++++++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='+ ++++*++*++ train-accuracy valid-accuracy 5 train-loss-x30 valid-loss-x30 0 10 15 20 25 30 35 40 45 50 55 60 65 70 75 08 85 90 95 100 105 110 115 130 135 140 145 160 180 185 190 195 200 the training epochRAF-DB AffectNet (7 cls) AffectNet (8 cls) CAER-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='0182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='0122 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='03100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content='0440 Conte Predicted label Predicted label Predicted labelthe accuracy/loss curve of train/val 100 95 90 85 80 75 10 65 60 55 45 40 35 30 卡 25 20 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' 10 - train-accuracy valid-accuracy 5 train-loss-x30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' valid-loss-x30 0 10 15 20 25 OE 35 40 45 55 60 65 70 80 90 95 100 105 110 120125 130 135140 145 150155 160 170175 180 185 190 195 200 the training epochthe accuracy/loss curve of train/val 100 95 90 85 80 75 70 65 60 55 50 45 35 30 - 25 20 15 train-accuracy valid-accuracy 5 valid-loss-x30 0 15 20 25 30 35404550 55 90 95 100 105 110 115 180 185 195 200 the training epochFigure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Comparison of POSTER V2 and POSTER V1 high-dimensional space t-SNE visualization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V1 t-SNE visual- ization results (first row), POSTER V2 t-SNE visualization results (second row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V2 cross-fusion stage attention visualization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' For each triplet, we show the input image (left), the landmark image (middle), and attention map (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Visualization T-SNE Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We visualized the high-dimensional features of POSTER V1 and POSTER V2 using t-SNE [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' As can be seen in Figure 12, both POSTER V2 and POSTER V1 present good t-SNE visualization results on RAF-DB and CAER-S datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' There is almost no signif- icant difference between the t-SNE visualization results of POSTER V1 and POSTER V2 on CAER-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' POSTER V2 has a closer intra-class distance than POSTER V1 on RAF- DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Although POSTER V1 and POSTER V2 have poor t- SNE visualization results on AffectNet (7 cls) and Affect- Net (8 cls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' But the inter-class distance between clusters in POSTER V2 is further than POSTER V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Above results indicates that POSTER V2 is better than POSTER V1 in al- leviating the issues of inter-class similarity and intra-class discrepancy in FER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' Attention Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' We visualize the attention map of the highest-level features of the POSTER V2 cross-fusion stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' From Figure 13, we observe that POSTER V2 suc- cessfully captures important facial expression features with the help of facial landmark features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} +page_content=' RAF-DB AffectNet (7 cls) AffectNet (8 cls) CAER-SNeutral Happy Sad Surprise Fear Disgust Angry Contempt' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NFLT4oBgHgl3EQfsy_c/content/2301.12149v1.pdf'} diff --git a/9dFRT4oBgHgl3EQfqjdh/content/2301.13617v1.pdf b/9dFRT4oBgHgl3EQfqjdh/content/2301.13617v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d07520d08d786ecbd4ce25abdef579c9195b8665 --- /dev/null +++ b/9dFRT4oBgHgl3EQfqjdh/content/2301.13617v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a74517c25d0ac22b61a1de92622aae50fb772eea528d30d6e16ffcf73a1a8fb4 +size 2040715 diff --git a/9dFRT4oBgHgl3EQfqjdh/vector_store/index.pkl b/9dFRT4oBgHgl3EQfqjdh/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d7b262c85a4a0611038717d9a4271b64809fc65e --- /dev/null +++ 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a/ANFQT4oBgHgl3EQf8jdP/content/tmp_files/2301.13447v1.pdf.txt b/ANFQT4oBgHgl3EQf8jdP/content/tmp_files/2301.13447v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ae0f9bfe5ffe93e1d985d959736597a09e30919 --- /dev/null +++ b/ANFQT4oBgHgl3EQf8jdP/content/tmp_files/2301.13447v1.pdf.txt @@ -0,0 +1,1150 @@ +A Data-Driven Modeling and Control +Framework for Physics-Based Building +Emulators +Chihyeon Song ⋆ Aayushman Sharma ⋆ Raman Goyal +Alejandro Brito Saman Mostafavi ⋆ +∗ Palo Alto Research Center, Inc. (PARC), Palo Alto, CA 94304 USA +(e-mail: {csong, asharma, rgoyal, abrito, smostafa}@parc.com). +Abstract: We present a data-driven modeling and control framework for physics-based building +emulators. Our approach comprises: (a) Offline training of differentiable surrogate models that +speed up model evaluations, provide cheap gradients, and have good predictive accuracy for the +receding horizon in Model Predictive Control (MPC) and (b) Formulating and solving nonlinear +building HVAC MPC problems. We extensively verify the modeling and control performance +using multiple surrogate models and optimization frameworks for different available test cases +in the Building Optimization Testing Framework (BOPTEST). The framework is compatible +with other modeling techniques and customizable with different control formulations. The +modularity makes the approach future-proof for test cases currently in development for physics- +based building emulators and provides a path toward prototyping predictive controllers in large +buildings. +Keywords: Data-driven control, Nonlinear model predictive control, Building emulator, +Surrogate modeling. +1. INTRODUCTION +According to recent estimates by United States Energy +Information Administration (2021), residential and com- +mercial buildings account for nearly 40% of energy usage +in the United States. A significant amount of this energy +consumption can be eliminated by improving the build- +ing’s HVAC control system, for example using predictive +control methods as has been shown in Drgoˇna et al. (2020). +Among these methods, model predictive control (MPC) is +a particularly powerful approach for handling constraints +for state and control inputs in nonlinear multivariable +control systems. While the gains are evident, the challenge +is to show that MPC can be implemented at scale in +a cost-friendly manner (O’Dwyer et al., 2022). It is well +understood that the main obstacle to this is the modeling +cost and according to one study (Atam and Helsen, 2016), +this can be as much as 70% of the total effort of setting +up an MPC-based building controller, mainly due to the +effort and expertise required to create realistically cali- +brated models. Recently, Building Optimization Testing +Framework (BOPTEST) (Blum et al., 2021) is developed +to facilitate simulation-based benchmarking of building +HVAC control algorithms. The emulator uses calibrated +Modelica models to emulate building physical dynamics +based on first principles. Models also output Key Perfor- +mance Indices (KPI) that represent occupant satisfaction, +energy cost and consumption, and carbon footprint. What +makes this platform even more impressive is the fact that +it is set up to replicate a real building control system with +⋆ Chihyeon Song and Aayushman Sharma contributed equally to +this paper. Saman Mostafavi is the corresponding author. +all its control limitations, e.g., there are realistic low-level +feedback control laws, box constraints on control inputs, +weather, occupancy profiles, economizer schedules, etc. +MOTIVATION While the value of BOPTEST, and +other physics-based emulators in creating a unified testing +platform for control is unquestionable, there are several +intrinsic obstacles to the implementation of predictive +control and its adoption by a broader audience 1 : (1) In +BOPTEST, and most other physic-based emulators, the +numerical solvers for scaled-up models will not be compu- +tationally efficient to run in iterative optimization loops. +(2) Solving the optimization problem requires gradient +calculations which, derived through perturbations, only +compounds the computational intensity. (3) Furthermore, +some optimal control methods, such as iterative Linear +Quadratic Regulator method (iLQR) (Todorov and Li, +2005), derive optimal solutions by exploring trajectories +that might be infeasible for the emulator to evaluate (due +to control input and state constraints), which can lead to +crashing the iterative algorithm prematurely. +While acknowledging the significant progress in deep +neural networks-based reinforcement learning (RL) ap- +proaches for controlling unknown dynamical systems, with +applications expanding from playing games (Silver et al., +2016), locomotion (Lillicrap et al., 2015) and robotic hand +manipulation (Levine et al., 2016), RL is still highly data +intensive. The training time for such algorithms is typically +1 It is worth mentioning that we consider these challenges to be +almost identical for a real building HVAC control system and, +therefore, addressing and solving them is a first step to deploying +such control algorithms in the field. +arXiv:2301.13447v1 [eess.SY] 31 Jan 2023 + +very large, and high variance and reproducibility issues +mar the performance Henderson et al. (2018). At the mo- +ment, RL algorithms remain intractable for adjustable and +reproducible implementations at scale. On the other hand, +most of the the building MPC work (Sturzenegger et al., +2015; Mostafavi et al., 2022; Oei et al., 2020; Walker et al., +2017) consider either simple low-fidelity RC-based models, +bilinear models with low accuracy, Machine Learning (ML) +approaches that cannot be directly used for fast MPC +implementation, or directly use Modelica-based models +with hand-tuned cost functions for nonlinear optimiza- +tion of energy consumption. Such modeling and control +approaches require a lot of customization for high-fidelity +models with complex, hybrid, and constrained systems +that use external inputs and therefore, are not suited to a +robust control framework. +CONTRIBUTIONS The main contribution of this pa- +per is the development of a modeling and control frame- +work for building HVAC control based on identifying dif- +ferentiable models that are compatible with optimization- +based nonlinear optimal control methods. We address +these limitations by the following two-fold approach: first, +in an off-line round, we identify a differentiable surrogate +model for the following nonlinear mapping: +xt+1 = f(xt, ut, dt) +(1) +where x represent the state of the model, u the control +inputs, and d the external time-varying disturbances, asso- +ciated with the weather and occupancy conditions. Second, +we use automatic differentiation (AD) (Paszke et al., 2017) +to compute gradients for solving nonlinear model predic- +tive control (NMPC) with box constraints for state and +inputs. The details for modeling and control approaches +are discussed in Section 2 and Section 3. The individual +contributions of the paper are as follows: we demonstrate +how to identify a suitable Neural Network (NN) to capture +the dynamics of building envelope and HVAC control sys- +tem. We investigate several choices of lags for states, con- +trols, and disturbances and provide insight into best prac- +tices. We also present different MPC formulations, assisted +by using AD, to maintain occupants’ comfort constraints +while minimizing KPIs for HVAC energy consumption. We +show the customizability of the framework through the +ease of using two different control approaches to solve the +MPC problem. We show that the proposed approach can +be used to warm-start the receding horizon replanning for +the MPC problem. In the result section, we also provide a +performance comparison between different approaches for +modeling and solving NMPC when operating on compu- +tationally intensive hybrid system models. We also discuss +potential best practices based on desired control criteria +(speed, optimality, etc.). Finally, to the best of our knowl- +edge, the NMPC control of the BOPTEST five-zone model +is the first of its kind. We believe this framework is scalable +for data-driven NMPC control of the BOPTEST, and +potentially other physics-based building emulators, that +are being developed for prototyping controllers in large +building HVAC systems. +2. SURROGATE MODELING FOR BUILDING +EMULATOR +Our aim is to replace the computationally expensive non- +linear numerical simulations with alternative, fast repre- +sentations for model-based control. In the context of using +NNs for MPC, we believe that one should include the +following criteria in their surrogate modeling process: +• Computing cost: Small computing cost for fast +iterative evaluations. +• Predictive accuracy: Good prediction accuracy for +MPC’s horizon. +• Differentiability: Fast and accurate gradient infor- +mation for successive linearization, nonlinear solvers, +etc., for different MPC formulations. +We leverage Pytorch (Paszke et al., 2019) modeling li- +brary to meet these goals. In this study, we consider +the following cases: Linear, MLP, and Long short-term +memory (LSTM). MLP has a fast forward computation +and good expressivity to approximate complex functions +Hornik et al. (1989). On the other, since BOPTEST is +Partially Observable MDP (POMDP), it requires lag infor- +mation from states, actions, and time-varying disturbance +for model fitting. This can be curtailed by using LSTM +which has proven to work well for nonlinear mappings with +autoregressive features Siami-Namini et al. (2018). While +fairly simple, the linear model has the advantage of fastest +model evaluations and plug-and-play viability for fast QP +solvers. +2.1 Linear +The surrogate model takes its input as the states x, control +inputs u, time-varying disturbances d, and their lags of +past time-steps. The output of the surrogate model is the +future state prediction {xt+1}, i.e.,: +xt+1 = f(xt−Mx:t, ut−Mu:t, dt−Md:t) +(2) +where Mx, Mu, Md are state, input and disturbance lags, +respectively. Since the choices of lags are application +dependent, we discuss this further in the result section. +Here, f is linearized as follows: +xt+1 = +Mx +� +k=0 +Akxt−k + +Mu +� +k=0 +Bkut−k + +Md +� +k=0 +Ckdt−k +(3) +where Ak = ∇xf ∈ RNx×Nx, Bk = ∇uf ∈ RNx×Nu and +Ck = ∇df ∈ RNx×Nd are learnable parameter matrices for +state, control input and disturbance, respectively. +2.2 MLP +The linearized model given by Equation 3 also applies +here. The forward computation in MLP is written as the +following: +h0 = [xt−Mx, ut−Mu, dt−Md] +hk+1 = tanh(Wkhk + bk), +k = {0, ..., K − 1} +xt+1 = ot+1 = WKhK + bK +(4) +where hk ∈ Rl is a hidden unit of the layer k, Wk and bk +are weight parameters of the layer k. +2.3 LSTM +The forward computation of LSTM is written as the +following: + +ht, ct = MLPenc(xt−Mx:t, ut−Mu:t−1, dt−Mu:t) +it = σ(Wiiut + bii + Whiht−1 + bhi) +ft = σ(Wifut + bif + Wwhfht−1 + bhf) +gt = tanh(Wigut + big + Whght + bhg) +ot+1 = σ(Wiout + bio + Whoht + bho) +ct+1 = ft ⊙ ct + it ⊙ gt +ht+1 = ot ⊙ tanh(ct+1) +xt+1 = MLPdec(ht+1) +(5) +where ht is the hidden state, ct is the cell state, it, ft, gt and +ot are the input, forget, cell, and output gates, respectively. +σ(·) is the sigmoid function, ⊙ is the Hadamard product, +and MLPenc and MLPdec are a MLP encoder and decoder, +respectively. +3. CONTROL PROBLEM FORMULATION +Consider the discrete-time nonlinear dynamical system: +xt+1 = f(xt, ut, dt), +(6) +where xt ∈ Rnx and ut ∈ Rnu correspond to the state +and control vectors at time t and dt +∈ Rnd is the +set of contextual variables/external inputs. The optimal +control problem is to find the optimal control policy that +minimizes the cumulative cost: +min +ut +T +� +t=0 +ct(xt, ut, dt) +(7) +Subject to : xt+1 = f(xt, ut, dt), +(8) +Subject to : ul +t ≤ ut ≤ uu +t , +(9) +for given x0, and where ct(·) is the instantaneous cost +function given as: +ct(·) = Pc + Ph + Lk + γPx, +(10) +where Pc and Ph are total cooling and heating cost, +Lk = ∥˜ut+1 − ˜ut∥2 +R is a regularizer term, which penalizes +large changes in the control inputs to avoid undesirable +oscillations, and Px = max(xl +t − xt, 0) + max(xt − xu +t , 0) +enforces the occupant comfort constraints implemented +with ReLU function with a penalty coefficient γ. The +problem also considers input box constraints with lower +and upper bound given as [ul +t, uu +t ]. +3.1 Gradient Descent Method +The gradient descent method is one of the widely-used +algorithms to optimize a differentiable objective function. +At each iteration, the gradient of the objective function +is computed and the decision variables are updated in +direction of the computed gradient. Gradient descent al- +gorithms have a precedent across domains such as training +neural networks Schmidhuber (2015) and solving optimal +control problems (Lin et al., 2014). In this paper, we use +Adam (Kingma and Ba, 2014), which has shown promising +results in deep learning applications. For input constraint +(9), we use projected gradient descent, a common method +in solving constrained optimization: after each gradient +update, we project the control vector ut into a feasible +region [ul +t, uu +t ]. Since the feasible region is a box constraint, +the projected control vector is easily computed by using a +clamp function after each update of the algorithm. +3.2 Sequential Quadratic Programming +There have been numerous tools and methods developed +to solve specific nonlinear optimization problems with par- +ticular structures of cost functions, equality, and inequal- +ity constraint functions. However, Sequential Quadratic +Programming (SQP) remains one of the most efficient +approaches to solving any general constrained-nonlinear +optimization problem. For the SQP approach, we utilize +the optimization subroutine originally proposed by Dieter +Kraft Kraft (1988) and as implemented in SciPy Virtanen +et al. (2020) to solve the control optimization problem +described in Eqns. (7-9). The algorithm is a quasi-Newton +method (using BFGS) applied to a Lagrange function +consisting of a loss function and equality and inequality +constraints. In our implementation, we provide the func- +tion evaluations, which are calculated using Equation 10, +and it’s Jacobian using automatic differentiation. Instead +of clamping, we pass bounds for control inputs directly to +the solver. +4. RESULTS +We demonstrate the effectiveness of our control framework +for controlling building models in BOPTEST (Blum et al., +2021), a software for simulation-based benchmarking of +building HVAC control algorithms. The rest of this sec- +tion details two test cases that demonstrate the results of +deriving different surrogate models and discusses the sub- +sequent control results for the control algorithms described +in Section 3. +4.1 Model Description +BOPTEST emulators use Modelica (Wetter et al., 2014) +to represent realistic physical dynamics. Embedded in +these models are baseline control algorithms that can +be overwritten using supervisory and local-loop control +signals. BOPTEST uses a containerized run-time environ- +ment (RTE) which enables rapid, repeatable deployment +of models. Using this feature, we stand up several instances +of models on servers and query these models to speed-up +data generation at scale for surrogate modeling. We also +test controls on the same containers, representing digital- +twins of real buildings. We consider the following case +studies: +BESTEST Case 900 model +This test case is a single +room with floor dimensions of 6m x 8m and a floor-to- +ceiling height of 2.7m. The building is assumed to be oc- +cupied by two people from 8 am to 6 pm each day. Heating +and cooling are provided to the office using an idealized +four-pipe fan coil unit (FCU), presented in Figure 1. The +FCU contains a fan, cooling coil, and heating coil. The +fan draws room air into the HVAC unit and supplies the +conditioned air back to the room. No outside air is mixed +during this process. The fan has a variable speed drive +serving the fan motor. The cooling coil is served by chilled +water produced by a chiller and the heating coil is served +by hot water produced by a gas boiler. Two different PI +controllers for heating and cooling modulate the supply +air temperature and fan speed to provide cooling and +heating load to the room. The schematics and control + +(a) +(b) +Fig. +1. +Control +schematics +of +the +BESTEST +Case +900 +model. +Source: +https://ibpsa.github.io/ +project1-boptest/ +mapping are shown in Figure 1. For our supervisory MPC +controller, we manipulate supply air temperature and fan +speed as control inputs to minimize the combined cooling, +heating, and fan power consumption while maintaining the +occupant comfort bounds. Assuming the building to be in +a climate close to Denver, CO, USA, the state and input +box constraints are as follows: +21oC ≤ xTzone,occ ≤ 24oC +(11) +15oC ≤ xTzone,unocc ≤ 30oC +(12) +0.0 ≤ ufan ≤ 1.0 +(13) +12oC ≤ uTsupp ≤ 40oC +(14) +Multi-zone office (ASHRAE 2006 VAVReaheat) +The +test case represents the middle floor of an office build- +ing located in Chicago, IL, as described in the set of +DOE Commercial Building Benchmarks for new construc- +tion (Deru et al., 2011) with weather data from TMY3 +for Chicago O’Hare International Airport. The represented +floor has five zones, with four perimeter zones and one +core zone. The occupied time for the HVAC system is +between 6 AM and 7 PM each day. The HVAC system is a +multi-zone single-duct Variable Air Volume (VAV) system +with pressure-independent terminal boxes with reheat. A +schematic of the system is shown in Figure 2. The cooling +and heating coils are water-based, served by an air-cooled +chiller and air-to-water heat pump respectively. A number +of low-level, local-loop controllers are used to maintain the +desired setpoints using the available actuators. The pri- +mary local-loop controllers are specified on the diagrams +of Figure 3 as C1 to C3. C1 is responsible for maintaining +the zone temperature setpoints as determined by the oper- +ating mode of the system and implements dual-maximum +logic. C2 is responsible for maintaining the duct static +pressure setpoint and implements a duct static pressure +reset strategy. C3 is responsible for maintaining the supply +air temperature setpoint as well as the minimum outside +air flow rate as determined by the operating mode of +the system. In this case, we assume the fan speed to be +constant and our supervisory MPC controller manipulates +the damper position and reheat control signal to control +the airflow and zone supply air temperature respectively +(at each zone). In addition, the central HVAC cooling and +heating units are manipulated to control the central supply +air temperature. The optimization objective is to minimize +the overall cooling and heating loads while maintaining the +occupant comfort bounds and central supply air tempera- +ture. The state and input box constraints are as follows: +21oC ≤ xTzonei,occ ≤ 24oC +(15) +15oC ≤ xTzonei,unocc ≤ 30oC +(16) +0.0 ≤ udami ≤ 1.0 +(17) +0.0 ≤ uyReaHeai ≤ 1.0 +(18) +∀i ∈ {1, 2, 3, 4, 5} +5oC ≤ xTsupp ≤ 20oC +(19) +0.0 ≤ uyHea ≤ 1.0 +(20) +0.0 ≤ uyCoo ≤ 1.0 +(21) +4.2 System Identification +We consider the three choices of models as described +in Section 2 for the single zone and multi-zone case. +We describe how we sufficiently excite the system to +generate data and report the training and out-of-training +performance of each model. +Data generation +For each time-step t = 0, ..., T − 1, +we sample a random control input ut from a uniform +distribution of the feasible input space and pass the +sampled control input to BOPTEST simulation to get +the next observation and disturbance. We collect the data +up to time-step T, and repeat this procedure K times +using different initial conditions. In the BESTEST case, we +choose K = 120, T = 500, and use 100 distinct trajectories +as training data, 10 for validation and 10 for test. In the +(a) +(b) +Fig. +2. +Envelope, +Floorplan +and +control +schematics +of multi zone office air simple emulator model +of BOPTEST. Source: https://ibpsa.github.io/ +project1-boptest/ + +C1 +C2 +Zone +0PI +Map +PI50.0 m +4.57 m +North +33.25 m +West +East +Core +Middle Floor +Southcorezone +southzone +eastzone +northzone +west zone +heating cooling +coil +coil +Motor +Sensor +Point +Ccontrol +Point +Sensor +Point +Airflow Rate +Sensor +Point +Differentia +Pressure +SensorTable 1. +MSE (×10−5) +for +dif- +fer- +ent +model +choices +in +BESTEST +case +Model +Train MSE +Val MSE +Test MSE +Linear +699.5 +566.8 +780.3 +MLP +8.846 +12.70 +17.56 +LSTM +1.418 +1.726 +2.145 +multi-zone office case, we choose K = 600, T = 1000, and +use 500 trajectories as the training dataset, and keep 50 +for validation and 50 for test purposes. It is evident that +test data, which all results are reported on, is the data +that the model has never been trained on. +Hyperparameters +The MLP framework consists of 4 +layers with 256 nodes in each layer, and tanh(·) activation +layers in-between the MLP layers. For the LSTM model, +we implement 2 layers with 256 nodes for MLPenc and +MLPdec and choose the dimension of hidden and cell state +as 256. Mean squared error (MSE) is used for computing +training loss. For all surrogate models, we choose Adam +to optimize the parameters with learning rate=0.001, and +epoch=1000. +Predictive performance +Table 1 and Table 2 show the +results of test performance for single-zone and five-zone +(a) +(b) +(c) +Fig. +3. +Lower-level +control +schematics +for +five-zone +model. +Source: +https://ibpsa.github.io/ +project1-boptest/ +Table 2. +MSE (×10−5) +for +dif- +fer- +ent +MLP +hy- +per- +pa- +ram- +e- +ter +choices +in +multi- +zone +of- +fice +case +(Mx, Mu, Md) +Train MSE +Val MSE +Test MSE +(1, 1, 1) +511.6 +623.9 +618.6 +(1, 1, 5) +476.0 +623.8 +624.3 +(1, 5, 1) +20.46 +21.74 +24.35 +(5, 1, 1) +82.43 +98.92 +103.8 +(1, 5, 5) +14.71 +17.76 +18.47 +(5, 1, 5) +78.38 +98.17 +100.06 +(5, 5, 1) +21.20 +23.67 +26.87 +(5, 5, 5) +10.37 +14.80 +14.82 +models respectively. Losses are calculated using average +prediction error for 40 steps.For multi-step ahead predic- +tion, a for-loop is implemented in the forward propaga- +tion of the ML models. The results for single-zone and +multi-zone models demonstrate the superiority of LSTM +in prediction accuracy, although, MLP performance is +comparable in the five-zone case as depicted in Figure 4. +In Table 2, we compare the performance of different MLP +model choices with different lag values of the state, input, +and time-varying disturbances. (5,5,5) is the best model +among all choices but (1,5,5) model comes very close +with fewer model inputs. This model depends on lags +of weather data and control inputs, which we speculate +is not unrelated to the lags associated with lower-level +controllers in this system. We chose (1,5,5) as a more +simple, equally accurate choice. Figure 5 is a visual de- +piction of the predictive accuracy of the chosen MLP for +surrogate modeling of the five-zone model during three +distinct weather events (January, May, and August) for +the core zone. Each orange trajectory is a 50-step ahead +prediction (12.5 hours) starting from the leftmost point of +the trajectory. These results appear to be conclusive for +deploying the model in MPC. +4.3 Control Results +For all control algorithms, we deploy a receding-horizon +controller, wherein a 10-step ”look-ahead” trajectory is +generated using the optimization algorithm, and only +the first step of the optimization solution is passed to +BOPTEST model to obtain new measurements. The new +data point is then used as the initial condition for the +next iteration of the control optimization. In addition, to + ++ +PI +Map +PIScale +PI +& +PI +Limit +max> yHea +TSupSet +Map +yOA1 +> yCoo +TSup +Max +yOA +supFanSpe +Map +yOA2speed up convergence, the previously optimized control +trajectory is used as the initial trajectory for warm- +starting the receding horizon replanning for the MPC +problem. +The control results for single-zone and multi-zone cases are +reported in Table 3 and Table 4, respectively. In the single- +zone case, LSTM model performs best for control. This +is expected from the superior predictive accuracy of the +model. fIt also has the best average computation time. As +or the control algorithm, Gradient-based approach finds a +better local minima for the problem. In the multi-zone +case, LSTM performs poorly (unexpectedly) and MLP +outperforms all models. Here, in contrast to the previous +case, SLSQP finds a better local minima. Next, we discuss +the significance of these results. +4.4 Discussion +The modeling results indicate that it is possible to derive +accurate ML models from the building emulators. It is +worth mentioning that the bottleneck in this process is +data generation which is not always trivial for hybrid +systems with many if-else conditions, low-level control +loops and system constraints, and finely-tuned numerical +solvers. +On the control side, we have run extensive tests using +SLSQP and Gradient-based approaches from different ini- +tial conditions. In the one-zone case, the gradient-based +approach with the LSTM model shows the lowest power +consumption with an acceptable discomfort level. How- +ever, in the multi-zone case, SLSQP with MLP model +reaches the lowest power consumption, even though LSTM +model shows better predictive performance. This can hap- +pen when the optimization problem in the control for- +mulation is highly non-convex. The complexity of the +surrogate model likely creates many additional local min- +ima, which in turn, depreciates the control performance. +This, somewhat contradictory, implies that better predic- +tive performance does not always guarantee better control +performance. We believe that based on this experiment, a +middle-ground between model complexity and predictive +performance should be considered for these types of MPC +problems. Alternatively, better control formulations might +help to curb this issue. Since we have found little precedent +in the literature, we are running more tests to find better +Fig. 4. Test MSE for different choices of surrogate models +in multi-zone test case. LSTM and MLP have compa- +rable performance and outperform the Linear model. +Table 3. +Av- +er- +age +of +to- +tal +power(kWh/m2), +ther- +mal +dis- +com- +fort +(kh/zone) +and +com- +pu- +ta- +tion +time +(sec) +on +BESTEST +case +Model +Solver +Power +Discomfort +Time +Linear +GDM +0.0189 +1556 +1.607 +Linear +SLSQP +0.2551 +1528 +0.933 +MLP +GDM +4.804 +2.935 +1.694 +MLP +SLSQP +5.059 +5.207 +1.684 +LSTM +GDM +4.818 +2.081 +0.620 +LSTM +SLSQP +4.943 +4.415 +0.661 +and more definitive answers. It is also worth pointing out +that the framework is working as designed, helping to +frame new hypotheses based on experimentation. +Computation Time +By comparing the average compu- +tation time between several methods, we make the fol- +lowing interesting observations: First, both the gradient- +based approach and SLSQP show comparable computa- +tion time, though the computation time of both solvers +depends on their stopping criteria. For example, after +running extensive tests, we decided that 100 iterations was +a good stopping criteria for the gradient-based approach. +We expect this hyperparameter tuning to be problem +specific. Second, for the surrogate model, it is obvious to +us that MLP should take longer than the Linear model to +run. Surprisingly, the LSTM model, which has the most +complex structure among the three candidates, shows the +fastest computation time. We think that this computation +time gap most likely comes from a difference in the imple- +mentation language. Each surrogate model has a for-loop +to predict the multi-steps. Although all surrogate models +are implemented in Pytorch, the linear and MLP model +conduct their for-loops in python, while LSTM model uses +C++. +5. CONCLUSION AND FUTURE WORK +We presented a modeling and control framework for con- +trolling physics-based building emulators. We have shown +that our approach is successful in reducing cooling and +heating loads in the BOPTEST emulator while satisfying + +1e-3 +Linear +4 +MLP +Prediction MSE +LSTM +m +0 +0 +10 +20 +30 +40 +Prediction StepsFig. 5. The set of figures show the results of out-of-training predictive performance for five zone model during three +distinct weather events (January, May, and August) for core zone (top). The ambient temperature trajectories is +depicted in red (bottom). The orange lines represent the 50-step ahead predictions (12.5 hours) starting from the +left most point of the trajectory. The full MSEs are reported in Table 2. +(a) +(b) +Fig. 6. Result comparison for different choices of models and control algorithms. The top figure represents the temperate. +The bottom figure is the relevant weather data, and the middle figures are the corresponding control inputs. +The results are divided into a cold (Jan) and hot (Aug) weather events. (a) Result for control of core-zone in +the multi-zone test case using SLSQP with Linear, MLP, and LSTM models. Using MLP model, the control +outperforms LSTM and Linear model-based implementation. (b) MLP-based control results with SLSQP solver +slightly outperform the Gradient-based approach. +occupant comfort and adhering to control system con- +straints. The approach is modular, meaning that it will +be compatible with various other choices of models and +control algorithms. For example, while we did not succeed +in training a good LSTM model for the five-zone case, we +anticipate that the right hyperparameter tuning should +address this issue and we are actively working on it. +The same is true for control. For example, we tested the +framework with an iLQR controller which failed to satisfy +constraints. While we did not manage to get the results +we expected, we anticipate that significantly better control +results are possible with iLQR and we are currently fixing +our implementation of the algorithm. This is especially +important since iLQR has shown superior performance +for nonlinear optimal control problems (Li and Todorov, +2007). We are also exploring other fast first-order solvers +with alternative control formulations. For example, we +are considering OSQP (Stellato et al., 2020), which will +significantly speed up the optimization while producing +high-quality solutions, or distributed ADMM (Boyd et al., + +Linear, SLSQP +MLP, SLSQP +LSTM, SLSOP +30 +Room Temp.(° C) +NNAN +25 +20 +15 +Heater Valve +1.0 +0.5 +0.0 +Cooler Valve +1.0 +0.5 +0.0 +Reheat Signal +1.0 +0.5 +..AA +0.0 +30 +Solar Irradiation (kW/m2) +Ambient +Irradiation +Ambient Temp.(° C) +10 +2 +10 +1 +-30 +0 +Jan 4 +Jan 7 +Jan 10 +Aug 8 +Aug 11 +Aug 14MLP, +GDM +MLP, SLSQP +30 +Room Temp.(° C) +25 +20 +15 +Heater Valve +1.0 +0.5 +0.0 +Cooler Valve +1.0 +0.5 +0.0 +Reheat Signal +1.0 +0.5 +0.0 +30 +Solar Irradiation (kW/m2) +Ambient +Irradiation +Ambient Temp.(° C) +10 +2 +10 +1 +-30 +0 +Jan 4 +Jan 7 +Jan 10 +Aug 8 +Aug 11 +Aug 1435 +(。) +30 +(0。) +Jan 29th +May 2nd +Aug 3rd +Ground Truth + Surrogate ModelTable 4. +Av- +er- +age +of +to- +tal +power(kWh/m2), +ther- +mal +dis- +com- +fort +(kh/zone) +and +com- +pu- +ta- +tion +time +(sec) +on +multi- +zone +of- +fice +case +Model +Solver +Power +Discomfort +Time +Linear +GDM +2.807 +10.44 +1.504 +Linear +SLSQP +2.487 +11.40 +1.600 +MLP +GDM +3.458 +4.054 +1.782 +MLP +SLSQP +2.778 +3.154 +2.144 +LSTM +GDM +2.222 +124.7 +0.570 +LSTM +SLSQP +2.880 +35.48 +0.818 +2011) for district-level problems. In addition, We are ac- +tively working with the developers of BOPTEST to control +scaled-up models, including multiple coupled buildings, +with the framework. +The main bottleneck for scaling the current approach is +the customized nature of the data generation process. In +the current process, many trials and errors are required to +find a feasible input space that does not break the emu- +lator in forward simulations. Latest studies(Chakrabarty +et al., 2022) provide some promising insight into more +robust sampling procedures. We are currently working on +incorporating similar approaches into our process. +Last but not least, while in this paper we focused on con- +trol as an application, we firmly believe that system design, +fault diagnosis, and reliability are other applications that +will benefit from the proposed modeling approach, and we +are actively investigating problems in these domains. +REFERENCES +Atam, E. and Helsen, L. (2016). Control-oriented thermal +modeling of multizone buildings: Methods and issues: +Intelligent control of a building system. IEEE Control +systems magazine, 36(3), 86–111. +Blum, D., Arroyo, J., Huang, S., Drgoˇna, J., Jorissen, +F., Walnum, H.T., Chen, Y., Benne, K., Vrabie, D., +Wetter, M., et al. (2021). Building optimization testing +framework (boptest) for simulation-based benchmarking +of control strategies in buildings. Journal of Building +Performance Simulation, 14(5), 586–610. +Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J., +et al. (2011). Distributed optimization and statistical +learning via the alternating direction method of multi- +pliers. Foundations and Trends® in Machine learning, +3(1), 1–122. +Chakrabarty, A., Bortoff, S.A., and Laughman, C.R. +(2022). Simulation failure-robust bayesian optimization +for data-driven parameter estimation. IEEE Transac- +tions on Systems, Man, and Cybernetics: Systems. +Deru, M., Field, K., Studer, D., Benne, K., Griffith, B., +Torcellini, P., Liu, B., Halverson, M., Winiarski, D., +Rosenberg, M., et al. (2011). Us department of energy +commercial reference building models of the national +building stock. National Renewable Energy Laboratory. +Drgoˇna, J., Arroyo, J., Figueroa, I.C., Blum, D., Arendt, +K., Kim, D., Oll´e, E.P., Oravec, J., Wetter, M., Vrabie, +D.L., et al. (2020). All you need to know about model +predictive control for buildings. +Annual Reviews in +Control. +Henderson, P., Islam, R., Bachman, P., Pineau, J., Precup, +D., and Meger, D. (2018). Deep reinforcement learning +that matters. +In Thirty-Second AAAI Conference on +Artificial Intelligence. +Hornik, K., Stinchcombe, M., and White, H. (1989). Multi- +layer feedforward networks are universal approximators. +Neural networks, 2(5), 359–366. +Kingma, +D.P. +and +Ba, +J. +(2014). +Adam: +A +method for stochastic optimization. +arXiv preprint +arXiv:1412.6980. +Kraft, D. (1988). +A software package for sequential +quadratic programming. +Forschungsbericht- Deutsche +Forschungs- und Versuchsanstalt fur Luft- und Raum- +fahrt. +Levine, S., Finn, C., Darrell, T., and Abbeel, P. (2016). +End-to-end training of deep visuomotor policies. The +Journal of Machine Learning Research, 17(1), 1334– +1373. +Li, W. and Todorov, E. (2007). +Iterative linearization +methods for approximately optimal control and esti- +mation of non-linear stochastic system. +International +Journal of Control, 80(9), 1439–1453. +Lillicrap, T.P., Hunt, J.J., Pritzel, A., Heess, N., Erez, +T., Tassa, Y., Silver, D., and Wierstra, D. (2015). +Continuous control with deep reinforcement learning. +arXiv preprint arXiv:1509.02971. +Lin, Q., Loxton, R., and Teo, K.L. (2014). The control +parameterization method for nonlinear optimal control: +a survey. Journal of Industrial and management opti- +mization, 10(1), 275–309. +Mostafavi, S., Doddi, H., Kalyanam, K., and Schwartz, +D. (2022). +Nonlinear moving horizon estimation and +model predictive control for buildings with unknown +hvac dynamics. Ifac-Papersonline, Accepted. +Oei, M., Guenther, J., B¨ohm, M., Park, S., and Sawodny, +O. (2020). +A bilinear approach to model predictive +control for thermal conditioning of adaptive buildings. +IFAC-PapersOnLine, 53(2), 8383–8388. +O’Dwyer, +E., +Atam, +E., +Falugi, +P., +Kerrigan, +E., +Zagorowska, M., and Shah, N. (2022). +A modelling +workflow for predictive control in residential buildings. + +In Active Building Energy Systems, 99–128. Springer. +Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, +E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., and +Lerer, A. (2017). Automatic differentiation in pytorch. +31st Conference on Neural Information Processing Sys- +tems (NIPS 2017), Long Beach, CA, USA. +Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., +Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, +L., et al. (2019). Pytorch: An imperative style, high- +performance deep learning library. Advances in neural +information processing systems, 32. +Schmidhuber, J. (2015). Deep learning in neural networks: +An overview. Neural networks, 61, 85–117. +Siami-Namini, S., Tavakoli, N., and Namin, A.S. (2018). A +comparison of arima and lstm in forecasting time series. +In 2018 17th IEEE international conference on machine +learning and applications (ICMLA), 1394–1401. IEEE. +Silver, D., Huang, A., Maddison, C.J., Guez, A., Sifre, L., +Van Den Driessche, G., Schrittwieser, J., Antonoglou, +I., Panneershelvam, V., Lanctot, M., et al. (2016). +Mastering the game of go with deep neural networks +and tree search. nature, 529(7587), 484. +Stellato, B., Banjac, G., Goulart, P., Bemporad, A., and +Boyd, S. (2020). +Osqp: An operator splitting solver +for quadratic programs. +Mathematical Programming +Computation, 12(4), 637–672. +Sturzenegger, D., Gyalistras, D., Morari, M., and Smith, +R.S. (2015). +Model predictive climate control of a +swiss office building: Implementation, results, and cost– +benefit analysis. IEEE Transactions on Control Systems +Technology, 24(1), 1–12. +Todorov, E. and Li, W. (2005). +A generalized iterative +LQG method for locally-optimal feedback control of +constrained nonlinear stochastic systems. In Proceedings +of American Control Conference, 300 – 306. +United States Energy Information Administration (2021). +Total energy monthly data. +URL https://www.eia. +gov/totalenergy/data/monthly/. +Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, +M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, +P., Weckesser, W., Bright, J., et al. (2020). +Scipy +1.0: fundamental algorithms for scientific computing in +python. Nature methods, 17(3), 261–272. +Walker, S.S., Lombardi, W., Lesecq, S., and Roshany- +Yamchi, S. (2017). +Application of distributed model +predictive approaches to temperature and co2 concen- +tration control in buildings. IFAC-PapersOnLine, 50(1), +2589–2594. +Wetter, M., Zuo, W., Nouidui, T.S., and Pang, X. (2014). +Modelica buildings library. Journal of Building Perfor- +mance Simulation, 7(4), 253–270. + diff --git a/ANFQT4oBgHgl3EQf8jdP/content/tmp_files/load_file.txt b/ANFQT4oBgHgl3EQf8jdP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7bf4c93f6545e71f911029c7c71093db141982e5 --- /dev/null +++ b/ANFQT4oBgHgl3EQf8jdP/content/tmp_files/load_file.txt @@ -0,0 +1,654 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf,len=653 +page_content='A Data-Driven Modeling and Control Framework for Physics-Based Building Emulators Chihyeon Song ⋆ Aayushman Sharma ⋆ Raman Goyal Alejandro Brito Saman Mostafavi ⋆ ∗ Palo Alto Research Center, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (PARC), Palo Alto, CA 94304 USA (e-mail: {csong, asharma, rgoyal, abrito, smostafa}@parc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Abstract: We present a data-driven modeling and control framework for physics-based building emulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Our approach comprises: (a) Offline training of differentiable surrogate models that speed up model evaluations, provide cheap gradients, and have good predictive accuracy for the receding horizon in Model Predictive Control (MPC) and (b) Formulating and solving nonlinear building HVAC MPC problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We extensively verify the modeling and control performance using multiple surrogate models and optimization frameworks for different available test cases in the Building Optimization Testing Framework (BOPTEST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The framework is compatible with other modeling techniques and customizable with different control formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The modularity makes the approach future-proof for test cases currently in development for physics- based building emulators and provides a path toward prototyping predictive controllers in large buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Keywords: Data-driven control, Nonlinear model predictive control, Building emulator, Surrogate modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' INTRODUCTION According to recent estimates by United States Energy Information Administration (2021), residential and com- mercial buildings account for nearly 40% of energy usage in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A significant amount of this energy consumption can be eliminated by improving the build- ing’s HVAC control system, for example using predictive control methods as has been shown in Drgoˇna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Among these methods, model predictive control (MPC) is a particularly powerful approach for handling constraints for state and control inputs in nonlinear multivariable control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' While the gains are evident, the challenge is to show that MPC can be implemented at scale in a cost-friendly manner (O’Dwyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' It is well understood that the main obstacle to this is the modeling cost and according to one study (Atam and Helsen, 2016), this can be as much as 70% of the total effort of setting up an MPC-based building controller, mainly due to the effort and expertise required to create realistically cali- brated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Recently, Building Optimization Testing Framework (BOPTEST) (Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2021) is developed to facilitate simulation-based benchmarking of building HVAC control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The emulator uses calibrated Modelica models to emulate building physical dynamics based on first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Models also output Key Perfor- mance Indices (KPI) that represent occupant satisfaction, energy cost and consumption, and carbon footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' What makes this platform even more impressive is the fact that it is set up to replicate a real building control system with ⋆ Chihyeon Song and Aayushman Sharma contributed equally to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Saman Mostafavi is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' all its control limitations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', there are realistic low-level feedback control laws, box constraints on control inputs, weather, occupancy profiles, economizer schedules, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' MOTIVATION While the value of BOPTEST, and other physics-based emulators in creating a unified testing platform for control is unquestionable, there are several intrinsic obstacles to the implementation of predictive control and its adoption by a broader audience 1 : (1) In BOPTEST, and most other physic-based emulators, the numerical solvers for scaled-up models will not be compu- tationally efficient to run in iterative optimization loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2) Solving the optimization problem requires gradient calculations which, derived through perturbations, only compounds the computational intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (3) Furthermore, some optimal control methods, such as iterative Linear Quadratic Regulator method (iLQR) (Todorov and Li, 2005), derive optimal solutions by exploring trajectories that might be infeasible for the emulator to evaluate (due to control input and state constraints), which can lead to crashing the iterative algorithm prematurely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' While acknowledging the significant progress in deep neural networks-based reinforcement learning (RL) ap- proaches for controlling unknown dynamical systems, with applications expanding from playing games (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2016), locomotion (Lillicrap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2015) and robotic hand manipulation (Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2016), RL is still highly data intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The training time for such algorithms is typically 1 It is worth mentioning that we consider these challenges to be almost identical for a real building HVAC control system and, therefore, addressing and solving them is a first step to deploying such control algorithms in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='13447v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='SY] 31 Jan 2023 very large, and high variance and reproducibility issues mar the performance Henderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' At the mo- ment, RL algorithms remain intractable for adjustable and reproducible implementations at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' On the other hand, most of the the building MPC work (Sturzenegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Mostafavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Oei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2017) consider either simple low-fidelity RC-based models, bilinear models with low accuracy, Machine Learning (ML) approaches that cannot be directly used for fast MPC implementation, or directly use Modelica-based models with hand-tuned cost functions for nonlinear optimiza- tion of energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Such modeling and control approaches require a lot of customization for high-fidelity models with complex, hybrid, and constrained systems that use external inputs and therefore, are not suited to a robust control framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' CONTRIBUTIONS The main contribution of this pa- per is the development of a modeling and control frame- work for building HVAC control based on identifying dif- ferentiable models that are compatible with optimization- based nonlinear optimal control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We address these limitations by the following two-fold approach: first, in an off-line round, we identify a differentiable surrogate model for the following nonlinear mapping: xt+1 = f(xt, ut, dt) (1) where x represent the state of the model, u the control inputs, and d the external time-varying disturbances, asso- ciated with the weather and occupancy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Second, we use automatic differentiation (AD) (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2017) to compute gradients for solving nonlinear model predic- tive control (NMPC) with box constraints for state and inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The details for modeling and control approaches are discussed in Section 2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The individual contributions of the paper are as follows: we demonstrate how to identify a suitable Neural Network (NN) to capture the dynamics of building envelope and HVAC control sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We investigate several choices of lags for states, con- trols, and disturbances and provide insight into best prac- tices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We also present different MPC formulations, assisted by using AD, to maintain occupants’ comfort constraints while minimizing KPIs for HVAC energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We show the customizability of the framework through the ease of using two different control approaches to solve the MPC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We show that the proposed approach can be used to warm-start the receding horizon replanning for the MPC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the result section, we also provide a performance comparison between different approaches for modeling and solving NMPC when operating on compu- tationally intensive hybrid system models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We also discuss potential best practices based on desired control criteria (speed, optimality, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Finally, to the best of our knowl- edge, the NMPC control of the BOPTEST five-zone model is the first of its kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We believe this framework is scalable for data-driven NMPC control of the BOPTEST, and potentially other physics-based building emulators, that are being developed for prototyping controllers in large building HVAC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' SURROGATE MODELING FOR BUILDING EMULATOR Our aim is to replace the computationally expensive non- linear numerical simulations with alternative, fast repre- sentations for model-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the context of using NNs for MPC, we believe that one should include the following criteria in their surrogate modeling process: Computing cost: Small computing cost for fast iterative evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Predictive accuracy: Good prediction accuracy for MPC’s horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Differentiability: Fast and accurate gradient infor- mation for successive linearization, nonlinear solvers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', for different MPC formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We leverage Pytorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2019) modeling li- brary to meet these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In this study, we consider the following cases: Linear, MLP, and Long short-term memory (LSTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' MLP has a fast forward computation and good expressivity to approximate complex functions Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' On the other, since BOPTEST is Partially Observable MDP (POMDP), it requires lag infor- mation from states, actions, and time-varying disturbance for model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' This can be curtailed by using LSTM which has proven to work well for nonlinear mappings with autoregressive features Siami-Namini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' While fairly simple, the linear model has the advantage of fastest model evaluations and plug-and-play viability for fast QP solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='1 Linear The surrogate model takes its input as the states x, control inputs u, time-varying disturbances d, and their lags of past time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The output of the surrogate model is the future state prediction {xt+1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=',: xt+1 = f(xt−Mx:t, ut−Mu:t, dt−Md:t) (2) where Mx, Mu, Md are state, input and disturbance lags, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Since the choices of lags are application dependent, we discuss this further in the result section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Here, f is linearized as follows: xt+1 = Mx � k=0 Akxt−k + Mu � k=0 Bkut−k + Md � k=0 Ckdt−k (3) where Ak = ∇xf ∈ RNx×Nx, Bk = ∇uf ∈ RNx×Nu and Ck = ∇df ∈ RNx×Nd are learnable parameter matrices for state, control input and disturbance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='2 MLP The linearized model given by Equation 3 also applies here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The forward computation in MLP is written as the following: h0 = [xt−Mx, ut−Mu, dt−Md] hk+1 = tanh(Wkhk + bk), k = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', K − 1} xt+1 = ot+1 = WKhK + bK (4) where hk ∈ Rl is a hidden unit of the layer k, Wk and bk are weight parameters of the layer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='3 LSTM The forward computation of LSTM is written as the following: ht, ct = MLPenc(xt−Mx:t, ut−Mu:t−1, dt−Mu:t) it = σ(Wiiut + bii + Whiht−1 + bhi) ft = σ(Wifut + bif + Wwhfht−1 + bhf) gt = tanh(Wigut + big + Whght + bhg) ot+1 = σ(Wiout + bio + Whoht + bho) ct+1 = ft ⊙ ct + it ⊙ gt ht+1 = ot ⊙ tanh(ct+1) xt+1 = MLPdec(ht+1) (5) where ht is the hidden state, ct is the cell state, it, ft, gt and ot are the input, forget, cell, and output gates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' σ(·) is the sigmoid function, ⊙ is the Hadamard product, and MLPenc and MLPdec are a MLP encoder and decoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' CONTROL PROBLEM FORMULATION Consider the discrete-time nonlinear dynamical system: xt+1 = f(xt, ut, dt), (6) where xt ∈ Rnx and ut ∈ Rnu correspond to the state and control vectors at time t and dt ∈ Rnd is the set of contextual variables/external inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The optimal control problem is to find the optimal control policy that minimizes the cumulative cost: min ut T � t=0 ct(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' dt) (7) Subject to : xt+1 = f(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' dt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (8) Subject to : ul t ≤ ut ≤ uu t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (9) for given x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' and where ct(·) is the instantaneous cost function given as: ct(·) = Pc + Ph + Lk + γPx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (10) where Pc and Ph are total cooling and heating cost,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Lk = ∥˜ut+1 − ˜ut∥2 R is a regularizer term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' which penalizes large changes in the control inputs to avoid undesirable oscillations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' and Px = max(xl t − xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 0) + max(xt − xu t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 0) enforces the occupant comfort constraints implemented with ReLU function with a penalty coefficient γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The problem also considers input box constraints with lower and upper bound given as [ul t, uu t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='1 Gradient Descent Method The gradient descent method is one of the widely-used algorithms to optimize a differentiable objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' At each iteration, the gradient of the objective function is computed and the decision variables are updated in direction of the computed gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Gradient descent al- gorithms have a precedent across domains such as training neural networks Schmidhuber (2015) and solving optimal control problems (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In this paper, we use Adam (Kingma and Ba, 2014), which has shown promising results in deep learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For input constraint (9), we use projected gradient descent, a common method in solving constrained optimization: after each gradient update, we project the control vector ut into a feasible region [ul t, uu t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Since the feasible region is a box constraint, the projected control vector is easily computed by using a clamp function after each update of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='2 Sequential Quadratic Programming There have been numerous tools and methods developed to solve specific nonlinear optimization problems with par- ticular structures of cost functions, equality, and inequal- ity constraint functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' However, Sequential Quadratic Programming (SQP) remains one of the most efficient approaches to solving any general constrained-nonlinear optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For the SQP approach, we utilize the optimization subroutine originally proposed by Dieter Kraft Kraft (1988) and as implemented in SciPy Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2020) to solve the control optimization problem described in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (7-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The algorithm is a quasi-Newton method (using BFGS) applied to a Lagrange function consisting of a loss function and equality and inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In our implementation, we provide the func- tion evaluations, which are calculated using Equation 10, and it’s Jacobian using automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Instead of clamping, we pass bounds for control inputs directly to the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' RESULTS We demonstrate the effectiveness of our control framework for controlling building models in BOPTEST (Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2021), a software for simulation-based benchmarking of building HVAC control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The rest of this sec- tion details two test cases that demonstrate the results of deriving different surrogate models and discusses the sub- sequent control results for the control algorithms described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='1 Model Description BOPTEST emulators use Modelica (Wetter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2014) to represent realistic physical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Embedded in these models are baseline control algorithms that can be overwritten using supervisory and local-loop control signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' BOPTEST uses a containerized run-time environ- ment (RTE) which enables rapid, repeatable deployment of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Using this feature, we stand up several instances of models on servers and query these models to speed-up data generation at scale for surrogate modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We also test controls on the same containers, representing digital- twins of real buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We consider the following case studies: BESTEST Case 900 model This test case is a single room with floor dimensions of 6m x 8m and a floor-to- ceiling height of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='7m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The building is assumed to be oc- cupied by two people from 8 am to 6 pm each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Heating and cooling are provided to the office using an idealized four-pipe fan coil unit (FCU), presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The FCU contains a fan, cooling coil, and heating coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The fan draws room air into the HVAC unit and supplies the conditioned air back to the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' No outside air is mixed during this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The fan has a variable speed drive serving the fan motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The cooling coil is served by chilled water produced by a chiller and the heating coil is served by hot water produced by a gas boiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Two different PI controllers for heating and cooling modulate the supply air temperature and fan speed to provide cooling and heating load to the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The schematics and control (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Control schematics of the BESTEST Case 900 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Source: https://ibpsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='io/ project1-boptest/ mapping are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For our supervisory MPC controller, we manipulate supply air temperature and fan speed as control inputs to minimize the combined cooling, heating, and fan power consumption while maintaining the occupant comfort bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Assuming the building to be in a climate close to Denver, CO, USA, the state and input box constraints are as follows: 21oC ≤ xTzone,occ ≤ 24oC (11) 15oC ≤ xTzone,unocc ≤ 30oC (12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 ≤ ufan ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 (13) 12oC ≤ uTsupp ≤ 40oC (14) Multi-zone office (ASHRAE 2006 VAVReaheat) The test case represents the middle floor of an office build- ing located in Chicago, IL, as described in the set of DOE Commercial Building Benchmarks for new construc- tion (Deru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2011) with weather data from TMY3 for Chicago O’Hare International Airport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The represented floor has five zones, with four perimeter zones and one core zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The occupied time for the HVAC system is between 6 AM and 7 PM each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The HVAC system is a multi-zone single-duct Variable Air Volume (VAV) system with pressure-independent terminal boxes with reheat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A schematic of the system is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The cooling and heating coils are water-based, served by an air-cooled chiller and air-to-water heat pump respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A number of low-level, local-loop controllers are used to maintain the desired setpoints using the available actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The pri- mary local-loop controllers are specified on the diagrams of Figure 3 as C1 to C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' C1 is responsible for maintaining the zone temperature setpoints as determined by the oper- ating mode of the system and implements dual-maximum logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' C2 is responsible for maintaining the duct static pressure setpoint and implements a duct static pressure reset strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' C3 is responsible for maintaining the supply air temperature setpoint as well as the minimum outside air flow rate as determined by the operating mode of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In this case, we assume the fan speed to be constant and our supervisory MPC controller manipulates the damper position and reheat control signal to control the airflow and zone supply air temperature respectively (at each zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In addition, the central HVAC cooling and heating units are manipulated to control the central supply air temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The optimization objective is to minimize the overall cooling and heating loads while maintaining the occupant comfort bounds and central supply air tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The state and input box constraints are as follows: 21oC ≤ xTzonei,occ ≤ 24oC (15) 15oC ≤ xTzonei,unocc ≤ 30oC (16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 ≤ udami ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 (17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 ≤ uyReaHeai ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 (18) ∀i ∈ {1, 2, 3, 4, 5} 5oC ≤ xTsupp ≤ 20oC (19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 ≤ uyHea ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 (20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 ≤ uyCoo ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 (21) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='2 System Identification We consider the three choices of models as described in Section 2 for the single zone and multi-zone case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We describe how we sufficiently excite the system to generate data and report the training and out-of-training performance of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Data generation For each time-step t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', T − 1, we sample a random control input ut from a uniform distribution of the feasible input space and pass the sampled control input to BOPTEST simulation to get the next observation and disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We collect the data up to time-step T, and repeat this procedure K times using different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the BESTEST case, we choose K = 120, T = 500, and use 100 distinct trajectories as training data, 10 for validation and 10 for test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Envelope, Floorplan and control schematics of multi zone office air simple emulator model of BOPTEST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Source: https://ibpsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='io/ project1-boptest/ C1 C2 Zone 0PI Map PI50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='57 m North 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='25 m West East Core Middle Floor Southcorezone southzone eastzone northzone west zone heating cooling coil coil Motor Sensor Point Ccontrol Point Sensor Point Airflow Rate Sensor Point Differentia Pressure SensorTable 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' MSE (×10−5) for dif- fer- ent model choices in BESTEST case Model Train MSE Val MSE Test MSE Linear 699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='8 780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='3 MLP 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='846 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='70 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='56 LSTM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='418 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='726 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='145 multi-zone office case, we choose K = 600, T = 1000, and use 500 trajectories as the training dataset, and keep 50 for validation and 50 for test purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' It is evident that test data, which all results are reported on, is the data that the model has never been trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Hyperparameters The MLP framework consists of 4 layers with 256 nodes in each layer, and tanh(·) activation layers in-between the MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For the LSTM model, we implement 2 layers with 256 nodes for MLPenc and MLPdec and choose the dimension of hidden and cell state as 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Mean squared error (MSE) is used for computing training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For all surrogate models, we choose Adam to optimize the parameters with learning rate=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='001, and epoch=1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Predictive performance Table 1 and Table 2 show the results of test performance for single-zone and five-zone (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Lower-level control schematics for five-zone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Source: https://ibpsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='io/ project1-boptest/ Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' MSE (×10−5) for dif- fer- ent MLP hy- per- pa- ram- e- ter choices in multi- zone of- fice case (Mx, Mu, Md) Train MSE Val MSE Test MSE (1, 1, 1) 511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='6 623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='9 618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='6 (1, 1, 5) 476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='8 624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='3 (1, 5, 1) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='46 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='74 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='35 (5, 1, 1) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='43 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='92 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='8 (1, 5, 5) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='71 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='76 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='47 (5, 1, 5) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='38 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='17 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='06 (5, 5, 1) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='20 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='67 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='87 (5, 5, 5) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='37 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='80 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='82 models respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Losses are calculated using average prediction error for 40 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='For multi-step ahead predic- tion, a for-loop is implemented in the forward propaga- tion of the ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The results for single-zone and multi-zone models demonstrate the superiority of LSTM in prediction accuracy, although, MLP performance is comparable in the five-zone case as depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In Table 2, we compare the performance of different MLP model choices with different lag values of the state, input, and time-varying disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (5,5,5) is the best model among all choices but (1,5,5) model comes very close with fewer model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' This model depends on lags of weather data and control inputs, which we speculate is not unrelated to the lags associated with lower-level controllers in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We chose (1,5,5) as a more simple, equally accurate choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Figure 5 is a visual de- piction of the predictive accuracy of the chosen MLP for surrogate modeling of the five-zone model during three distinct weather events (January, May, and August) for the core zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Each orange trajectory is a 50-step ahead prediction (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 hours) starting from the leftmost point of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' These results appear to be conclusive for deploying the model in MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='3 Control Results For all control algorithms, we deploy a receding-horizon controller, wherein a 10-step ”look-ahead” trajectory is generated using the optimization algorithm, and only the first step of the optimization solution is passed to BOPTEST model to obtain new measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The new data point is then used as the initial condition for the next iteration of the control optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In addition, to + PI Map PIScale PI & PI Limit max> yHea TSupSet Map yOA1 > yCoo TSup Max yOA supFanSpe Map yOA2speed up convergence, the previously optimized control trajectory is used as the initial trajectory for warm- starting the receding horizon replanning for the MPC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The control results for single-zone and multi-zone cases are reported in Table 3 and Table 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the single- zone case, LSTM model performs best for control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' This is expected from the superior predictive accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' fIt also has the best average computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' As or the control algorithm, Gradient-based approach finds a better local minima for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the multi-zone case, LSTM performs poorly (unexpectedly) and MLP outperforms all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Here, in contrast to the previous case, SLSQP finds a better local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Next, we discuss the significance of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='4 Discussion The modeling results indicate that it is possible to derive accurate ML models from the building emulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' It is worth mentioning that the bottleneck in this process is data generation which is not always trivial for hybrid systems with many if-else conditions, low-level control loops and system constraints, and finely-tuned numerical solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' On the control side, we have run extensive tests using SLSQP and Gradient-based approaches from different ini- tial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the one-zone case, the gradient-based approach with the LSTM model shows the lowest power consumption with an acceptable discomfort level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' How- ever, in the multi-zone case, SLSQP with MLP model reaches the lowest power consumption, even though LSTM model shows better predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' This can hap- pen when the optimization problem in the control for- mulation is highly non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The complexity of the surrogate model likely creates many additional local min- ima, which in turn, depreciates the control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' This, somewhat contradictory, implies that better predic- tive performance does not always guarantee better control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We believe that based on this experiment, a middle-ground between model complexity and predictive performance should be considered for these types of MPC problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Alternatively, better control formulations might help to curb this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Since we have found little precedent in the literature, we are running more tests to find better Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Test MSE for different choices of surrogate models in multi-zone test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' LSTM and MLP have compa- rable performance and outperform the Linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Av- er- age of to- tal power(kWh/m2), ther- mal dis- com- fort (kh/zone) and com- pu- ta- tion time (sec) on BESTEST case Model Solver Power Discomfort Time Linear GDM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0189 1556 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='607 Linear SLSQP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='2551 1528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='933 MLP GDM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='804 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='935 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='694 MLP SLSQP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='059 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='207 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='684 LSTM GDM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='818 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='620 LSTM SLSQP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='943 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='661 and more definitive answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' It is also worth pointing out that the framework is working as designed, helping to frame new hypotheses based on experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Computation Time By comparing the average compu- tation time between several methods, we make the fol- lowing interesting observations: First, both the gradient- based approach and SLSQP show comparable computa- tion time, though the computation time of both solvers depends on their stopping criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For example, after running extensive tests, we decided that 100 iterations was a good stopping criteria for the gradient-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We expect this hyperparameter tuning to be problem specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Second, for the surrogate model, it is obvious to us that MLP should take longer than the Linear model to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Surprisingly, the LSTM model, which has the most complex structure among the three candidates, shows the fastest computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We think that this computation time gap most likely comes from a difference in the imple- mentation language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Each surrogate model has a for-loop to predict the multi-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Although all surrogate models are implemented in Pytorch, the linear and MLP model conduct their for-loops in python, while LSTM model uses C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK We presented a modeling and control framework for con- trolling physics-based building emulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We have shown that our approach is successful in reducing cooling and heating loads in the BOPTEST emulator while satisfying 1e-3 Linear 4 MLP Prediction MSE LSTM m 0 0 10 20 30 40 Prediction StepsFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The set of figures show the results of out-of-training predictive performance for five zone model during three distinct weather events (January, May, and August) for core zone (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The ambient temperature trajectories is depicted in red (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The orange lines represent the 50-step ahead predictions (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 hours) starting from the left most point of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The full MSEs are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Result comparison for different choices of models and control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The top figure represents the temperate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The bottom figure is the relevant weather data, and the middle figures are the corresponding control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The results are divided into a cold (Jan) and hot (Aug) weather events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (a) Result for control of core-zone in the multi-zone test case using SLSQP with Linear, MLP, and LSTM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Using MLP model, the control outperforms LSTM and Linear model-based implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (b) MLP-based control results with SLSQP solver slightly outperform the Gradient-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' occupant comfort and adhering to control system con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The approach is modular, meaning that it will be compatible with various other choices of models and control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For example, while we did not succeed in training a good LSTM model for the five-zone case, we anticipate that the right hyperparameter tuning should address this issue and we are actively working on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The same is true for control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For example, we tested the framework with an iLQR controller which failed to satisfy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' While we did not manage to get the results we expected, we anticipate that significantly better control results are possible with iLQR and we are currently fixing our implementation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' This is especially important since iLQR has shown superior performance for nonlinear optimal control problems (Li and Todorov, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We are also exploring other fast first-order solvers with alternative control formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' For example, we are considering OSQP (Stellato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2020), which will significantly speed up the optimization while producing high-quality solutions, or distributed ADMM (Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Linear, SLSQP MLP, SLSQP LSTM, SLSOP 30 Room Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (° C) NNAN 25 20 15 Heater Valve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 Cooler Valve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 Reheat Signal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='.AA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 30 Solar Irradiation (kW/m2) Ambient Irradiation Ambient Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (° C) 10 2 10 1 30 0 Jan 4 Jan 7 Jan 10 Aug 8 Aug 11 Aug 14MLP, GDM MLP, SLSQP 30 Room Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (° C) 25 20 15 Heater Valve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 Cooler Valve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 Reheat Signal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0 30 Solar Irradiation (kW/m2) Ambient Irradiation Ambient Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (° C) 10 2 10 1 30 0 Jan 4 Jan 7 Jan 10 Aug 8 Aug 11 Aug 1435 (。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=') 30 (0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=') Jan 29th May 2nd Aug 3rd Ground Truth Surrogate ModelTable 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Av- er- age of to- tal power(kWh/m2), ther- mal dis- com- fort (kh/zone) and com- pu- ta- tion time (sec) on multi- zone of- fice case Model Solver Power Discomfort Time Linear GDM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='807 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='504 Linear SLSQP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='487 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='600 MLP GDM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='458 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='054 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='782 MLP SLSQP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='778 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='154 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='144 LSTM GDM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='222 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='570 LSTM SLSQP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='880 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='818 2011) for district-level problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In addition, We are ac- tively working with the developers of BOPTEST to control scaled-up models, including multiple coupled buildings, with the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The main bottleneck for scaling the current approach is the customized nature of the data generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In the current process, many trials and errors are required to find a feasible input space that does not break the emu- lator in forward simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Latest studies(Chakrabarty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', 2022) provide some promising insight into more robust sampling procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' We are currently working on incorporating similar approaches into our process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Last but not least, while in this paper we focused on con- trol as an application, we firmly believe that system design, fault diagnosis, and reliability are other applications that will benefit from the proposed modeling approach, and we are actively investigating problems in these domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' REFERENCES Atam, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' and Helsen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Control-oriented thermal modeling of multizone buildings: Methods and issues: Intelligent control of a building system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' IEEE Control systems magazine, 36(3), 86–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Blum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Arroyo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Drgoˇna, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Jorissen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Walnum, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Benne, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Vrabie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Wetter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Building optimization testing framework (boptest) for simulation-based benchmarking of control strategies in buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Journal of Building Performance Simulation, 14(5), 586–610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Boyd, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Parikh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Chu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Peleato, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Eckstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Distributed optimization and statistical learning via the alternating direction method of multi- pliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Foundations and Trends® in Machine learning, 3(1), 1–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Chakrabarty, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Bortoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Laughman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Simulation failure-robust bayesian optimization for data-driven parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' IEEE Transac- tions on Systems, Man, and Cybernetics: Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Deru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Field, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Studer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Benne, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Griffith, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Torcellini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Halverson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Winiarski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Rosenberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Us department of energy commercial reference building models of the national building stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' National Renewable Energy Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Drgoˇna, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Arroyo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Figueroa, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Blum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Arendt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Oll´e, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Oravec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Wetter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Vrabie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' All you need to know about model predictive control for buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Annual Reviews in Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Henderson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Islam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Bachman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Pineau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Precup, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Meger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Deep reinforcement learning that matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In Thirty-Second AAAI Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Hornik, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Stinchcombe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and White, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Multi- layer feedforward networks are universal approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Neural networks, 2(5), 359–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' and Ba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='6980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Kraft, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A software package for sequential quadratic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raum- fahrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Finn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Abbeel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' End-to-end training of deep visuomotor policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The Journal of Machine Learning Research, 17(1), 1334– 1373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' and Todorov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Iterative linearization methods for approximately optimal control and esti- mation of non-linear stochastic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' International Journal of Control, 80(9), 1439–1453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Lillicrap, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Hunt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Pritzel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Heess, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Erez, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Tassa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Silver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Wierstra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Continuous control with deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' arXiv preprint arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='02971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Lin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Loxton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Teo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' The control parameterization method for nonlinear optimal control: a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Journal of Industrial and management opti- mization, 10(1), 275–309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Mostafavi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Doddi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Kalyanam, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Schwartz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Nonlinear moving horizon estimation and model predictive control for buildings with unknown hvac dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Ifac-Papersonline, Accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Oei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Guenther, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', B¨ohm, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Sawodny, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A bilinear approach to model predictive control for thermal conditioning of adaptive buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' IFAC-PapersOnLine, 53(2), 8383–8388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' O’Dwyer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Atam, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Falugi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Kerrigan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Zagorowska, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Shah, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A modelling workflow for predictive control in residential buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In Active Building Energy Systems, 99–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Gross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Chintala, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Chanan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Yang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', DeVito, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Desmaison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Antiga, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Lerer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Automatic differentiation in pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' 31st Conference on Neural Information Processing Sys- tems (NIPS 2017), Long Beach, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Gross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Massa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Lerer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Bradbury, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Chanan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Killeen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Gimelshein, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Antiga, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Pytorch: An imperative style, high- performance deep learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Advances in neural information processing systems, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Schmidhuber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Deep learning in neural networks: An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Neural networks, 61, 85–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Siami-Namini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Tavakoli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Namin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A comparison of arima and lstm in forecasting time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In 2018 17th IEEE international conference on machine learning and applications (ICMLA), 1394–1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Silver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Huang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Maddison, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Guez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Sifre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Van Den Driessche, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Schrittwieser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Antonoglou, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Panneershelvam, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Lanctot, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Mastering the game of go with deep neural networks and tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' nature, 529(7587), 484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Stellato, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Banjac, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Goulart, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Bemporad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Boyd, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Osqp: An operator splitting solver for quadratic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Mathematical Programming Computation, 12(4), 637–672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Sturzenegger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Gyalistras, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Morari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Smith, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Model predictive climate control of a swiss office building: Implementation, results, and cost– benefit analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' IEEE Transactions on Control Systems Technology, 24(1), 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Todorov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' and Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' In Proceedings of American Control Conference, 300 – 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' United States Energy Information Administration (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Total energy monthly data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='eia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' gov/totalenergy/data/monthly/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Virtanen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Gommers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Haberland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Reddy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Cournapeau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Burovski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Peterson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Weckesser, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Bright, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Scipy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='0: fundamental algorithms for scientific computing in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Nature methods, 17(3), 261–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Walker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Lombardi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Lesecq, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Roshany- Yamchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Application of distributed model predictive approaches to temperature and co2 concen- tration control in buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' IFAC-PapersOnLine, 50(1), 2589–2594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Wetter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Zuo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', Nouidui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=', and Pang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Modelica buildings library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} +page_content=' Journal of Building Perfor- mance Simulation, 7(4), 253–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQf8jdP/content/2301.13447v1.pdf'} diff --git a/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf b/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..434d384198f19f33a92118bb3d5760f762ac1732 --- /dev/null +++ b/AdAzT4oBgHgl3EQfF_tg/content/2301.01020v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3fd13189bc5602af2662f53db8efafa8bfccb215962a33654c754674721c063e +size 161342 diff --git a/AdAzT4oBgHgl3EQfF_tg/vector_store/index.faiss 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0000000000000000000000000000000000000000..2a1115c37603273dd214735eb570f68de3db413e --- /dev/null +++ b/AtE2T4oBgHgl3EQf8QmS/content/tmp_files/2301.04217v1.pdf.txt @@ -0,0 +1,418 @@ +arXiv:2301.04217v1 [math.CO] 10 Jan 2023 +Neighbourhood complexity of graphs of bounded twin-width∗ +´Edouard Bonnet† +Florent Foucaud‡ § +Tuomo Lehtil¨a¶ ‖ +Aline Parreau∗∗ +January 12, 2023 +Abstract +We give essentially tight bounds for, ν(d, k), the maximum number of distinct neighbourhoods +on a set X of k vertices in a graph with twin-width at most d. Using the celebrated Marcus-Tardos +theorem, two independent works [Bonnet et al., Algorithmica ’22; Przybyszewski ’22] have shown the +upper bound ν(d, k) ⩽ exp(exp(O(d)))k, with a double-exponential dependence in the twin-width. +We give a short self-contained proof that for every d and k, +ν(d, k) ⩽ (d + 2)2d+1k = 2d+O(log d)k, +and build a bipartite graph implying ν(d, k) ⩾ 2d+log d+O(1)k, in the regime when k is large enough +compared to d. +1 +Introduction +The aim of this paper is to refine our understanding of how complex the neighbourhoods of graphs of +bounded twin-width can be. We provide an improved bound on the neighbourhood complexity of such +graphs, complemented by a construction showing that our bound is essentially tight. The improvements in +the bounds for neighbourhood complexities translate directly to better structural bounds and algorithms, +in some contexts which are explained below. +Twin-width. +Twin-width is a recently introduced graph invariant [10]; see Section 2 for a definition. +It can be naturally extended to matrices over finite alphabets and binary structures [10, 7, 12]. Although +classes of bounded twin-width are broad and diverse, they allow (most of the time, provided a witness +is given as an input) improved algorithms, compared to what is possible on general graphs or binary +structures. +Most prominently, it was shown [10] that, on n-vertex graphs given with a d-sequence (a witness that +their twin-width is at most d), deciding if a first-order sentence ϕ holds can be solved in time f(d, ϕ)n, for +some computable function f. In some special cases, such as for k-Independent Set or k-Dominating +Set1, single-exponential parameterised algorithms running in time 2Od(k)n are possible [5]. In the same +setting, the triangles of an n-vertex m-edge graph can be counted in time O(d2n+m) [19]. See [8, 18, 25] +for more applications of twin-width with an algorithmic flavour. +Classes of binary structures with bounded twin-width include bounded treewidth, and more gener- +ally, bounded clique-width classes, proper minor-closed classes, posets of bounded width (that is, whose +antichains are of bounded size), hereditary subclasses of permutations, as well as Ω(log n)-subdivisions of +∗Florent Foucaud was financed by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) and by +the ANR project GRALMECO (ANR-21-CE48-0004). Tuomo Lehtil¨a’s research was supported by the Finnish Cultural +Foundation and by the Academy of Finland grant 338797. +†Univ Lyon, CNRS, ENS de Lyon, Universit´e Claude Bernard Lyon 1, LIP UMR5668, France. +‡Universit´e Clermont-Auvergne, CNRS, Mines de Saint-´Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont- +Ferrand, France. +§Univ. Orl´eans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orl´eans Cedex 2, France. +¶Univ Lyon, UCBL, CNRS, LIRIS - UMR 5205, F69622, France +‖University of Turku, Department of Mathematics and Statistics, Turku, Finland +∗∗Univ Lyon, CNRS, INSA Lyon, UCBL, Centrale Lyon, Univ Lyon 2, LIRIS, UMR5205, F-69622 Villeurbanne, France +1That is, the problems of deciding whether in an input graph, there are k vertices that are pairwise non-adjacent or +whose closed neighbourhood is the entire vertex set, respectively. +1 + +n-vertex graphs [10], and particular classes of (bounded-degree) expanders [6]. A rich range of geometric +graph classes have bounded twin-width such as map graphs, bounded-degree string graphs [10], classes +with bounded queue number or bounded stack number [6], segment graphs with no Kt,t subgraph, and +visibility graphs of simple polygons without large independent sets [4], to give a few examples. +If efficiently approximating the twin-width is a challenging open question in general, this is known +to be possible for the above-mentioned classes (albeit a representation may be needed for the geometric +classes) and for ordered graphs [7]. By that, we mean that there are two computable functions f, g and +an algorithm that, for an input n-vertex graph G from the class and an integer k, and in time g(k)nO(1), +either outputs an f(k)-sequence (again, witnessing that the twin-width is at most f(k)) or correctly +reports that the twin-width of G is larger than k. +Structural properties of graph classes of bounded twin-width include χ-boundedness [5], even with +a quasipolynomial binding function [24], smallness (i.e., containing up to isomorphism 2O(n) n-vertex +graphs) [6, 12], and Vapnik-Chervonenkis (VC) density at most 1 [9, 26]. The latter property is the topic +of the current article. +VC density and neighbourhood complexity. +VC density is related to the celebrated VC dimen- +sion [29]. Given a set-system (or hypergraph) S on a domain X, the shatter function πS : N → N is +defined as +πS(n) = max +A∈(X +n) +|{Y ⊆ A | ∃S ∈ S, Y = A ∩ S}|. +The Perles-Sauer-Shelah lemma states that πS(n) = O(nd) if the VC dimension of S (i.e., the supremum of +{n | πS(n) = 2n}) is a finite integer d. Then the VC density of S is defined as inf{c ∈ R | πS(n) = O(nc)}, +and as +∞ if the VC dimension is unbounded. +We define the VC density of an infinite class C of finite graphs as the VC density of the infinite +set-system formed by the neighbourhood hypergraph of the disjoint union of the graphs of C, that is, +{NG(v) | v ∈ V (⊎G∈CG)}, where NG(v) denotes the set of neighbours of v in G. +The VC density +is an important measure in finite model theory, often more tractable than the VC dimension (see for +instance [1, 2]). Tight bounds have been obtained for the VC density of (logically) definable hypergraphs +from graph classes of bounded clique-width [23] (with monadic second-order logic), and more recently, of +bounded twin-width [18] (with first-order logic). +In structural graph theory and kernelisation [16] (a subarea of parameterised complexity [14]) the +function πN(G), where N(G) is the neighbourhood hypergraph of G, is often1 called neighbourhood com- +plexity. (See [3] for an algorithmic study of the computation of this notion.) In these contexts, obtaining +the best possible upper bound for πN(G) (and not just the exponent matching the VC density) translates +to qualitatively better structural bounds and algorithms; see for instance [9, 11, 15, 28]. +The r-neighbourhood complexity of G is the neighbourhood complexity of Gr, with same vertex set +as G, and an edge between two vertices at distance at most r in G. Reidl et al. [28] showed that among +subgraph-closed classes, bounded expansion2 is equivalent to linear r-neighbourhood complexity. Indeed, +the more general nowhere dense classes [21] (another invention of the Sparsity program [22]) have almost +linear r-neighbourhood complexity [15]: there is a function f : N × N → N such that for every ε > 0, +πN(Gr)(n) ⩽ f(r, ε)n1+ε for all n. On hereditary classes, i.e., closed under taking induced subgraphs, +there is no known characterisation of linear neighbourhood complexity. +As we already mentioned in a different language, bounded twin-width classes have been proven to have +linear neighbourhood complexity. See [9, Lemma 3] or [26, Section 3] for two independent proofs, both +using the Marcus-Tardos theorem [20]. However, the dependence in the twin-width is doubly exponential +in both papers. Setting ν(d, k) as the maximum number of distinct neighbourhoods on a set of size k +within a graph of twin-width at most d, i.e., max{πN(G)(k) | G has twin-width at most d}, they show +that ν(d, k) ⩽ exp(exp(O(d)))k. +Our results. +In this note, we give in Section 3 a self-contained proof (not using the Marcus-Tardos +theorem) that ν(d, k) ⩽ 2d+O(log d)k. In Section 4, we complement that proof with a construction of a +1Some authors define the neighbourhood complexity as n �→ +πN (G)(n) +n +. +2A notion from the Sparsity theory of Neˇsetˇril and Ossona de Mendez [22] extending bounded degree and proper minor- +free classes. +2 + +bipartite graph witnessing that ν(d, k) ⩾ 2d+log d+O(1)k, which makes our single-exponential upper bound +in twin-width essentially tight. +2 +Preliminaries +We use the standard graph-theoretic notations: V (G), E(G), G[S], G − S respectively denote the vertex +set, edge set, subgraph of G induced by S, and subgraph of G induced by V (G) \ S. If v ∈ V (G), then +NG(v) (or N(v) if G is clear from the context) denotes the set of neighbours of v in G. If X ⊆ V (G), +then an X-neighbourhood is a set N(v) ∩ X for some v ∈ V (G). +We now define the twin-width of a graph, following the definition of [10]. +A trigraph is a triple G = (V (G), E(G), R(G)) where E(G) and R(G) are two disjoint sets of edges +on V (G): the usual edges (also called black edges) and the red edges. Informally, a red edge between two +vertices u and v means that some errors have been made between u and v. The red degree of a trigraph +is the maximum degree of the graph (V (G), R(G)). +Any graph G can be interepreted as a trigraph +G = (V (G), E(G), ∅). Given a trigraph and two vertices u, v ∈ V (G) (not necessarily adjacent), the +trigraph G/u, v = G′ is obtained by contracting u and v in a new vertex w such that: +• V (G′) = {w} ∪ V (G) \ {u, v}; +• the edges between vertices of V (G) \ {u, v} are the same in G′; +• the following edges are incident to w: +– wx ∈ E(G′) if xu ∈ E(G) and xv ∈ E(G); +– wx /∈ E(G′) ∪ R(G′) if xu /∈ E(G) ∪ R(G) and xv /∈ E(G) ∪ R(G); +– wx ∈ R(G′) otherwise. +In other words, the common black neighbours of u and v are black neighbours of w. All the other +neighbours of u or v are red neighbours of w. Red edges stay red, black edges stay black, red and black +edges become red. We say that G/u, v is a contraction of G. A d-sequence of an n-vertex graph G is +a sequence of n trigraphs Gn = G, Gn−1, ...., G1 such that each trigraph Gi is obtained from Gi+1 by a +contraction and has red degree at most d. The twin-width of G, denoted by tww(G), is the minimum +integer d such that G admits a d-sequence. Note that an induced subgraph of G has a twin-width smaller +or equal to the twin-width of G [10]. +If u ∈ Gi, then u(G) denotes the set of vertices of G eventually contracted to u in Gi. Instead of +considering the trigraphs Gi, we might prefer to deal with the partitions induced by the sets u(G) for u +in Gi: Pi = {u(G) | u ∈ V (Gi)}. We say that there is a red edge between two parts u(G) and v(G) of Pi +if uv is red in Gi. +3 +Upper bound on the number of distinct neighbourhoods +We state and prove our upper bound on the maximum number of distinct X-neighbourhoods in bounded +twin-width graphs. +Theorem 1. Let G be an n-vertex graph of twin-width d, and X ⊆ V (G). Then the number of distinct +X-neighbourhoods in G is at most (d + 2)2d+1|X| = 2d+O(log d)|X|. +Proof. Fix X ⊆ V (G). First of all, for all vertices of V (G) \ X with the same X-neighbourhood, we keep +only one representative. Note that the new graph G′′ is an induced subgraph of G, thus its twin-width +is at most d. We further modify graph G′′ by adding for each v ∈ X a new vertex u to G′′ so that +N(u) = N(v) if such vertex does not exist in V (G′′) \ X. The new graph is called G′ and it has the same +twin-width as G′′. +Let M = (d + 2)2d+1 + 1. We prove by induction on n that an n-vertex graph of twin-width at +most d with a set X of k vertices, where all vertices outside X have a distinct X-neighbourhood, satisfies +n ⩽ kM. This will prove that G′ has at most kM vertices, and thus that in G, there are at most (M −1)k +distinct X-neighbourhoods. +The statement is trivially true for n ⩽ 5 since M ⩾ 5, for all d ⩾ 0. +3 + +Thus, assume n ⩾ 6. In particular, we have k > 1. Let x ∈ X. Let X′ = X \ {x} and let Tx be the +set of pairs of vertices outside X that are twins with respect to X′, i.e. +Tx = +� +{u, v} ∈ +�V (G′) \ X +2 +� +| N(u) ∩ X′ = N(v) ∩ X′ +� +. +Since every vertex of V (G′) \ X has a distinct neighbourhood in X, there are at most two vertices of +V (G′) \ X with the same (possibly empty) neighbourhood N in X′; namely the vertices u, v ∈ V (G′) \ X +with N(u) ∩ X = N and N(v) ∩ X = N ∪ {x} (if they exist). Hence, Tx consists of pairwise-disjoint pairs +of vertices. +We prove the following claim. +Claim A. There exists a vertex x of X such that Tx comprises at most M − 1 pairs, in G′. +Proof of claim. By contradiction, assume this is not the case: for every x in X, Tx has size at least M. +Consider a d-sequence of contractions G′ +n, . . . , G′ +1 of G′. Consider the last step G′ +i of the sequence where +all the parts of Pi contain at most one vertex of X (that is, contrary to Pi, some part of Pi−1 contains +two vertices of X). +Let P be a part of Pi. Let x be the unique (if there exists one) element of P ∩X. Then we claim that +|P \ X| ⩽ 2d+1. Indeed, any two vertices of P \ X have some vertex in the symmetric difference of their +X-neighbourhoods, either it is x, or some vertex x′ of X outside P. If that distinguishing vertex is some +x′ that is not in P, then there has to be a red edge between P and the part that contains x′. There are +at most d red edges with P as an extremity. Since all the elements of X are in distinct parts in G′ +i, it +means that d + 1 vertices of X are enough to distinguish all the X-neighbourhoods of vertices of P \ X, +and thus |P \ X| ⩽ 2d+1. +We now consider the next contraction in the sequence, which leads to G′ +i−1. By definition of G′ +i, it +must contract two vertices corresponding to two parts of Pi that both contain an element of X. Let +x1 and x2 be these two elements of X. Let Q be the part of Pi−1 that contains both x1 and x2. By +our assumption, Tx1 has size at least M. Let {u, v} be a pair of Tx1. Since u and v have the same +neighbourhood in X \ {x1}, it means that they are either both adjacent or both non-adjacent to x2, and +exactly one of them is adjacent to x1. Thus, necessarily, one vertex among the pair {u, v} is adjacent +to exactly one vertex among {x1, x2}. In particular, if this vertex is not in Q, then there has to be a +red edge between the part containing this vertex and the part Q in G′ +i−1. Since Tx1 contains at least M +pairs (which are disjoint) and Q has at most 2d+2 vertices not in X, there are at least M − 2d+2 vertices +not in X whose part in G′ +i−1 has a red edge to Q. Since each other part has at most 2d+1 vertices not +in X, it makes at least M−2d+2 +2d+1 +red edges incident to Q. Thus, we must have M−2d+2 +2d+1 +⩽ d, leading to +M ⩽ 2d+1(d + 2), a contradiction that proves the claim. (□) +By Claim A, there exists a vertex x ∈ X such that |Tx| ⩽ M − 1. Let Y be a set of |Tx| vertices that +intersects each pair of Tx exactly once. Let GY = G′ − (Y ∪ {x}). Then, X′ = X \ {x} is a vertex set +of size k − 1 such that all X′-neighbourhoods of vertices outside X′ are distinct. The graph GY has at +least n − M vertices, and twin-width at most d. By induction, we have n − M ⩽ |V (GY )| ⩽ (k − 1)M +and thus, n ⩽ kM. Hence, once we recall that no vertex in X has unique X-neighbourhood, there are at +most (M − 1)k distinct X-neighbourhoods, which completes the proof. +4 +Lower bound on the number of distinct neighbourhoods +Notice that when |X| and tww(G) are roughly the same, the bound from Theorem 1 cannot be sharp, +since G′ has at most 2|X| + |X| vertices. However, when |X| is large enough compared to tww(G), we +next show that the bound is sharp up to a constant factor. +Proposition 2. There is a positive constant c, such that for any integer d, there is a bipartite graph G of +twin-width at most d, and a large enough set X ⊆ V (G), with at least c·d2d|X| distinct X-neighbourhoods +in G. +Proof. Observe that the claim is clearly true for any small d. Thus, we do not need to consider separately +graphs with small twin-width upper bounded by a constant. Hence, we assume from now on that d ≥ d′ +where d′ is some positive constant. +4 + +We construct the graph G as follows. Let A, B, C ∈ Z be three constants that will be given later +(A and B will be roughly equal to +√ +d and C will be roughly equal to d). Let X = {x1, ..., xk} be an +independent set of k vertices. Our goal is that each vertex in V (G) \ X has a unique X-neighbourhood. +For any integers i, j, t with 1 ⩽ i ⩽ j ⩽ i + A − 1, j + 2 ⩽ t ⩽ j + 1 + B and t ⩽ k − C, we create +a set Vi,j,t of vertices as follows. Consider the set Xt = {xt+1, ..., xt+C}. For every subset Y of Xt, let +Y ′ = {xi, ..., xj, xt} ∪ Y and add a vertex vY ′ to Vi,j,t, making it adjacent to the vertices of Y ′. Each set +Vi,j,t has size 2C and there are Θ(kAB) (for fixed A and B and growing k) such sets. Thus there are +Θ(kAB2C) vertices in the graph. +Any two vertices not in X have distinct X-neighbourhoods. +Indeed, by considering the natural +ordering of X induced by the indices, any vertex not in X is first adjacent to a consecutive interval +of vertices from xi to xj, then is not adjacent to vertices from xj+1 to xt−1 (which is not empty since +t ⩾ j + 2), and then adjacent to xt. Thus, if two vertices have the same X-neighbourhood, they must be +in the same set Vi,j,t. But then, they have a distinct neighbourhood in {xt+1, ..., xt+C}. +We now prove that the twin-width of G is at most M = max{AB, C}+2. For that, we give a sequence +of contractions with red degree at most M. +The contraction sequence is split into k − C steps, for each vertex of X. Let 0 ≤ i ≤ k − C − 1. Step 0 +corresponds to the starting point, where each vertex is alone. Let i ⩾ 1. After Step i, there will be the +following parts in the corresponding partition (vertices not in any part have not yet been contracted): +• For each j, t such that i ⩽ j ⩽ i + A − 1 and j + 2 ⩽ t ⩽ j + 1 + B, there is a part Bj,t. The parts +Bi,t (parts with j = i), contain all the vertices of the sets Vi′,j′,t such that j′ ≤ i. The parts Bj,t +with j > i contain all the vertices of the sets Vi′,j′,t such that i′ ⩽ i and j′ = j. Note that there +are AB non-empty Bj,t parts in total. +• There is a part X0 that contains vertices from x1 to xi of X. +• There is a part T (for “trash”) that contains all the vertices of the sets Vi′,j,t with t ⩽ i + 1. +All the other vertices are not yet contracted. This corresponds to the vertices from xi+1 to xk of X +and to the vertices of the sets Vi′,j,t with i′ > i. Indeed, if i′ ⩽ i and t ⩽ i + 1, then the vertices of Vi′,j,t +are in T . If t ⩾ i + 2 but j ⩽ i, then they are in the part Bi,t. If j > i, then they are in the part Bj,t. +We first prove that the red degree after Step i is at most M. Then, we explain how to get from Step +i to Step i + 1 by keeping the red degree at most M. +Consider the part Bj,t at the end of Step i. A vertex in this part belongs to some set Vi′,j′,t with +i′ ⩽ i and j′ = j if j > i or j′ ⩽ i otherwise. In particular, two vertices of Bj,t are adjacent to all the +vertices between xi+1 and xj, to no vertex between xj+1 and xt−1, to xt, and to no vertex after xt+C. +Thus, there is a red edge between the parts Bj,t and X0, and C red edges between the part Bj,t and the +vertices {xt+1, ..., xt+C}. Therefore, the number of red edges incident with Bj,t is at most C + 1. +Consider now the part T . Vertices in T are adjacent only to vertices of X up to xi+C+1. Since vertices +x1 to xi are all in the part X0, the red degree of T is at most C + 2. +Single vertices not in X have no incident red edges: indeed, they are all in some sets Vi′,j,t for i′ > i +and thus are not adjacent to any vertex of X0. For the same reason, there are red edges incident to X0 +only to T and to the parts Bj,t. Hence, the red degree of X0 is at most AB + 1. Similarly, the red degree +of xi′, i′ > i + 1 is at most AB + 1. Moreover, the red degree of xi+1 is at most one. Indeed, the only +red edge is between xi+1 and T . +Finally, the red degree after step i is at most max{AB + 1, C + 2} ⩽ M. +Let i ≥ 0. We now explain how we perform the contractions to go from step i to step i + 1. +1. (only if i ≥ 1) For any i + 3 ⩽ t ⩽ i + 2 + B, merge the part Bi,t with the part Bi+1,t. The only +new red edge this merging may lead to, when Bi,t is non-empty, is between Bi+1,t and xi+1. Thus, +we add only one red edge between xi+1 and Bi+1,t. Thus, the red degree of Bi+1,t is at most C + 2 +and the red degree of xi+1 is at most 2. +2. Add all the vertices of Vi+1,j,t for some j, t to the part (that might be empty at this point) Bj,t. +The red degree of Bj,t is at most C + 2 since we might have a red edge between Bj,t and xi+1. The +number of nonempty parts Bj,t at this point is AB + 1 (there is still the part Bi,i+2). Adding T , +this gives AB + 2 red edges incident to a vertex in X (or from part X0). +5 + +3. Add xi+1 to X0. The part X0 has red edges only to parts Bi+1,t, to Bi,i+2 and to T , but no edges +to the single vertices. Thus, it has red degree at most AB + 2. +4. Put the part Bi,i+2 into T . This part is only adjacent to vertices up to xi+2+C, and thus has C + 2 +red edges. +Thus, at each point, the red degree is always at most M = max{AB, C} + 2. +The process ends at step i = k − C − 1. Then, all the vertices not in X are in some parts, and there +are at most AB + 1 such parts. On the other side of bipartition, we have part X0 and C + 1 single +vertices. Thus, the graph is bipartite with both sides of size at most M. One can contract each part +independently to finish the contraction sequence. +To conclude, taking C = d − 2 and A = B = ⌊ +√ +d − 2⌋, we have M ⩽ d and kAB2C = Θ(kd2d). +Notice that we may assume that A, B and C are positive since d ≥ d′ where d′ was some well chosen +positive constant. This concludes the proof. +5 +Conclusion +We have given an improved and tight upper bound for the neighbourhood complexity of graphs of bounded +twin-width. Unlike the previously known (weaker) bounds, our method is simple and avoids the use of +the Marcus-Tardos theorem. We hope that it can inspire future works in the area. +It is known that the twin-width of Gr can be upper-bounded by a function of the twin-width of G +and r [10]. Thus, graphs of twin-width at most d have linear r-neighbourhood complexity. We leave as an +interesting open problem to obtain an essentially tight twin-width dependence for the r-neighbourhood +complexity. +We remark that the neighbourhood complexity is also related to identification problems on graphs +such as identifying codes or locating-dominating sets, where one seeks a (small) set A of vertices of a graph +such that all other vertices have a distinct neighbourhood in A [17]. Some works in this area about specific +graph classes, are equivalent to the study of the neighbourhood complexity of these graph classes: see for +example [13, 17, 27]. Moreover, we note that for graph classes with VC density 1, since any solution has +linear size, the natural minimisation versions of the above identification problems have a polynomial-time +constant-factor approximation algorithm (trivially select the whole vertex set), while such an algorithm +is unlikely to exist in the general case [13]. Thus, our work implies a better approximation ratio for these +problems, when restricted to input graph classes of bounded twin-width. +References +[1] Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson, and Sergei Starchenko. +Vapnik–chervonenkis density in some theories without the independence property, II. Notre Dame +Journal of Formal Logic, 54(3-4):311–363, 2013. +[2] Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson, and Sergei Starchenko. +Vapnik-chervonenkis density in some theories without the independence property, I. Transactions +of the American Mathematical Society, 368(8):5889–5949, 2016. +[3] Cristina Bazgan, Florent Foucaud, and Florian Sikora. Parameterized and approximation complexity +of partial VC dimension. Theor. Comput. Sci., 766:1–15, 2019. +[4] ´Edouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen K¨ohler, Raul Lopes, and St´ephan +Thomass´e. Twin-width VIII: delineation and win-wins. In Holger Dell and Jesper Nederlof, editors, +17th International Symposium on Parameterized and Exact Computation, IPEC 2022, September +7-9, 2022, Potsdam, Germany, volume 249 of LIPIcs, pages 9:1–9:18. Schloss Dagstuhl - Leibniz- +Zentrum f¨ur Informatik, 2022. +[5] ´Edouard Bonnet, Colin Geniet, Eun Jung Kim, St´ephan Thomass´e, and R´emi Watrigant. Twin- +width III: max independent set, min dominating set, and coloring. In Nikhil Bansal, Emanuela +Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and +Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 +of LIPIcs, pages 35:1–35:20. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2021. +6 + +[6] ´Edouard Bonnet, Colin Geniet, Eun Jung Kim, St´ephan Thomass´e, and R´emi Watrigant. Twin- +width II: small classes. Combinatorial Theory, 2 (2), 2022. +[7] ´Edouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, St´ephan Thomass´e, and +Szymon Toru´nczyk. Twin-width IV: ordered graphs and matrices. In Stefano Leonardi and Anupam +Gupta, editors, STOC ’22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, +Italy, June 20 - 24, 2022, pages 924–937. ACM, 2022. +[8] ´Edouard Bonnet, Eun Jung Kim, Amadeus Reinald, and St´ephan Thomass´e. Twin-width VI: the +lens of contraction sequences. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete +Algorithms (SODA), pages 1036–1056. SIAM, 2022. +[9] ´Edouard Bonnet, Eun Jung Kim, Amadeus Reinald, St´ephan Thomass´e, and R´emi Watrigant. Twin- +width and polynomial kernels. Algorithmica, 84:3300–3337, 2022. +[10] ´Edouard Bonnet, Eun Jung Kim, St´ephan Thomass´e, and R´emi Watrigant. Twin-width I: tractable +FO model checking. J. ACM, 69(1):3:1–3:46, 2022. +[11] ´Edouard Bonnet, O-joung Kwon, and David R. Wood. Reduced bandwidth: a qualitative strength- +ening of twin-width in minor-closed classes (and beyond). CoRR, abs/2202.11858, 2022. +[12] ´Edouard Bonnet, Jaroslav Neˇsetˇril, Patrice Ossona de Mendez, Sebastian Siebertz, and St´ephan +Thomass´e. Twin-width and permutations. CoRR, abs/2102.06880, 2021. +[13] Nicolas Bousquet, Aur´elie Lagoutte, Zhentao Li, Aline Parreau, and St´ephan Thomass´e. Identifying +codes in hereditary classes of graphs and VC-dimension. SIAM J. Discret. Math., 29(4):2047–2064, +2015. +[14] Marek Cygan, Fedor V Fomin, �Lukasz Kowalik, Daniel Lokshtanov, D´aniel Marx, Marcin Pilipczuk, +Micha�l Pilipczuk, and Saket Saurabh. Parameterized algorithms, volume 4. Springer, 2015. +[15] Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Micha�l Pilipczuk, +Roman Rabinovich, and Sebastian Siebertz. Neighborhood complexity and kernelization for nowhere +dense classes of graphs. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, +editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, +July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 63:1–63:14. Schloss Dagstuhl - +Leibniz-Zentrum f¨ur Informatik, 2017. +[16] Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of +parameterized preprocessing. Cambridge University Press, 2019. +[17] Florent Foucaud, George B. Mertzios, Reza Naserasr, Aline Parreau, and Petru Valicov. Identifi- +cation, location-domination and metric dimension on interval and permutation graphs. I. bounds. +Theor. Comput. Sci., 668:43–58, 2017. +[18] Jakub Gajarsk´y, Micha�l Pilipczuk, Wojciech Przybyszewski, and Szymon Toru´nczyk. Twin-width +and types. +In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th In- +ternational Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, +Paris, France, volume 229 of LIPIcs, pages 123:1–123:21. Schloss Dagstuhl - Leibniz-Zentrum f¨ur +Informatik, 2022. +[19] Stefan Kratsch, Florian Nelles, and Alexandre Simon. On triangle counting parameterized by twin- +width. CoRR, abs/2202.06708, 2022. +[20] Adam Marcus and G´abor Tardos. Excluded permutation matrices and the Stanley-Wilf conjecture. +J. Comb. Theory, Ser. A, 107(1):153–160, 2004. +[21] Jaroslav Neˇsetˇril and Patrice Ossona de Mendez. +On nowhere dense graphs. +Eur. J. Comb., +32(4):600–617, 2011. +[22] Jaroslav Neˇsetˇril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, +volume 28 of Algorithms and combinatorics. Springer, 2012. +7 + +[23] Adam Paszke and Micha�l Pilipczuk. VC density of set systems definable in tree-like graphs. In Javier +Esparza and Daniel Kr´al’, editors, 45th International Symposium on Mathematical Foundations of +Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, +pages 78:1–78:13. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2020. +[24] Micha�l Pilipczuk and Marek Soko�lowski. +Graphs of bounded twin-width are quasi-polynomially +χ-bounded. CoRR, abs/2202.07608, 2022. +[25] Micha�l Pilipczuk, Marek Soko�lowski, and Anna Zych-Pawlewicz. Compact representation for matri- +ces of bounded twin-width. In Petra Berenbrink and Benjamin Monmege, editors, 39th International +Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15-18, 2022, Mar- +seille, France (Virtual Conference), volume 219 of LIPIcs, pages 52:1–52:14. Schloss Dagstuhl - +Leibniz-Zentrum f¨ur Informatik, 2022. +[26] Wojciech Przybyszewski. VC-density and abstract cell decomposition for edge relation in graphs of +bounded twin-width. CoRR, abs/2202.04006, 2022. +[27] Douglas Rall and Peter J. Slater. On location-domination numbers for certain classes of graphs. +Congressus Numerantium, 45:97–106, 1984. +[28] Felix Reidl, Fernando S´anchez Villaamil, and Konstantinos S. Stavropoulos. Characterising bounded +expansion by neighbourhood complexity. Eur. J. Comb., 75:152–168, 2019. +[29] Vladimir N. Vapnik and Alexey Y. Chervonenkis. On the uniform convergence of relative frequencies +of events to their probabilities. In Measures of complexity, pages 11–30. Springer, 2015. +8 + diff --git a/AtE2T4oBgHgl3EQf8QmS/content/tmp_files/load_file.txt b/AtE2T4oBgHgl3EQf8QmS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77198212c469bb485674a459308569f42484ebc3 --- /dev/null +++ b/AtE2T4oBgHgl3EQf8QmS/content/tmp_files/load_file.txt @@ -0,0 +1,385 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf,len=384 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='04217v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='CO] 10 Jan 2023 Neighbourhood complexity of graphs of bounded twin-width∗ ´Edouard Bonnet† Florent Foucaud‡ § Tuomo Lehtil¨a¶ ‖ Aline Parreau∗∗ January 12, 2023 Abstract We give essentially tight bounds for, ν(d, k), the maximum number of distinct neighbourhoods on a set X of k vertices in a graph with twin-width at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Using the celebrated Marcus-Tardos theorem, two independent works [Bonnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', Algorithmica ’22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Przybyszewski ’22] have shown the upper bound ν(d, k) ⩽ exp(exp(O(d)))k, with a double-exponential dependence in the twin-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We give a short self-contained proof that for every d and k, ν(d, k) ⩽ (d + 2)2d+1k = 2d+O(log d)k, and build a bipartite graph implying ν(d, k) ⩾ 2d+log d+O(1)k, in the regime when k is large enough compared to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 1 Introduction The aim of this paper is to refine our understanding of how complex the neighbourhoods of graphs of bounded twin-width can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We provide an improved bound on the neighbourhood complexity of such graphs, complemented by a construction showing that our bound is essentially tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The improvements in the bounds for neighbourhood complexities translate directly to better structural bounds and algorithms, in some contexts which are explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width is a recently introduced graph invariant [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' see Section 2 for a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' It can be naturally extended to matrices over finite alphabets and binary structures [10, 7, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Although classes of bounded twin-width are broad and diverse, they allow (most of the time, provided a witness is given as an input) improved algorithms, compared to what is possible on general graphs or binary structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Most prominently, it was shown [10] that, on n-vertex graphs given with a d-sequence (a witness that their twin-width is at most d), deciding if a first-order sentence ϕ holds can be solved in time f(d, ϕ)n, for some computable function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In some special cases, such as for k-Independent Set or k-Dominating Set1, single-exponential parameterised algorithms running in time 2Od(k)n are possible [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In the same setting, the triangles of an n-vertex m-edge graph can be counted in time O(d2n+m) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' See [8, 18, 25] for more applications of twin-width with an algorithmic flavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Classes of binary structures with bounded twin-width include bounded treewidth, and more gener- ally, bounded clique-width classes, proper minor-closed classes, posets of bounded width (that is, whose antichains are of bounded size), hereditary subclasses of permutations, as well as Ω(log n)-subdivisions of ∗Florent Foucaud was financed by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) and by the ANR project GRALMECO (ANR-21-CE48-0004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Tuomo Lehtil¨a’s research was supported by the Finnish Cultural Foundation and by the Academy of Finland grant 338797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' †Univ Lyon, CNRS, ENS de Lyon, Universit´e Claude Bernard Lyon 1, LIP UMR5668, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' ‡Universit´e Clermont-Auvergne, CNRS, Mines de Saint-´Etienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont- Ferrand, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' §Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Orl´eans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orl´eans Cedex 2, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' ¶Univ Lyon, UCBL, CNRS, LIRIS - UMR 5205, F69622, France ‖University of Turku, Department of Mathematics and Statistics, Turku, Finland ∗∗Univ Lyon, CNRS, INSA Lyon, UCBL, Centrale Lyon, Univ Lyon 2, LIRIS, UMR5205, F-69622 Villeurbanne, France 1That is, the problems of deciding whether in an input graph, there are k vertices that are pairwise non-adjacent or whose closed neighbourhood is the entire vertex set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 1 n-vertex graphs [10], and particular classes of (bounded-degree) expanders [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' A rich range of geometric graph classes have bounded twin-width such as map graphs, bounded-degree string graphs [10], classes with bounded queue number or bounded stack number [6], segment graphs with no Kt,t subgraph, and visibility graphs of simple polygons without large independent sets [4], to give a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' If efficiently approximating the twin-width is a challenging open question in general, this is known to be possible for the above-mentioned classes (albeit a representation may be needed for the geometric classes) and for ordered graphs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' By that, we mean that there are two computable functions f, g and an algorithm that, for an input n-vertex graph G from the class and an integer k, and in time g(k)nO(1), either outputs an f(k)-sequence (again, witnessing that the twin-width is at most f(k)) or correctly reports that the twin-width of G is larger than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Structural properties of graph classes of bounded twin-width include χ-boundedness [5], even with a quasipolynomial binding function [24], smallness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', containing up to isomorphism 2O(n) n-vertex graphs) [6, 12], and Vapnik-Chervonenkis (VC) density at most 1 [9, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The latter property is the topic of the current article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' VC density and neighbourhood complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' VC density is related to the celebrated VC dimen- sion [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Given a set-system (or hypergraph) S on a domain X, the shatter function πS : N → N is defined as πS(n) = max A∈(X n) |{Y ⊆ A | ∃S ∈ S, Y = A ∩ S}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The Perles-Sauer-Shelah lemma states that πS(n) = O(nd) if the VC dimension of S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', the supremum of {n | πS(n) = 2n}) is a finite integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Then the VC density of S is defined as inf{c ∈ R | πS(n) = O(nc)}, and as +∞ if the VC dimension is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We define the VC density of an infinite class C of finite graphs as the VC density of the infinite set-system formed by the neighbourhood hypergraph of the disjoint union of the graphs of C, that is, {NG(v) | v ∈ V (⊎G∈CG)}, where NG(v) denotes the set of neighbours of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The VC density is an important measure in finite model theory, often more tractable than the VC dimension (see for instance [1, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Tight bounds have been obtained for the VC density of (logically) definable hypergraphs from graph classes of bounded clique-width [23] (with monadic second-order logic), and more recently, of bounded twin-width [18] (with first-order logic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In structural graph theory and kernelisation [16] (a subarea of parameterised complexity [14]) the function πN(G), where N(G) is the neighbourhood hypergraph of G, is often1 called neighbourhood com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' (See [3] for an algorithmic study of the computation of this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=') In these contexts, obtaining the best possible upper bound for πN(G) (and not just the exponent matching the VC density) translates to qualitatively better structural bounds and algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' see for instance [9, 11, 15, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The r-neighbourhood complexity of G is the neighbourhood complexity of Gr, with same vertex set as G, and an edge between two vertices at distance at most r in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Reidl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [28] showed that among subgraph-closed classes, bounded expansion2 is equivalent to linear r-neighbourhood complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Indeed, the more general nowhere dense classes [21] (another invention of the Sparsity program [22]) have almost linear r-neighbourhood complexity [15]: there is a function f : N × N → N such that for every ε > 0, πN(Gr)(n) ⩽ f(r, ε)n1+ε for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' On hereditary classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', closed under taking induced subgraphs, there is no known characterisation of linear neighbourhood complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' As we already mentioned in a different language, bounded twin-width classes have been proven to have linear neighbourhood complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' See [9, Lemma 3] or [26, Section 3] for two independent proofs, both using the Marcus-Tardos theorem [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' However, the dependence in the twin-width is doubly exponential in both papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Setting ν(d, k) as the maximum number of distinct neighbourhoods on a set of size k within a graph of twin-width at most d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', max{πN(G)(k) | G has twin-width at most d}, they show that ν(d, k) ⩽ exp(exp(O(d)))k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In this note, we give in Section 3 a self-contained proof (not using the Marcus-Tardos theorem) that ν(d, k) ⩽ 2d+O(log d)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Section 4, we complement that proof with a construction of a 1Some authors define the neighbourhood complexity as n �→ πN (G)(n) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 2A notion from the Sparsity theory of Neˇsetˇril and Ossona de Mendez [22] extending bounded degree and proper minor- free classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 2 bipartite graph witnessing that ν(d, k) ⩾ 2d+log d+O(1)k, which makes our single-exponential upper bound in twin-width essentially tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 2 Preliminaries We use the standard graph-theoretic notations: V (G), E(G), G[S], G − S respectively denote the vertex set, edge set, subgraph of G induced by S, and subgraph of G induced by V (G) \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' If v ∈ V (G), then NG(v) (or N(v) if G is clear from the context) denotes the set of neighbours of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' If X ⊆ V (G), then an X-neighbourhood is a set N(v) ∩ X for some v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We now define the twin-width of a graph, following the definition of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' A trigraph is a triple G = (V (G), E(G), R(G)) where E(G) and R(G) are two disjoint sets of edges on V (G): the usual edges (also called black edges) and the red edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Informally, a red edge between two vertices u and v means that some errors have been made between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The red degree of a trigraph is the maximum degree of the graph (V (G), R(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Any graph G can be interepreted as a trigraph G = (V (G), E(G), ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Given a trigraph and two vertices u, v ∈ V (G) (not necessarily adjacent), the trigraph G/u, v = G′ is obtained by contracting u and v in a new vertex w such that: V (G′) = {w} ∪ V (G) \\ {u, v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' the edges between vertices of V (G) \\ {u, v} are the same in G′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' the following edges are incident to w: – wx ∈ E(G′) if xu ∈ E(G) and xv ∈ E(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' – wx /∈ E(G′) ∪ R(G′) if xu /∈ E(G) ∪ R(G) and xv /∈ E(G) ∪ R(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' – wx ∈ R(G′) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In other words, the common black neighbours of u and v are black neighbours of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' All the other neighbours of u or v are red neighbours of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Red edges stay red, black edges stay black, red and black edges become red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We say that G/u, v is a contraction of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' A d-sequence of an n-vertex graph G is a sequence of n trigraphs Gn = G, Gn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='., G1 such that each trigraph Gi is obtained from Gi+1 by a contraction and has red degree at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The twin-width of G, denoted by tww(G), is the minimum integer d such that G admits a d-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Note that an induced subgraph of G has a twin-width smaller or equal to the twin-width of G [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' If u ∈ Gi, then u(G) denotes the set of vertices of G eventually contracted to u in Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Instead of considering the trigraphs Gi, we might prefer to deal with the partitions induced by the sets u(G) for u in Gi: Pi = {u(G) | u ∈ V (Gi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We say that there is a red edge between two parts u(G) and v(G) of Pi if uv is red in Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 3 Upper bound on the number of distinct neighbourhoods We state and prove our upper bound on the maximum number of distinct X-neighbourhoods in bounded twin-width graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let G be an n-vertex graph of twin-width d, and X ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Then the number of distinct X-neighbourhoods in G is at most (d + 2)2d+1|X| = 2d+O(log d)|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Fix X ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' First of all, for all vertices of V (G) \\ X with the same X-neighbourhood, we keep only one representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Note that the new graph G′′ is an induced subgraph of G, thus its twin-width is at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We further modify graph G′′ by adding for each v ∈ X a new vertex u to G′′ so that N(u) = N(v) if such vertex does not exist in V (G′′) \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The new graph is called G′ and it has the same twin-width as G′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let M = (d + 2)2d+1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We prove by induction on n that an n-vertex graph of twin-width at most d with a set X of k vertices, where all vertices outside X have a distinct X-neighbourhood, satisfies n ⩽ kM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' This will prove that G′ has at most kM vertices, and thus that in G, there are at most (M −1)k distinct X-neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The statement is trivially true for n ⩽ 5 since M ⩾ 5, for all d ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 3 Thus, assume n ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In particular, we have k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let X′ = X \\ {x} and let Tx be the set of pairs of vertices outside X that are twins with respect to X′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Tx = � {u, v} ∈ �V (G′) \\ X 2 � | N(u) ∩ X′ = N(v) ∩ X′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Since every vertex of V (G′) \\ X has a distinct neighbourhood in X, there are at most two vertices of V (G′) \\ X with the same (possibly empty) neighbourhood N in X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' namely the vertices u, v ∈ V (G′) \\ X with N(u) ∩ X = N and N(v) ∩ X = N ∪ {x} (if they exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Hence, Tx consists of pairwise-disjoint pairs of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We prove the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' There exists a vertex x of X such that Tx comprises at most M − 1 pairs, in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' By contradiction, assume this is not the case: for every x in X, Tx has size at least M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Consider a d-sequence of contractions G′ n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' , G′ 1 of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Consider the last step G′ i of the sequence where all the parts of Pi contain at most one vertex of X (that is, contrary to Pi, some part of Pi−1 contains two vertices of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let P be a part of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let x be the unique (if there exists one) element of P ∩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Then we claim that |P \\ X| ⩽ 2d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Indeed, any two vertices of P \\ X have some vertex in the symmetric difference of their X-neighbourhoods, either it is x, or some vertex x′ of X outside P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' If that distinguishing vertex is some x′ that is not in P, then there has to be a red edge between P and the part that contains x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' There are at most d red edges with P as an extremity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Since all the elements of X are in distinct parts in G′ i, it means that d + 1 vertices of X are enough to distinguish all the X-neighbourhoods of vertices of P \\ X, and thus |P \\ X| ⩽ 2d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We now consider the next contraction in the sequence, which leads to G′ i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' By definition of G′ i, it must contract two vertices corresponding to two parts of Pi that both contain an element of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let x1 and x2 be these two elements of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let Q be the part of Pi−1 that contains both x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' By our assumption, Tx1 has size at least M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let {u, v} be a pair of Tx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Since u and v have the same neighbourhood in X \\ {x1}, it means that they are either both adjacent or both non-adjacent to x2, and exactly one of them is adjacent to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, necessarily, one vertex among the pair {u, v} is adjacent to exactly one vertex among {x1, x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In particular, if this vertex is not in Q, then there has to be a red edge between the part containing this vertex and the part Q in G′ i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Since Tx1 contains at least M pairs (which are disjoint) and Q has at most 2d+2 vertices not in X, there are at least M − 2d+2 vertices not in X whose part in G′ i−1 has a red edge to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Since each other part has at most 2d+1 vertices not in X, it makes at least M−2d+2 2d+1 red edges incident to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, we must have M−2d+2 2d+1 ⩽ d, leading to M ⩽ 2d+1(d + 2), a contradiction that proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' (□) By Claim A, there exists a vertex x ∈ X such that |Tx| ⩽ M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let Y be a set of |Tx| vertices that intersects each pair of Tx exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let GY = G′ − (Y ∪ {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Then, X′ = X \\ {x} is a vertex set of size k − 1 such that all X′-neighbourhoods of vertices outside X′ are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The graph GY has at least n − M vertices, and twin-width at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' By induction, we have n − M ⩽ |V (GY )| ⩽ (k − 1)M and thus, n ⩽ kM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Hence, once we recall that no vertex in X has unique X-neighbourhood, there are at most (M − 1)k distinct X-neighbourhoods, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 4 Lower bound on the number of distinct neighbourhoods Notice that when |X| and tww(G) are roughly the same, the bound from Theorem 1 cannot be sharp, since G′ has at most 2|X| + |X| vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' However, when |X| is large enough compared to tww(G), we next show that the bound is sharp up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' There is a positive constant c, such that for any integer d, there is a bipartite graph G of twin-width at most d, and a large enough set X ⊆ V (G), with at least c·d2d|X| distinct X-neighbourhoods in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Observe that the claim is clearly true for any small d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, we do not need to consider separately graphs with small twin-width upper bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Hence, we assume from now on that d ≥ d′ where d′ is some positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 4 We construct the graph G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let A, B, C ∈ Z be three constants that will be given later (A and B will be roughly equal to √ d and C will be roughly equal to d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', xk} be an independent set of k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Our goal is that each vertex in V (G) \\ X has a unique X-neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' For any integers i, j, t with 1 ⩽ i ⩽ j ⩽ i + A − 1, j + 2 ⩽ t ⩽ j + 1 + B and t ⩽ k − C, we create a set Vi,j,t of vertices as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Consider the set Xt = {xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', xt+C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' For every subset Y of Xt, let Y ′ = {xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', xj, xt} ∪ Y and add a vertex vY ′ to Vi,j,t, making it adjacent to the vertices of Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Each set Vi,j,t has size 2C and there are Θ(kAB) (for fixed A and B and growing k) such sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus there are Θ(kAB2C) vertices in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Any two vertices not in X have distinct X-neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Indeed, by considering the natural ordering of X induced by the indices, any vertex not in X is first adjacent to a consecutive interval of vertices from xi to xj, then is not adjacent to vertices from xj+1 to xt−1 (which is not empty since t ⩾ j + 2), and then adjacent to xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, if two vertices have the same X-neighbourhood, they must be in the same set Vi,j,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' But then, they have a distinct neighbourhood in {xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', xt+C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We now prove that the twin-width of G is at most M = max{AB, C}+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' For that, we give a sequence of contractions with red degree at most M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The contraction sequence is split into k − C steps, for each vertex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let 0 ≤ i ≤ k − C − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Step 0 corresponds to the starting point, where each vertex is alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let i ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' After Step i, there will be the following parts in the corresponding partition (vertices not in any part have not yet been contracted): For each j, t such that i ⩽ j ⩽ i + A − 1 and j + 2 ⩽ t ⩽ j + 1 + B, there is a part Bj,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The parts Bi,t (parts with j = i), contain all the vertices of the sets Vi′,j′,t such that j′ ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The parts Bj,t with j > i contain all the vertices of the sets Vi′,j′,t such that i′ ⩽ i and j′ = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Note that there are AB non-empty Bj,t parts in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' There is a part X0 that contains vertices from x1 to xi of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' There is a part T (for “trash”) that contains all the vertices of the sets Vi′,j,t with t ⩽ i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' All the other vertices are not yet contracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' This corresponds to the vertices from xi+1 to xk of X and to the vertices of the sets Vi′,j,t with i′ > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Indeed, if i′ ⩽ i and t ⩽ i + 1, then the vertices of Vi′,j,t are in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' If t ⩾ i + 2 but j ⩽ i, then they are in the part Bi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' If j > i, then they are in the part Bj,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We first prove that the red degree after Step i is at most M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Then, we explain how to get from Step i to Step i + 1 by keeping the red degree at most M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Consider the part Bj,t at the end of Step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' A vertex in this part belongs to some set Vi′,j′,t with i′ ⩽ i and j′ = j if j > i or j′ ⩽ i otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In particular, two vertices of Bj,t are adjacent to all the vertices between xi+1 and xj, to no vertex between xj+1 and xt−1, to xt, and to no vertex after xt+C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, there is a red edge between the parts Bj,t and X0, and C red edges between the part Bj,t and the vertices {xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', xt+C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Therefore, the number of red edges incident with Bj,t is at most C + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Consider now the part T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Vertices in T are adjacent only to vertices of X up to xi+C+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Since vertices x1 to xi are all in the part X0, the red degree of T is at most C + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Single vertices not in X have no incident red edges: indeed, they are all in some sets Vi′,j,t for i′ > i and thus are not adjacent to any vertex of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' For the same reason, there are red edges incident to X0 only to T and to the parts Bj,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Hence, the red degree of X0 is at most AB + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Similarly, the red degree of xi′, i′ > i + 1 is at most AB + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Moreover, the red degree of xi+1 is at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Indeed, the only red edge is between xi+1 and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Finally, the red degree after step i is at most max{AB + 1, C + 2} ⩽ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Let i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We now explain how we perform the contractions to go from step i to step i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' (only if i ≥ 1) For any i + 3 ⩽ t ⩽ i + 2 + B, merge the part Bi,t with the part Bi+1,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The only new red edge this merging may lead to, when Bi,t is non-empty, is between Bi+1,t and xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, we add only one red edge between xi+1 and Bi+1,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, the red degree of Bi+1,t is at most C + 2 and the red degree of xi+1 is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Add all the vertices of Vi+1,j,t for some j, t to the part (that might be empty at this point) Bj,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The red degree of Bj,t is at most C + 2 since we might have a red edge between Bj,t and xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The number of nonempty parts Bj,t at this point is AB + 1 (there is still the part Bi,i+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Adding T , this gives AB + 2 red edges incident to a vertex in X (or from part X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Add xi+1 to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The part X0 has red edges only to parts Bi+1,t, to Bi,i+2 and to T , but no edges to the single vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, it has red degree at most AB + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Put the part Bi,i+2 into T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' This part is only adjacent to vertices up to xi+2+C, and thus has C + 2 red edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, at each point, the red degree is always at most M = max{AB, C} + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' The process ends at step i = k − C − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Then, all the vertices not in X are in some parts, and there are at most AB + 1 such parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' On the other side of bipartition, we have part X0 and C + 1 single vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, the graph is bipartite with both sides of size at most M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' One can contract each part independently to finish the contraction sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' To conclude, taking C = d − 2 and A = B = ⌊ √ d − 2⌋, we have M ⩽ d and kAB2C = Θ(kd2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Notice that we may assume that A, B and C are positive since d ≥ d′ where d′ was some well chosen positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 5 Conclusion We have given an improved and tight upper bound for the neighbourhood complexity of graphs of bounded twin-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Unlike the previously known (weaker) bounds, our method is simple and avoids the use of the Marcus-Tardos theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We hope that it can inspire future works in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' It is known that the twin-width of Gr can be upper-bounded by a function of the twin-width of G and r [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, graphs of twin-width at most d have linear r-neighbourhood complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We leave as an interesting open problem to obtain an essentially tight twin-width dependence for the r-neighbourhood complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' We remark that the neighbourhood complexity is also related to identification problems on graphs such as identifying codes or locating-dominating sets, where one seeks a (small) set A of vertices of a graph such that all other vertices have a distinct neighbourhood in A [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Some works in this area about specific graph classes, are equivalent to the study of the neighbourhood complexity of these graph classes: see for example [13, 17, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Moreover, we note that for graph classes with VC density 1, since any solution has linear size, the natural minimisation versions of the above identification problems have a polynomial-time constant-factor approximation algorithm (trivially select the whole vertex set), while such an algorithm is unlikely to exist in the general case [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Thus, our work implies a better approximation ratio for these problems, when restricted to input graph classes of bounded twin-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' References [1] Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson, and Sergei Starchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Vapnik–chervonenkis density in some theories without the independence property, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Notre Dame Journal of Formal Logic, 54(3-4):311–363, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [2] Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson, and Sergei Starchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Vapnik-chervonenkis density in some theories without the independence property, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Transactions of the American Mathematical Society, 368(8):5889–5949, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [3] Cristina Bazgan, Florent Foucaud, and Florian Sikora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Parameterized and approximation complexity of partial VC dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', 766:1–15, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [4] ´Edouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen K¨ohler, Raul Lopes, and St´ephan Thomass´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width VIII: delineation and win-wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Holger Dell and Jesper Nederlof, editors, 17th International Symposium on Parameterized and Exact Computation, IPEC 2022, September 7-9, 2022, Potsdam, Germany, volume 249 of LIPIcs, pages 9:1–9:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz- Zentrum f¨ur Informatik, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [5] ´Edouard Bonnet, Colin Geniet, Eun Jung Kim, St´ephan Thomass´e, and R´emi Watrigant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin- width III: max independent set, min dominating set, and coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 35:1–35:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 6 [6] ´Edouard Bonnet, Colin Geniet, Eun Jung Kim, St´ephan Thomass´e, and R´emi Watrigant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin- width II: small classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Combinatorial Theory, 2 (2), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [7] ´Edouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, St´ephan Thomass´e, and Szymon Toru´nczyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width IV: ordered graphs and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Stefano Leonardi and Anupam Gupta, editors, STOC ’22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 924–937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' ACM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [8] ´Edouard Bonnet, Eun Jung Kim, Amadeus Reinald, and St´ephan Thomass´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width VI: the lens of contraction sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1036–1056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' SIAM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [9] ´Edouard Bonnet, Eun Jung Kim, Amadeus Reinald, St´ephan Thomass´e, and R´emi Watrigant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin- width and polynomial kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Algorithmica, 84:3300–3337, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [10] ´Edouard Bonnet, Eun Jung Kim, St´ephan Thomass´e, and R´emi Watrigant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width I: tractable FO model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' ACM, 69(1):3:1–3:46, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [11] ´Edouard Bonnet, O-joung Kwon, and David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Reduced bandwidth: a qualitative strength- ening of twin-width in minor-closed classes (and beyond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' CoRR, abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='11858, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [12] ´Edouard Bonnet, Jaroslav Neˇsetˇril, Patrice Ossona de Mendez, Sebastian Siebertz, and St´ephan Thomass´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width and permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' CoRR, abs/2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='06880, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [13] Nicolas Bousquet, Aur´elie Lagoutte, Zhentao Li, Aline Parreau, and St´ephan Thomass´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Identifying codes in hereditary classes of graphs and VC-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Discret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', 29(4):2047–2064, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [14] Marek Cygan, Fedor V Fomin, �Lukasz Kowalik, Daniel Lokshtanov, D´aniel Marx, Marcin Pilipczuk, Micha�l Pilipczuk, and Saket Saurabh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Parameterized algorithms, volume 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [15] Kord Eickmeyer, Archontia C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Giannopoulou, Stephan Kreutzer, O-joung Kwon, Micha�l Pilipczuk, Roman Rabinovich, and Sebastian Siebertz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Neighborhood complexity and kernelization for nowhere dense classes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 63:1–63:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [16] Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Kernelization: theory of parameterized preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Cambridge University Press, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [17] Florent Foucaud, George B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Mertzios, Reza Naserasr, Aline Parreau, and Petru Valicov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Identifi- cation, location-domination and metric dimension on interval and permutation graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', 668:43–58, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [18] Jakub Gajarsk´y, Micha�l Pilipczuk, Wojciech Przybyszewski, and Szymon Toru´nczyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Twin-width and types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Mikolaj Bojanczyk, Emanuela Merelli, and David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Woodruff, editors, 49th In- ternational Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 123:1–123:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [19] Stefan Kratsch, Florian Nelles, and Alexandre Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' On triangle counting parameterized by twin- width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' CoRR, abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='06708, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [20] Adam Marcus and G´abor Tardos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Excluded permutation matrices and the Stanley-Wilf conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Theory, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' A, 107(1):153–160, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [21] Jaroslav Neˇsetˇril and Patrice Ossona de Mendez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' On nowhere dense graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', 32(4):600–617, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [22] Jaroslav Neˇsetˇril and Patrice Ossona de Mendez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 7 [23] Adam Paszke and Micha�l Pilipczuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' VC density of set systems definable in tree-like graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Javier Esparza and Daniel Kr´al’, editors, 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic, volume 170 of LIPIcs, pages 78:1–78:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [24] Micha�l Pilipczuk and Marek Soko�lowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Graphs of bounded twin-width are quasi-polynomially χ-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' CoRR, abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='07608, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [25] Micha�l Pilipczuk, Marek Soko�lowski, and Anna Zych-Pawlewicz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Compact representation for matri- ces of bounded twin-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Petra Berenbrink and Benjamin Monmege, editors, 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15-18, 2022, Mar- seille, France (Virtual Conference), volume 219 of LIPIcs, pages 52:1–52:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [26] Wojciech Przybyszewski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' VC-density and abstract cell decomposition for edge relation in graphs of bounded twin-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' CoRR, abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content='04006, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [27] Douglas Rall and Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Slater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' On location-domination numbers for certain classes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Congressus Numerantium, 45:97–106, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [28] Felix Reidl, Fernando S´anchez Villaamil, and Konstantinos S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Stavropoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Characterising bounded expansion by neighbourhood complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=', 75:152–168, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' [29] Vladimir N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Vapnik and Alexey Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Chervonenkis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' On the uniform convergence of relative frequencies of events to their probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' In Measures of complexity, pages 11–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQf8QmS/content/2301.04217v1.pdf'} diff --git a/BtE1T4oBgHgl3EQfpgUG/content/tmp_files/2301.03331v1.pdf.txt b/BtE1T4oBgHgl3EQfpgUG/content/tmp_files/2301.03331v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..47faa8e5bb4f12c448314dde13a12170463c45dc --- /dev/null +++ b/BtE1T4oBgHgl3EQfpgUG/content/tmp_files/2301.03331v1.pdf.txt @@ -0,0 +1,890 @@ +1 +A Specific Task-oriented Semantic Image +Communication System for substation patrol +inspection +Senran Fan, Haotai Liang, Chen Dong*, Xiaodong Xu, Geng Liu +Abstract—Intelligent inspection robots are widely used in +substation patrol inspection, which can help check potential +safety hazards by patrolling the substation and sending back +scene images. However, when patrolling some marginal areas with +weak signal, the scene images cannot be sucessfully transmissted +to be used for hidden danger elimination, which greatly reduces +the quality of robots’ daily work. To solve such problem, +a Specific Task-oriented Semantic Communication System for +Image—–STSCI is designed, which involves the semantic features +extraction, transmission, restoration and enhancement to get +clearer images sent by intelligent robots under weak signals. +Inspired by that only some specific details of the image are +needed in such substation patrol inspection task, we proposed +a new paradigm of semantic enhancement in such specific +task to ensure the clarity of key semantic information when +facing a lower bit rate or a low signal-to-noise ratio situation. +Across the reality-based simulation, experiments show our STSCI +can generally surpass traditional image-compression-based and +channel-coding-based or other semantic communication system +in the substation patrol inspection task with a lower bit rate even +under a low signal-to-noise ratio situation. +Index Terms—Semantic Communication, substation patrol +robot, STSCI +I. INTRODUCTION +W +Ith the development of Internet Technology especially +in intelligent applications like IoT fields, the fierce +demand for tremendous amount of information transmissions +is becoming inevitable, which urges people to continuously +improve the efficiency in communication process. However, +the transmission rate based on traditional communication +system in physical layer has already been approaching the +Shannon limit under most situations, so researchers are willing +to explore new theories and new forms of communication +systems. +Based on this, the concept of semantic communication has +attracted more and more attention. First mentioned in Shannon +Senran Fan, Haotai Liang are with the State Key Laboratory of Network- +ing and Switching Technology, Beijing University of Posts and Telecom- +munications, Beijing, 100876, China. (E-mail: FSR@bupt.edu.cn; lianghao- +tai@bupt.edu.cn) +Xiaodong Xu is with the State Key Laboratory of Networking and +Switching Technology, Beijing University of Posts and Telecommunications, +Beijing, China, and also with the Department of Broad-band Communica- +tion, Peng Cheng Laboratory, Shenzhen, Guangdong, China. (E-mail: xuxi- +aodong@bupt.edu.cn) +Geng Liu is with the Beijing Smart-chip Microelectronics Technology +Co.,Ltd. (E-mail: liugeng@sgchip.sgcc.com.cn) +*Chen Dong is the corresponding author and with the State Key +Laboratory of Networking and Switching Technology, Beijing Univer- +sity of Posts and Telecommunications, Beijing, 100876, China. (E-mail: +dongchen@bupt.edu.cn) +and Weaver’s paper [1], the semantic-based communication +system is believed to be a new and bright direction for +communication feilds. The explosion of data requires the +communication systems to greatly upgrade their ability in +data compression, while the semantic-based communication +system is suitable for it. Considering that when transmitting +the information, large amount of task-irrelevant information +is involved especially in some specific communication scenes, +which leads to massive waste in communication resources. +Especially in this task, intelligent substation patrol inspection, +what really cared about is only the key semantic contents +such as the areas with key units of the image. Introduced +in [2], [3], by transmitting the information through semantic +feature extracting, transmission and reconstruction, semantic +communication system only keeps the effective information +which achieves extremely compression and high efficiency +communication. +Deep learning can be an answer to reciously extract the se- +mantic features from the image. Indeed, using neural networks +to make semantic analysis from images is a large subject in +the computer vision fields. Semantic segmentation networks +[4]–[6] as well as target detection networks [7]–[10] show +great power in semantic features extracting and analysis. At the +same time, GAN-based [11] networks is possessed of ability +in handling semantic features. GAN-based networks can gen- +erator images from semantic vectors. Moverover in InfoGAN +[12] and StyleGAN [13], semantic vector can be edited to +control the features of the generated images. And comes into +unsupervised feilds, auto-encoder [14] is an inspiring archi- +tecture for semantic feature extracting. Compressing the high- +dimensional data into a low-dimensional latent which is used +for data reconstruction. Auto-encoder forced the reconstruction +results to get close enough to the original ones. Combined with +semantic-related networks such as GANs and the structure +of auto-encoder, the system can realize the aim to decrease +distortion in semantic contents of images during the process +of extreme compression as well as transmission. +The traditional communication systems involve source cod- +ing and channel coding. We replace the former part with +neural networks, and to resist against noise in channels we +decide to use the Joint Source-Channel Coding(JSCC) [15], +[16]. Leading the channel simulation models into the deep +networks, the networks can perform well in real-world channel +conditions and even better than traditional channel coding +methods like LDPC especially in low bite rate or low signal- +to-noise ratio situation. +arXiv:2301.03331v1 [cs.CV] 9 Jan 2023 + +2 +Figure 1. The framework of STSCI. +The above studies shows the possiblity in applying the +semantic communication system into the subtsation patrol +inspection task. At the same time, substations do have toubles +in intelligent patrol task. When the robots patrolling the +marginal areas of substation with weak signals, the images +sent back by robots can be too blurred to be used for +security check. Considering that semantic communication has +potential to be the answer for solving such problem and +no literature been published before is trying to apply the +semantic communication technology to the substation patrol +task, a specific task-oriented semantic communication system +STSCI is proposed for solving this specific task and the +similar communication tasks featured by the fixed image +source, fixed channel conditions and focusing only on some +specific task-oriented semantic contents of the image. The +system is mainly a GAN-based auto-encoder-structure network +for image’ compressing, transmission and reconstruction. In +addition, a yolo-net is involved to locate the images’ specific +semantic contents, which will then be embedded and sent to +the semantic enhancement models to improve the transmission +quality of the important semantic contents of the images to +make sure there is no errors or missing when making security +check with the transmitted images. The main contributions of +this paper are summarized as follows. +(1) A specific task-oriented semantic communication system +for image is proposed for the transmission of images +obtained by intelligent robots in the substation patrol +inspection task. A new paradigm of key semantic contents +extraction and preservation for such specific tasks is +proposed. A Yolo networks is involved to locate the +key semantic contents which is the task exactly cares +about, while the located part will be sent into a semantic +enhancement models to enhance the transmission quality +of the located areas. +(2) A GAN-based auto-encoder structure network is de- +signed. Combined with RRDB blocks, channel normal- +ization, idea of conditional gan and some other tricks, +the network can extremely compress the images into the +semantic feature latent and reconstruct them after the +transmission. +(3) Through simulations and experiments, this paper show +the application and performance of the semantic commu- +nication system in haddling the specific task. By present +the metrics, semantic communication system is proven +to be superior to the traditional communication systems +in such specific tasks with fixed image source and fixed +channel conditions. As a practice, the STSCI has better +transmission quality especially under low bit rate or low +signal-to-noise ratio channel conditions compared with +the traditional communication systems, which signifi- +cantly enlarge the areas covered by effective signal to +ensure the proper work of the intelligent robots when +patrolling the marginal areas of the substation with weak +signal. +This paper is arranged as follows. In section II, we review +the structure of the specific task-oriented semantic commu- +nication system for image STSCI, and show details in the +model architectures and training flow path of two parts of +STSCI. Then, in section III, a direct comparison between the +STSCI and other image communication systems is provided to +quantify the performance of STSCI with the proposed method. +Finally, conclusions of this paper are drawn in section IV. +II. SPECIFC TASK-ORIENTED SEMANTIC COMMUNICATION +SYSTEM +Shown in Fig. 1, the specific task-oriented semantic com- +munication system for image(STSCI) is mainly composed of +two parallel parts: the base system and semantic enhance- +ment system. The base system is mainly a GAN-based auto- +encoder network to achieve images’ compression, transmission +and reconstruction through semantic features. Meanwhile, the +semantic enhancement system locates the areas with key +semantic contents of the image and improves these areas’ +quality during transmission. Both of the two parts will be +introduced in detail in the following contents. + +Semantic +Semantic +Channels +Encoder +Decoder +Base system +Semantic enhancement system +Enhancement +个 +个 +Yolo-Net +Model +Receiver3 +Figure 2. The architecture of the base system. +A. Base System +Shown in Fig. 2, the base system is mainly a neural network +consists of three parts: an Encoder network, simulated channel +models and a GAN-based Decoder network. The images +gained by substation patrol inspection robots will be com- +pressed by Encoder network, sent to receiver through physical +channels simulated by channel models and reconstructed by +the Decoder network. +The most frequently proposed semantic-based communica- +tion systems used the structure of auto-encoder to achieve +image compression, however traditional CNN and loss in +auto-encoder have difficulties in acquiring high quality re- +constructed images. In pace with development in image en- +hancement task especially image de-noising and image super- +resolution, GANs have been proven to be possessed of strong +talents in high-quality image generation, which were pre- +viously employed to improve the visual quality of image +compression systems [17], [18]. Inspired by these previous +studies, we decide to use GAN to replace the traditional CNN +as decoder network in auto-encoder to significantly improve +the quality and similarity of images transmitted by semantic +communication system. +Meanwhile, considering that structure of auto-encoder in- +volved in semantic communication system is highly consistent +with the information communication process, Joint Source- +Channel Coding(JSCC) was proposed in [15]. No longer need +additional channel coding like LDPC to resist against noise +in channels, adding noise through simulation channel models +when training auto-encoder networks, an anti-noise communi- +cation system is formed, which can ensure high-quality image +transmission even under low signal-to-noise ratio situation. +Though JSCC methods has its limitation for being constrained +by specific source, specific scene and specific task, which +lead to deep-based semantic communication system’s lack of +generalization. However, in this task, the information source +and channels are fixed, such constrains can be ignored. In +addition, the data of channel conditions in the substation +can be collected continuously in the practical application to +fine-tune simulation channel model to improve the system’s +performance in this specific task. +In terms of the loss functions which plays a decisive role in +training the networks, MSE loss and LPIPS loss is chosen +to measure the distortion between the original images and +the generated ones. MSE loss measures the difference per +pixel and shows their distance in the high-dimension space, +which helps keep the similarity. At the same time, LPIPS +loss proposed in [19] is calculated through a VGG-net which +has been trained previously. Having special model structure +and trained with tricks, the pre-trained VGG-net gives more +attention to the structure and texture of the images and does +well in telling such kind of difference between images. It’s +the difference in structure and texture that is of importance +but hard to measure through tradition losses such as L1 loss +or MSE loss. LPIPS loss helps supply this gap, and makes the +generated images more close to the original ones in visual. +In fact, before the final training, the encoder and GAN-based +decoder is trained by only using the two mentioned losses +instead of involving the adversarial loss at the beginning. Such +tricks were also applied in [17], [20], [21]. Initializing the +generator net in this way helps the generator performs better +in the final training process so the discriminator can learn +more useful information and the adversarial loss can be more +rational. Otherwise if skip this process, the images generated +by the generator is far from the ground truth and easy for +discriminator to tell, which may lead to the vanishing gradient +of the generator. +Speaking of the adversarial loss, the structure of our dis- +criminator is abnormal. Inspired by [17], the discriminator +shares the structure of that in conditional GAN. Receiving +not only the generated images as well as the ground truth but +also the latent which puts into the generator, the discriminator +no longer only focus on the quality of generator images. +With limitation from the different latent involved in, the +discriminator is forced to take attention to the connections +between the latent and the image and the difference between +images with different latent, so the adversarial loss covers +more useful information to help the network performs better +in reconstruction process. +According to all above introductions and the structure +shown in Fig. 2, the complete process of the base system is +as follows. +The image 𝑋 to be transmitted is sent in to the Encoder +first to get the semantic features 𝑌, +𝑌 = 𝐸(𝑥). +(1) + +Encoder +Gan-based +D +Residual-in-Residual +Decodel +Conv Norm ReLU + Conv Norm ReLU +Upsample Conv +Upsample Conv +Upsample Conv +X' +Channel +Vgg networks +X- +Conv +Dense Blocks +Model + concat4 +The nearest neighbor quantization operation is then performed +on the extracted semantic features 𝑌, +𝑌𝑞(𝑖) = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑗||𝑌𝑖 − 𝐼 𝑗||. +(2) +Where the set 𝐼 of quantization centers is: +𝐼 = {𝐼0, 𝐼1, ..., 𝐼 𝑗, ..., 𝐼𝑙}. +(3) +According to JSCC, then the quantized semantic feature 𝑌𝑞 is +sent to the simulated channel models. In this paper, AWGN +model is chosen as the simulated channel model, +𝑌𝑞 +′ = ℎ · 𝑌𝑞 + 𝑛. +(4) +In this formula, ℎ represents the channel gain, while 𝑛 rep- +resents the independent identically distributed Gaussian noise. +Such model simulate the feature’s distortion transmitted in the +real-world channel and give the base model the ability to resist +the noise. +The image 𝑋 +′ is generated by the Generator(the Decoder +network) from the processed latent 𝑌𝑞 +′ at the receiver, +𝑋 +′ = 𝐺(𝑌𝑞 +′). +(5) +The Encoder maps the source image 𝑋 to a specific distribution +𝑃𝑋. The generator G tries to map samples 𝑌 from a fixed +known distribution 𝑃𝑌 to 𝑃𝑋, while the Discriminator D is +learned to tell the difference between such two distributions +using the sampled data 𝑋 and the generated 𝑋 +′.A properly +trained Discriminator helps the Generator to find and simulate +the distribution 𝑃𝑋 more preciously. Involving the idea of +conditional GANs as mentioned before, the adversarial loss +is as follows. +𝐿𝐺 = −𝑙𝑜𝑔(𝐷(𝑋 +′,𝑌𝑞 +′)), +(6) +𝐿𝐷 = −𝑙𝑜𝑔(1 − 𝐷(𝑋 +′,𝑌𝑞 +′)) − 𝑙𝑜𝑔(𝐷(𝑋,𝑌𝑞 +′)). +(7) +Besides, when optimizing the Encoder and the Generator, the +MSE loss and the LPIPS loss are also involved to measure the +texture and perception distance between the source image X +and the generator image 𝑋 +′. Moverover, helping to initialize +these two networks, these two kinds of loss guide the Gener- +ator and Discriminator to be trained on the right direction. So +the final loss for the Encoder and the Generator are as follows. +In the initial training: +𝐿𝐸𝐺 = ||𝑋 − 𝑋 +′|| + 𝛼𝐿𝑃𝐼𝑃𝑆(𝑋, 𝑋 +′). +(8) +In the final training: +𝐿𝐸𝐺 = ||𝑋−𝑋 +′||+𝛼𝐿𝑃𝐼𝑃𝑆(𝑋, 𝑋 +′)+𝛽[−𝑙𝑜𝑔(𝐷(𝑋 +′,𝑌𝑞 +′)]. (9) +B. Semantic Enhancement System +The Semantic Enhance System is designed to enhance +transmission quality of the key semantic contents which is +cared about in the specific task such as the panels or electrical +insulators in the intelligent substation patrol inspection task. +Figure 3. The process of the semantic enhancement system. +The system consists of two parts: a Yolo net to locate the +area with key semantic contents which will be sent into the +base model as well as a enhancement network which can get +more precious and high quality images at the receiver with the +input of the transmitted image and the areas with key semantic +contents. +In this paper, target detection network yolo-net is involved +to locate the key semantic contents instead of the semantic +segmentation network such as Unet or FCN in some other se- +mantic communication systems like [18], the principal reason +is as follows. +The pre-trained Yolo-net has the ability in finding and locat- +ing the objects which need to be shoot during the patrol task, +which is not only used to locate and mark the area containing +key semantic information during the semantic communication +process, but can also help the intelligent robots to judge +whether there exists objects in the patrol list to shoot and +how to change the position, angle and focal length of camera +to get a sharper image. Under the constraint of storage space +in the patrol robot, Yolo-net which can do multiple jobs is a +rather cost-effective choice. +As shown in Fig. 3, semantic enhancement system’s process +is as follows. +The area 𝑋𝑠𝑢𝑏 with key semantic contents is located by the +yolo-net with input of source image 𝑋, + +Yolo-net +不 +BASE +SYSTEM +个 +Semantic +Enhancement +Model +不 +Xsub1 +Xsub +X final5 +𝑋𝑠𝑢𝑏 = 𝑌𝑜𝑙𝑜 𝑛𝑒𝑡(𝑋). +(10) +After sent into the base model, 𝑋𝑠𝑢𝑏 is encoded, transmitted +and finally reconstructed as the 𝑋𝑠𝑢𝑏 +′ at the receiver, +𝑋𝑠𝑢𝑏 +′ = 𝐵𝑎𝑠𝑒 𝑠𝑦𝑠𝑡𝑒𝑚(𝑋𝑠𝑢𝑏). +(11) +At the same time, the whole image 𝑋 is transmitted through +the base model to get another area 𝑋𝑠𝑢𝑏1 +′ with key semantic +contents cut from the reconstructed image 𝑋 +′. The difference +between these two sub-images is calculated as follows. +𝑋𝑑𝑖 𝑓 𝑓 = 𝑋𝑠𝑢𝑏1 +′ − 𝑋 +′ +𝑠𝑢𝑏. +(12) +The DIFF image is sent to the semantic enhancement model +whose job is to balance the difference between two sub- +image to make full use of these extra information to let the +transmitted image as close as the original one in the area with +key semantic contents, +𝑋𝑑𝑖 𝑓 𝑓 +′ = 𝐸𝑛ℎ𝑎𝑛𝑐𝑒𝑚𝑒𝑛𝑡 𝑀𝑜𝑑𝑒𝑙(𝑋𝑑𝑖 𝑓 𝑓 ). +(13) +The final image is formed as follows. +𝑋 𝑓 𝑖𝑛𝑎𝑙 = 𝑋 +′ + 𝑋𝑑𝑖 𝑓 𝑓 +′. +(14) +In this task, the similarity between the final image 𝑋 𝑓 𝑖𝑛𝑎𝑙 +and the original image 𝑋 is focused on, which can help +decrease the possibility of errors or missing during analyzing +the images. So we choose the MSE loss and SSIM loss to +optimize the semantic enhancement models, and parameters +in the yolonet as well as the base model are fixed during the +optimization, +𝐿𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑚𝑒𝑛𝑡 = ||𝑋 − 𝑋 𝑓 𝑖𝑛𝑎𝑙|| + 𝛼𝑆𝑆𝐼𝑀(𝑋, 𝑋 𝑓 𝑖𝑛𝑎𝑙). +(15) +In the end of Section II, the details of networks involved in +the STSCI is shown in table I and table II. +III. EXPERIMENTAL RESULTS +This section is mainly introduced the relevant testing set- +tings, including the dataset for STSCI’s train and test, the +introduction of baseline as well as evalation metrics and the +performance for the STSCI in different metrics. +Discription and figures are given to show how the STSCI +surpass the traditional image communication system or other +semantic system under some specific situations. +A. Dataset for train and test +The training dataset is formed of 10000 images sampled +from the COCO2014 dataset while 200 images of substation +are used to fine-tune the base system to improve the STSCI’s +performance in the intelligent substation patrol inspection task. +During the testing process, the images from COCO2014 +testset which are not involved in training process are sampled +to measure the metrics of the communication systems. +B. Baseline and Evaluation metrics +The widely used image compression technology JPEG and +JPEG2000 are used as baseline for the image compression. +Table I +BASE SYSTEM +Model +Layers +Encoder +Conv2d,kernel=(7,7),stride=(1,1),channels=64 +Conv2d,kernel=(3,3),stride=(2,2),channels=128 +Conv2d,kernel=(3,3),stride=(2,2),channels=256 +Conv2d,kernel=(3,3),stride=(2,2),channels=512 +Conv2d,kernel=(3,3),stride=(2,2),channels=1024 +Conv2d,kernel=(3,3),stride=(1,1),channels=3 +Decoder +Conv2d,kernel=(3,3),stride=(1,1),channels=1024 +RRDB(1024, 1024) x 9 +ConvT,kernel=(3,3),stride=(2,2),channels=1024 +ConvT,kernel=(3,3),stride=(2,2),channels=512 +ConvT,kernel=(3,3),stride=(2,2),channels=256 +ConvT,kernel=(3,3),stride=(2,2),channels=128 +Conv2d,kernel=(7,7),stride=(1,1),channels=3 +Discriminator +For latent Y: nearest neighbor upsampling 16x +concat[upsampled latent Y, input image X or X’] +Conv2d,kernel=(3,3),stride=(2,2),channels=64 +Conv2d,kernel=(3,3),stride=(2,2),channels=128 +Conv2d,kernel=(3,3),stride=(2,2),channels=256 +Conv2d,kernel=(3,3),stride=(2,2),channels=512 +Conv2d,kernel=(1,1),stride=(1,1),channels=1 +Table II +SEMANTIC ENHANCEMENT MODEL +Model +Layers +Enhancement +Conv2d,kernel=(7,7),stride=(1,1),channels=64 +Conv2d,kernel=(3,3),stride=(1,1),channels=128 +Conv2d,kernel=(3,3),stride=(1,1),channels=256 +Conv2d,kernel=(3,3),stride=(1,1),channels=512 +Conv2d,kernel=(3,3),stride=(1,1),channels=1024 +Conv2d,kernel=(3,3),stride=(1,1),channels=512 +Conv2d,kernel=(3,3),stride=(1,1),channels=256 +Conv2d,kernel=(3,3),stride=(1,1),channels=128 +Conv2d,kernel=(3,3),stride=(1,1),channels=64 +Conv2d,kernel=(7,7),stride=(1,1),channels=3 +Both of the compression methods are the target for the base +model in STSCI to substitute for in the patrol task. The +LSCI proposed in [18] is also involved in the comparison. We +draw lessons from some tricks proposed in that paper, so it’s +necessary to show how we surpass it especially in the specific +task. + +6 +Figure 4. The performance of the reconstructed image of JPEG, JPEG2000, LSCI and STSCI. +Figure 5. Visual example of images produced by LSCI along with the corresponding results for JPEG and JPEG2000. +Meanwhile, the LDPC channel coding is used to make +comparison with JSCC methods under simulated channel +conditions of the wireless transmission channels. +SSIM as well as PSNR is chosen as evaluation metrics to +measure both the quality of images at the recevier and the +similarity between the transmitted ones with the original ones, +which can help comprehensively describe the performance of +the communication systems. +C. Analysis for results in image compression +We visualize the outcome of the comparison between JPEG, +JPEG2000, LSCI and STSCI in image compression task in Fig. +4. The x coordinate represents the average bits per pixel (bpp) +on the images, while the y coordinate individually show the +value of metrics of SSIM and PSNR. +From the Fig. 4, it’s obvious that STSCI is always preferred +to other image compression methods at equal bitrates. In the +bitrate around 0.15, the STSCI is 0.75 higher than the LSCI +and JPEG2000 in value of SSIM and 0.75 is a enormous +number which means the reconstructed image gained by +STSCI is much more resemble to the original ones. +And that is extatly the truth, visual examples presented in +Fig. 5 shows how clear the imge compressed by the STSCI. +Even using only half bpp of JPEG2000 and one of three bpp +of JPEG, image handled by STSCI is 0.1 higher in SSIM +and around 8dB higher in PSNR metrics. It’s esay for us +to see noises and distortions in images compressed by JPEG +and JPEG2000, compared to which, the STSCI’s job is much +better. Such results in compressing and transmitting the image +shows that STSCI can be equal to the specific patrol task with +higher quality and less bpp. +Considering that the base system is fine-tuned with some + +34 +0.900 +0.875 +32 +0.850 +上 +0.825 +30 +SSIM +PSNR +0.800 +SSIMvsbpp +PSNRvsbpp +28 +STSCI +STSCI +0.775 +LSCI +LSCI +0.750 +JPEG2000 +26 +JPEG2000 +JPEG +JPEG +0.725 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +bpp +bpp80 +60 +10 +Ob +40 +OB +40 +120 +120 +120 +20 +140 +20 +140 +20 +OC +140 +0 +160 +U +160 +160 +SSCI: +JPEG: +JPEG2000: +bpp = 0.13 +bpp = 0.35 +bpp = 0.21 +SSIM = 0.92 +SSIM = 0.79 +SSIM = 0.82 +PSNR = 32.6 +PSNR = 26.1 +PSNR = 26.57 +Figure 6. Training details and visual example of the yolonet. +substation and industral images, and that’s why in this visual +sample, the STSCI’s SSIM and PSNR metrics are higher than +the average values in 0.13bpp. Indeed, in the substation patrol +task, the images of substation can be collected continuously to +fine-tune or even retrain the networks of Base system, which +can leads to better performance in the specific task. +Figure 7. visual example of the semantic enhancement model. +D.Analysis for semantic enhancement system +For example, taking the panel as the key semantic informa- +tion, a yolo-net is trained with 200 images of panels. Both the +details and the example of trained yolonet is shown in Fig. 6. +With pre-trained checkpoints involved, after 200 images’ +training, the yolo-net is precious enough for the daily patorl +task with making errors or missing in low frequency. +Meanwhile Fig. 7 shows the effect of the semantic enhance- +ment model. The enhanced area in Fig. 7 has the high SSIM at +0.946 and PSNR at 34.4dB. Through the enhancement model, +we can still see the direction of the hand on the panel, which +is of great meaningful information for the patrol task. +E.Simulated results for channel communication +In the experiments, we choose AWGN model to make +channel simulation. As shown in Fig. 7, when the SNR is +larger than 5dB, the value of SSIM and PSNR gained by +STSCI+LDPC is a bit higher than STSCI+JSCC, but when +the channel conditions gets bad and the SNR is close or +even lower than 0db, the quality of image transmitted through +JSCC metheds doesn’t decrease very fast and becomes much +higher than that of LDPC methods.And that’s what we want in +solving the specific task. One of the most importance mission +for STSCI in this task is to ensure the quality of image +sent back by robots when patrolling some marginal areas + +train/box_loss +train/obj_loss +train/cls_loss +metrics/precision +metrics/recall +0.12 +0.035 +results +1.0 +1.0 +0.030 +0.04 +0.10 +0.8 +0.8 +0.025 +0.02 +0.08 +0.6 +0.6 +0.020 +0.00 +0.06 +0.4 +0.4 +0.015 +0.02 +0.04 +0.2 +0.010 +0.2 +0.04 +0.02 +0.005 +0.0 +0.0 +0 +200 +0 +200 +200 +0 +200 +0 +200 +val/box_loss +val/obj_loss +val/cls_loss +metrics/mAP_0.5 +metrics/mAP_0.5:0.95 +1.0 +0.10 +0.020 +0.04 +0.8 +0.6 +0.08 +0.02 +0.015 +0.6 +0.00 +0.4 +0.06 +0.4 +0.010 +0.02 +0.2 +0.04 +0.2 +0.04 +0.005 +0.02 +0.0 +0.0 +0 +200 +200 +0 +200 +0 +200 +0 +200 +90%bpp:0.15 +Enhance Part: +PSNR:32.4 +SSIM:0.9468 +Figure 8. Comparison between STSCI and LSCI with JSCC or channel slice models and traditional channel coding LDPC with SSIM and PSNR metrics. +with weak signal or under low signal-to-noise ratio channel +conditions. And unlike LSCI whose Encder and Decoder is +not optimized when involving the noise by using channel slice +models, STSCI’s performance in good channel conditions can +get closer and closer to the LDPC metheds. +IV. CONCLUSION +In this paper, a specific task-oriented semantic image com- +munication system STSCI is proposed for intelligent substa- +tion patorl inspection, which is mainly composed of a base +system and a semantic enhancemant system. To haddle the +task of ensuring the quality of images sent back by robots in +singal-weak areas of substation. We designed a GAN-based +networks in structure of auto-encoders to extremely compress +the images. And to preserve the key semantic contents during +transmission to decrease the posibility of errors or missing +of the inspection, a yolo-net is involved to locate the areas +with key semantic information, and a semantic enhancement +model is designed to make full use of these extra information +to make these areas clearer. Meanwhile, technology of JSCC +is involved to improve the performance of STSCI under low +signal-to-noise ratio channel conditions. +With all metheds taken, expriments show the specific task- +oriented semantic image communication system, the STSCI +has the ability in solving this inspection task. +V. ACKNOWLEDGEMENTS +This work is supported in part by the National Key R&D +Program of China under Grant 2022YFB2902102.The work +of Chen Dong is supported by The Academician expert Open +Fund of Beijing Smart-chip Microelectronics Technology Co., +Ltd under project SGITZXDTKJJS2201045. +REFERENCES +[1] C. E. Shannon, “A mathematical theory of communication,” The Bell +System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948. I +[2] M. Kountouris and N. Pappas, “Semantics-empowered communication +for networked intelligent systems,” IEEE Communications Magazine, +vol. 59, no. 6, pp. 96–102, 2021. I +[3] P. Zhang, W. Xu, H. Gao, K. Niu, X. Xu, X. Qin, C. Yuan, Z. Qin, +H. Zhao, J. Wei, and F. Zhang, “Toward wisdom-evolutionary and +primitive-concise 6g: A new paradigm of semantic communication +networks,” Engineering, vol. 8, pp. 60–73, 2022. I +[4] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks +for semantic segmentation,” in Proceedings of the IEEE conference on +computer vision and pattern recognition, 2015, pp. 3431–3440. I +[5] V. Badrinarayanan, A. Handa, and R. Cipolla, “Segnet: A deep con- +volutional encoder-decoder architecture for robust semantic pixel-wise +labelling,” arXiv preprint arXiv:1505.07293, 2015. I +[6] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks +for biomedical image segmentation,” in International Conference on +Medical image computing and computer-assisted intervention. Springer, +2015, pp. 234–241. I +[7] S. Ren, K. He, R. Girshick, and J. Sun, “Faster r-cnn: Towards real-time +object detection with region proposal networks,” Advances in neural +information processing systems, vol. 28, 2015. I +[8] W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C.-Y. Fu, and A. C. +Berg, “Ssd: Single shot multibox detector,” in European conference on +computer vision. +Springer, 2016, pp. 21–37. I +[9] J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, “You only look +once: Unified, real-time object detection,” in Proceedings of the IEEE +conference on computer vision and pattern recognition, 2016, pp. 779– +788. I +[10] J. Redmon and A. Farhadi, “Yolo9000: better, faster, stronger,” in +Proceedings of the IEEE conference on computer vision and pattern +recognition, 2017, pp. 7263–7271. I +[11] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, +S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial networks,” +Communications of the ACM, vol. 63, no. 11, pp. 139–144, 2020. I +[12] X. Chen, Y. Duan, R. Houthooft, J. Schulman, I. Sutskever, and +P. Abbeel, “Infogan: Interpretable representation learning by information +maximizing generative adversarial nets,” Advances in neural information +processing systems, vol. 29, 2016. I +[13] T. Karras, S. Laine, and T. Aila, “A style-based generator architecture +for generative adversarial networks,” in Proceedings of the IEEE/CVF +conference on computer vision and pattern recognition, 2019, pp. 4401– +4410. I +[14] U. Michelucci, “An introduction to autoencoders,” arXiv preprint +arXiv:2201.03898, 2022. I +[15] E. Bourtsoulatze, D. B. Kurka, and D. G¨und¨uz, “Deep joint source- +channel coding for wireless image transmission,” IEEE Transactions on +Cognitive Communications and Networking, vol. 5, no. 3, pp. 567–579, +2019. I, II +[16] D. B. Kurka and D. G¨und¨uz, “Deepjscc-f: Deep joint source-channel +coding of images with feedback,” IEEE Journal on Selected Areas in +Information Theory, vol. 1, no. 1, pp. 178–193, 2020. I +[17] F. Mentzer, G. D. Toderici, M. Tschannen, and E. Agustsson, “High- +fidelity generative image compression,” Advances in Neural Information +Processing Systems, vol. 33, pp. 11 913–11 924, 2020. II, II +[18] C. Dong, H. Liang, X. Xu, S. Han, B. Wang, and P. Zhang, “Semantic + +0.85 +30 +28 +0.80 +PSNR(dB) +SSIM +26 +0.75 +AWGN +AWGN +24 +STSCI-JSCC +STSCI-JSCC +STSCI-LDPC +STSCI-LDPC +0.70 +22 +LSCI-ChannelModel +. +LSCI-ChannelModel +LSCI-LDPC +LSCI-LDPC +0.65 +20 +5 +0 +5 +10 +15 +5 +0 +5 +10 +15 +SNR(dB) +AWGN +SNR(dB)9 +communication system based on semantic slice models propagation,” +IEEE Journal on Selected Areas in Communications, 2022. II, II, III +[19] J. Johnson, A. Alahi, and L. Fei-Fei, “Perceptual losses for real-time +style transfer and super-resolution,” in European conference on computer +vision. +Springer, 2016, pp. 694–711. II +[20] C. Ledig, L. Theis, F. Husz´ar, J. Caballero, A. Cunningham, A. Acosta, +A. Aitken, A. Tejani, J. Totz, Z. Wang et al., “Photo-realistic single +image super-resolution using a generative adversarial network,” in +Proceedings of the IEEE conference on computer vision and pattern +recognition, 2017, pp. 4681–4690. II +[21] X. Wang, K. Yu, S. Wu, J. Gu, Y. Liu, C. Dong, Y. Qiao, and +C. Change Loy, “Esrgan: Enhanced super-resolution generative adversar- +ial networks,” in Proceedings of the European conference on computer +vision (ECCV) workshops, 2018, pp. 0–0. II + diff --git a/BtE1T4oBgHgl3EQfpgUG/content/tmp_files/load_file.txt b/BtE1T4oBgHgl3EQfpgUG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..53a22b3de2c79769b8a6c065d9e493391f5c8dd7 --- /dev/null +++ b/BtE1T4oBgHgl3EQfpgUG/content/tmp_files/load_file.txt @@ -0,0 +1,611 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf,len=610 +page_content='1 A Specific Task-oriented Semantic Image Communication System for substation patrol inspection Senran Fan, Haotai Liang, Chen Dong*, Xiaodong Xu, Geng Liu Abstract—Intelligent inspection robots are widely used in substation patrol inspection, which can help check potential safety hazards by patrolling the substation and sending back scene images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' However, when patrolling some marginal areas with weak signal, the scene images cannot be sucessfully transmissted to be used for hidden danger elimination, which greatly reduces the quality of robots’ daily work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' To solve such problem, a Specific Task-oriented Semantic Communication System for Image—–STSCI is designed, which involves the semantic features extraction, transmission, restoration and enhancement to get clearer images sent by intelligent robots under weak signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Inspired by that only some specific details of the image are needed in such substation patrol inspection task, we proposed a new paradigm of semantic enhancement in such specific task to ensure the clarity of key semantic information when facing a lower bit rate or a low signal-to-noise ratio situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Across the reality-based simulation, experiments show our STSCI can generally surpass traditional image-compression-based and channel-coding-based or other semantic communication system in the substation patrol inspection task with a lower bit rate even under a low signal-to-noise ratio situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Index Terms—Semantic Communication, substation patrol robot, STSCI I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' INTRODUCTION W Ith the development of Internet Technology especially in intelligent applications like IoT fields, the fierce demand for tremendous amount of information transmissions is becoming inevitable, which urges people to continuously improve the efficiency in communication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' However, the transmission rate based on traditional communication system in physical layer has already been approaching the Shannon limit under most situations, so researchers are willing to explore new theories and new forms of communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Based on this, the concept of semantic communication has attracted more and more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' First mentioned in Shannon Senran Fan, Haotai Liang are with the State Key Laboratory of Network- ing and Switching Technology, Beijing University of Posts and Telecom- munications, Beijing, 100876, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (E-mail: FSR@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' lianghao- tai@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='cn) Xiaodong Xu is with the State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, China, and also with the Department of Broad-band Communica- tion, Peng Cheng Laboratory, Shenzhen, Guangdong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (E-mail: xuxi- aodong@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='cn) Geng Liu is with the Beijing Smart-chip Microelectronics Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=',Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (E-mail: liugeng@sgchip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='sgcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='cn) Chen Dong is the corresponding author and with the State Key Laboratory of Networking and Switching Technology, Beijing Univer- sity of Posts and Telecommunications, Beijing, 100876, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (E-mail: dongchen@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='cn) and Weaver’s paper [1], the semantic-based communication system is believed to be a new and bright direction for communication feilds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The explosion of data requires the communication systems to greatly upgrade their ability in data compression, while the semantic-based communication system is suitable for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Considering that when transmitting the information, large amount of task-irrelevant information is involved especially in some specific communication scenes, which leads to massive waste in communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Especially in this task, intelligent substation patrol inspection, what really cared about is only the key semantic contents such as the areas with key units of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Introduced in [2], [3], by transmitting the information through semantic feature extracting, transmission and reconstruction, semantic communication system only keeps the effective information which achieves extremely compression and high efficiency communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Deep learning can be an answer to reciously extract the se- mantic features from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Indeed, using neural networks to make semantic analysis from images is a large subject in the computer vision fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Semantic segmentation networks [4]–[6] as well as target detection networks [7]–[10] show great power in semantic features extracting and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' At the same time, GAN-based [11] networks is possessed of ability in handling semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' GAN-based networks can gen- erator images from semantic vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Moverover in InfoGAN [12] and StyleGAN [13], semantic vector can be edited to control the features of the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' And comes into unsupervised feilds, auto-encoder [14] is an inspiring archi- tecture for semantic feature extracting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Compressing the high- dimensional data into a low-dimensional latent which is used for data reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Auto-encoder forced the reconstruction results to get close enough to the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Combined with semantic-related networks such as GANs and the structure of auto-encoder, the system can realize the aim to decrease distortion in semantic contents of images during the process of extreme compression as well as transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The traditional communication systems involve source cod- ing and channel coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' We replace the former part with neural networks, and to resist against noise in channels we decide to use the Joint Source-Channel Coding(JSCC) [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Leading the channel simulation models into the deep networks, the networks can perform well in real-world channel conditions and even better than traditional channel coding methods like LDPC especially in low bite rate or low signal- to-noise ratio situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='03331v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='CV] 9 Jan 2023 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The framework of STSCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The above studies shows the possiblity in applying the semantic communication system into the subtsation patrol inspection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' At the same time, substations do have toubles in intelligent patrol task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' When the robots patrolling the marginal areas of substation with weak signals, the images sent back by robots can be too blurred to be used for security check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Considering that semantic communication has potential to be the answer for solving such problem and no literature been published before is trying to apply the semantic communication technology to the substation patrol task,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' a specific task-oriented semantic communication system STSCI is proposed for solving this specific task and the similar communication tasks featured by the fixed image source,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' fixed channel conditions and focusing only on some specific task-oriented semantic contents of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The system is mainly a GAN-based auto-encoder-structure network for image’ compressing, transmission and reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In addition, a yolo-net is involved to locate the images’ specific semantic contents, which will then be embedded and sent to the semantic enhancement models to improve the transmission quality of the important semantic contents of the images to make sure there is no errors or missing when making security check with the transmitted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (1) A specific task-oriented semantic communication system for image is proposed for the transmission of images obtained by intelligent robots in the substation patrol inspection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' A new paradigm of key semantic contents extraction and preservation for such specific tasks is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' A Yolo networks is involved to locate the key semantic contents which is the task exactly cares about, while the located part will be sent into a semantic enhancement models to enhance the transmission quality of the located areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (2) A GAN-based auto-encoder structure network is de- signed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Combined with RRDB blocks, channel normal- ization, idea of conditional gan and some other tricks, the network can extremely compress the images into the semantic feature latent and reconstruct them after the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (3) Through simulations and experiments, this paper show the application and performance of the semantic commu- nication system in haddling the specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' By present the metrics, semantic communication system is proven to be superior to the traditional communication systems in such specific tasks with fixed image source and fixed channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' As a practice, the STSCI has better transmission quality especially under low bit rate or low signal-to-noise ratio channel conditions compared with the traditional communication systems, which signifi- cantly enlarge the areas covered by effective signal to ensure the proper work of the intelligent robots when patrolling the marginal areas of the substation with weak signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' This paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In section II, we review the structure of the specific task-oriented semantic commu- nication system for image STSCI, and show details in the model architectures and training flow path of two parts of STSCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Then, in section III, a direct comparison between the STSCI and other image communication systems is provided to quantify the performance of STSCI with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Finally, conclusions of this paper are drawn in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' SPECIFC TASK-ORIENTED SEMANTIC COMMUNICATION SYSTEM Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 1, the specific task-oriented semantic com- munication system for image(STSCI) is mainly composed of two parallel parts: the base system and semantic enhance- ment system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The base system is mainly a GAN-based auto- encoder network to achieve images’ compression, transmission and reconstruction through semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Meanwhile, the semantic enhancement system locates the areas with key semantic contents of the image and improves these areas’ quality during transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Both of the two parts will be introduced in detail in the following contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Semantic Semantic Channels Encoder Decoder Base system Semantic enhancement system Enhancement 个 个 Yolo-Net Model Receiver3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The architecture of the base system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Base System Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 2, the base system is mainly a neural network consists of three parts: an Encoder network, simulated channel models and a GAN-based Decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The images gained by substation patrol inspection robots will be com- pressed by Encoder network, sent to receiver through physical channels simulated by channel models and reconstructed by the Decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The most frequently proposed semantic-based communica- tion systems used the structure of auto-encoder to achieve image compression, however traditional CNN and loss in auto-encoder have difficulties in acquiring high quality re- constructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In pace with development in image en- hancement task especially image de-noising and image super- resolution, GANs have been proven to be possessed of strong talents in high-quality image generation, which were pre- viously employed to improve the visual quality of image compression systems [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Inspired by these previous studies, we decide to use GAN to replace the traditional CNN as decoder network in auto-encoder to significantly improve the quality and similarity of images transmitted by semantic communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Meanwhile, considering that structure of auto-encoder in- volved in semantic communication system is highly consistent with the information communication process, Joint Source- Channel Coding(JSCC) was proposed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' No longer need additional channel coding like LDPC to resist against noise in channels, adding noise through simulation channel models when training auto-encoder networks, an anti-noise communi- cation system is formed, which can ensure high-quality image transmission even under low signal-to-noise ratio situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Though JSCC methods has its limitation for being constrained by specific source, specific scene and specific task, which lead to deep-based semantic communication system’s lack of generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' However, in this task, the information source and channels are fixed, such constrains can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In addition, the data of channel conditions in the substation can be collected continuously in the practical application to fine-tune simulation channel model to improve the system’s performance in this specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In terms of the loss functions which plays a decisive role in training the networks, MSE loss and LPIPS loss is chosen to measure the distortion between the original images and the generated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' MSE loss measures the difference per pixel and shows their distance in the high-dimension space, which helps keep the similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' At the same time, LPIPS loss proposed in [19] is calculated through a VGG-net which has been trained previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Having special model structure and trained with tricks, the pre-trained VGG-net gives more attention to the structure and texture of the images and does well in telling such kind of difference between images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' It’s the difference in structure and texture that is of importance but hard to measure through tradition losses such as L1 loss or MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' LPIPS loss helps supply this gap, and makes the generated images more close to the original ones in visual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In fact, before the final training, the encoder and GAN-based decoder is trained by only using the two mentioned losses instead of involving the adversarial loss at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Such tricks were also applied in [17], [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Initializing the generator net in this way helps the generator performs better in the final training process so the discriminator can learn more useful information and the adversarial loss can be more rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Otherwise if skip this process, the images generated by the generator is far from the ground truth and easy for discriminator to tell, which may lead to the vanishing gradient of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Speaking of the adversarial loss, the structure of our dis- criminator is abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Inspired by [17], the discriminator shares the structure of that in conditional GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Receiving not only the generated images as well as the ground truth but also the latent which puts into the generator, the discriminator no longer only focus on the quality of generator images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' With limitation from the different latent involved in, the discriminator is forced to take attention to the connections between the latent and the image and the difference between images with different latent, so the adversarial loss covers more useful information to help the network performs better in reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' According to all above introductions and the structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 2, the complete process of the base system is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The image 𝑋 to be transmitted is sent in to the Encoder first to get the semantic features 𝑌, 𝑌 = 𝐸(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=" (1) Encoder Gan-based D Residual-in-Residual Decodel Conv Norm ReLU Conv Norm ReLU Upsample Conv Upsample Conv Upsample Conv X' Channel Vgg networks X- Conv Dense Blocks Model concat4 The nearest neighbor quantization operation is then performed on the extracted semantic features 𝑌, 𝑌𝑞(𝑖) = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑗||𝑌𝑖 − 𝐼 𝑗||." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (2) Where the set 𝐼 of quantization centers is: 𝐼 = {𝐼0, 𝐼1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=', 𝐼 𝑗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=', 𝐼𝑙}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (3) According to JSCC, then the quantized semantic feature 𝑌𝑞 is sent to the simulated channel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In this paper, AWGN model is chosen as the simulated channel model, 𝑌𝑞 ′ = ℎ · 𝑌𝑞 + 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (4) In this formula, ℎ represents the channel gain, while 𝑛 rep- resents the independent identically distributed Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Such model simulate the feature’s distortion transmitted in the real-world channel and give the base model the ability to resist the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The image 𝑋 ′ is generated by the Generator(the Decoder network) from the processed latent 𝑌𝑞 ′ at the receiver, 𝑋 ′ = 𝐺(𝑌𝑞 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (5) The Encoder maps the source image 𝑋 to a specific distribution 𝑃𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The generator G tries to map samples 𝑌 from a fixed known distribution 𝑃𝑌 to 𝑃𝑋, while the Discriminator D is learned to tell the difference between such two distributions using the sampled data 𝑋 and the generated 𝑋 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='A properly trained Discriminator helps the Generator to find and simulate the distribution 𝑃𝑋 more preciously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Involving the idea of conditional GANs as mentioned before, the adversarial loss is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 𝐿𝐺 = −𝑙𝑜𝑔(𝐷(𝑋 ′,𝑌𝑞 ′)), (6) 𝐿𝐷 = −𝑙𝑜𝑔(1 − 𝐷(𝑋 ′,𝑌𝑞 ′)) − 𝑙𝑜𝑔(𝐷(𝑋,𝑌𝑞 ′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (7) Besides, when optimizing the Encoder and the Generator, the MSE loss and the LPIPS loss are also involved to measure the texture and perception distance between the source image X and the generator image 𝑋 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Moverover, helping to initialize these two networks, these two kinds of loss guide the Gener- ator and Discriminator to be trained on the right direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' So the final loss for the Encoder and the Generator are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In the initial training: 𝐿𝐸𝐺 = ||𝑋 − 𝑋 ′|| + 𝛼𝐿𝑃𝐼𝑃𝑆(𝑋, 𝑋 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (8) In the final training: 𝐿𝐸𝐺 = ||𝑋−𝑋 ′||+𝛼𝐿𝑃𝐼𝑃𝑆(𝑋, 𝑋 ′)+𝛽[−𝑙𝑜𝑔(𝐷(𝑋 ′,𝑌𝑞 ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (9) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Semantic Enhancement System The Semantic Enhance System is designed to enhance transmission quality of the key semantic contents which is cared about in the specific task such as the panels or electrical insulators in the intelligent substation patrol inspection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The process of the semantic enhancement system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The system consists of two parts: a Yolo net to locate the area with key semantic contents which will be sent into the base model as well as a enhancement network which can get more precious and high quality images at the receiver with the input of the transmitted image and the areas with key semantic contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In this paper, target detection network yolo-net is involved to locate the key semantic contents instead of the semantic segmentation network such as Unet or FCN in some other se- mantic communication systems like [18], the principal reason is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The pre-trained Yolo-net has the ability in finding and locat- ing the objects which need to be shoot during the patrol task, which is not only used to locate and mark the area containing key semantic information during the semantic communication process, but can also help the intelligent robots to judge whether there exists objects in the patrol list to shoot and how to change the position, angle and focal length of camera to get a sharper image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Under the constraint of storage space in the patrol robot, Yolo-net which can do multiple jobs is a rather cost-effective choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 3, semantic enhancement system’s process is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The area 𝑋𝑠𝑢𝑏 with key semantic contents is located by the yolo-net with input of source image 𝑋, Yolo-net 不 BASE SYSTEM 个 Semantic Enhancement Model 不 Xsub1 Xsub X final5 𝑋𝑠𝑢𝑏 = 𝑌𝑜𝑙𝑜 𝑛𝑒𝑡(𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (10) After sent into the base model, 𝑋𝑠𝑢𝑏 is encoded, transmitted and finally reconstructed as the 𝑋𝑠𝑢𝑏 ′ at the receiver, 𝑋𝑠𝑢𝑏 ′ = 𝐵𝑎𝑠𝑒 𝑠𝑦𝑠𝑡𝑒𝑚(𝑋𝑠𝑢𝑏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (11) At the same time, the whole image 𝑋 is transmitted through the base model to get another area 𝑋𝑠𝑢𝑏1 ′ with key semantic contents cut from the reconstructed image 𝑋 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The difference between these two sub-images is calculated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 𝑋𝑑𝑖 𝑓 𝑓 = 𝑋𝑠𝑢𝑏1 ′ − 𝑋 ′ 𝑠𝑢𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (12) The DIFF image is sent to the semantic enhancement model whose job is to balance the difference between two sub- image to make full use of these extra information to let the transmitted image as close as the original one in the area with key semantic contents, 𝑋𝑑𝑖 𝑓 𝑓 ′ = 𝐸𝑛ℎ𝑎𝑛𝑐𝑒𝑚𝑒𝑛𝑡 𝑀𝑜𝑑𝑒𝑙(𝑋𝑑𝑖 𝑓 𝑓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (13) The final image is formed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 𝑋 𝑓 𝑖𝑛𝑎𝑙 = 𝑋 ′ + 𝑋𝑑𝑖 𝑓 𝑓 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (14) In this task, the similarity between the final image 𝑋 𝑓 𝑖𝑛𝑎𝑙 and the original image 𝑋 is focused on, which can help decrease the possibility of errors or missing during analyzing the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' So we choose the MSE loss and SSIM loss to optimize the semantic enhancement models, and parameters in the yolonet as well as the base model are fixed during the optimization, 𝐿𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑚𝑒𝑛𝑡 = ||𝑋 − 𝑋 𝑓 𝑖𝑛𝑎𝑙|| + 𝛼𝑆𝑆𝐼𝑀(𝑋, 𝑋 𝑓 𝑖𝑛𝑎𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' (15) In the end of Section II, the details of networks involved in the STSCI is shown in table I and table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' EXPERIMENTAL RESULTS This section is mainly introduced the relevant testing set- tings, including the dataset for STSCI’s train and test, the introduction of baseline as well as evalation metrics and the performance for the STSCI in different metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Discription and figures are given to show how the STSCI surpass the traditional image communication system or other semantic system under some specific situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Dataset for train and test The training dataset is formed of 10000 images sampled from the COCO2014 dataset while 200 images of substation are used to fine-tune the base system to improve the STSCI’s performance in the intelligent substation patrol inspection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' During the testing process, the images from COCO2014 testset which are not involved in training process are sampled to measure the metrics of the communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Baseline and Evaluation metrics The widely used image compression technology JPEG and JPEG2000 are used as baseline for the image compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Table I BASE SYSTEM Model Layers Encoder Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=64 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=128 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=256 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=512 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=1024 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=3 Decoder Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=1024 RRDB(1024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 1024) x 9 ConvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=1024 ConvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=512 ConvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=256 ConvT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=128 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=3 Discriminator For latent Y: nearest neighbor upsampling 16x concat[upsampled latent Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' input image X or X’] Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=64 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=128 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=256 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=512 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=1 Table II SEMANTIC ENHANCEMENT MODEL Model Layers Enhancement Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=64 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=128 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=256 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=512 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=1024 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=512 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=256 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=128 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=64 Conv2d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='kernel=(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='stride=(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='channels=3 Both of the compression methods are the target for the base model in STSCI to substitute for in the patrol task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The LSCI proposed in [18] is also involved in the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' We draw lessons from some tricks proposed in that paper, so it’s necessary to show how we surpass it especially in the specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 6 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The performance of the reconstructed image of JPEG, JPEG2000, LSCI and STSCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Visual example of images produced by LSCI along with the corresponding results for JPEG and JPEG2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Meanwhile, the LDPC channel coding is used to make comparison with JSCC methods under simulated channel conditions of the wireless transmission channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' SSIM as well as PSNR is chosen as evaluation metrics to measure both the quality of images at the recevier and the similarity between the transmitted ones with the original ones, which can help comprehensively describe the performance of the communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Analysis for results in image compression We visualize the outcome of the comparison between JPEG, JPEG2000, LSCI and STSCI in image compression task in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The x coordinate represents the average bits per pixel (bpp) on the images, while the y coordinate individually show the value of metrics of SSIM and PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' From the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 4, it’s obvious that STSCI is always preferred to other image compression methods at equal bitrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' In the bitrate around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='15, the STSCI is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='75 higher than the LSCI and JPEG2000 in value of SSIM and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='75 is a enormous number which means the reconstructed image gained by STSCI is much more resemble to the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' And that is extatly the truth, visual examples presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 5 shows how clear the imge compressed by the STSCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Even using only half bpp of JPEG2000 and one of three bpp of JPEG, image handled by STSCI is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1 higher in SSIM and around 8dB higher in PSNR metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' It’s esay for us to see noises and distortions in images compressed by JPEG and JPEG2000, compared to which, the STSCI’s job is much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Such results in compressing and transmitting the image shows that STSCI can be equal to the specific patrol task with higher quality and less bpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Considering that the base system is fine-tuned with some 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='875 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='850 上 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='825 30 SSIM PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='800 SSIMvsbpp PSNRvsbpp 28 STSCI STSCI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='775 LSCI LSCI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='750 JPEG2000 26 JPEG2000 JPEG JPEG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='35 bpp bpp80 60 10 Ob 40 OB 40 120 120 120 20 140 20 140 20 OC 140 0 160 U 160 160 SSCI: JPEG: JPEG2000: bpp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='13 bpp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='35 bpp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='21 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='92 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='79 SSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='82 PSNR = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='6 PSNR = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='1 PSNR = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='57 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Training details and visual example of the yolonet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' substation and industral images, and that’s why in this visual sample, the STSCI’s SSIM and PSNR metrics are higher than the average values in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='13bpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Indeed, in the substation patrol task, the images of substation can be collected continuously to fine-tune or even retrain the networks of Base system, which can leads to better performance in the specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' visual example of the semantic enhancement model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='Analysis for semantic enhancement system For example, taking the panel as the key semantic informa- tion, a yolo-net is trained with 200 images of panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Both the details and the example of trained yolonet is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' With pre-trained checkpoints involved, after 200 images’ training, the yolo-net is precious enough for the daily patorl task with making errors or missing in low frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Meanwhile Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 7 shows the effect of the semantic enhance- ment model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' The enhanced area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 7 has the high SSIM at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='946 and PSNR at 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='4dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Through the enhancement model, we can still see the direction of the hand on the panel, which is of great meaningful information for the patrol task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='Simulated results for channel communication In the experiments, we choose AWGN model to make channel simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 7, when the SNR is larger than 5dB, the value of SSIM and PSNR gained by STSCI+LDPC is a bit higher than STSCI+JSCC, but when the channel conditions gets bad and the SNR is close or even lower than 0db, the quality of image transmitted through JSCC metheds doesn’t decrease very fast and becomes much higher than that of LDPC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='And that’s what we want in solving the specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' One of the most importance mission for STSCI in this task is to ensure the quality of image sent back by robots when patrolling some marginal areas train/box_loss train/obj_loss train/cls_loss metrics/precision metrics/recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='035 results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='04 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='0 0 200 200 0 200 0 200 0 200 90%bpp:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='15 Enhance Part: PSNR:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='4 SSIM:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='9468 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Comparison between STSCI and LSCI with JSCC or channel slice models and traditional channel coding LDPC with SSIM and PSNR metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' with weak signal or under low signal-to-noise ratio channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' And unlike LSCI whose Encder and Decoder is not optimized when involving the noise by using channel slice models, STSCI’s performance in good channel conditions can get closer and closer to the LDPC metheds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' CONCLUSION In this paper, a specific task-oriented semantic image com- munication system STSCI is proposed for intelligent substa- tion patorl inspection, which is mainly composed of a base system and a semantic enhancemant system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' To haddle the task of ensuring the quality of images sent back by robots in singal-weak areas of substation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' We designed a GAN-based networks in structure of auto-encoders to extremely compress the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' And to preserve the key semantic contents during transmission to decrease the posibility of errors or missing of the inspection, a yolo-net is involved to locate the areas with key semantic information, and a semantic enhancement model is designed to make full use of these extra information to make these areas clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Meanwhile, technology of JSCC is involved to improve the performance of STSCI under low signal-to-noise ratio channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' With all metheds taken, expriments show the specific task- oriented semantic image communication system, the STSCI has the ability in solving this inspection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is supported in part by the National Key R&D Program of China under Grant 2022YFB2902102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='The work of Chen Dong is supported by The Academician expert Open Fund of Beijing Smart-chip Microelectronics Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=', Ltd under project SGITZXDTKJJS2201045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 379–423, 1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Kountouris and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Pappas, “Semantics-empowered communication for networked intelligent systems,” IEEE Communications Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 96–102, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Gao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Niu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Qin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Yuan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Qin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Wei, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Zhang, “Toward wisdom-evolutionary and primitive-concise 6g: A new paradigm of semantic communication networks,” Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 60–73, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Long, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Shelhamer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Darrell, “Fully convolutional networks for semantic segmentation,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 3431–3440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Badrinarayanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Handa, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Cipolla, “Segnet: A deep con- volutional encoder-decoder architecture for robust semantic pixel-wise labelling,” arXiv preprint arXiv:1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='07293, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [6] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Ronneberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Fischer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Brox, “U-net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical image computing and computer-assisted intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Springer, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Ren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' He, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Girshick, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Sun, “Faster r-cnn: Towards real-time object detection with region proposal networks,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 28, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Anguelov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Erhan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Szegedy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Reed, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Fu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Berg, “Ssd: Single shot multibox detector,” in European conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Springer, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 21–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Redmon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Divvala, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Girshick, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Farhadi, “You only look once: Unified, real-time object detection,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 779– 788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Redmon and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Farhadi, “Yolo9000: better, faster, stronger,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 7263–7271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [11] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Pouget-Abadie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Mirza, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Warde-Farley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Ozair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Courville, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Bengio, “Generative adversarial networks,” Communications of the ACM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 63, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 139–144, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Duan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Houthooft, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Schulman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Sutskever, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Abbeel, “Infogan: Interpretable representation learning by information maximizing generative adversarial nets,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Karras, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Laine, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Aila, “A style-based generator architecture for generative adversarial networks,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 4401– 4410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [14] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Michelucci, “An introduction to autoencoders,” arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='03898, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Bourtsoulatze, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Kurka, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' G¨und¨uz, “Deep joint source- channel coding for wireless image transmission,” IEEE Transactions on Cognitive Communications and Networking, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 567–579, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I, II [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Kurka and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' G¨und¨uz, “Deepjscc-f: Deep joint source-channel coding of images with feedback,” IEEE Journal on Selected Areas in Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 178–193, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' I [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Mentzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Toderici, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Tschannen, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Agustsson, “High- fidelity generative image compression,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 11 913–11 924, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' II, II [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Dong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Liang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Han, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Wang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Zhang, “Semantic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='85 30 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='80 PSNR(dB) SSIM 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='75 AWGN AWGN 24 STSCI-JSCC STSCI-JSCC STSCI-LDPC STSCI-LDPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='70 22 LSCI-ChannelModel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' LSCI-ChannelModel LSCI-LDPC LSCI-LDPC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content='65 20 5 0 5 10 15 5 0 5 10 15 SNR(dB) AWGN SNR(dB)9 communication system based on semantic slice models propagation,” IEEE Journal on Selected Areas in Communications, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' II, II, III [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Johnson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Alahi, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Fei-Fei, “Perceptual losses for real-time style transfer and super-resolution,” in European conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Springer, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 694–711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' II [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Ledig, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Theis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Husz´ar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Caballero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Cunningham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Acosta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Aitken, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Tejani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Totz, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=', “Photo-realistic single image super-resolution using a generative adversarial network,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' 4681–4690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' II [21] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Gu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Dong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf'} +page_content=' Qiao, and C.' metadata={'source': 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b/HNE1T4oBgHgl3EQfrQW4/content/tmp_files/2301.03353v1.pdf.txt @@ -0,0 +1,1029 @@ +Learning Bidirectional Action-Language Translation with Limited +Supervision and Incongruent Extra Input +Ozan ¨Ozdemira, Matthias Kerzela, Cornelius Webera, Jae Hee Leea, Muhammad +Burhan Hafeza, Patrick Brunsb, Stefan Wermtera +aKnowledge Technology, Department of Informatics, University of Hamburg, +Vogt-Koelln-Str. 30, 22527 Hamburg, Germany +bBiological Psychology and Neuropsychology, University of Hamburg, Von-Melle-Park 11, +20146 Hamburg, Germany +ARTICLE HISTORY +Compiled January 10, 2023 +ABSTRACT +Human infant learning happens during exploration of the environment, by interac- +tion with objects, and by listening to and repeating utterances casually, which is +analogous to unsupervised learning. Only occasionally, a learning infant would re- +ceive a matching verbal description of an action it is committing, which is similar to +supervised learning. Such a learning mechanism can be mimicked with deep learn- +ing. We model this weakly supervised learning paradigm using our Paired Gated +Autoencoders (PGAE) model, which combines an action and a language autoen- +coder. After observing a performance drop when reducing the proportion of super- +vised training, we introduce the Paired Transformed Autoencoders (PTAE) model, +using Transformer-based crossmodal attention. PTAE achieves significantly higher +accuracy in language-to-action and action-to-language translations, particularly in +realistic but difficult cases when only few supervised training samples are available. +We also test whether the trained model behaves realistically with conflicting multi- +modal input. In accordance with the concept of incongruence in psychology, conflict +deteriorates the model output. Conflicting action input has a more severe impact +than conflicting language input, and more conflicting features lead to larger interfer- +ence. PTAE can be trained on mostly unlabelled data where labeled data is scarce, +and it behaves plausibly when tested with incongruent input. +KEYWORDS +Unsupervised learning; weak supervision; autoencoders; object manipulation; +robot action; language grounding; Transformers; bidirectional translation +1. Introduction +Embodiment, i.e., action-taking in the environment, is considered essential for lan- +guage learning (Bisk et al. 2020). Recently, language grounding with robotic object +manipulation has received considerable attention from the research community. Most +approaches proposed in this domain cover robotic action execution based on linguistic +input (Hatori et al. 2018; Shridhar, Mittal, and Hsu 2020; Shao et al. 2020; Lynch +and Sermanet 2021), i.e., language-to-action translation. Others cover language pro- +duction based on the actions done on objects (Heinrich et al. 2020; Eisermann et al. +CONTACT Ozan ¨Ozdemir. Email: ozan.oezdemir@uni-hamburg.de +arXiv:2301.03353v1 [cs.CL] 9 Jan 2023 + +2021), i.e., action-to-language translation. However, only few approaches (Ogata et al. +2007; Yamada et al. 2018; Antunes et al. 2019; Abramson et al. 2020; ¨Ozdemir, Kerzel, +and Wermter 2021) handle both directions by being able to not just execute actions +according to given instructions but also to describe those actions, i.e., bidirectional +translation. +Moreover, as infants learn, the actions that they are performing are not permanently +being labeled by matching words from their caretakers, hence, supervised learning with +labels must be considered rare. Instead, infants rather explore the objects around them +and listen to utterances, which may not frequently relate to their actions, hence, un- +supervised learning without matching labels is abundant. Nevertheless, most language +grounding approaches do not make use of unsupervised learning except those that use +some unsupervised loss terms (Yamada et al. 2018; Abramson et al. 2020; ¨Ozdemir, +Kerzel, and Wermter 2021), while large language models (LLMs) (Devlin et al. 2019; +Radford et al. 2019; Brown et al. 2020) introduced for various unimodal downstream +language tasks rely on unsupervised learning for pretraining objectives. +In order to reduce this dependence on labeled data during training, we introduce +a new training procedure, in which we limit the amount of training data used for +supervised learning. More precisely, we only use a certain portion of training samples +for crossmodal action-to-language and language-to-action translations whilst training +unimodally on the rest of the training samples. As crossmodal translation requires +each sample modality to be labeled with the other modality (e.g., an action sequence +must be paired with a corresponding language description), we artificially simulate the +realistic conditions where there is a large amount of unlabelled (unimodal) data but +a much smaller amount of labeled (crossmodal) data. +slide blue quickly +Figure 1. +Our table-top object manipulation sce- +nario in the simulation environment: the NICO robot +is moving the blue cube on the table. The performed +action is labeled as “slide blue quickly”. Our approach +can translate from language to action and vice versa; +i.e., we perform actions that are described in language +and also describe the given actions using language. +Another aspect of human language +learning is that it takes place in an envi- +ronment and while using different modal- +ities such as vision and proprioception. +Concepts such as weight, softness, and +size cannot be grounded without being +in the environment and interacting with +objects. Language learning approaches +that use multiple modalities and take ac- +tion in an environment into account are +preferable to those that use a unimodal +approach to process large amounts of +text. Hence we strive to devise embodied +multimodal models that tackle language +grounding. To this end, our robotic object +manipulation dataset is generated from a +simulation setup as seen in Figure 1. We +use a humanoid child-size robot Neuro- +Inspired COmpanion (NICO) (Kerzel et +al. 2017; Kerzel et al. 2020) to perform +various actions on cubes on a table and label those actions with language descriptions. +We introduce further details of our setup in Section 4. +Different from other approaches, our previous Paired Gated Autoencoders (PGAE) +model (¨Ozdemir, Kerzel, Weber, Lee, and Wermter 2022) can bidirectionally trans- +late between language and action, which enables an agent not only to execute actions +according to given instructions but also to recognize and verbalize its own actions +2 + +or actions executed by another agent. As the desired translation task is communi- +cated to the network through an additional signal word in the language input, PGAE +can flexibly translate between and within modalities during inference. However, when +trained under limited supervision conditions, PGAE performs poorly on the action-to- +language translation task, under two conditions: Firstly, we experiment with reducing +the number of supervised training iterations while using the whole data set for super- +vised training. Secondly, we experiment with reducing the number of training samples +used with the supervised signals. In both instances, though the first is more trivial than +the second, the action-to-language performance of PGAE suffers as the proportion of +supervision decreases. +To overcome this hurdle, we present a novel model, Paired Transformed Au- +toencoders (PTAE), in this follow-up paper. Inspired by the successful application +of the Crossmodal Transformer in vision-language navigation by the Hierarchical +Cross-Modal Agent (HCM) architecture (Irshad, Ma, and Kira 2021), PTAE replaces +PGAE’s gated multimodal fusion mechanism and optionally the LSTM-based (long +short-term memory) (Hochreiter and Schmidhuber 1997) encoders with a Crossmodal +Transformer. Thanks to its more efficient and sequence-retaining crossmodal attention +mechanism, PTAE achieves superior performance even when an overwhelming major- +ity of training iterations (e.g., 98 or 99%) consist of unsupervised learning. When the +majority of training samples are used for unsupervised learning, PTAE still maintains +its perfect action-to-language performance up to 80% of training samples learned uni- +modally and performs relatively well for the 90% case (over 80% sentence accuracy). +Even for the cases where only 1 or 2% of the training samples are used in a super- +vised fashion, which is analogous to few-shot learning, PTAE describes actions well +over chance level with up to 50% success rate. Our results hint that PTAE precludes +the need for large amounts of expensive labeled data, which is required for supervised +learning, as the new architecture with the Crossmodal Transformer as the multimodal- +ity fusion technique significantly outperforms PGAE (¨Ozdemir et al. 2022) under the +limited supervision training conditions. +Furthermore, inspired by the concept of incongruence in psychology and to test +the robustness of the trained model to noise, for each task we introduce an extra +input that is contradictory to the expected output of the model. For example, for +language-to-action translation, we introduce extra conflicting action input showing an +action that is different from the expected action from the model. The intertwined +processing of language and action input in the Crossmodal Transformer resembles +the tight interconnection between language and sensorimotor processes that has been +observed in the human brain (Hauk, Johnsrude, and Pulverm¨uller 2004; van Elk et al. +2010). Embodied accounts of human language comprehension assume that linguistic +information induces mental simulations of relevant sensorimotor experiences. As a +direct consequence of embodied language processing, conflicts between linguistic input +and sensorimotor processes have been shown to result in bidirectional impairments +of language comprehension on the one hand and perceptual judgments and motor +responses on the other hand (Aravena et al. 2010; Glenberg and Kaschak 2002; Kaschak +et al. 2005; Meteyard, Bahrami, and Vigliocco 2007), although the strength of these +behavioral effects has recently been debated (Winter et al. 2022). In our PTAE model, +we found asymmetry in terms of the impact of the action and language modalities on +the performance of the model. Regardless of the output modality, introducing extra +contradictory action input affects the model performance much more than introducing +it in the language modality. +Our contributions in this work can be summarised as: +3 + +(1) We introduce PTAE that handles realistic learning conditions that mainly in- +clude unsupervised/unpaired language and action experiences while requiring +minimal use of labeled data, which is expensive to collect. +(2) We show plausible behavior of the model when testing it with psychology- +inspired contradictory information. +The remainder of this paper is as follows: in Section 2, we summarise different +approaches in language grounding with robotic object manipulation. In Section 3, we +define our PTAE in detail. Section 4 introduces the experiments and their results. In +Section 5, we discuss these results, while Section 6 concludes the paper. +2. Related Work +There are several approaches toward intelligent agents that combine language learning +with interactions in a 3D environment. A comprehensive research program (Abramson +et al. 2020) proposed combining supervised learning, reinforcement learning (RL), +and imitation learning. In the environment, two agents communicate with each other +as one agent (setter) asks questions to or instructs the other (solver) that answers +questions and interacts with objects accordingly. However, the scenario is abstract +with unrealistic object interaction. Hence, proprioception is not used as the actions +are high level, and a transfer of the approach from simulation to the real world would +be non-trivial. +Jang et al. (2021) proposed BC-Z which leverages a large multi-task dataset (100 +tasks) to train a single policy, which is supervised with behavior cloning to match +the actions demonstrated by humans in the dataset. To generalize to new tasks, the +policy is conditioned on a task description; a joint embedding of a video demonstra- +tion, and a language instruction. This allows passing either the video command or +the language command to the policy when being trained to match the actions in a +demonstration. BC-Z generalizes to different tasks, but requires a large collection of +human demonstrations, which is expensive. It also relies on human intervention to +avoid unsafe situations and to correct mistakes. +Inspired by Yamada et al. (2018), we introduced the bidirectional Paired Varia- +tional Autoencoders (PVAE) (¨Ozdemir et al. 2021) that is capable of modeling both +language-to-action and action-to-language translation in a simple table-top setting +where a humanoid robot interacts with small cubes. The approach can pair each robotic +action sample (a sequence of joint values and visual features) with multiple language +descriptions involving alternative words replacing original words. The two variational +autoencoder networks of the model do not share any connections but are aligned with +a binding loss term. Due to the lack of common multimodal representations, PVAE +needs to be prepared for each translation task in advance. To overcome this issue, we +proposed a bidirectional attention-based multimodal network, PGAE (¨Ozdemir et al. +2022), which can flexibly translate between the two modalities with the help of a signal +phrase. +Another approach, CLIPort (Shridhar, Manuelli, and Fox 2021), combines the CLIP +model (Radford et al. 2021) for pretrained vision-language representations with the +Transporter model (Zeng et al. 2020) for robotic manipulation tasks. Transporter takes +an action-centric approach to perception by detecting actions, rather than objects, and +then learns a policy, which allows CLIPort to exploit geometric symmetries for efficient +representation learning. On multiple object manipulation tasks, CLIPort outperforms +4 + +CLIP and Transporter alone. Further, CLIPort trained on multiple tasks performs +better in most cases than CLIPort trained only on particular tasks. This supports +the hypothesis that language-conditioned task-learning skills can be transferred from +one task to another. However, the approach is only realized with a relatively simple +gripper as it does not output joint angle values but 2D pixel affordance predictions. +The actual action execution relies on the calibration between the robotic arm base +and the RGB-D camera. +More recently, the same authors introduced Perceiver-Actor (PERACT) (Shridhar, +Manuelli, and Fox 2022), which is designed to efficiently learn multi-task robotic ma- +nipulations according to given language input by utilizing voxel grids extracted from +RGB-D images. The backbone of the model is the Transformer-based Perceiver IO +(Jaegle et al. 2021) that uses latent vectors to tackle the processing of very long se- +quences. After the processing of appended language and voxel encodings by Perceiver +IO, the voxels are decoded again to generate discrete actions by using linear trans- +formations. PERACT achieves promising results in multiple tasks such as opening a +drawer, turning a tap, and sliding blocks. However, as it only produces discrete actions, +it relies on a random motion planner to execute instructions. +SayCan (Ahn et al. 2022), utilizes LLMs to provide task-grounding capabilities to +the agent, which is capable of executing short-horizon commands. The use of LLMs +helps to ground these capabilities in the real world using value functions of the agent +in order to produce feasible and useful instructions. However, the approach is limited +to the set of skills that the agent can possess in the environment. An LLM is utilized +to assign affordance probabilities to these skills according to a given high-level user +instruction. The way these skills are defined in language (the wording, the length, +etc.) can affect the performance of the whole system, e.g., LLMs tend to favor shorter +phrases over longer ones. +GATO (Reed et al. 2022) is a single multi-task, multi-embodiment model that is +general and performs well on hundreds of tasks in various domains such as playing Atari +games, manipulating objects, image captioning, etc. Regardless of the modality (e.g., +vision, proprioception, language, etc.), the input is flattened and embedded before it +is provided to the model. The model is a large Transformer decoder that has the same +weights and architecture for all tasks and is trained solely in a supervised manner. +However, despite performing moderately in each task, the approach cannot compete +with specialized approaches in various tasks. +The encoder-decoder-based VisuoMotor Attention model, VIMA for short, (Jiang +et al. 2022) is another object manipulation approach. It deals with robot action gen- +eration from multimodal prompts by interleaving language and image or video frame +tokens at the input level. VIMA uses an object detection module to extract objects +and bounding boxes from visual input to use as object tokens. The object tokens +are then interleaved with the language tokens and processed by the pretrained T5 +model (Raffel et al. 2020) which is used as the encoder. On the decoder end, the ap- +proach uses a causal Transformer decoder which consists of cross- and self-attention +layers and autoregressively generates actions based on the history of previous actions +and the multimodal prompt. It is shown that VIMA outperforms state-of-the-art ap- +proaches, including GATO, on a number of increasingly difficult object manipulation +tasks involving zero-shot generalization with unseen objects and their combinations. +An apparent weakness of VIMA is that it relies on the performance of off-the-self +object detectors. +Different from most of the aforementioned approaches, our model is bidirectional: it +can not only produce actions according to given language descriptions but also recog- +5 + + +pull +red + +pull +fast +j1 +v1 +jM +vM +j1 + ĵ2 +y1 +yN-1 +v1 +vM-1 +v2 + ĵ2 +ĵM-1 +y2 +ĵM +x1 +y1 +y3 +'execute: +pull red +fast' +LSTM +LSTM +LSTM + ĵ3 +LSTM +LSTM +LSTM +Crossmodal Transformer +FFW +FFW +h +Lfeats +Afeats +hdec +hdec +A +L +Figure 2. +The architecture of the PTAE model. The inputs are a language description (incl. a task signal) +and a sequence of visual features (extracted using the channel-separated convolutional autoencoder) and joint +values, while the outputs are a description and a sequence of joint values. Language encoder can be an LSTM, +the BERT Base model (Devlin et al. 2019), or the descriptions can be directly passed to the transformer word +by word. The action encoder can be an LSTM or the action sequence can be passed directly to the transformer. +Both decoders are LSTMs - we show unfolded versions of the LSTMs. The bottleneck, where the two streams +are connected, is based on the Crossmodal Transformer. h is the shared representation vector. +nize actions and produce their descriptions. As our model is based on an autoencoder- +like architecture, it can be trained in a mostly unsupervised way by asking the model +to reproduce the given language or proprioception input. Moreover, our approach is +flexible during inference since it does not need to be reconfigured for the translation +task: due to the inclusion of the task signal in the language input, our PTAE can +reliably execute the desired task on the go, whether it is a translation from language +to action or vice versa. This is an essential step towards an autonomous agent that +can interact within the environment as well as communicate with humans. +3. Paired Transformed Autoencoder +Our model, named PTAE, is an encoder-decoder architecture that is capable of bidi- +rectional translation between robot actions and language. It consists of a Crossmodal +Transformer that is the backbone and multimodality fusion mechanism of the architec- +ture, and LSTM-based decoders that output language and joint values respectively. As +input, PTAE accepts language descriptions of actions including the task signal, which +defines the translation direction, as well as a sequence of the concatenation of joint +values and visual features. According to the task signal, PTAE outputs joint values +required for executing a particular action or it outputs language descriptions of an +action. +As shown in Figure 2, PTAE is composed of a Crossmodal Transformer, which ac- +cepts multimodal input (i.e., language, proprioception, and vision), and language and +action decoders that output language descriptions and joint values respectively. The +language and action input can optionally be preprocessed by LSTM-based encoders +as in the case of PGAE1. However, after some initial trials with both cases, in this +paper, we do not use any extra encoding layers before the Crossmodal Transformer +1For exact definitions of LSTM-based language and action encoder, readers may refer to the PGAE paper +(¨Ozdemir et al. 2022). +6 + +Scaled Dot Product Attention +Lfeats +Afeats +Input +Emb. +V +Conc. +K +Q +Pos. +Emb. +Lfeats +Input +Emb. +FFW +h +FFW +FFW +FFW +Scal. +Dot +Prod. +Att. +Figure 3. +The architecture of the Crossmodal Transformer: Language features are embedded and used as +the query vector (Q), whereas the embedded action features are used as the key (K) and value (V) vectors. +The positional embedding is applied only to the language features. The multi-head attention (MHA) involves +the Q-, K- and V-specific feedforward (FFW) and scaled dot product attention layer following the original +Transformer architecture. The multiple heads are then concatenated and fed to the final FFW, which outputs +the common hidden representation vector h. +for the sake of simplicity and model size as we do not see any significant change in the +performance. +3.1. Crossmodal Transformer +The Crossmodal Transformer replaces the Gated Multimodal Unit (GMU) (Arevalo +et al. 2020) in our previous PGAE model (¨Ozdemir et al. 2022) and can be employed +essentially as language and action encoders. The simplified architecture of the Cross- +modal Transformer can be seen in Figure 3. The functionality of the Crossmodal +Transformer is to extract the common latent representations of paired language and +action sequences. Following the HCM architecture (Irshad et al. 2021), we use the lan- +guage modality as queries (Q vectors) and the action modality (concatenated visual +features and joint values) as keys (K vectors) and values (V vectors). The language de- +scriptions are represented as one-hot encoded vectors, whilst action input is composed +of joint values of NICO’s left arm and the visual features from images recorded by +the camera in NICO’s eye. As in PGAE, we use a channel-separated convolutional au- +toencoder (CAE) to extract visual features from images. The Crossmodal Transformer +encodes the common latent representations as follows: +Q = ReLU +� +W token · xt + btoken� ++ PE(xt) +(1 ≤ t ≤ N + 1), +K, V = ReLU +� +W act · [vt; jt] + bact� +(1 ≤ t ≤ M), +At = MHA(Q, K, V ) +(1 ≤ t ≤ N + 1), +ht = PWFF(At) +(1 ≤ t ≤ N + 1), +h = AvgPool(ht) +(1 ≤ t ≤ N + 1), +where x, v and j are linguistic, visual, and proprioceptive inputs respectively – note +that when no language or action encoder is used, x corresponds to Lfeats, while the +concatenation of visual features and joint values [vt; jt] corresponds to Afeats in Figure +3. ReLU is the rectified linear unit activation function while PE, MHA, and PWFF are +the positional encodings, multi-head attention layer, and the position-wise feedforward +7 + +layer as used in the original Transformer paper (Vaswani, Shazeer, Parmar, Uszkoreit, +Jones, Gomez, Kaiser, and Polosukhin 2017). At is the crossmodal attention vector +for time step t, whereas ht is the hidden vector for time step t. AvgPool is the average +pooling applied on the time axis to the sequential hidden vector to arrive at the +common latent representation vector h. For our experiments, we employ a single-layer +Crossmodal Transformer with 4 parallel attention heads. +3.2. Language Decoder +We use an LSTM as the language decoder in order to autoregressively generate the +descriptions word by word by expanding the common latent representation vector h +produced by the Crossmodal Transformer: +hdec +0 , cdec +0 += W dec · h + bdec, +hdec +t +, cdec +t += LSTM(yt−1, hdec +t−1, cdec +t−1) (1 ≤ t ≤ N − 1), +yt = soft(W out · hdec +t ++ bout) +(1 ≤ t ≤ N − 1), +where soft represents the softmax activation function. y0 is the vector for the symbol +indicating the beginning of the sentence, the tag. +3.3. Action Decoder +Similarly, an LSTM is employed as the action decoder to output joint angle values at +each time step with the help of the common representation vector h: +hdec +0 , cdec +0 += W dec · h + bdec, +hdec +t +, cdec +t += LSTM(vt, ˆȷt, hdec +t−1, cdec +t−1) +(1 ≤ t ≤ M − 1), +ˆȷt+1 = tanh(W out · hdec +t ++ bout) +(1 ≤ t ≤ M − 1), +where ˆȷt is the predicted joint values for time step t and tanh is the hyperbolic tangent +activation function. We take ˆȷ1 as j1, i.e., ground-truth joint angle values corresponding +to the initial position of the arm. The visual features used as input v are extracted +from the ground-truth images and used similarly to teacher forcing, whereas the joint +angle values ˆȷt are used autoregressively. +3.4. Visual Feature Extraction +Following the PGAE pipeline (¨Ozdemir et al. 2022), the channel-separated convolu- +tional autoencoder (CAE) is used to extract visual features from first-person images +from the eye cameras of NICO recorded in the simulation. We utilize channel sep- +aration when extracting visual features: an instance of the CAE is trained for each +RGB color channel. In a previous paper (¨Ozdemir et al. 2021), we show that channel +separation distinguishes object colors more accurately than the regular CAE without +channel separation. +We feed each instance of the channel-separated CAE with the corresponding chan- +nel of RGB images of size 120 × 160. The channel-separated CAE is made up of a +convolutional encoder, a fully-connected bottleneck, and a deconvolutional decoder. +8 + +Each RGB channel is trained separately, after which we extract the channel-specific +visual features from the bottleneck and concatenate them to arrive at composite visual +features. These visual features make up v which is used as vision input to PTAE. For +further details on the visual feature extraction process, readers may refer to (¨Ozdemir +et al. 2021). +3.5. Loss Function +We use two loss functions to calculate the deviation from the ground-truth language +descriptions and joint values. The language loss, Llang, is calculated as the cross entropy +between input and output words, while the action loss, Lact, is the mean squared error +(MSE) between original and predicted joint values: +Llang = +1 +N − 1 +N−1 +� +t=1 +� +− +V −1 +� +i=0 +x[i] +t+1 log y[i] +t +� +, +Lact = +1 +M − 1 +M−1 +� +t=1 +∥jt+1 − ˆȷt+1∥2 +2 , +where V is the vocabulary size, N is the number of words per description, and M is the +sequence length for action trajectories. The total loss is then the sum of the language +and action losses: +Lall = αLlang + βLact +where α and β are weighting factors for language and action terms in the loss function. +In our experiments, we take both α and β as 1.0. We use the identical loss functions +as PGAE except for the weight vector used in the language loss. +3.6. Training Details +Visual features are extracted in advance by the channel-separated CAE before train- +ing PTAE and PGAE. Visual features are necessary to execute actions according to +language instructions since cube arrangements are decisive in manipulating the left or +right object, i.e., determining whether to manipulate the left or right cube depends on +the position of the target cube. After extracting visual features, both PGAE and PTAE +are trained end-to-end with all three modalities. After initial experiments, PGAE is +trained for 6,000 epochs, while PTAE is trained for 2,500 epochs using the gradient +descent algorithm and Adam optimizer (Kingma and Ba 2015). For PTAE, we decided +that h has 256 dimensions, whereas the same vector has 50 dimensions in PGAE. x +has 28 dimensions, j has 5 dimensions, N is equal to 5, while M is 50 for fast and +100 for slow actions. For both PGAE and PTAE, we take the learning rate as 10−5 +with a batch size of 6 samples after determining them as optimal hyperparameters. +PTAE has approximately 1.5M parameters compared to PGAE’s a little over 657K +parameters. +9 + +4. Experiments +We use the same dataset (¨Ozdemir et al. 2021) as in the PGAE paper (¨Ozdemir +et al. 2022), except that in this paper we exclude experiments with another agent +from the opposite side of the table. The dataset encompasses 864 samples of sequences +of images and joint values alongside their textual descriptions. It consists of robot +actions on two cubes of different colors on the table by the NICO robot, generated +using inverse kinematics and created in the simulation environment using Blender +software2. The NICO robot has a camera in each eye, which is used to record a sequence +of egocentric images. According to the scenario, NICO manipulates one of the two +cubes on the table with its left arm at a time. In total, the dataset includes 12 distinct +actions, 6 cube colors, 288 descriptions, and 144 patterns (action-description-cube +arrangement combinations). The 144 patterns are randomly varied six times in terms +of action execution in simulation: we arrive at a dataset of 864 samples in total. Out +of 864 samples, 216 samples that involve every unique description and action type are +excluded and used as the test set. The remaining 648 samples make up the training +set. The vocabulary consists of the following words divided into 3 categories: +• 6 action words (3 original/3 alternative): “push/move-up”, “pull/move-down”, +“slide/move-sideways” +• 12 colour words (6 original/6 alternative): “red/scarlet”, “green/harlequin”, +“blue/azure”, “yellow/blonde”, “cyan/greenish-blue”, “violet/purple” +• 4 speed words (2 original/2 alternative): “slowly/unhurriedly”, “fast/quickly” +The sentences consist of a word from each category: therefore, our textual descriptions +are 3-word sentences. For more details on the dataset, readers may consult our pre- +vious work (¨Ozdemir et al. 2021). PGAE and PTAE are trained on this dataset and +their performances are tested in terms of action-to-language and language-to-action +translations under different amounts of supervision. +Task signals. We use four signals to train PTAE. According to the given signal, the +input and output of the model change. The signals are: +• Describe: action-to-language translation +• Execute: language-to-action translation +• Repeat Action: action-to-action translation +• Repeat Language: language-to-language translation +According to the latter two “repeat” signals, the network uses mainly unimodal infor- +mation. The “describe” and “execute” signals, on the other hand, involve crossmodal +translation from one modality to the other. The unimodal signals are used in the +unsupervised learning of an autoencoder, whereas the crossmodal signals are used +in supervised learning, where coordinated action values and language labels must be +available. In the case of PGAE training, an additional “repeat both” signal is also +used, which also requires coordinated labels, and leads to slightly better performance +(¨Ozdemir et al. 2022). For the PTAE, however, this was found unnecessary. +Reduction of supervised training. We restrict the amount of supervision by in- +creasing the ratio of unsupervised learning iterations, i.e., training with the unimodal +“repeat” signals, in the overall training iterations. Thereby the ratio of supervised +2https://www.blender.org/ +10 + +Figure 4. +Sentence accuracy for action-to-language translation on the test set wrt. supervised training itera- +tions. Supervised training refers to crossmodal translation cases “describe” and “execute”. The two crossmodal +signals receive the same number of iterations between them out of the supervised iterations. We report the +results for 1%, 2%, 10%, 20%, 50%, and 66.6% (the regular training case) crossmodal (supervised) iterations. +These percentages correspond to the fraction of supervised training iterations for PGAE and PTAE. Note that +the 100% case is not shown here, since the models need unsupervised iterations (unimodal repeat signals) to +be able to perform the “repeat language” and “repeat action” tasks. +learning iterations, i.e., training with the crossmodal signals, decreases. The resulting +training paradigm is analogous to developmental language learning, where an infant +is exposed only to a limited amount of supervision. We train both PTAE and PGAE +with varying ratios of unimodal/total training iterations. For another set of experi- +ments, we restrict the amount of supervision by limiting the proportion of training +samples used for crossmodal translation tasks. We test the performance of both mod- +els with varying degrees of unsupervised training under different schemes (limiting the +percentage of iterations or samples) on the crossmodal translation tasks. +In this work, we investigate action-to-language and language-to-action translations +because they are the more important and difficult tasks. For the “repeat” tasks, the +results match our previous work; therefore, the readers can refer to our publication +(¨Ozdemir et al. 2022). Figure 4 shows the results of PGAE and PTAE on action- +to-language translation with different percentages of training iterations used in a su- +pervised fashion. Both PGAE and PTAE with different training regimes based on +different proportions of supervised training iterations achieve accuracies higher than +chance level (2.78%), which we calculate based on our grammar (action, color, speed): +1÷(3×6×2). The action-to-language translation performance of PGAE falls when the +ratio of crossmodal (viz. supervised) training iterations is low, particularly when 10% +or a smaller proportion of the iterations are supervised. Even though the description +accuracy slightly increases to over 95% when supervised training amounts to only 20% +of all training iterations, it sharply drops to well below 50% when the rate is decreased +to 2%. PGAE is able to describe 36% of the test samples when only 1% of the training +iterations are used to learn crossmodal translations between action and language. In +contrast, PTAE maintains its perfect description accuracy even when it has only been +11 + +Action-to-Language Performance wrt. Ratio of Supervised Training Iterations +80 +(%) +Sentence Accuracy +60 +PGAE +PTAE +chance +40 +20 +2 +10 +20 +50 +66 +(Crossmodal Training Iterations)-(Total Training Iterations) (%)Figure 5. +Sentence accuracy for action-to-language translation on the test set wrt. supervised training sam- +ples. Supervised training refers to crossmodal translation cases “describe” and “execute”. We limit the number +of training samples for the supervised tasks. We report the results for the 1%, 2%, 5% 10%, 20%, 50%, and +66.6% cases as well as the 100% regular training case. These percentages correspond to the fraction of training +samples used exclusively for the supervised training for PGAE and PTAE, i.e., both “execute” and “describe” +signals are trained with only a limited number of samples corresponding to the percentages. +trained with 1% supervised training iterations. While there is a detrimental impact of +reduced supervision, i.e., the limitation on the percentage of crossmodal training itera- +tions, on the action-to-language translation performance of PGAE, transformer-based +PTAE is not affected by the same phenomenon. For space reasons, we do not report +language-to-action results wrt. different percentages of supervised iterations, but we +observed a similar trend comparable with Figure 4. +In order to further investigate the performance of PTAE with limited supervision, +we introduce a more challenging training regime. We limit the number of training +samples shown to supervised signals, “describe” and “execute”, and show the rest of +the training samples only on “repeat action” and “repeat language” modes. We train +both PGAE and PTAE with varying percentages of supervised training samples. The +results can be seen in Figure 5. In all cases with different proportions of supervised +training samples, both PGAE and PTAE outperform the chance level. While main- +taining perfect sentence accuracy down to 20% supervised training and keeping up +its performance for 10% supervised training for the “describe” signal, PTAE’s perfor- +mance drops sharply when the ratio of training samples used for crossmodal signals +is 2% and below. Nevertheless, PTAE beats PGAE in each case when trained on dif- +ferent percentages of supervised training samples. PGAE’s performance suffers even +when 50% of training samples are used for supervised signals; it drops below 80% - +PTAE retains 100% for the same case. It takes more than 90% of the training samples +to be exclusively used in the unsupervised signals for PTAE’s performance to decrease +meaningfully (from 100% to 81%), while this ratio is much lower for PGAE as its per- +formance already drops significantly at 50%. Even for 1% supervised training samples +which amount to only 7 training samples, PTAE manages to translate one-third of the +test samples from action to sentences. +12 + +Action-to-Language Performance wrt. Ratio of Supervised Training Samples +100 +PGAE +-PTAE +chance +80- +Sentence Accuracy (%) +60 +40 +20 +0- +12 +5 +20 +50 +1o +66 +100 +(Crossmodal Training Samples)-(Total Training Samples) (%)Figure 6. +Joint value prediction error in language-to-action translation on the test set wrt. supervised training +samples. Supervised training refers to crossmodal translation cases “describe” and “execute”. We limit the +number of training samples for the supervised tasks. We report the results for the 1%, 2%, 5% 10%, 20%, 50%, +and 66.6% cases as well as the 100% regular training case. These percentages correspond to the fraction of +training samples used exclusively for the supervised training for PGAE and PTAE. “execute” and “describe” +translations are shown the same limited number of samples. +Language-to-action translation results with respect to different percentages of su- +pervised training samples for PGAE and PTAE are shown in Figure 6. We show the +deviation of the produced joint values from the original ones in terms of the normalized +root-mean-squared error (NRMSE), which we obtain by normalizing the root-mean- +squared error (RMSE) between the predicted and ground-truth values by the range of +joint values – the lower percentages indicate better prediction (0% NRMSE meaning +predicted values are identical with ground-truth values), whereas the higher percent- +ages indicate worse prediction (100% NRMSE meaning the RMSE between predicted +and ground-truth values is equal to the range of possible values). We can see a similar +trend as in action-to-language translation apart from the regular case (100%) when +PGAE has a lower error than PTAE, which is probably due to the fact that PGAE +is trained for more than two times the number of iterations than PTAE since it takes +longer for PGAE’s training loss to reach a global minimum. In all other cases, limiting +the ratio of training samples to be used in the supervised modes impacts PGAE’s +language-to-action performance heavily: the NRMSE rises from less than 0.5% to al- +most 8% when the percentage of supervised samples is reduced to two-thirds of the +training samples. The error rate increases further as the number of training samples +used in the crossmodal training modes decreases. The NRMSE for PTAE is also in- +versely proportional to the ratio of supervised training samples. However, the impact +of limiting the number of training samples for supervised modes on PTAE is much +lower than on PGAE. When the percentage of supervised training samples is reduced +to 1%, the deviation from the ground-truth joint values is only a little more than 4% +for PTAE, whereas the same statistic for PGAE is almost 14%. +13 + +14 +★-PGAE +PTAE +12 +10 +2 +01 +12 +5 +10 +20 +50 +66 +100 +(Crossmodal Training Samples)-(Total Training Samples) (%)Figure 7. +Model performance on the test set wrt. no. of conflicts introduced in the extra input. For action- +to-language and language-to-language (the top row), we show the predicted sentence accuracies. For language- +to-action and action-to-action, we show the normalized root-mean-squared error (NRMSE) for predicted joint +values. The modality in which the conflicts are introduced is given in the x-axis. For each signal, we add extra +conflicting inputs either in the action or language input. When the conflict is introduced in action, we also +test having the conflict only in the vision and only in the proprioception submodality - in this case, the other +submodality has the matching input. +Exposure to conflicting input modalities. We also investigate the impact of +contradictory extra input on the performance of PTAE. For this, we use PTAE-regular +that is trained with 33% unsupervised training iterations and no contradictory input. +We test the robustness of our approach to varying numbers of conflicts (up to 3) in +the extra input. The definitions of the added conflict per task signal are: +• “describe”: Here, we add a conflicting description to the language input (conflict +in language). +• “execute”: Here, we use a conflicting sequence of vision and proprioception input +(conflict in action). +• “repeat action”: Here, we add a conflicting description to the language input +(conflict in language). +• “repeat language”: Here, we use a conflicting sequence of vision and propriocep- +tion input (conflict in action). +The conflicts are introduced using the following scheme: +• for the conflict in the extra language input; one, two, or all of the action, color, +and speed words that constitute a description, do not match with the action. +• for the conflict in the extra action input; one, two, or all of the action-type, +position, and speed aspects, which form distinct actions, do not match with the +language description. +The results of this experiment are given in Figure 7. In the case of the “describe” and +“repeat action” signals, the action supplies the relevant input whereas the language +14 + +Action-to-Language Performance +Language-to-Language Performance +100 +100 +Action (Vis.+Prop.) Conf. +Only Vis. Conf. +Only Prop. Conf. +Sentence Accuracy (%) +80 +80 +60 - +60 - +40 +40 +20 - +20 +0 +No. of conflicts in extra language input +No.of conflicts in extra action input +Action-to-Action Performance +Language-to-Action Performance +Action (Vis.+Prop.) Conf. +Only Vis. Conf. +Only Prop. Conf. +2 +2 +No. of conflicts in extra language input +No. of conflicts in extra action inputis the conflicting distractor. Here, we observe only a slight decrease in performance. +In the case of action-to-language translation (“describe”) the sentence accuracy goes +down from 100% to 95% when there are three conflicting input elements (action type, +color, speed). Action-to-action (“repeat action”) translation manages to retain its +performance as the error in joint values only slightly increases from 1.03% to 1.09% +for the case with 3 conflicts. +In the case of “execute” and “repeat language” signals, the language supplies the +relevant input while the action is the conflicting distractor. Here, we observe a big +performance drop. Language-to-action translation (“execute”) suffers heavily as the +deviation of the predicted joint values from the ground-truth joint values increases from +0.99% to 4.95%. In the language-to-language translation case (“repeat language”), +PTAE loses its ability to repeat the given language description when one or more +conflicting elements (action type, position, speed) are introduced with the extra input: +the sentence accuracy decreases from 100% to 0%. +Therefore, we can see the asymmetric impact of conflicts in the two modalities, +namely, when language input is introduced as a contradictory element, the perfor- +mance drops slightly, whereas when the contradictory input is introduced in the action +stream, the model is affected heavily and performs poorly. The output modality has +no significant impact on the result; for example, we can see that both “describe” and +“repeat language” output language at large, but they are affected very differently by +the conflicting input. To test whether the bigger impact of conflicting action input +is due to the involvement of two modalities in action (vision and proprioception), we +also tried introducing the conflict either only in vision or only in proprioception (the +relatively brighter bars in the two charts on the right in Figure 7). In either case, the +performance is still substantially negatively affected, although the drop in performance +is naturally not as severe as introducing the conflict in both modalities. +5. Discussion +The experimental results on action-to-language and language-to-action translations +show the superior performance and efficiency of our novel PTAE model under lim- +ited supervision. Limiting the percentage of supervised crossmodal iterations during +training has no adverse effect on PTAE as it maintains its perfect sentence accuracy +when translating from action to language. In contrast, the previous PGAE model’s +action-to-language translation accuracy drops substantially to around 40% when only +1 or 2% of the training iterations are supervised. +When we challenge both models more by limiting the number of training samples for +the supervised crossmodal “execute” and “describe” signals, we see a similar pattern: +when 50% or less of the training samples are used for supervised signals, action-to- +language sentence accuracy for PGAE decreases directly proportional to the ratio of +supervised samples. PTAE, on the other hand, retains its action-to-language perfor- +mance up until the case where only 5% of the training samples are used in a supervised +fashion. Even after being trained with 2% supervised training, which amounts to only +13 samples out of 648, PTAE is able to describe more than half of the action sequences +correctly. All in all, PTAE shows superior action-to-language performance than PGAE +for varied levels of limited supervision. +The adverse effect of limiting the number of supervised training samples on the +language-to-action performance can already be seen for PGAE even when only one- +third of the samples are excluded (66% supervised case). The NRMSE between pre- +15 + +dicted and ground-truth joint values rises significantly from around 0.5% to around +8%. It continues to increase gradually after reducing the level of supervision to 20%. +On the contrary, PTAE is robust against the limited supervision with respect to the +ratio of crossmodal training samples until the supervised percentage is brought down +to 10%. After that, it can be seen that the error rate gradually increases, albeit only +just over 4% for PTAE when only 7 samples are used for the supervised signals. Over- +all, these results indicate the clear superiority of Transformer-based multimodal fusion +over a simpler attention mechanism by GMU in terms of performance and efficiency. +Although it is relatively larger than PGAE, PTAE is trained much faster and reaches +a global optimum in less than half of the training iterations of PGAE. +When introducing a conflicting modality input during testing, we observed an asym- +metry in that a conflicting action input leads to a larger disturbance than a conflicting +language input. One possible reason is that the Crossmodal Transformer architecture +is asymmetric: As input, we are using action input as two input vectors (K and V: +keys and values), whereas language as one input vector (Q: queries). This setting was +chosen because the opposite setup (with action as queries) was found less performant. +Our setup can be interpreted as language-conditioned action attention. A computa- +tionally more expensive architecture could combine both asymmetric setups, as has +been done for learning vision and language representations (Lu et al. 2019). +Another possible reason for the larger impact of a conflicting action could be that +the action input combines two submodalities, vision, and proprioception, and therefore +involves more information than the language input. However, limiting the conflict to +one of the submodalities did not completely remove the asymmetry as introducing the +conflict only in one action submodality (vision or proprioception) still had a stronger +effect on the model performance than a conflicting language input. Unlike language, +vision contains the complete information to perform a task. Consider the example “pull +red slowly” for language-to-action translation. Here, the language does not contain +any information about whether the object is on the left or right side, so the agent can +only execute this correctly when also taking visual input into account during action +execution. In contrast, in the opposite direction (action-to-language translation) and +in action repetition, the visual input has the complete information. +6. Conclusion +In this paper, we introduced a paired Transformer-based autoencoder, PTAE, which we +trained largely by unsupervised learning with additional, but reduced supervision. The +PTAE achieves significantly better action-to-language and language-to-action transla- +tion performance under limited supervision conditions compared to the former GMU- +based model, PGAE. Furthermore, we tested the robustness of our new approach +against contradictory extra input. In line with the concept of incongruence in psy- +chology, these experiments show that conflict deteriorates the output of our model, +and more conflicting features lead to higher interference. We also found an asymmetry +between the action and language modalities in terms of their conflicting impact: the +action modality has significantly more influence over the performance of the model +regardless of the main output modality. +Our novel bidirectional embodied language learning model is flexible in performing +multiple tasks and it is efficient and robust against the scarcity of labeled data. Hence, +it is a step towards an autonomous agent that can communicate with humans while +performing various tasks in the real world. In the future, we will expand our approach +16 + +with reinforcement learning to reduce the need for expert-defined action trajectories. +Furthermore, a reinforcement learner may explore more dexterous object manipula- +tion with diversified action trajectories. With more realistic action execution, we will +attempt to tackle the problem of sim-to-real transfer. Lastly, diversifying our action +repertoire will inevitably lead to more diverse natural language descriptions, which we +can tackle by employing a pretrained Transformer-based large language model as a +language encoder. +Disclosure statement +The authors report there are no competing interests to declare. +Funding +This work was supported by the German Research Foundation (DFG) under Project +TRR 169 Crossmodal Learning (CML), LeCareBot, IDEAS, and MoReSpace. +References +Abramson, J., A. Ahuja, A. Brussee, F. Carnevale, M. Cassin, S. Clark, A. Dudzik, P. Georgiev, +A. Guy, T. Harley, F. Hill, A. Hung, Z. Kenton, J. Landon, T. P. Lillicrap, K. W. Mathewson, +A. Muldal, A. Santoro, N. Savinov, V. Varma, G. Wayne, N. Wong, C. Yan, and R. Zhu. +2020. Imitating interactive intelligence. arXiv preprint arXiv:2012.05672. +Ahn, M., A. Brohan, N. Brown, Y. Chebotar, O. Cortes, B. David, C. Finn, C. Fu, K. Gopalakr- +ishnan, K. Hausman, A. Herzog, D. Ho, J. Hsu, J. Ibarz, B. Ichter, A. Irpan, E. Jang, R. J. +Ruano, K. Jeffrey, S. Jesmonth, N. Joshi, R. Julian, D. Kalashnikov, Y. Kuang, K.-H. Lee, +S. Levine, Y. Lu, L. Luu, C. Parada, P. Pastor, J. Quiambao, K. Rao, J. Rettinghouse, +D. Reyes, P. Sermanet, N. Sievers, C. Tan, A. Toshev, V. Vanhoucke, F. Xia, T. Xiao, +P. Xu, S. Xu, M. Yan, and A. Zeng. 2022. Do as I can and not as I say: Grounding language +in robotic affordances. arXiv preprint arXiv:2204.01691. +Antunes, A., A. Laflaquiere, T. Ogata, and A. Cangelosi. 2019. +A bi-directional multiple +timescales LSTM model for grounding of actions and verbs. In 2019 IEEE/RSJ International +Conference on Intelligent Robots and Systems (IROS), pp. 2614–2621. +Aravena, P., E. Hurtado, R. Riveros, J. F. Cardona, F. Manes, and A. Ib´a˜nez. 2010. Applauding +with closed hands: neural signature of action-sentence compatibility effects. PloS ONE 5(7), +e11751. +Arevalo, J., T. Solorio, M. Montes-y G´omez, and F. A. Gonz´alez. 2020. Gated multimodal +networks. Neural Computing and Applications 32(14), 10209–10228. +Bisk, Y., A. Holtzman, J. Thomason, J. Andreas, Y. Bengio, J. Chai, M. Lapata, A. Lazari- +dou, J. May, A. Nisnevich, N. Pinto, and J. Turian. 2020, November. Experience grounds +language. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language +Processing, pp. 8718–8735. Association for Computational Linguistics. +Brown, T., B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, +P. Shyam, G. Sastry, A. Askell, et al. 2020. Language models are few-shot learners. Advances +in Neural Information Processing Systems 33, 1877–1901. +Devlin, J., M.-W. Chang, K. Lee, and K. Toutanova. 2019, June. BERT: Pre-training of deep +bidirectional transformers for language understanding. In Proceedings of the 2019 Confer- +ence of the North American Chapter of the Association for Computational Linguistics: Hu- +man Language Technologies, Volume 1 (Long and Short Papers), Minneapolis, Minnesota, +pp. 4171–4186. Association for Computational Linguistics. +17 + +Eisermann, A., J. H. Lee, C. Weber, and S. Wermter. 2021, Jul. Generalization in multimodal +language learning from simulation. In Proceedings of the International Joint Conference on +Neural Networks (IJCNN 2021). +Glenberg, A. M. and M. P. Kaschak. 2002. +Grounding language in action. +Psychonomic +Bulletin & Review 9(3), 558–565. +Hatori, J., Y. Kikuchi, S. Kobayashi, K. Takahashi, Y. Tsuboi, Y. Unno, W. Ko, and J. Tan. +2018. Interactively picking real-world objects with unconstrained spoken language instruc- +tions. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. +3774–3781. IEEE. +Hauk, O., I. Johnsrude, and F. Pulverm¨uller. 2004. Somatotopic representation of action words +in human motor and premotor cortex. Neuron 41(2), 301–307. +Heinrich, S., Y. Yao, T. Hinz, Z. Liu, T. Hummel, M. Kerzel, C. Weber, and S. Wermter. +2020. Crossmodal language grounding in an embodied neurocognitive model. Frontiers in +Neurorobotics 14, 52. +Hochreiter, S. and J. Schmidhuber. 1997. Long short-term memory. Neural Computation 9(8), +1735–1780. +Irshad, M. Z., C.-Y. Ma, and Z. Kira. 2021. +Hierarchical cross-modal agent for robotics +vision-and-language navigation. In 2021 IEEE International Conference on Robotics and +Automation (ICRA), pp. 13238–13246. +Jaegle, A., S. Borgeaud, J.-B. Alayrac, C. Doersch, C. Ionescu, D. Ding, S. Koppula, D. Zoran, +A. Brock, E. Shelhamer, et al. 2021. Perceiver io: A general architecture for structured inputs +& outputs. In International Conference on Learning Representations. +Jang, E., A. Irpan, M. Khansari, D. Kappler, F. Ebert, C. Lynch, S. Levine, and C. Finn. +2021. BC-z: Zero-shot task generalization with robotic imitation learning. In 5th Annual +Conference on Robot Learning. +Jiang, Y., A. Gupta, Z. Zhang, G. Wang, Y. Dou, Y. Chen, L. Fei-Fei, A. Anandkumar, Y. Zhu, +and L. Fan. 2022. VIMA: General robot manipulation with multimodal prompts. +Kaschak, M. P., C. J. Madden, D. J. Therriault, R. H. Yaxley, M. Aveyard, A. A. Blanchard, +and R. A. Zwaan. 2005. Perception of motion affects language processing. Cognition 94(3), +B79–B89. +Kerzel, M., T. Pekarek-Rosin, E. Strahl, S. Heinrich, and S. Wermter. 2020. Teaching NICO +how to grasp: an empirical study on crossmodal social interaction as a key factor for robots +learning from humans. Frontiers in Neurorobotics 14, 28. +Kerzel, M., E. Strahl, S. Magg, N. Navarro-Guerrero, S. Heinrich, and S. Wermter. 2017. +NICO—Neuro-Inspired COmpanion: A developmental humanoid robot platform for mul- +timodal interaction. In 2017 26th IEEE International Symposium on Robot and Human +Interactive Communication (RO-MAN), pp. 113–120. IEEE. +Kingma, D. P. and J. Ba. 2015. Adam: A method for stochastic optimization. In 3rd Interna- +tional Conference on Learning Representations, ICLR, San Diego, CA, USA, May 7-9. +Lu, J., D. Batra, D. Parikh, and S. Lee. 2019. +ViLBERT: Pretraining task-agnostic visi- +olinguistic representations for vision-and-language tasks. +In H. Wallach, H. Larochelle, +A. Beygelzimer, F. d'Alch´e-Buc, E. Fox, and R. Garnett (Eds.), Advances in Neural Infor- +mation Processing Systems, Volume 32. Curran Associates, Inc. +Lynch, C. and P. Sermanet. 2021. Language conditioned imitation learning over unstructured +data. In D. A. Shell, M. Toussaint, and M. A. Hsieh (Eds.), Robotics: Science and System +XVII. +Meteyard, L., B. Bahrami, and G. Vigliocco. 2007. Motion detection and motion verbs: Lan- +guage affects low-level visual perception. Psychological Science 18(11), 1007–1013. PMID: +17958716. +Ogata, T., M. Murase, J. Tani, K. Komatani, and H. G. Okuno. 2007. Two-way translation +of compound sentences and arm motions by recurrent neural networks. In 2007 IEEE/RSJ +International Conference on Intelligent Robots and Systems, pp. 1858–1863. +¨Ozdemir, O., M. Kerzel, C. Weber, J. H. Lee, and S. Wermter. 2022. Learning flexible transla- +tion between robot actions and language descriptions. In E. Pimenidis, P. Angelov, C. Jayne, +18 + +A. Papaleonidas, and M. Aydin (Eds.), Artificial Neural Networks and Machine Learning – +ICANN 2022, Cham, pp. 246–257. Springer Nature Switzerland. +¨Ozdemir, O., M. Kerzel, and S. Wermter. 2021, Aug. Embodied language learning with paired +variational autoencoders. +In 2021 IEEE International Conference on Development and +Learning (ICDL), pp. 1–6. +Radford, A., J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, +P. Mishkin, J. Clark, et al. 2021. Learning transferable visual models from natural language +supervision. In International Conference on Machine Learning, pp. 8748–8763. PMLR. +Radford, A., J. Wu, R. Child, D. Luan, D. Amodei, and I. Sutskever. 2019. Language models +are unsupervised multitask learners. +Raffel, C., N. Shazeer, A. Roberts, K. Lee, S. Narang, M. Matena, Y. Zhou, W. Li, and P. J. +Liu. 2020. Exploring the limits of transfer learning with a unified text-to-text transformer. +Journal of Machine Learning Research 21(140), 1–67. +Reed, S., K. Zolna, E. Parisotto, S. G. Colmenarejo, A. Novikov, G. Barth-Maron, M. Gimenez, +Y. Sulsky, J. Kay, J. T. Springenberg, T. Eccles, J. Bruce, A. Razavi, A. Edwards, N. Heess, +Y. Chen, R. Hadsell, O. Vinyals, M. Bordbar, and N. de Freitas. 2022. A generalist agent. +Shao, L., T. Migimatsu, Q. Zhang, K. Yang, and J. Bohg. 2020. Concept2robot: Learning +manipulation concepts from instructions and human demonstrations. +In Proceedings of +Robotics: Science and Systems (RSS). +Shridhar, M., L. Manuelli, and D. Fox. 2021. CLIPort: What and where pathways for robotic +manipulation. In Proceedings of the 5th Conference on Robot Learning (CoRL). +Shridhar, M., L. Manuelli, and D. Fox. 2022. Perceiver-Actor: A multi-task transformer for +robotic manipulation. In Proceedings of the 6th Conference on Robot Learning (CoRL). +Shridhar, M., D. Mittal, and D. Hsu. 2020. INGRESS: Interactive visual grounding of referring +expressions. The International Journal of Robotics Research 39(2-3), 217–232. +van Elk, M., H. T. van Schie, R. A. Zwaan, and H. Bekkering. 2010. The functional role of +motor activation in language processing: Motor cortical oscillations support lexical-semantic +retrieval. Neuroimage 50(2), 665–677. +Vaswani, A., N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, �L. Kaiser, and +I. Polosukhin. 2017. Attention is all you need. In Advances in Neural Information Processing +Systems, pp. 5998–6008. +Winter, A., C. Dudschig, J. Miller, R. Ulrich, and B. Kaup. 2022. The action-sentence com- +patibility effect (ace): Meta-analysis of a benchmark finding for embodiment. Acta Psycho- +logica 230, 103712. +Yamada, T., H. Matsunaga, and T. Ogata. 2018. Paired recurrent autoencoders for bidirec- +tional translation between robot actions and linguistic descriptions. IEEE Robotics and +Automation Letters 3(4), 3441–3448. +Zeng, A., P. Florence, J. Tompson, S. Welker, J. Chien, M. Attarian, T. Armstrong, I. Krasin, +D. Duong, V. Sindhwani, and J. Lee. 2020. Transporter networks: Rearranging the visual +world for robotic manipulation. In J. Kober, F. Ramos, and C. J. Tomlin (Eds.), 4th Con- +ference on Robot Learning, CoRL 2020, 16-18 November 2020, Virtual Event / Cambridge, +MA, USA, Volume 155 of Proceedings of Machine Learning Research, pp. 726–747. PMLR. +19 + diff --git a/HNE1T4oBgHgl3EQfrQW4/content/tmp_files/load_file.txt b/HNE1T4oBgHgl3EQfrQW4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a407022d505acf39641730e545b52d8e4f5ecde --- /dev/null +++ b/HNE1T4oBgHgl3EQfrQW4/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf,len=1041 +page_content='Learning Bidirectional Action-Language Translation with Limited Supervision and Incongruent Extra Input Ozan ¨Ozdemira, Matthias Kerzela, Cornelius Webera, Jae Hee Leea, Muhammad Burhan Hafeza, Patrick Brunsb, Stefan Wermtera aKnowledge Technology, Department of Informatics, University of Hamburg, Vogt-Koelln-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 30, 22527 Hamburg, Germany bBiological Psychology and Neuropsychology, University of Hamburg, Von-Melle-Park 11, 20146 Hamburg, Germany ARTICLE HISTORY Compiled January 10, 2023 ABSTRACT Human infant learning happens during exploration of the environment, by interac- tion with objects, and by listening to and repeating utterances casually, which is analogous to unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Only occasionally, a learning infant would re- ceive a matching verbal description of an action it is committing, which is similar to supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Such a learning mechanism can be mimicked with deep learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We model this weakly supervised learning paradigm using our Paired Gated Autoencoders (PGAE) model, which combines an action and a language autoen- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' After observing a performance drop when reducing the proportion of super- vised training, we introduce the Paired Transformed Autoencoders (PTAE) model, using Transformer-based crossmodal attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PTAE achieves significantly higher accuracy in language-to-action and action-to-language translations, particularly in realistic but difficult cases when only few supervised training samples are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We also test whether the trained model behaves realistically with conflicting multi- modal input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In accordance with the concept of incongruence in psychology, conflict deteriorates the model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Conflicting action input has a more severe impact than conflicting language input, and more conflicting features lead to larger interfer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PTAE can be trained on mostly unlabelled data where labeled data is scarce, and it behaves plausibly when tested with incongruent input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' KEYWORDS Unsupervised learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' weak supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' autoencoders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' object manipulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' robot action;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' language grounding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Transformers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' bidirectional translation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Introduction Embodiment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', action-taking in the environment, is considered essential for lan- guage learning (Bisk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Recently, language grounding with robotic object manipulation has received considerable attention from the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Most approaches proposed in this domain cover robotic action execution based on linguistic input (Hatori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shridhar, Mittal, and Hsu 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lynch and Sermanet 2021), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', language-to-action translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Others cover language pro- duction based on the actions done on objects (Heinrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Eisermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' CONTACT Ozan ¨Ozdemir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Email: ozan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='oezdemir@uni-hamburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='de arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='03353v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='CL] 9 Jan 2023 2021), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', action-to-language translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, only few approaches (Ogata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Antunes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Abramson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ¨Ozdemir, Kerzel, and Wermter 2021) handle both directions by being able to not just execute actions according to given instructions but also to describe those actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', bidirectional translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Moreover, as infants learn, the actions that they are performing are not permanently being labeled by matching words from their caretakers, hence, supervised learning with labels must be considered rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Instead, infants rather explore the objects around them and listen to utterances, which may not frequently relate to their actions, hence, un- supervised learning without matching labels is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Nevertheless, most language grounding approaches do not make use of unsupervised learning except those that use some unsupervised loss terms (Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Abramson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ¨Ozdemir, Kerzel, and Wermter 2021), while large language models (LLMs) (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020) introduced for various unimodal downstream language tasks rely on unsupervised learning for pretraining objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In order to reduce this dependence on labeled data during training, we introduce a new training procedure, in which we limit the amount of training data used for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' More precisely, we only use a certain portion of training samples for crossmodal action-to-language and language-to-action translations whilst training unimodally on the rest of the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' As crossmodal translation requires each sample modality to be labeled with the other modality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', an action sequence must be paired with a corresponding language description), we artificially simulate the realistic conditions where there is a large amount of unlabelled (unimodal) data but a much smaller amount of labeled (crossmodal) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' slide blue quickly Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Our table-top object manipulation sce- nario in the simulation environment: the NICO robot is moving the blue cube on the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The performed action is labeled as “slide blue quickly”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Our approach can translate from language to action and vice versa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', we perform actions that are described in language and also describe the given actions using language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Another aspect of human language learning is that it takes place in an envi- ronment and while using different modal- ities such as vision and proprioception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Concepts such as weight, softness, and size cannot be grounded without being in the environment and interacting with objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language learning approaches that use multiple modalities and take ac- tion in an environment into account are preferable to those that use a unimodal approach to process large amounts of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hence we strive to devise embodied multimodal models that tackle language grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' To this end, our robotic object manipulation dataset is generated from a simulation setup as seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We use a humanoid child-size robot Neuro- Inspired COmpanion (NICO) (Kerzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kerzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020) to perform various actions on cubes on a table and label those actions with language descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We introduce further details of our setup in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Different from other approaches, our previous Paired Gated Autoencoders (PGAE) model (¨Ozdemir, Kerzel, Weber, Lee, and Wermter 2022) can bidirectionally trans- late between language and action, which enables an agent not only to execute actions according to given instructions but also to recognize and verbalize its own actions 2 or actions executed by another agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' As the desired translation task is communi- cated to the network through an additional signal word in the language input, PGAE can flexibly translate between and within modalities during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, when trained under limited supervision conditions, PGAE performs poorly on the action-to- language translation task, under two conditions: Firstly, we experiment with reducing the number of supervised training iterations while using the whole data set for super- vised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Secondly, we experiment with reducing the number of training samples used with the supervised signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In both instances, though the first is more trivial than the second, the action-to-language performance of PGAE suffers as the proportion of supervision decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' To overcome this hurdle, we present a novel model, Paired Transformed Au- toencoders (PTAE), in this follow-up paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Inspired by the successful application of the Crossmodal Transformer in vision-language navigation by the Hierarchical Cross-Modal Agent (HCM) architecture (Irshad, Ma, and Kira 2021), PTAE replaces PGAE’s gated multimodal fusion mechanism and optionally the LSTM-based (long short-term memory) (Hochreiter and Schmidhuber 1997) encoders with a Crossmodal Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Thanks to its more efficient and sequence-retaining crossmodal attention mechanism, PTAE achieves superior performance even when an overwhelming major- ity of training iterations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', 98 or 99%) consist of unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' When the majority of training samples are used for unsupervised learning, PTAE still maintains its perfect action-to-language performance up to 80% of training samples learned uni- modally and performs relatively well for the 90% case (over 80% sentence accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Even for the cases where only 1 or 2% of the training samples are used in a super- vised fashion, which is analogous to few-shot learning, PTAE describes actions well over chance level with up to 50% success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Our results hint that PTAE precludes the need for large amounts of expensive labeled data, which is required for supervised learning, as the new architecture with the Crossmodal Transformer as the multimodal- ity fusion technique significantly outperforms PGAE (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022) under the limited supervision training conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Furthermore, inspired by the concept of incongruence in psychology and to test the robustness of the trained model to noise, for each task we introduce an extra input that is contradictory to the expected output of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For example, for language-to-action translation, we introduce extra conflicting action input showing an action that is different from the expected action from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The intertwined processing of language and action input in the Crossmodal Transformer resembles the tight interconnection between language and sensorimotor processes that has been observed in the human brain (Hauk, Johnsrude, and Pulverm¨uller 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' van Elk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Embodied accounts of human language comprehension assume that linguistic information induces mental simulations of relevant sensorimotor experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' As a direct consequence of embodied language processing, conflicts between linguistic input and sensorimotor processes have been shown to result in bidirectional impairments of language comprehension on the one hand and perceptual judgments and motor responses on the other hand (Aravena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Glenberg and Kaschak 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kaschak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Meteyard, Bahrami, and Vigliocco 2007), although the strength of these behavioral effects has recently been debated (Winter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In our PTAE model, we found asymmetry in terms of the impact of the action and language modalities on the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Regardless of the output modality, introducing extra contradictory action input affects the model performance much more than introducing it in the language modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Our contributions in this work can be summarised as: 3 (1) We introduce PTAE that handles realistic learning conditions that mainly in- clude unsupervised/unpaired language and action experiences while requiring minimal use of labeled data, which is expensive to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' (2) We show plausible behavior of the model when testing it with psychology- inspired contradictory information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The remainder of this paper is as follows: in Section 2, we summarise different approaches in language grounding with robotic object manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Section 3, we define our PTAE in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Section 4 introduces the experiments and their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Section 5, we discuss these results, while Section 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Related Work There are several approaches toward intelligent agents that combine language learning with interactions in a 3D environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A comprehensive research program (Abramson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020) proposed combining supervised learning, reinforcement learning (RL), and imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In the environment, two agents communicate with each other as one agent (setter) asks questions to or instructs the other (solver) that answers questions and interacts with objects accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, the scenario is abstract with unrealistic object interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hence, proprioception is not used as the actions are high level, and a transfer of the approach from simulation to the real world would be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' (2021) proposed BC-Z which leverages a large multi-task dataset (100 tasks) to train a single policy, which is supervised with behavior cloning to match the actions demonstrated by humans in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' To generalize to new tasks, the policy is conditioned on a task description;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' a joint embedding of a video demonstra- tion, and a language instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' This allows passing either the video command or the language command to the policy when being trained to match the actions in a demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' BC-Z generalizes to different tasks, but requires a large collection of human demonstrations, which is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' It also relies on human intervention to avoid unsafe situations and to correct mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Inspired by Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' (2018), we introduced the bidirectional Paired Varia- tional Autoencoders (PVAE) (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021) that is capable of modeling both language-to-action and action-to-language translation in a simple table-top setting where a humanoid robot interacts with small cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The approach can pair each robotic action sample (a sequence of joint values and visual features) with multiple language descriptions involving alternative words replacing original words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The two variational autoencoder networks of the model do not share any connections but are aligned with a binding loss term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Due to the lack of common multimodal representations, PVAE needs to be prepared for each translation task in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' To overcome this issue, we proposed a bidirectional attention-based multimodal network, PGAE (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022), which can flexibly translate between the two modalities with the help of a signal phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Another approach, CLIPort (Shridhar, Manuelli, and Fox 2021), combines the CLIP model (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021) for pretrained vision-language representations with the Transporter model (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020) for robotic manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Transporter takes an action-centric approach to perception by detecting actions, rather than objects, and then learns a policy, which allows CLIPort to exploit geometric symmetries for efficient representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' On multiple object manipulation tasks, CLIPort outperforms 4 CLIP and Transporter alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Further, CLIPort trained on multiple tasks performs better in most cases than CLIPort trained only on particular tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' This supports the hypothesis that language-conditioned task-learning skills can be transferred from one task to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, the approach is only realized with a relatively simple gripper as it does not output joint angle values but 2D pixel affordance predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The actual action execution relies on the calibration between the robotic arm base and the RGB-D camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' More recently, the same authors introduced Perceiver-Actor (PERACT) (Shridhar, Manuelli, and Fox 2022), which is designed to efficiently learn multi-task robotic ma- nipulations according to given language input by utilizing voxel grids extracted from RGB-D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The backbone of the model is the Transformer-based Perceiver IO (Jaegle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021) that uses latent vectors to tackle the processing of very long se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' After the processing of appended language and voxel encodings by Perceiver IO, the voxels are decoded again to generate discrete actions by using linear trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PERACT achieves promising results in multiple tasks such as opening a drawer, turning a tap, and sliding blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, as it only produces discrete actions, it relies on a random motion planner to execute instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' SayCan (Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022), utilizes LLMs to provide task-grounding capabilities to the agent, which is capable of executing short-horizon commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The use of LLMs helps to ground these capabilities in the real world using value functions of the agent in order to produce feasible and useful instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, the approach is limited to the set of skills that the agent can possess in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' An LLM is utilized to assign affordance probabilities to these skills according to a given high-level user instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The way these skills are defined in language (the wording, the length, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=') can affect the performance of the whole system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', LLMs tend to favor shorter phrases over longer ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' GATO (Reed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022) is a single multi-task, multi-embodiment model that is general and performs well on hundreds of tasks in various domains such as playing Atari games, manipulating objects, image captioning, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Regardless of the modality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', vision, proprioception, language, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ), the input is flattened and embedded before it is provided to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The model is a large Transformer decoder that has the same weights and architecture for all tasks and is trained solely in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, despite performing moderately in each task, the approach cannot compete with specialized approaches in various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The encoder-decoder-based VisuoMotor Attention model, VIMA for short, (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022) is another object manipulation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' It deals with robot action gen- eration from multimodal prompts by interleaving language and image or video frame tokens at the input level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' VIMA uses an object detection module to extract objects and bounding boxes from visual input to use as object tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The object tokens are then interleaved with the language tokens and processed by the pretrained T5 model (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020) which is used as the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' On the decoder end, the ap- proach uses a causal Transformer decoder which consists of cross- and self-attention layers and autoregressively generates actions based on the history of previous actions and the multimodal prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' It is shown that VIMA outperforms state-of-the-art ap- proaches, including GATO, on a number of increasingly difficult object manipulation tasks involving zero-shot generalization with unseen objects and their combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' An apparent weakness of VIMA is that it relies on the performance of off-the-self object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=" Different from most of the aforementioned approaches, our model is bidirectional: it can not only produce actions according to given language descriptions but also recog- 5 pull red pull fast j1 v1 jM vM j1 ĵ2 y1 yN-1 v1 vM-1 v2 ĵ2 ĵM-1 y2 ĵM x1 y1 y3 'execute: pull red fast' LSTM LSTM LSTM ĵ3 LSTM LSTM LSTM Crossmodal Transformer FFW FFW h Lfeats Afeats hdec hdec A L Figure 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The architecture of the PTAE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The inputs are a language description (incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' a task signal) and a sequence of visual features (extracted using the channel-separated convolutional autoencoder) and joint values, while the outputs are a description and a sequence of joint values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language encoder can be an LSTM, the BERT Base model (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019), or the descriptions can be directly passed to the transformer word by word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The action encoder can be an LSTM or the action sequence can be passed directly to the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Both decoders are LSTMs - we show unfolded versions of the LSTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The bottleneck, where the two streams are connected, is based on the Crossmodal Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' h is the shared representation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' nize actions and produce their descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' As our model is based on an autoencoder- like architecture, it can be trained in a mostly unsupervised way by asking the model to reproduce the given language or proprioception input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Moreover, our approach is flexible during inference since it does not need to be reconfigured for the translation task: due to the inclusion of the task signal in the language input, our PTAE can reliably execute the desired task on the go, whether it is a translation from language to action or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' This is an essential step towards an autonomous agent that can interact within the environment as well as communicate with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Paired Transformed Autoencoder Our model, named PTAE, is an encoder-decoder architecture that is capable of bidi- rectional translation between robot actions and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' It consists of a Crossmodal Transformer that is the backbone and multimodality fusion mechanism of the architec- ture, and LSTM-based decoders that output language and joint values respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' As input, PTAE accepts language descriptions of actions including the task signal, which defines the translation direction, as well as a sequence of the concatenation of joint values and visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' According to the task signal, PTAE outputs joint values required for executing a particular action or it outputs language descriptions of an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' As shown in Figure 2, PTAE is composed of a Crossmodal Transformer, which ac- cepts multimodal input (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', language, proprioception, and vision), and language and action decoders that output language descriptions and joint values respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The language and action input can optionally be preprocessed by LSTM-based encoders as in the case of PGAE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, after some initial trials with both cases, in this paper, we do not use any extra encoding layers before the Crossmodal Transformer 1For exact definitions of LSTM-based language and action encoder, readers may refer to the PGAE paper (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 6 Scaled Dot Product Attention Lfeats Afeats Input Emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' V Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' K Q Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lfeats Input Emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' FFW h FFW FFW FFW Scal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Dot Prod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The architecture of the Crossmodal Transformer: Language features are embedded and used as the query vector (Q), whereas the embedded action features are used as the key (K) and value (V) vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The positional embedding is applied only to the language features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The multi-head attention (MHA) involves the Q-, K- and V-specific feedforward (FFW) and scaled dot product attention layer following the original Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The multiple heads are then concatenated and fed to the final FFW, which outputs the common hidden representation vector h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' for the sake of simplicity and model size as we do not see any significant change in the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Crossmodal Transformer The Crossmodal Transformer replaces the Gated Multimodal Unit (GMU) (Arevalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020) in our previous PGAE model (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022) and can be employed essentially as language and action encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The simplified architecture of the Cross- modal Transformer can be seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The functionality of the Crossmodal Transformer is to extract the common latent representations of paired language and action sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Following the HCM architecture (Irshad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021), we use the lan- guage modality as queries (Q vectors) and the action modality (concatenated visual features and joint values) as keys (K vectors) and values (V vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The language de- scriptions are represented as one-hot encoded vectors, whilst action input is composed of joint values of NICO’s left arm and the visual features from images recorded by the camera in NICO’s eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' As in PGAE, we use a channel-separated convolutional au- toencoder (CAE) to extract visual features from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The Crossmodal Transformer encodes the common latent representations as follows: Q = ReLU � W token · xt + btoken� + PE(xt) (1 ≤ t ≤ N + 1), K, V = ReLU � W act · [vt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' jt] + bact� (1 ≤ t ≤ M), At = MHA(Q, K, V ) (1 ≤ t ≤ N + 1), ht = PWFF(At) (1 ≤ t ≤ N + 1), h = AvgPool(ht) (1 ≤ t ≤ N + 1), where x, v and j are linguistic, visual, and proprioceptive inputs respectively – note that when no language or action encoder is used, x corresponds to Lfeats, while the concatenation of visual features and joint values [vt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' jt] corresponds to Afeats in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ReLU is the rectified linear unit activation function while PE, MHA, and PWFF are the positional encodings, multi-head attention layer, and the position-wise feedforward 7 layer as used in the original Transformer paper (Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, Kaiser, and Polosukhin 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' At is the crossmodal attention vector for time step t, whereas ht is the hidden vector for time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' AvgPool is the average pooling applied on the time axis to the sequential hidden vector to arrive at the common latent representation vector h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For our experiments, we employ a single-layer Crossmodal Transformer with 4 parallel attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language Decoder We use an LSTM as the language decoder in order to autoregressively generate the descriptions word by word by expanding the common latent representation vector h produced by the Crossmodal Transformer: hdec 0 , cdec 0 = W dec · h + bdec, hdec t , cdec t = LSTM(yt−1, hdec t−1, cdec t−1) (1 ≤ t ≤ N − 1), yt = soft(W out · hdec t + bout) (1 ≤ t ≤ N − 1), where soft represents the softmax activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' y0 is the vector for the symbol indicating the beginning of the sentence, the tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Action Decoder Similarly, an LSTM is employed as the action decoder to output joint angle values at each time step with the help of the common representation vector h: hdec 0 , cdec 0 = W dec · h + bdec, hdec t , cdec t = LSTM(vt, ˆȷt, hdec t−1, cdec t−1) (1 ≤ t ≤ M − 1), ˆȷt+1 = tanh(W out · hdec t + bout) (1 ≤ t ≤ M − 1), where ˆȷt is the predicted joint values for time step t and tanh is the hyperbolic tangent activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We take ˆȷ1 as j1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', ground-truth joint angle values corresponding to the initial position of the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The visual features used as input v are extracted from the ground-truth images and used similarly to teacher forcing, whereas the joint angle values ˆȷt are used autoregressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Visual Feature Extraction Following the PGAE pipeline (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022), the channel-separated convolu- tional autoencoder (CAE) is used to extract visual features from first-person images from the eye cameras of NICO recorded in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We utilize channel sep- aration when extracting visual features: an instance of the CAE is trained for each RGB color channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In a previous paper (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021), we show that channel separation distinguishes object colors more accurately than the regular CAE without channel separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We feed each instance of the channel-separated CAE with the corresponding chan- nel of RGB images of size 120 × 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The channel-separated CAE is made up of a convolutional encoder, a fully-connected bottleneck, and a deconvolutional decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 8 Each RGB channel is trained separately, after which we extract the channel-specific visual features from the bottleneck and concatenate them to arrive at composite visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' These visual features make up v which is used as vision input to PTAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For further details on the visual feature extraction process, readers may refer to (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Loss Function We use two loss functions to calculate the deviation from the ground-truth language descriptions and joint values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The language loss, Llang, is calculated as the cross entropy between input and output words, while the action loss, Lact, is the mean squared error (MSE) between original and predicted joint values: Llang = 1 N − 1 N−1 � t=1 � − V −1 � i=0 x[i] t+1 log y[i] t � , Lact = 1 M − 1 M−1 � t=1 ∥jt+1 − ˆȷt+1∥2 2 , where V is the vocabulary size, N is the number of words per description, and M is the sequence length for action trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The total loss is then the sum of the language and action losses: Lall = αLlang + βLact where α and β are weighting factors for language and action terms in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In our experiments, we take both α and β as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We use the identical loss functions as PGAE except for the weight vector used in the language loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Training Details Visual features are extracted in advance by the channel-separated CAE before train- ing PTAE and PGAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Visual features are necessary to execute actions according to language instructions since cube arrangements are decisive in manipulating the left or right object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', determining whether to manipulate the left or right cube depends on the position of the target cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' After extracting visual features, both PGAE and PTAE are trained end-to-end with all three modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' After initial experiments, PGAE is trained for 6,000 epochs, while PTAE is trained for 2,500 epochs using the gradient descent algorithm and Adam optimizer (Kingma and Ba 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For PTAE, we decided that h has 256 dimensions, whereas the same vector has 50 dimensions in PGAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' x has 28 dimensions, j has 5 dimensions, N is equal to 5, while M is 50 for fast and 100 for slow actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For both PGAE and PTAE, we take the learning rate as 10−5 with a batch size of 6 samples after determining them as optimal hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PTAE has approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='5M parameters compared to PGAE’s a little over 657K parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Experiments We use the same dataset (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021) as in the PGAE paper (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022), except that in this paper we exclude experiments with another agent from the opposite side of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The dataset encompasses 864 samples of sequences of images and joint values alongside their textual descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' It consists of robot actions on two cubes of different colors on the table by the NICO robot, generated using inverse kinematics and created in the simulation environment using Blender software2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The NICO robot has a camera in each eye, which is used to record a sequence of egocentric images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' According to the scenario, NICO manipulates one of the two cubes on the table with its left arm at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In total, the dataset includes 12 distinct actions, 6 cube colors, 288 descriptions, and 144 patterns (action-description-cube arrangement combinations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The 144 patterns are randomly varied six times in terms of action execution in simulation: we arrive at a dataset of 864 samples in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Out of 864 samples, 216 samples that involve every unique description and action type are excluded and used as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The remaining 648 samples make up the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The vocabulary consists of the following words divided into 3 categories: 6 action words (3 original/3 alternative): “push/move-up”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “pull/move-down”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “slide/move-sideways” 12 colour words (6 original/6 alternative): “red/scarlet”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “green/harlequin”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “blue/azure”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “yellow/blonde”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “cyan/greenish-blue”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “violet/purple” 4 speed words (2 original/2 alternative): “slowly/unhurriedly”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “fast/quickly” The sentences consist of a word from each category: therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' our textual descriptions are 3-word sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For more details on the dataset, readers may consult our pre- vious work (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PGAE and PTAE are trained on this dataset and their performances are tested in terms of action-to-language and language-to-action translations under different amounts of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Task signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We use four signals to train PTAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' According to the given signal, the input and output of the model change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The signals are: Describe: action-to-language translation Execute: language-to-action translation Repeat Action: action-to-action translation Repeat Language: language-to-language translation According to the latter two “repeat” signals, the network uses mainly unimodal infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The “describe” and “execute” signals, on the other hand, involve crossmodal translation from one modality to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The unimodal signals are used in the unsupervised learning of an autoencoder, whereas the crossmodal signals are used in supervised learning, where coordinated action values and language labels must be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In the case of PGAE training, an additional “repeat both” signal is also used, which also requires coordinated labels, and leads to slightly better performance (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For the PTAE, however, this was found unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Reduction of supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We restrict the amount of supervision by in- creasing the ratio of unsupervised learning iterations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', training with the unimodal “repeat” signals, in the overall training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Thereby the ratio of supervised 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='blender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='org/ 10 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sentence accuracy for action-to-language translation on the test set wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' supervised training itera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Supervised training refers to crossmodal translation cases “describe” and “execute”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The two crossmodal signals receive the same number of iterations between them out of the supervised iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We report the results for 1%, 2%, 10%, 20%, 50%, and 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='6% (the regular training case) crossmodal (supervised) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' These percentages correspond to the fraction of supervised training iterations for PGAE and PTAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Note that the 100% case is not shown here, since the models need unsupervised iterations (unimodal repeat signals) to be able to perform the “repeat language” and “repeat action” tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' learning iterations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', training with the crossmodal signals, decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The resulting training paradigm is analogous to developmental language learning, where an infant is exposed only to a limited amount of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We train both PTAE and PGAE with varying ratios of unimodal/total training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For another set of experi- ments, we restrict the amount of supervision by limiting the proportion of training samples used for crossmodal translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We test the performance of both mod- els with varying degrees of unsupervised training under different schemes (limiting the percentage of iterations or samples) on the crossmodal translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In this work, we investigate action-to-language and language-to-action translations because they are the more important and difficult tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For the “repeat” tasks, the results match our previous work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' therefore, the readers can refer to our publication (¨Ozdemir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Figure 4 shows the results of PGAE and PTAE on action- to-language translation with different percentages of training iterations used in a su- pervised fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Both PGAE and PTAE with different training regimes based on different proportions of supervised training iterations achieve accuracies higher than chance level (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='78%), which we calculate based on our grammar (action, color, speed): 1÷(3×6×2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The action-to-language translation performance of PGAE falls when the ratio of crossmodal (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' supervised) training iterations is low, particularly when 10% or a smaller proportion of the iterations are supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Even though the description accuracy slightly increases to over 95% when supervised training amounts to only 20% of all training iterations, it sharply drops to well below 50% when the rate is decreased to 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PGAE is able to describe 36% of the test samples when only 1% of the training iterations are used to learn crossmodal translations between action and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In contrast, PTAE maintains its perfect description accuracy even when it has only been 11 Action-to-Language Performance wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ratio of Supervised Training Iterations 80 (%) Sentence Accuracy 60 PGAE PTAE chance 40 20 2 10 20 50 66 (Crossmodal Training Iterations)-(Total Training Iterations) (%)Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sentence accuracy for action-to-language translation on the test set wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' supervised training sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Supervised training refers to crossmodal translation cases “describe” and “execute”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We limit the number of training samples for the supervised tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We report the results for the 1%, 2%, 5% 10%, 20%, 50%, and 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='6% cases as well as the 100% regular training case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' These percentages correspond to the fraction of training samples used exclusively for the supervised training for PGAE and PTAE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', both “execute” and “describe” signals are trained with only a limited number of samples corresponding to the percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' trained with 1% supervised training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' While there is a detrimental impact of reduced supervision, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', the limitation on the percentage of crossmodal training itera- tions, on the action-to-language translation performance of PGAE, transformer-based PTAE is not affected by the same phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For space reasons, we do not report language-to-action results wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' different percentages of supervised iterations, but we observed a similar trend comparable with Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In order to further investigate the performance of PTAE with limited supervision, we introduce a more challenging training regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We limit the number of training samples shown to supervised signals, “describe” and “execute”, and show the rest of the training samples only on “repeat action” and “repeat language” modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We train both PGAE and PTAE with varying percentages of supervised training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The results can be seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In all cases with different proportions of supervised training samples, both PGAE and PTAE outperform the chance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' While main- taining perfect sentence accuracy down to 20% supervised training and keeping up its performance for 10% supervised training for the “describe” signal, PTAE’s perfor- mance drops sharply when the ratio of training samples used for crossmodal signals is 2% and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Nevertheless, PTAE beats PGAE in each case when trained on dif- ferent percentages of supervised training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PGAE’s performance suffers even when 50% of training samples are used for supervised signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' it drops below 80% - PTAE retains 100% for the same case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' It takes more than 90% of the training samples to be exclusively used in the unsupervised signals for PTAE’s performance to decrease meaningfully (from 100% to 81%), while this ratio is much lower for PGAE as its per- formance already drops significantly at 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Even for 1% supervised training samples which amount to only 7 training samples, PTAE manages to translate one-third of the test samples from action to sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 12 Action-to-Language Performance wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ratio of Supervised Training Samples 100 PGAE PTAE chance 80- Sentence Accuracy (%) 60 40 20 0- 12 5 20 50 1o 66 100 (Crossmodal Training Samples)-(Total Training Samples) (%)Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Joint value prediction error in language-to-action translation on the test set wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' supervised training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Supervised training refers to crossmodal translation cases “describe” and “execute”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We limit the number of training samples for the supervised tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We report the results for the 1%, 2%, 5% 10%, 20%, 50%, and 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='6% cases as well as the 100% regular training case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' These percentages correspond to the fraction of training samples used exclusively for the supervised training for PGAE and PTAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “execute” and “describe” translations are shown the same limited number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language-to-action translation results with respect to different percentages of su- pervised training samples for PGAE and PTAE are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We show the deviation of the produced joint values from the original ones in terms of the normalized root-mean-squared error (NRMSE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' which we obtain by normalizing the root-mean- squared error (RMSE) between the predicted and ground-truth values by the range of joint values – the lower percentages indicate better prediction (0% NRMSE meaning predicted values are identical with ground-truth values),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' whereas the higher percent- ages indicate worse prediction (100% NRMSE meaning the RMSE between predicted and ground-truth values is equal to the range of possible values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We can see a similar trend as in action-to-language translation apart from the regular case (100%) when PGAE has a lower error than PTAE, which is probably due to the fact that PGAE is trained for more than two times the number of iterations than PTAE since it takes longer for PGAE’s training loss to reach a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In all other cases, limiting the ratio of training samples to be used in the supervised modes impacts PGAE’s language-to-action performance heavily: the NRMSE rises from less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='5% to al- most 8% when the percentage of supervised samples is reduced to two-thirds of the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The error rate increases further as the number of training samples used in the crossmodal training modes decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The NRMSE for PTAE is also in- versely proportional to the ratio of supervised training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, the impact of limiting the number of training samples for supervised modes on PTAE is much lower than on PGAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' When the percentage of supervised training samples is reduced to 1%, the deviation from the ground-truth joint values is only a little more than 4% for PTAE, whereas the same statistic for PGAE is almost 14%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 13 14 ★-PGAE PTAE 12 10 2 01 12 5 10 20 50 66 100 (Crossmodal Training Samples)-(Total Training Samples) (%)Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Model performance on the test set wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' of conflicts introduced in the extra input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For action- to-language and language-to-language (the top row), we show the predicted sentence accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For language- to-action and action-to-action, we show the normalized root-mean-squared error (NRMSE) for predicted joint values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The modality in which the conflicts are introduced is given in the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For each signal, we add extra conflicting inputs either in the action or language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' When the conflict is introduced in action, we also test having the conflict only in the vision and only in the proprioception submodality - in this case, the other submodality has the matching input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Exposure to conflicting input modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We also investigate the impact of contradictory extra input on the performance of PTAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' For this, we use PTAE-regular that is trained with 33% unsupervised training iterations and no contradictory input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We test the robustness of our approach to varying numbers of conflicts (up to 3) in the extra input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The definitions of the added conflict per task signal are: “describe”: Here, we add a conflicting description to the language input (conflict in language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “execute”: Here, we use a conflicting sequence of vision and proprioception input (conflict in action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “repeat action”: Here, we add a conflicting description to the language input (conflict in language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' “repeat language”: Here, we use a conflicting sequence of vision and propriocep- tion input (conflict in action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The conflicts are introduced using the following scheme: for the conflict in the extra language input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' one, two, or all of the action, color, and speed words that constitute a description, do not match with the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' for the conflict in the extra action input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' one, two, or all of the action-type, position, and speed aspects, which form distinct actions, do not match with the language description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The results of this experiment are given in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In the case of the “describe” and “repeat action” signals, the action supplies the relevant input whereas the language 14 Action-to-Language Performance Language-to-Language Performance 100 100 Action (Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='+Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=') Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Only Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Only Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sentence Accuracy (%) 80 80 60 - 60 - 40 40 20 - 20 0 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' of conflicts in extra language input No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='of conflicts in extra action input Action-to-Action Performance Language-to-Action Performance Action (Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='+Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=') Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Only Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Only Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2 2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' of conflicts in extra language input No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' of conflicts in extra action inputis the conflicting distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Here, we observe only a slight decrease in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In the case of action-to-language translation (“describe”) the sentence accuracy goes down from 100% to 95% when there are three conflicting input elements (action type, color, speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Action-to-action (“repeat action”) translation manages to retain its performance as the error in joint values only slightly increases from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='03% to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='09% for the case with 3 conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In the case of “execute” and “repeat language” signals, the language supplies the relevant input while the action is the conflicting distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Here, we observe a big performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language-to-action translation (“execute”) suffers heavily as the deviation of the predicted joint values from the ground-truth joint values increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='99% to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In the language-to-language translation case (“repeat language”), PTAE loses its ability to repeat the given language description when one or more conflicting elements (action type, position, speed) are introduced with the extra input: the sentence accuracy decreases from 100% to 0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Therefore, we can see the asymmetric impact of conflicts in the two modalities, namely, when language input is introduced as a contradictory element, the perfor- mance drops slightly, whereas when the contradictory input is introduced in the action stream, the model is affected heavily and performs poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The output modality has no significant impact on the result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' for example, we can see that both “describe” and “repeat language” output language at large, but they are affected very differently by the conflicting input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' To test whether the bigger impact of conflicting action input is due to the involvement of two modalities in action (vision and proprioception), we also tried introducing the conflict either only in vision or only in proprioception (the relatively brighter bars in the two charts on the right in Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In either case, the performance is still substantially negatively affected, although the drop in performance is naturally not as severe as introducing the conflict in both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Discussion The experimental results on action-to-language and language-to-action translations show the superior performance and efficiency of our novel PTAE model under lim- ited supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Limiting the percentage of supervised crossmodal iterations during training has no adverse effect on PTAE as it maintains its perfect sentence accuracy when translating from action to language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In contrast, the previous PGAE model’s action-to-language translation accuracy drops substantially to around 40% when only 1 or 2% of the training iterations are supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' When we challenge both models more by limiting the number of training samples for the supervised crossmodal “execute” and “describe” signals, we see a similar pattern: when 50% or less of the training samples are used for supervised signals, action-to- language sentence accuracy for PGAE decreases directly proportional to the ratio of supervised samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PTAE, on the other hand, retains its action-to-language perfor- mance up until the case where only 5% of the training samples are used in a supervised fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Even after being trained with 2% supervised training, which amounts to only 13 samples out of 648, PTAE is able to describe more than half of the action sequences correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' All in all, PTAE shows superior action-to-language performance than PGAE for varied levels of limited supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The adverse effect of limiting the number of supervised training samples on the language-to-action performance can already be seen for PGAE even when only one- third of the samples are excluded (66% supervised case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The NRMSE between pre- 15 dicted and ground-truth joint values rises significantly from around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='5% to around 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' It continues to increase gradually after reducing the level of supervision to 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' On the contrary, PTAE is robust against the limited supervision with respect to the ratio of crossmodal training samples until the supervised percentage is brought down to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' After that, it can be seen that the error rate gradually increases, albeit only just over 4% for PTAE when only 7 samples are used for the supervised signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Over- all, these results indicate the clear superiority of Transformer-based multimodal fusion over a simpler attention mechanism by GMU in terms of performance and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Although it is relatively larger than PGAE, PTAE is trained much faster and reaches a global optimum in less than half of the training iterations of PGAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' When introducing a conflicting modality input during testing, we observed an asym- metry in that a conflicting action input leads to a larger disturbance than a conflicting language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' One possible reason is that the Crossmodal Transformer architecture is asymmetric: As input, we are using action input as two input vectors (K and V: keys and values), whereas language as one input vector (Q: queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' This setting was chosen because the opposite setup (with action as queries) was found less performant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Our setup can be interpreted as language-conditioned action attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A computa- tionally more expensive architecture could combine both asymmetric setups, as has been done for learning vision and language representations (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Another possible reason for the larger impact of a conflicting action could be that the action input combines two submodalities, vision, and proprioception, and therefore involves more information than the language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' However, limiting the conflict to one of the submodalities did not completely remove the asymmetry as introducing the conflict only in one action submodality (vision or proprioception) still had a stronger effect on the model performance than a conflicting language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Unlike language, vision contains the complete information to perform a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Consider the example “pull red slowly” for language-to-action translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Here, the language does not contain any information about whether the object is on the left or right side, so the agent can only execute this correctly when also taking visual input into account during action execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In contrast, in the opposite direction (action-to-language translation) and in action repetition, the visual input has the complete information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Conclusion In this paper, we introduced a paired Transformer-based autoencoder, PTAE, which we trained largely by unsupervised learning with additional, but reduced supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The PTAE achieves significantly better action-to-language and language-to-action transla- tion performance under limited supervision conditions compared to the former GMU- based model, PGAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Furthermore, we tested the robustness of our new approach against contradictory extra input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In line with the concept of incongruence in psy- chology, these experiments show that conflict deteriorates the output of our model, and more conflicting features lead to higher interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' We also found an asymmetry between the action and language modalities in terms of their conflicting impact: the action modality has significantly more influence over the performance of the model regardless of the main output modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Our novel bidirectional embodied language learning model is flexible in performing multiple tasks and it is efficient and robust against the scarcity of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hence, it is a step towards an autonomous agent that can communicate with humans while performing various tasks in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In the future, we will expand our approach 16 with reinforcement learning to reduce the need for expert-defined action trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Furthermore, a reinforcement learner may explore more dexterous object manipula- tion with diversified action trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' With more realistic action execution, we will attempt to tackle the problem of sim-to-real transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lastly, diversifying our action repertoire will inevitably lead to more diverse natural language descriptions, which we can tackle by employing a pretrained Transformer-based large language model as a language encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Disclosure statement The authors report there are no competing interests to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Funding This work was supported by the German Research Foundation (DFG) under Project TRR 169 Crossmodal Learning (CML), LeCareBot, IDEAS, and MoReSpace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' References Abramson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ahuja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Brussee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Carnevale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Cassin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Clark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Dudzik, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Georgiev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Guy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Harley, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hill, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hung, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kenton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Landon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lillicrap, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Mathewson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Muldal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Santoro, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Savinov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Varma, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wayne, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Yan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Imitating interactive intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='05672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ahn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Brohan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Brown, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Chebotar, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Cortes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' David, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Finn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Fu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Gopalakr- ishnan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hausman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Herzog, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hsu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ibarz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ichter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Irpan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ruano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jeffrey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jesmonth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Joshi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Julian, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kalashnikov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kuang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Levine, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Luu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Parada, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Pastor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Quiambao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Rao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Rettinghouse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Reyes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sermanet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sievers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Tan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Toshev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Vanhoucke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Xia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Xiao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Yan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Do as I can and not as I say: Grounding language in robotic affordances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='01691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Antunes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Laflaquiere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ogata, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Cangelosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A bi-directional multiple timescales LSTM model for grounding of actions and verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2614–2621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Aravena, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hurtado, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Riveros, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Cardona, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Manes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ib´a˜nez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Applauding with closed hands: neural signature of action-sentence compatibility effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PloS ONE 5(7), e11751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Arevalo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Solorio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Montes-y G´omez, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Gonz´alez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Gated multimodal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Neural Computing and Applications 32(14), 10209–10228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Bisk, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Holtzman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Thomason, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Andreas, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Bengio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Chai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lapata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lazari- dou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' May, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Nisnevich, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Pinto, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Turian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020, November.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Experience grounds language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 8718–8735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Brown, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Mann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ryder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Subbiah, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kaplan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Dhariwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Neelakantan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shyam, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sastry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Askell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 33, 1877–1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Devlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Chang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lee, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019, June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' BERT: Pre-training of deep bidirectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Proceedings of the 2019 Confer- ence of the North American Chapter of the Association for Computational Linguistics: Hu- man Language Technologies, Volume 1 (Long and Short Papers), Minneapolis, Minnesota, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 4171–4186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 17 Eisermann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Weber, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wermter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Generalization in multimodal language learning from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Proceedings of the International Joint Conference on Neural Networks (IJCNN 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Glenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kaschak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Grounding language in action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Psychonomic Bulletin & Review 9(3), 558–565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hatori, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kikuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Takahashi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Tsuboi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Unno, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ko, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Interactively picking real-world objects with unconstrained spoken language instruc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 2018 IEEE International Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 3774–3781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hauk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Johnsrude, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Pulverm¨uller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Somatotopic representation of action words in human motor and premotor cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Neuron 41(2), 301–307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Heinrich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Yao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hinz, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hummel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kerzel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Weber, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wermter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Crossmodal language grounding in an embodied neurocognitive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Frontiers in Neurorobotics 14, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hochreiter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Schmidhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Long short-term memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Neural Computation 9(8), 1735–1780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Irshad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ma, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hierarchical cross-modal agent for robotics vision-and-language navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 2021 IEEE International Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 13238–13246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jaegle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Borgeaud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Alayrac, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Doersch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ionescu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ding, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Koppula, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zoran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Brock, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shelhamer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Perceiver io: A general architecture for structured inputs & outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Irpan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Khansari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kappler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ebert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lynch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Levine, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Finn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' BC-z: Zero-shot task generalization with robotic imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 5th Annual Conference on Robot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Gupta, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Dou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Fei-Fei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Anandkumar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zhu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' VIMA: General robot manipulation with multimodal prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kaschak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Madden, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Therriault, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Yaxley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Aveyard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Blanchard, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zwaan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Perception of motion affects language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Cognition 94(3), B79–B89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kerzel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Pekarek-Rosin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Strahl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Heinrich, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wermter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Teaching NICO how to grasp: an empirical study on crossmodal social interaction as a key factor for robots learning from humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Frontiers in Neurorobotics 14, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kerzel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Strahl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Magg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Navarro-Guerrero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Heinrich, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wermter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' NICO—Neuro-Inspired COmpanion: A developmental humanoid robot platform for mul- timodal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 2017 26th IEEE International Symposium on Robot and Human Interactive Communication (RO-MAN), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 113–120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 3rd Interna- tional Conference on Learning Representations, ICLR, San Diego, CA, USA, May 7-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Batra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Parikh, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ViLBERT: Pretraining task-agnostic visi- olinguistic representations for vision-and-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Larochelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Beygelzimer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=" d'Alch´e-Buc, E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Fox, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Garnett (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ), Advances in Neural Infor- mation Processing Systems, Volume 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lynch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sermanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language conditioned imitation learning over unstructured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Toussaint, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hsieh (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ), Robotics: Science and System XVII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Meteyard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Bahrami, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Vigliocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Motion detection and motion verbs: Lan- guage affects low-level visual perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Psychological Science 18(11), 1007–1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PMID: 17958716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ogata, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Murase, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Tani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Komatani, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Okuno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Two-way translation of compound sentences and arm motions by recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 1858–1863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ¨Ozdemir, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kerzel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Weber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lee, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wermter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Learning flexible transla- tion between robot actions and language descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Pimenidis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Angelov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jayne, 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Papaleonidas, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Aydin (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ), Artificial Neural Networks and Machine Learning – ICANN 2022, Cham, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 246–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Springer Nature Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ¨Ozdemir, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kerzel, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wermter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Embodied language learning with paired variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In 2021 IEEE International Conference on Development and Learning (ICDL), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hallacy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ramesh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Goh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Agarwal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sastry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Askell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Mishkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Clark, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Learning transferable visual models from natural language supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 8748–8763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Child, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Luan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Amodei, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sutskever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Language models are unsupervised multitask learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Raffel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shazeer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Roberts, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Narang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Matena, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Li, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Exploring the limits of transfer learning with a unified text-to-text transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Journal of Machine Learning Research 21(140), 1–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Reed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zolna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Parisotto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Colmenarejo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Novikov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Barth-Maron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Gimenez, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sulsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Springenberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Eccles, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Bruce, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Razavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Edwards, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Heess, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hadsell, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Vinyals, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Bordbar, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' de Freitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A generalist agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Migimatsu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Yang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Bohg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Concept2robot: Learning manipulation concepts from instructions and human demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Proceedings of Robotics: Science and Systems (RSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shridhar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Manuelli, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Fox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' CLIPort: What and where pathways for robotic manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Proceedings of the 5th Conference on Robot Learning (CoRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shridhar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Manuelli, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Fox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Perceiver-Actor: A multi-task transformer for robotic manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Proceedings of the 6th Conference on Robot Learning (CoRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shridhar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Mittal, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Hsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' INGRESS: Interactive visual grounding of referring expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The International Journal of Robotics Research 39(2-3), 217–232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' van Elk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' van Schie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zwaan, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Bekkering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The functional role of motor activation in language processing: Motor cortical oscillations support lexical-semantic retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Neuroimage 50(2), 665–677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Vaswani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Gomez, �L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Winter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Dudschig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Miller, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ulrich, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kaup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' The action-sentence com- patibility effect (ace): Meta-analysis of a benchmark finding for embodiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Acta Psycho- logica 230, 103712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Yamada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Matsunaga, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ogata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Paired recurrent autoencoders for bidirec- tional translation between robot actions and linguistic descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' IEEE Robotics and Automation Letters 3(4), 3441–3448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Zeng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Florence, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Tompson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Welker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Chien, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Attarian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Armstrong, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Krasin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Duong, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Sindhwani, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Transporter networks: Rearranging the visual world for robotic manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' In J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Kober, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Ramos, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' Tomlin (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' ), 4th Con- ference on Robot Learning, CoRL 2020, 16-18 November 2020, Virtual Event / Cambridge, MA, USA, Volume 155 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 726–747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf'} diff --git a/HdE1T4oBgHgl3EQfFgNH/vector_store/index.pkl b/HdE1T4oBgHgl3EQfFgNH/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..c47c469f11e15dd2a65a9ad835d44b1af4c6a561 --- /dev/null +++ b/HdE1T4oBgHgl3EQfFgNH/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6f6e71d495dec2fa4efa815b19b38c09dcf61c85fa6f328b80d5922c6affba03 +size 94039 diff --git a/I9E2T4oBgHgl3EQfpAhk/content/2301.04024v1.pdf b/I9E2T4oBgHgl3EQfpAhk/content/2301.04024v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c1dbb7ee2904cd9c75fccb5ea689c20849eaf248 --- /dev/null +++ b/I9E2T4oBgHgl3EQfpAhk/content/2301.04024v1.pdf @@ -0,0 +1,3 @@ +version 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and Mathematics +Lebanese American University +Byblos Campus, P.O. Box 36 +Byblos, Lebanon +E-mail: cnour@lau.edu.lb +Jean Takche +Department of Computer Science and Mathematics +Lebanese American University +Byblos Campus, P.O. Box 36 +Byblos, Lebanon +E-mail: jtakchi@lau.edu.lb +Abstract. We prove that the complement of a closed set S satisfying an extended +exterior sphere condition is nothing but the union of closed balls with common radius. +This generalizes [11, Theorem 3] where the set S is assumed to be prox-regular, a +property stronger than the extended exterior sphere condition. We also provide a +sufficient condition for the equivalence between prox-regularity and the extended +exterior sphere condition that generalizes [13, Corollary 3.12] to the case in which S +is not necessarily regular closed. +1. Introduction. Let S ⊂ Rn be a closed set. For x ∈ S, a vector ζ ∈ Rn is said to +be proximal normal to S at x if there exists σ = σ(x, ζ) ≥ 0 such that +⟨ζ, s − x⟩ ≤ σ∥s − x∥2, +∀s ∈ S, +(1) +where ⟨·, ·⟩ and ∥·∥ denote the standard inner product and Euclidean norm, respectively. +The relation (1) is commonly referred to as the proximal normal inequality. Note that +for ζ ̸= 0 and σ ̸= 0, the proximal normal inequality (1) is equivalent to +B +� +x + 1 +2σ +ζ +∥ζ∥; 1 +2σ +� +∩ S = ∅, +where B(y; ρ) denotes the open ball of radius ρ centered at y. In that case, we say that +ζ is realized by a +1 +2σ-sphere. On the other hand, for ζ ̸= 0 and σ = 0, the proximal +normal inequality (1) is equivalent to B(x + ρζ; ρ) ∩ S = ∅ for all ρ > 0. In that +case, ζ is realized by an ρ-sphere for any ρ > 0. Now, in view of (1), the set of all +proximal normals to S at x is a convex cone containing 0, and we denote it by N P +S (x). +Since no nonzero ζ satisfying (1) exists if x ∈ int S (the interior of S), we deduce that +Key words and phrases. Prox-regularity, exterior sphere condition, union of closed balls property, +proximal analysis, nonsmooth analysis. +∗ Corresponding author. +1 +arXiv:2301.01051v1 [math.MG] 3 Jan 2023 + +2 +C. NOUR AND J. TAKCHE +N P +S (x) = {0} for all x ∈ int S. This may also occur for x ∈ bdry S, the boundary of S, +as is the case when S is the epigraph of the function f(z) = −|z| and x is the origin. +More information about N P +S (·) and proximal analysis can be found in [6, 10, 19, 22]. +Now we fix r > 0. +We recall that the set S is said to be r-prox-regular if for +any x ∈ bdry S and for any 0 ̸= ζ ∈ N P +S (x), ζ is realized by an r-sphere. +For +more information about prox-regularity, and related properties such as positive reach, +proximal smoothness, p-convexity and ϕ0-convexity, see [1, 7, 8, 9, 20, 23]. On the +other hand, the set S is said to be satisfying the exterior r-sphere condition if for any +x ∈ bdry S, there exists 0 ̸= ζ ∈ N P +S (x) such that ζ is realized by an r-sphere. Clearly +if S is r-prox-regular, then it satisfies the exterior r-sphere condition. For the converse, +it does not hold in general as it is shown via counterexamples in [13]. In this latter, +Nour, Stern and Takche provided sufficient condition for the equivalence between prox- +regularity and the exterior sphere condition. More precisely, they proved that the two +properties are equivalent if S is epi-Lipschitz (or wedged ) and has compact boundary, +see [13, Corollary 3.12].1 Recall that a closed set S is said to be epi-Lipschitz at a +point x ∈ S if the set S can be viewed near x, after application of an orthogonal +matrix, as the epigraph of a Lipschitz continuous function. If this holds for all x ∈ S, +then we simply say that S is epi-Lipschitz. This geometric definition was introduced +by Rockafellar in [21]. The epi-Lipschitz property of S at x is also characterizable in +terms of the nonemptiness of the topological interior of the Clarke tangent cone of S +at x which is also equivalent to the pointedness of the Clarke normal cone of S at x, +see [5, Theorem 7.3.1]. +In their recent paper [11], Nacry and Thibault proved in [11, Theorem 3] that if S is +r-prox-regular, then the complement of S, denoted by Sc, is the union of closed balls +with common radius. What is remarkable in this result is that the set S can be any +closed set, including the sets that are not regular closed (that is, S ̸= cl (int S) the +closure of the interior of S). +In this paper, we generalize [11, Theorem 3] by replacing the r-prox-regulariy of S +by weaker conditions, including the exterior r-sphere condition if S is regular closed, +and a new extended version of the exterior r-sphere condition if S is not necessarily +regular closed. +When S is regular closed, the exterior r-sphere condition of S coincides with the +interior r-sphere condition of (int S)c, a well known condition in control theory, see +e.g., [2, 3, 4, 12].2 Using this fact and [16, Conjecture 1.2],3 we obtain the following +theorem which is the first main result of this paper. +Theorem 1.1. Let S ⊂ Rn be a closed set such that cl (int S) satisfies the exterior +r-sphere condition for some r > 0. Then (int S)c is the union of closed r +2-balls. If in +addition S is regular closed, then Sc is the union of closed r′-balls for any r′ < r +2. +1This equivalence result is generalized to the variable radius case in [15], see also [17]. +2In these references, the interior sphere condition is used to study the regularity of the unilateral +and the bilateral minimal time functions associated to a control system. +3This conjecture is introduced in [13, 14], and proved in [16]. A generalization of this result to the +variable radius case is given in [16, Theorem 3.1], see also [18]. + +THE EXTENDED EXTERIOR SPHERE CONDITION +3 +Note that the “In addition” part of Theorem 1.1 is not necessarily true when S is not +regular closed, see Example 2.2. On the other hand, the assumption “cl (int S) satisfies +the exterior r-sphere condition” is weaker than the assumption “S satisfies the exterior +r-sphere condition”, see Proposition 2.1 and Example 2.1. +In the following theorem, which is our second main result, we prove that a similar +result to [11, Theorem 3] can be obtained if the prox-regularity is replaced by a weaker +property, namely the following extended exterior r-sphere condition. We say that S +satisfies the extended exterior r-sphere condition if: +• For any x ∈ bdry (int S) ⊂ bdry S, there exists 0 ̸= ζ ∈ N P +S (x) such that ζ is +realized by an r-sphere. +• For any x ∈ (bdry S) \ bdry (int S) and for any 0 ̸= ζ ∈ N P +S (x), ζ is realized by +an r-sphere. +Theorem 1.2. Let S ⊂ Rn be a closed set satisfying the extended exterior r-sphere +condition for some r > 0. Then Sc is the union of closed r +2-balls. +One can easily see that if S satisfies the extended exterior r-sphere condition then S +satisfies the exterior r-sphere condition. The converse, which does not hold in general +(see Example 2.1), is valid if S is regular closed. Note that the extended exterior r- +sphere condition of Theorem 1.2 cannot be replaced by the exterior r-sphere condition +as it will be shown in Example 2.3. On the other hand, clearly if S is r-prox-regular +then S satisfies the extended exterior r-sphere condition. The converse does not hold +in general, even if S is regular closed, as it is proved in the counterexamples of [13, +Section 2]. In the following theorem which is our third and last main result, we provide +a sufficient condition for the equivalence between prox-regularity and the extended +exterior sphere condition. It generalizes [13, Corollary 3.12] to the case in which S is +not necessarily regular closed. +Theorem 1.3. Let S ⊂ Rn be a closed set such that cl (int S) is epi-Lipschitz and has +compact boundary. Then S is r-prox-regular for some r > 0 if and only if S satisfies +the extended exterior r′-sphere condition for some r′ > 0. +The details of the proofs of Theorems 1.1, 1.2 and 1.3 will be presented in the next +section after giving a brief description of the notation used in this paper, and providing +some auxiliary results. +2. Proofs of the main results. We denote by ∥ · ∥, ⟨·, ·⟩, B and ¯B, the Euclidean +norm, the usual inner product, the open unit ball and the closed unit ball, respectively. +For ρ > 0 and x ∈ Rn, we set B(x; ρ) := x + ρB and ¯B(x; ρ) := x + ρ ¯B. For a +set S ⊂ Rn, Sc, int S, bdry S and cl S are the complement (with respect to Rn), the +interior, the boundary and the closure of S, respectively. The closed segment joining +two points x and y in Rn is denoted by [x, y]. The distance from a point x to a set S is +denoted by dS(x). We also denote by proj S(x) the set of closest points in S to x, that +is, the set of points s ∈ S which satisfy dA(x) = ∥s − x∥. For A and B two subsets of +Rn, d(A, B) denotes the distance between A and B, that is, +d(A, B) := inf{∥a − b∥ : (a, b) ∈ A × B}. + +4 +C. NOUR AND J. TAKCHE +For S ⊂ Rn closed, the following proposition provides some useful properties of the +set cl (int S). +Proposition 2.1. Let S ⊂ Rn be closed. Then we have the following: +(i) cl (int S) ⊂ S, int (cl (int S)) = int S, and +bdry (cl (int S)) = bdry (int S) ⊂ bdry S. +(ii) S = [cl (int S)] ∪ [(bdry S) \ bdry (int S)] with +[cl (int S)] ∩ [(bdry S) \ bdry (int S)] = ∅. +(iii) If S = cl O, where O is open, then S is regular closed. +(iv) If for r > 0, S satisfies the exterior r-sphere condition, then cl (int S) also satisfies +the exterior r-sphere condition. +Proof. The properties (i)-(iii) are well known in metric topology. For (iv), let r > 0 +and assume that S satisfies the exterior r-sphere condition. Let x ∈ bdry (cl (int S)). +By (i) we have that x ∈ bdry S, and hence there exists 0 ̸= ζ ∈ N P +S (x) such that ζ is +realized by an r-sphere. This gives using the proximal normal inequality that +� ζ +∥ζ∥, s − x +� +≤ 1 +2r∥s − x∥2, ∀s ∈ S. +Since cl (int S) ⊂ S, we obtain that +� ζ +∥ζ∥, s − x +� +≤ 1 +2r∥s − x∥2, ∀s ∈ cl (int S). +Therefore, ζ ∈ N P +cl (int S)(x) and is realized by an r-sphere. +Example 2.1. The converse of Proposition 2.1(iv) is not necessarily true. Indeed, +consider in R2, S := ¯B(0; 1) ∪ [(1, 0), (2, 0)]. Clearly S does not satisfy the exterior +r-sphere condition for any r > 0, but cl (int S) = ¯B(0; 1) satisfies the exterior r-sphere +condition for any r > 0. +In the following proposition we list some well known characterizations and conse- +quences of prox-regular and epi-Lipshitz properties. For the proofs, see [5, 20, 22]. +Proposition 2.2. Let S ⊂ Rn be closed. Then we have the following: +(i) If S is prox-regular, then for each x ∈ bdry S we have +N P +S (x) = N L +S (x) = NS(x), +where N L +S (x) and NS(x) denotes the Mordukhovich (or limiting) and the Clarke +normal cones to S at x, respectively. +(ii) For x ∈ bdry S, S is epi-Lipschitz at x if and only if NS(x) is pointed, that is, +NS(x) ∩ −NS(x) = {0}. +(iii) If for x ∈ bdry S, S is epi-Lipschitz at x, then x ∈ cl (int S). + +THE EXTENDED EXTERIOR SPHERE CONDITION +5 +2.1. Proof of Theorem 1.1. Let S ⊂ Rn be a closed set such that cl (int S) satisfies +the exterior r-sphere condition for some r > 0. Since by Proposition 2.1(iii) we have +that cl (int S) is regular closed, we get that (int (cl (int S))c = (int S)c satisfies the +interior r-sphere condition. Hence by [16, Conjecture 1.2] we obtain that (int S)c is the +union of closed r +2-balls. +We proceed to prove the “In addition” part of Theorem 1.1. We assume that S is +regular closed, and we consider r′ < r +2. Let x ∈ Sc. Since Sc ⊂ (int S)c and (int S)c is +the union of closed r +2-balls, there exists yx ∈ (int S)c such that +x ∈ ¯B +� +yx; r +2 +� +⊂ (int S)c. +(2) +If ∥yx − x∥ < r′, then using (2) we get that +x ∈ ¯B(yx; r′) ⊂ B +� +yx; r +2 +� +⊂ ¯B +� +yx; r +2 +� +⊂ (int S)c. +This gives that +x ∈ ¯B(yx; r′) ⊂ int ((int S)c) = (cl (int S))c = Sc. +Now we assume that ∥yx − x∥ ≥ r′. Then by (2) and since x ∈ Sc, we have that +x ∈ ¯B +� +x + r′ yx − x +∥yx − x∥; r′ +� +⊂ +B (yx; ∥yx − x∥) ∪ {x} +⊂ +B +� +yx; r +2 +� +∪ {x} +⊂ +int ((int S)c) ∪ Sc = Sc. +Therefore, Sc is the union of closed r′-balls. +This terminated the proof of the “In +addition” part, and hence the proof of Theorem 1.1. +□ +In the following example, we prove that the “In addition” part of Theorem 1.1 is not +necessarily true when S is not regular closed. +Example 2.2. Let S := {(x, ex) : x ∈ R}∪{(x, −ex) : x ∈ R}. Clearly S is not regular +closed, and satisfies the exterior 3 +√ +3 +2 -sphere condition. On the other hand, Sc is not +the union of closed r-balls for any r > 0. Note that S does not satisfy the extended +exterior r-sphere condition for any r > 0. +2.2. Proof of Theorem 1.2. Let S ⊂ Rn be a closed set satisfying the extended +exterior r-sphere condition for some r > 0. We consider x ∈ Sc. Let s0 ∈ proj S(x) +and let r0 := ∥x − s0∥. Clearly we have s0 ∈ bdry S and s0 ̸= x, and hence r0 ̸= 0. +Moreover, we have +B(x; r0) ∩ S = ∅. +(3) +There are three cases to consider. +Case 1: r0 > r +2. +Then using (3), we obtain that +x ∈ ¯B +� +x; r +2 +� +⊂ B(x; r0) ⊂ Sc. +Case 2: r0 ≤ r +2 and s0 ̸∈ bdry (int S). + +6 +C. NOUR AND J. TAKCHE +Then the proximal normal vector ζ0 := x − s0 ∈ N P +S (s0) is realized by an r-sphere. +This gives that +B +� +s0 + r ζ0 +∥ζ0∥; r +� +⊂ Sc. +(4) +We claim that +¯B +� +x + r +2 +ζ0 +∥ζ0∥; r +2 +� +⊂ B +� +s0 + r ζ0 +∥ζ0∥; r +� +. +(5) +Indeed, for y ∈ ¯B +� +x + r +2 +ζ0 +∥ζ0∥; r +2 +� +we have +����y − s0 − r ζ0 +∥ζ0∥ +���� += +���� +� +y − x − r +2 +ζ0 +∥ζ0∥ +� ++ +� +x − s0 − r +2 +ζ0 +∥ζ0∥ +����� +≤ +����y − x − r +2 +ζ0 +∥ζ0∥ +���� + +����x − s0 − r +2 +ζ0 +∥ζ0∥ +���� +≤ +r +2 + +����r0 +ζ0 +∥ζ0∥ − r +2 +ζ0 +∥ζ0∥ +���� += +r +2 + r +2 − r0 = r − r0 < r. +Now combining (4) and (5), we obtain that +x ∈ ¯B +� +x + r +2 +ζ0 +∥ζ0∥; r +2 +� +⊂ B +� +s0 + r ζ0 +∥ζ0∥; r +� +⊂ Sc. +Case 3: r0 ≤ r +2 and s0 ∈ bdry (int S). +Then for every ε > 0, we have B(s0; ε)∩int S ̸= ∅. Let zε ∈ B(s0; ε)∩int S. We denote +by ξε := +zε−x +∥zε−x∥ and we consider sε ∈ [zε, x] ∩ bdry S, where this latter intersection is +nonempty since zε ∈ int S and x ∈ Sc. Let ζε ∈ N P +S (sε) be the unit vector realized by +an r-sphere. Hence, for yε := sε + rζε, we have +B (yε; r) ∩ S = ∅. +(6) +If yε = x, then +x ∈ ¯B +� +x; r +2 +� +⊂ B(yε; r) ⊂ Sc. +If 0 < ∥yε − x∥ < r, then for w ∈ ¯B +� +x + r +2 +yε−x +∥yε−x∥; r +2 +� +, we have +∥w − yε∥ +≤ +����w − x − r +2 +yε − x +∥yε − x∥ +���� + +����x + r +2 +yε − x +∥yε − x∥ − yε +���� +≤ +r +2 + +���� +r +2 +yε − x +∥yε − x∥ − (yε − x) +���� += +r +2 + +���r +2 − ∥yε − x∥ +��� < r. +This gives using (6) that +x ∈ ¯B +� +x + r +2 +yε − x +∥yε − x∥; r +2 +� +⊂ B (yε; r) ⊂ Sc. + +THE EXTENDED EXTERIOR SPHERE CONDITION +7 +x +s0 +zϵ +sϵ +yϵ +ξϵ +ζϵ +Figure 1. +Case 3: ∥yε − x∥ ≥ r +Now we assume that ∥yε − x∥ ≥ r and we denote by +rε := +r2 +0∥yε − x∥ +∥yε − x∥2 + r2 +0 − r2 > 0. +(7) +Claim 1. B +� +x + rε +yε−x +∥yε−x∥; rε +� +⊂ B(x; r0) ∪ B(yε; r). +To prove this claim, let w ∈ B +� +x + rε +yε−x +∥yε−x∥; rε +� +and assume that w ̸∈ B(x; r0). +Then +∥w − x∥ < 2rε +� w − x +∥w − x∥, yε − x +∥yε − x∥ +� +. +(8) +We have +∥w − yε∥2 += +∥(w − x) − (yε − x)∥2 += +∥w − x∥2 + ∥yε − x∥2 − 2⟨w − x, yε − x⟩ +(8) +< +∥w − x∥2 + ∥yε − x∥2 − ∥yε − x∥ +rε +∥w − x∥2 += +∥yε − x∥2 + ∥w − x∥2 +� +1 − ∥yε − x∥ +rε +� +(7) += +∥yε − x∥2 + ∥w − x∥2 +� +1 − ∥yε − x∥2 + r2 +0 − r2 +r2 +0 +� += +∥yε − x∥2 + ∥w − x∥2 +� +�� +� +≥r2 +0 +�r2 − ∥yε − x∥2 +r2 +0 +� +� +�� +� +≤0 +≤ +∥yε − x∥2 + r2 − ∥yε − x∥2 = r2. +This gives that w ∈ B(yε, r) which terminates the proof of the claim. + +8 +C. NOUR AND J. TAKCHE +Claim 2. ⟨ξε, ζε⟩ < +ε +2r. +To prove this claim, first we remark that since zε ∈ int S, we deduce from (6) that +zε ̸∈ ¯B(yε; r) = ¯B (sε + rζε; r). Hence, using the equality zε − sε = ∥zε − sε∥ξε, we +obtain that +∥zε − sε∥ > 2r ⟨ξε, ζε⟩ . +Therefore, using that s0 ∈ proj S(x), we get that +⟨ξε, ζε⟩ < 1 +2r∥zε − sε∥ += +1 +2r (∥zε − x∥ − ∥sε − x∥) +≤ +1 +2r +� +�∥zε − s0∥ + ∥s0 − x∥ − ∥sε − x∥ +� +�� +� +≤0 +� +� +≤ +1 +2r∥zε − s0∥ < ε +2r. +The proof of the claim is terminated. +Claim 3. If ε ∈ +� +0, r3 +0 +4r2 +� +then rε > r +2. +To prove the claim, first we remark using (7) that rε > r +2 is equivalent to +P(∥yε − x∥) := r∥yε − x∥2 − 2r2 +0∥yε − x∥ + r(r2 +0 − r2) < 0. +(9) +Since the discriminant ∆′ of P(∥yε − x∥) is (r2 − r2 +0)2 + r2r2 +0 > 0 and the product of +the roots is r2 +0 − r2 < 0, we conclude that (9) is equivalent to +∥yε − x∥ < r2 +0 + +√ +∆′ +r +. +(10) +We have +∥yε − x∥2 += +∥yε − sε∥2 + ∥sε − x∥2 + 2⟨yε − sε, sε − x⟩ += +r2 + ∥sε − x∥2 + 2r∥sε − x∥ ⟨ζε, ξε⟩ +Claim 2 +< +r2 + ∥sε − x∥2 + ε∥sε − x∥ +sε∈[zε,x] +≤ +r2 + ∥zε − x∥2 + ε∥zε − x∥ +≤ +r2 + (∥zε − s0∥ + ∥s0 − x∥)2 + ε(∥zε − s0∥ + ∥s0 − x∥) +≤ +r2 + (ε + r0)2 + ε(ε + r0) += +r2 + r2 +0 + 2ε2 + 3εr0 +ε∈ +� +0, +r3 +0 +4r2 +� +≤ +r2 + r2 +0 + r6 +0 +8r4 + 3r4 +0 +4r2 += +r2 + r2 +0 + r4 +0 +r2 +� r2 +0 +8r2 + 3 +4 +� +r0≤ r +2 +< +r2 + r2 +0 + r4 +0 +r2. +(11) + +THE EXTENDED EXTERIOR SPHERE CONDITION +9 +On the other hand, we have +(r2 +0 + +√ +∆′)2 += +2r4 +0 + r4 − r2r2 +0 + 2r2 +0 +�� +r2 − 1 +2r2 +0 +�2 ++ 3r4 +0 +4 +> +2r4 +0 + r4 − r2r2 +0 + 2r2 +0 +� +r2 − 1 +2r2 +0 +� += +r4 +0 + r4 + r2r2 +0. +This gives that +� +r2 +0 + +√ +∆′ +r +�2 +> r2 + r2 +0 + r4 +0 +r2. +Combining this latter with (11), we obtain inequality (10), and this terminates the +proof of Claim 3. +Now we fix ε ∈ +� +0, r3 +0 +4r2 +� +. By Claim 3 we have that r +2 < rε. Hence, using Claim 1, (3) +and (6), we get that +x ∈ ¯B +� +x + r +2 +yε − x +∥yε − x∥; r +2 +� +⊂ +B +� +x + rε +yε − x +∥yε − x∥; rε +� +∪ {x} +⊂ +B(x; r0) ∪ B(yε; r) ⊂ Sc. +In the three cases above, we have shown that x belongs to a closed r +2-ball included +in Sc. The proof of Theorem 1.2 is terminated. +□ +In the following example, we prove that the extended exterior r-sphere condition +assumption of Theorem 1.2 cannot be replaced by the exterior r-sphere condition. +Example 2.3. In Figure 2, S is the set that consists of both curves and the dashed +region between them. The set S satisfies the exterior r-sphere condition, but it does +not satisfy the extended exterior r-sphere condition since some of the boundary points +on the right-hand side have two normal vectors where only one of them is realized by +an r-sphere. Clearly, Sc is not the union of r′-balls for any r′ > 0. +S +Figure 2. +Example 2.3 +Remark 2.1. From Theorem 1.2, we deduce that if S is regular closed and satisfies +the exterior r-sphere condition for some r > 0, then Sc is the union of closed r +2-balls, +which is stronger than the “In addition” part of Theorem 1.1. Of course we can adjust + +10 +C. NOUR AND J. TAKCHE +O +c0 +c1 +c2 +Figure 3. +Example of Remark 2.2 +the proof of the “In addition” part of Theorem 1.1 and obtain the radius r +2 instead +of r′ < +r +2. We did not do this to not complicate the proof of Theorem 1.1, and to +show how its proof, when r′ < r +2, is a direct consequence of the union of closed balls +conjecture [16, Conjecture 1.2]. Note that since r +2 < +nr +2 +√ +n2−1 for all n ≥ 2, the same +techniques used in the proof of Theorem 1.1 combined with the strong version of the +union of closed balls conjecture [16, Conjecture 1.3], gives that Sc is the union of closed +r′-balls for any r′ < +nr +2 +√ +n2−1, including r′ = r +2. Unfortunately, this latter result cannot +be confirmed since the strong version of the union of closed balls conjecture remains +till now an open question. +Remark 2.2. The radius r +2 of Theorem 1.2 is the largest radius that works in all the +spaces Rn for any n ≥ 2. Indeed, assume that the radius r +2 of Theorem 1.2 is replaced +by an r′ > 0. We claim that r′ ≤ r +2. Indeed, inspired by [17, Example 2], we consider +in Rn the set S to be the complement of the (n + 1) open r-balls centered at Ci, +i = 0, . . . , n, where for (ei)n +i=1 the standard basis of Rn and e0 := 0, we have +Ci := +−nr +� +n(n − 1) +�� +i +i + 1 ei + +n +� +k=i+1 +−1 +� +k(k + 1) +ek +� += +−nr +� +n(n − 1) +� +0, . . . , 0, +√ +i +√i + 1, +−1 +� +(i + 1)(i + 2) +, +−1 +� +(i + 2)(i + 3) +, . . . , +−1 +� +n(n + 1) +� +. +The set S satisfies the extended exterior r-sphere condition (in fact, it is regular closed +and satisfies the exterior r-sphere condition), but the largest closed balls contained in +Sc and containing the origin 0 ∈ Sc is of radius +nr +2 +√ +n2−1, see Figure 3 for n = 2 and +n = 3. Hence, r′ ≤ +nr +2 +√ +n2−1 for all n ≥ 2. Taking n −→ ∞ in this latter, we conclude +that r′ ≤ r +2. + +THE EXTENDED EXTERIOR SPHERE CONDITION +11 +2.3. Proof of Theorem 1.3. Let S ⊂ Rn be a closed set such that cl (int S) is epi- +Lipschitz and has compact boundary. Since the first implication is straightforward +(take r′ = r), we focus on the second implication. So, we assume that S satisfies the +extended exterior r′-sphere condition for some r′ > 0. Then S satisfies the exterior +r′-sphere condition, and hence by Proposition 2.2(iv), cl (int S) satisfies the exterior +r′-sphere condition. Now using [13, Corollary 3.12], applied to cl (int S), we get that +cl (int S) is ρ-prox-regular for some ρ > 0. +Claim 1. d(bdry (cl (int S)), (bdry S) \ bdry (int S)) = rS > 0. +If not, then there exist two sequences xn and yn such that +xn ∈ bdry (cl (int S)), yn ∈ (bdry S) \ bdry (int S) and ∥xn − yn∥ ≤ 1 +n, ∀n ≥ 1. +Since cl (int S) has compact boundary, we get the existence of ¯x ∈ bdry (int S) and a +subsequence of yn, we do not relabel, such that yn −→ ¯x. Since yn ̸∈ cl (int S), we have +from Proposition 2.2(iii) that S is not epi-Lipschitz at yn. This gives using Proposition +2.2(ii) the existence of unit vector ζn such that +ζn ∈ NS(yn) ∩ −NS(yn). +(12) +Now since yn ∈ (bdry S) \ bdry (int S) and S satisfies the extended exterior r′-sphere +condition, we have that S is r′-prox-regular near yn, that is, in a (bdry S)-neighborhood +of yn. Hence, Proposition 2.2(i) and (12) yield that +ζn ∈ N P +S (yn) ∩ −N P +S (yn). +Using the proximal normal inequality and the fact that ζn is unit and realized by an +r′-sphere, we obtain that (ζn)n has a subsequence, we do not relabel, that converges to +a unit vector ζ satisfying +ζ ∈ N P +S (¯x) ∩ −N P +S (¯x). +Since N P +S (¯x) ⊂ N P +cl (int S)(¯x), we deduce that +ζ ∈ N P +cl (int S)(¯x) ∩ −N P +cl (int S)(¯x). +This yields using the prox-regularity of cl (int S) and Proposition 2.2(i), that +ζ ∈ Ncl (int S)(¯x) ∩ −Ncl (int S)(¯x), +which contradicts the fact that cl (int S) is epi-Lipschitz. The proof of the claim is +terminated. +Now we prove that S is r-prox-regular for r := min{ρ, r′, rS +4 }. +Let x ∈ bdry S. +If x ∈ (bdry S) \ bdry (cl (int S)), then from the definition of the extended r′-sphere +condition, we know that any 0 ̸= ζ ∈ N P +S (x) is realized by an r′-sphere. This gives that +any 0 ̸= ζ ∈ N P +S (x) is realized by an r-sphere. Now we assume that x ∈ bdry (cl (int S)). +Having cl (int S) ρ-prox-regular, it results that for any 0 ̸= ζ ∈ N P +S (x) ⊂ N P +cl (int S)(x), +we have +B +� +x + ρ ζ +∥ζ∥; ρ +� +∩ cl (int S) = ∅. + +12 +C. NOUR AND J. TAKCHE +This gives that for any 0 ̸= ζ ∈ N P +S (x), we have +B +� +x + r ζ +∥ζ∥; r +� +∩ cl (int S) = ∅. +(13) +On the other hand, since r < rS +4 and using Claim 1, we have for any 0 ̸= ζ ∈ N P +S (x) +that +B +� +x + r ζ +∥ζ∥; r +� +⊂ B +� +x; rS +2 +� +⊂ [(bdry S) \ bdry (int S)]c. +Combining this latter with Proposition 2.1(ii) and (13), we get for any 0 ̸= ζ ∈ N P +S (x) +that +B +� +x + r ζ +∥ζ∥; r +� +∩ Sc = ∅. +This terminates the proof of Theorem 1.3. +□ +REFERENCES +[1] A. Canino, On p-convex sets and geodesics, J. Diff. Equations 75/1 (1988) 118–157. +[2] P. Cannarsa, H. Frankowska: Interior sphere property of attainable sets and time optimal control +problems, ESAIM: Control Optim. Calc. Var. 12 (2006) 350–370. +[3] P. Cannarsa, C. Sinestrari: Convexity properties of the minimum time function, Calc. Var. 3 +(1995) 273–298. +[4] P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Con- +trol, Birkh¨aser, Boston (2004). +[5] F. H. Clarke: Optimization and Nonsmooth Analysis, John Wiley, New York (1983). +[6] F. H. Clarke, Yu. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Con trol Theory, +Graduate Texts in Mathematics 178, Springer, New York (1998). +[7] F. H. Clarke, R. J. Stern, P. R. Wolenski: Proximal smoothness and the lower-C2 property, J. +Convex Analysis 2/1-2 (1995) 117–144. +[8] G. Colombo, A. Marigonda: Differentiability properties for a class of non-convex functions, Calc. +Var. 25 (2005) 1–31. +[9] H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491. +[10] B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I: Basic Theory, +Springer, Berlin (2006). +[11] F. Nacry, L. Thibault: Distance Function Associated to a Prox-regular set, Set-Valued Var. Anal. +30 (2022) 731–750. +[12] C. Nour, R. J. Stern: Semiconcavity of the bilateral minimal time function: the linear case, +Systems & Control Letters 57 (2008) 863–866. +[13] C. Nour, R. J. Stern, J. Takche: Proximal smoothness and the exterior sphere condition, J. +Convex Analysis 16/2 (2009) 501–514. +[14] C. Nour, R. J. Stern, J. Takche: The union of uniform closed bal ls conjecture, Control and +Cybernetics 38/4B (2009) 1525–1534. +[15] C. Nour, R. J. Stern, J. Takche: The θ-exterior sphere condition, ϕ-convexity and local semicon- +cavity, Nonlinear Analysis 73/2 (2010) 573–589. +[16] C. Nour, R. J. Stern, J. Takche: Validity of the union of uniform closed bal ls conjecture, J. +Convex Analysis 18/2 (2011) 589–600. +[17] C. Nour, R. J. Stern, J. Takche: Generalized exterior sphere conditions and ϕ-convexity, Discrete +Contin. Dyn. Systems 29/2 (2011) 615–622. +[18] C. Nour, J. Takche: On the union of closed bal ls property, J. Optim. Theory Appl. 155 (2012) +376–389. + +THE EXTENDED EXTERIOR SPHERE CONDITION +13 +[19] J.-P.Penot: Calculus Without Derivatives, Graduate Texts in Mathematics 266, Springer, New +York (2013). +[20] R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. +Amer. Math. Soc. 352 (2000) 5231–5249. +[21] R. T. Rockafellar: Clarke’s tangent cones and the boundaries of closed sets in Rn, Nonlinear +Analysis Theory Math. Appl. 3 (1979) 145–154. +[22] R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wis- +senschaften 317, Springer, Berlin (1998). +[23] A. S. Shapiro: Existence and differentiability of metric projections in Hilbert spaces, SIAM J. +Optim. 4 (1994) 231–259. + diff --git a/INAzT4oBgHgl3EQfHvu0/content/tmp_files/load_file.txt b/INAzT4oBgHgl3EQfHvu0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d531a07400c0d6cbf05112068f277dce1e6d6712 --- /dev/null +++ b/INAzT4oBgHgl3EQfHvu0/content/tmp_files/load_file.txt @@ -0,0 +1,489 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf,len=488 +page_content='November 8, 2022 The Extended Exterior Sphere Condition Chadi Nour∗ Department of Computer Science and Mathematics Lebanese American University Byblos Campus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Box 36 Byblos, Lebanon E-mail: cnour@lau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='lb Jean Takche Department of Computer Science and Mathematics Lebanese American University Byblos Campus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Box 36 Byblos, Lebanon E-mail: jtakchi@lau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='lb Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We prove that the complement of a closed set S satisfying an extended exterior sphere condition is nothing but the union of closed balls with common radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This generalizes [11, Theorem 3] where the set S is assumed to be prox-regular, a property stronger than the extended exterior sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We also provide a sufficient condition for the equivalence between prox-regularity and the extended exterior sphere condition that generalizes [13, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='12] to the case in which S is not necessarily regular closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For x ∈ S, a vector ζ ∈ Rn is said to be proximal normal to S at x if there exists σ = σ(x, ζ) ≥ 0 such that ⟨ζ, s − x⟩ ≤ σ∥s − x∥2, ∀s ∈ S, (1) where ⟨·, ·⟩ and ∥·∥ denote the standard inner product and Euclidean norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The relation (1) is commonly referred to as the proximal normal inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Note that for ζ ̸= 0 and σ ̸= 0, the proximal normal inequality (1) is equivalent to B � x + 1 2σ ζ ∥ζ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 1 2σ � ∩ S = ∅, where B(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ρ) denotes the open ball of radius ρ centered at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In that case, we say that ζ is realized by a 1 2σ-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' On the other hand, for ζ ̸= 0 and σ = 0, the proximal normal inequality (1) is equivalent to B(x + ρζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ρ) ∩ S = ∅ for all ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In that case, ζ is realized by an ρ-sphere for any ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now, in view of (1), the set of all proximal normals to S at x is a convex cone containing 0, and we denote it by N P S (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since no nonzero ζ satisfying (1) exists if x ∈ int S (the interior of S), we deduce that Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Prox-regularity, exterior sphere condition, union of closed balls property, proximal analysis, nonsmooth analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='01051v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='MG] 3 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' NOUR AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' TAKCHE N P S (x) = {0} for all x ∈ int S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This may also occur for x ∈ bdry S, the boundary of S, as is the case when S is the epigraph of the function f(z) = −|z| and x is the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' More information about N P S (·) and proximal analysis can be found in [6, 10, 19, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now we fix r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We recall that the set S is said to be r-prox-regular if for any x ∈ bdry S and for any 0 ̸= ζ ∈ N P S (x), ζ is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For more information about prox-regularity, and related properties such as positive reach, proximal smoothness, p-convexity and ϕ0-convexity, see [1, 7, 8, 9, 20, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' On the other hand, the set S is said to be satisfying the exterior r-sphere condition if for any x ∈ bdry S, there exists 0 ̸= ζ ∈ N P S (x) such that ζ is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clearly if S is r-prox-regular, then it satisfies the exterior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For the converse, it does not hold in general as it is shown via counterexamples in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In this latter, Nour, Stern and Takche provided sufficient condition for the equivalence between prox- regularity and the exterior sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' More precisely, they proved that the two properties are equivalent if S is epi-Lipschitz (or wedged ) and has compact boundary, see [13, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1 Recall that a closed set S is said to be epi-Lipschitz at a point x ∈ S if the set S can be viewed near x, after application of an orthogonal matrix, as the epigraph of a Lipschitz continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' If this holds for all x ∈ S, then we simply say that S is epi-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This geometric definition was introduced by Rockafellar in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The epi-Lipschitz property of S at x is also characterizable in terms of the nonemptiness of the topological interior of the Clarke tangent cone of S at x which is also equivalent to the pointedness of the Clarke normal cone of S at x, see [5, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In their recent paper [11], Nacry and Thibault proved in [11, Theorem 3] that if S is r-prox-regular, then the complement of S, denoted by Sc, is the union of closed balls with common radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' What is remarkable in this result is that the set S can be any closed set, including the sets that are not regular closed (that is, S ̸= cl (int S) the closure of the interior of S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In this paper, we generalize [11, Theorem 3] by replacing the r-prox-regulariy of S by weaker conditions, including the exterior r-sphere condition if S is regular closed, and a new extended version of the exterior r-sphere condition if S is not necessarily regular closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' When S is regular closed, the exterior r-sphere condition of S coincides with the interior r-sphere condition of (int S)c, a well known condition in control theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=', [2, 3, 4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 Using this fact and [16, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2],3 we obtain the following theorem which is the first main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be a closed set such that cl (int S) satisfies the exterior r-sphere condition for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then (int S)c is the union of closed r 2-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' If in addition S is regular closed, then Sc is the union of closed r′-balls for any r′ < r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 1This equivalence result is generalized to the variable radius case in [15], see also [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 2In these references, the interior sphere condition is used to study the regularity of the unilateral and the bilateral minimal time functions associated to a control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 3This conjecture is introduced in [13, 14], and proved in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' A generalization of this result to the variable radius case is given in [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1], see also [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' THE EXTENDED EXTERIOR SPHERE CONDITION 3 Note that the “In addition” part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1 is not necessarily true when S is not regular closed, see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' On the other hand, the assumption “cl (int S) satisfies the exterior r-sphere condition” is weaker than the assumption “S satisfies the exterior r-sphere condition”, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In the following theorem, which is our second main result, we prove that a similar result to [11, Theorem 3] can be obtained if the prox-regularity is replaced by a weaker property, namely the following extended exterior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We say that S satisfies the extended exterior r-sphere condition if: For any x ∈ bdry (int S) ⊂ bdry S, there exists 0 ̸= ζ ∈ N P S (x) such that ζ is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For any x ∈ (bdry S) \\ bdry (int S) and for any 0 ̸= ζ ∈ N P S (x), ζ is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be a closed set satisfying the extended exterior r-sphere condition for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then Sc is the union of closed r 2-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' One can easily see that if S satisfies the extended exterior r-sphere condition then S satisfies the exterior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The converse, which does not hold in general (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1), is valid if S is regular closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Note that the extended exterior r- sphere condition of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 cannot be replaced by the exterior r-sphere condition as it will be shown in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' On the other hand, clearly if S is r-prox-regular then S satisfies the extended exterior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The converse does not hold in general, even if S is regular closed, as it is proved in the counterexamples of [13, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In the following theorem which is our third and last main result, we provide a sufficient condition for the equivalence between prox-regularity and the extended exterior sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' It generalizes [13, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='12] to the case in which S is not necessarily regular closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be a closed set such that cl (int S) is epi-Lipschitz and has compact boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then S is r-prox-regular for some r > 0 if and only if S satisfies the extended exterior r′-sphere condition for some r′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The details of the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3 will be presented in the next section after giving a brief description of the notation used in this paper, and providing some auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We denote by ∥ · ∥, ⟨·, ·⟩, B and ¯B, the Euclidean norm, the usual inner product, the open unit ball and the closed unit ball, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For ρ > 0 and x ∈ Rn, we set B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ρ) := x + ρB and ¯B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ρ) := x + ρ ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For a set S ⊂ Rn, Sc, int S, bdry S and cl S are the complement (with respect to Rn), the interior, the boundary and the closure of S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The closed segment joining two points x and y in Rn is denoted by [x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The distance from a point x to a set S is denoted by dS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We also denote by proj S(x) the set of closest points in S to x, that is, the set of points s ∈ S which satisfy dA(x) = ∥s − x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For A and B two subsets of Rn, d(A, B) denotes the distance between A and B, that is, d(A, B) := inf{∥a − b∥ : (a, b) ∈ A × B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' NOUR AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' TAKCHE For S ⊂ Rn closed, the following proposition provides some useful properties of the set cl (int S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then we have the following: (i) cl (int S) ⊂ S, int (cl (int S)) = int S, and bdry (cl (int S)) = bdry (int S) ⊂ bdry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (ii) S = [cl (int S)] ∪ [(bdry S) \\ bdry (int S)] with [cl (int S)] ∩ [(bdry S) \\ bdry (int S)] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (iii) If S = cl O, where O is open, then S is regular closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (iv) If for r > 0, S satisfies the exterior r-sphere condition, then cl (int S) also satisfies the exterior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The properties (i)-(iii) are well known in metric topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For (iv), let r > 0 and assume that S satisfies the exterior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let x ∈ bdry (cl (int S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' By (i) we have that x ∈ bdry S, and hence there exists 0 ̸= ζ ∈ N P S (x) such that ζ is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives using the proximal normal inequality that � ζ ∥ζ∥, s − x � ≤ 1 2r∥s − x∥2, ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since cl (int S) ⊂ S, we obtain that � ζ ∥ζ∥, s − x � ≤ 1 2r∥s − x∥2, ∀s ∈ cl (int S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Therefore, ζ ∈ N P cl (int S)(x) and is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The converse of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1(iv) is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Indeed, consider in R2, S := ¯B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 1) ∪ [(1, 0), (2, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clearly S does not satisfy the exterior r-sphere condition for any r > 0, but cl (int S) = ¯B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 1) satisfies the exterior r-sphere condition for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In the following proposition we list some well known characterizations and conse- quences of prox-regular and epi-Lipshitz properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' For the proofs, see [5, 20, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then we have the following: (i) If S is prox-regular, then for each x ∈ bdry S we have N P S (x) = N L S (x) = NS(x), where N L S (x) and NS(x) denotes the Mordukhovich (or limiting) and the Clarke normal cones to S at x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (ii) For x ∈ bdry S, S is epi-Lipschitz at x if and only if NS(x) is pointed, that is, NS(x) ∩ −NS(x) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (iii) If for x ∈ bdry S, S is epi-Lipschitz at x, then x ∈ cl (int S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' THE EXTENDED EXTERIOR SPHERE CONDITION 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be a closed set such that cl (int S) satisfies the exterior r-sphere condition for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1(iii) we have that cl (int S) is regular closed, we get that (int (cl (int S))c = (int S)c satisfies the interior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Hence by [16, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2] we obtain that (int S)c is the union of closed r 2-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We proceed to prove the “In addition” part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We assume that S is regular closed, and we consider r′ < r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let x ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since Sc ⊂ (int S)c and (int S)c is the union of closed r 2-balls, there exists yx ∈ (int S)c such that x ∈ ¯B � yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ (int S)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (2) If ∥yx − x∥ < r′, then using (2) we get that x ∈ ¯B(yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r′) ⊂ B � yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ ¯B � yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ (int S)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives that x ∈ ¯B(yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r′) ⊂ int ((int S)c) = (cl (int S))c = Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now we assume that ∥yx − x∥ ≥ r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then by (2) and since x ∈ Sc, we have that x ∈ ¯B � x + r′ yx − x ∥yx − x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r′ � ⊂ B (yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ∥yx − x∥) ∪ {x} ⊂ B � yx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ∪ {x} ⊂ int ((int S)c) ∪ Sc = Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Therefore, Sc is the union of closed r′-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This terminated the proof of the “In addition” part, and hence the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' □ In the following example, we prove that the “In addition” part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1 is not necessarily true when S is not regular closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S := {(x, ex) : x ∈ R}∪{(x, −ex) : x ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clearly S is not regular closed, and satisfies the exterior 3 √ 3 2 -sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' On the other hand, Sc is not the union of closed r-balls for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Note that S does not satisfy the extended exterior r-sphere condition for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be a closed set satisfying the extended exterior r-sphere condition for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We consider x ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let s0 ∈ proj S(x) and let r0 := ∥x − s0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clearly we have s0 ∈ bdry S and s0 ̸= x, and hence r0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Moreover, we have B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r0) ∩ S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (3) There are three cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Case 1: r0 > r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then using (3), we obtain that x ∈ ¯B � x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r0) ⊂ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Case 2: r0 ≤ r 2 and s0 ̸∈ bdry (int S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' NOUR AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' TAKCHE Then the proximal normal vector ζ0 := x − s0 ∈ N P S (s0) is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives that B � s0 + r ζ0 ∥ζ0∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r � ⊂ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (4) We claim that ¯B � x + r 2 ζ0 ∥ζ0∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ B � s0 + r ζ0 ∥ζ0∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (5) Indeed, for y ∈ ¯B � x + r 2 ζ0 ∥ζ0∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � we have ����y − s0 − r ζ0 ∥ζ0∥ ���� = ���� � y − x − r 2 ζ0 ∥ζ0∥ � + � x − s0 − r 2 ζ0 ∥ζ0∥ ����� ≤ ����y − x − r 2 ζ0 ∥ζ0∥ ���� + ����x − s0 − r 2 ζ0 ∥ζ0∥ ���� ≤ r 2 + ����r0 ζ0 ∥ζ0∥ − r 2 ζ0 ∥ζ0∥ ���� = r 2 + r 2 − r0 = r − r0 < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now combining (4) and (5), we obtain that x ∈ ¯B � x + r 2 ζ0 ∥ζ0∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ B � s0 + r ζ0 ∥ζ0∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r � ⊂ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Case 3: r0 ≤ r 2 and s0 ∈ bdry (int S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then for every ε > 0, we have B(s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ε)∩int S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let zε ∈ B(s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ε)∩int S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We denote by ξε := zε−x ∥zε−x∥ and we consider sε ∈ [zε, x] ∩ bdry S, where this latter intersection is nonempty since zε ∈ int S and x ∈ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let ζε ∈ N P S (sε) be the unit vector realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Hence, for yε := sε + rζε, we have B (yε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r) ∩ S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (6) If yε = x, then x ∈ ¯B � x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ B(yε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r) ⊂ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' If 0 < ∥yε − x∥ < r, then for w ∈ ¯B � x + r 2 yε−x ∥yε−x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � , we have ∥w − yε∥ ≤ ����w − x − r 2 yε − x ∥yε − x∥ ���� + ����x + r 2 yε − x ∥yε − x∥ − yε ���� ≤ r 2 + ���� r 2 yε − x ∥yε − x∥ − (yε − x) ���� = r 2 + ���r 2 − ∥yε − x∥ ��� < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives using (6) that x ∈ ¯B � x + r 2 yε − x ∥yε − x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ B (yε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r) ⊂ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' THE EXTENDED EXTERIOR SPHERE CONDITION 7 x s0 zϵ sϵ yϵ ξϵ ζϵ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Case 3: ∥yε − x∥ ≥ r Now we assume that ∥yε − x∥ ≥ r and we denote by rε := r2 0∥yε − x∥ ∥yε − x∥2 + r2 0 − r2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (7) Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' B � x + rε yε−x ∥yε−x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' rε � ⊂ B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r0) ∪ B(yε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' To prove this claim, let w ∈ B � x + rε yε−x ∥yε−x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' rε � and assume that w ̸∈ B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then ∥w − x∥ < 2rε � w − x ∥w − x∥, yε − x ∥yε − x∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (8) We have ∥w − yε∥2 = ∥(w − x) − (yε − x)∥2 = ∥w − x∥2 + ∥yε − x∥2 − 2⟨w − x, yε − x⟩ (8) < ∥w − x∥2 + ∥yε − x∥2 − ∥yε − x∥ rε ∥w − x∥2 = ∥yε − x∥2 + ∥w − x∥2 � 1 − ∥yε − x∥ rε � (7) = ∥yε − x∥2 + ∥w − x∥2 � 1 − ∥yε − x∥2 + r2 0 − r2 r2 0 � = ∥yε − x∥2 + ∥w − x∥2 � �� � ≥r2 0 �r2 − ∥yε − x∥2 r2 0 � � �� � ≤0 ≤ ∥yε − x∥2 + r2 − ∥yε − x∥2 = r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives that w ∈ B(yε, r) which terminates the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' NOUR AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' TAKCHE Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ⟨ξε, ζε⟩ < ε 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' To prove this claim, first we remark that since zε ∈ int S, we deduce from (6) that zε ̸∈ ¯B(yε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r) = ¯B (sε + rζε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Hence, using the equality zε − sε = ∥zε − sε∥ξε, we obtain that ∥zε − sε∥ > 2r ⟨ξε, ζε⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Therefore, using that s0 ∈ proj S(x), we get that ⟨ξε, ζε⟩ < 1 2r∥zε − sε∥ = 1 2r (∥zε − x∥ − ∥sε − x∥) ≤ 1 2r � �∥zε − s0∥ + ∥s0 − x∥ − ∥sε − x∥ � �� � ≤0 � � ≤ 1 2r∥zε − s0∥ < ε 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The proof of the claim is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' If ε ∈ � 0, r3 0 4r2 � then rε > r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' To prove the claim, first we remark using (7) that rε > r 2 is equivalent to P(∥yε − x∥) := r∥yε − x∥2 − 2r2 0∥yε − x∥ + r(r2 0 − r2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (9) Since the discriminant ∆′ of P(∥yε − x∥) is (r2 − r2 0)2 + r2r2 0 > 0 and the product of the roots is r2 0 − r2 < 0, we conclude that (9) is equivalent to ∥yε − x∥ < r2 0 + √ ∆′ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (10) We have ∥yε − x∥2 = ∥yε − sε∥2 + ∥sε − x∥2 + 2⟨yε − sε, sε − x⟩ = r2 + ∥sε − x∥2 + 2r∥sε − x∥ ⟨ζε, ξε⟩ Claim 2 < r2 + ∥sε − x∥2 + ε∥sε − x∥ sε∈[zε,x] ≤ r2 + ∥zε − x∥2 + ε∥zε − x∥ ≤ r2 + (∥zε − s0∥ + ∥s0 − x∥)2 + ε(∥zε − s0∥ + ∥s0 − x∥) ≤ r2 + (ε + r0)2 + ε(ε + r0) = r2 + r2 0 + 2ε2 + 3εr0 ε∈ � 0, r3 0 4r2 � ≤ r2 + r2 0 + r6 0 8r4 + 3r4 0 4r2 = r2 + r2 0 + r4 0 r2 � r2 0 8r2 + 3 4 � r0≤ r 2 < r2 + r2 0 + r4 0 r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (11) THE EXTENDED EXTERIOR SPHERE CONDITION 9 On the other hand, we have (r2 0 + √ ∆′)2 = 2r4 0 + r4 − r2r2 0 + 2r2 0 �� r2 − 1 2r2 0 �2 + 3r4 0 4 > 2r4 0 + r4 − r2r2 0 + 2r2 0 � r2 − 1 2r2 0 � = r4 0 + r4 + r2r2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives that � r2 0 + √ ∆′ r �2 > r2 + r2 0 + r4 0 r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Combining this latter with (11), we obtain inequality (10), and this terminates the proof of Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now we fix ε ∈ � 0, r3 0 4r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' By Claim 3 we have that r 2 < rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Hence, using Claim 1, (3) and (6), we get that x ∈ ¯B � x + r 2 yε − x ∥yε − x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r 2 � ⊂ B � x + rε yε − x ∥yε − x∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' rε � ∪ {x} ⊂ B(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r0) ∪ B(yε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r) ⊂ Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In the three cases above, we have shown that x belongs to a closed r 2-ball included in Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' □ In the following example, we prove that the extended exterior r-sphere condition assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 cannot be replaced by the exterior r-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' In Figure 2, S is the set that consists of both curves and the dashed region between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The set S satisfies the exterior r-sphere condition, but it does not satisfy the extended exterior r-sphere condition since some of the boundary points on the right-hand side have two normal vectors where only one of them is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clearly, Sc is not the union of r′-balls for any r′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' S Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2, we deduce that if S is regular closed and satisfies the exterior r-sphere condition for some r > 0, then Sc is the union of closed r 2-balls, which is stronger than the “In addition” part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Of course we can adjust 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' NOUR AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' TAKCHE O c0 c1 c2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Example of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 the proof of the “In addition” part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1 and obtain the radius r 2 instead of r′ < r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We did not do this to not complicate the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1, and to show how its proof, when r′ < r 2, is a direct consequence of the union of closed balls conjecture [16, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Note that since r 2 < nr 2 √ n2−1 for all n ≥ 2, the same techniques used in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1 combined with the strong version of the union of closed balls conjecture [16, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3], gives that Sc is the union of closed r′-balls for any r′ < nr 2 √ n2−1, including r′ = r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Unfortunately, this latter result cannot be confirmed since the strong version of the union of closed balls conjecture remains till now an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The radius r 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 is the largest radius that works in all the spaces Rn for any n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Indeed, assume that the radius r 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2 is replaced by an r′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' We claim that r′ ≤ r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Indeed, inspired by [17, Example 2], we consider in Rn the set S to be the complement of the (n + 1) open r-balls centered at Ci, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' , n, where for (ei)n i=1 the standard basis of Rn and e0 := 0, we have Ci := −nr � n(n − 1) �� i i + 1 ei + n � k=i+1 −1 � k(k + 1) ek � = −nr � n(n − 1) � 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' , 0, √ i √i + 1, −1 � (i + 1)(i + 2) , −1 � (i + 2)(i + 3) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' , −1 � n(n + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The set S satisfies the extended exterior r-sphere condition (in fact, it is regular closed and satisfies the exterior r-sphere condition), but the largest closed balls contained in Sc and containing the origin 0 ∈ Sc is of radius nr 2 √ n2−1, see Figure 3 for n = 2 and n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Hence, r′ ≤ nr 2 √ n2−1 for all n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Taking n −→ ∞ in this latter, we conclude that r′ ≤ r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' THE EXTENDED EXTERIOR SPHERE CONDITION 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let S ⊂ Rn be a closed set such that cl (int S) is epi- Lipschitz and has compact boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since the first implication is straightforward (take r′ = r), we focus on the second implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' So, we assume that S satisfies the extended exterior r′-sphere condition for some r′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Then S satisfies the exterior r′-sphere condition, and hence by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2(iv), cl (int S) satisfies the exterior r′-sphere condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now using [13, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='12], applied to cl (int S), we get that cl (int S) is ρ-prox-regular for some ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' d(bdry (cl (int S)), (bdry S) \\ bdry (int S)) = rS > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' If not, then there exist two sequences xn and yn such that xn ∈ bdry (cl (int S)), yn ∈ (bdry S) \\ bdry (int S) and ∥xn − yn∥ ≤ 1 n, ∀n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since cl (int S) has compact boundary, we get the existence of ¯x ∈ bdry (int S) and a subsequence of yn, we do not relabel, such that yn −→ ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since yn ̸∈ cl (int S), we have from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2(iii) that S is not epi-Lipschitz at yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2(ii) the existence of unit vector ζn such that ζn ∈ NS(yn) ∩ −NS(yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (12) Now since yn ∈ (bdry S) \\ bdry (int S) and S satisfies the extended exterior r′-sphere condition, we have that S is r′-prox-regular near yn, that is, in a (bdry S)-neighborhood of yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Hence, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2(i) and (12) yield that ζn ∈ N P S (yn) ∩ −N P S (yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Using the proximal normal inequality and the fact that ζn is unit and realized by an r′-sphere, we obtain that (ζn)n has a subsequence, we do not relabel, that converges to a unit vector ζ satisfying ζ ∈ N P S (¯x) ∩ −N P S (¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Since N P S (¯x) ⊂ N P cl (int S)(¯x), we deduce that ζ ∈ N P cl (int S)(¯x) ∩ −N P cl (int S)(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This yields using the prox-regularity of cl (int S) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='2(i), that ζ ∈ Ncl (int S)(¯x) ∩ −Ncl (int S)(¯x), which contradicts the fact that cl (int S) is epi-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' The proof of the claim is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now we prove that S is r-prox-regular for r := min{ρ, r′, rS 4 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Let x ∈ bdry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' If x ∈ (bdry S) \\ bdry (cl (int S)), then from the definition of the extended r′-sphere condition, we know that any 0 ̸= ζ ∈ N P S (x) is realized by an r′-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This gives that any 0 ̸= ζ ∈ N P S (x) is realized by an r-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Now we assume that x ∈ bdry (cl (int S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Having cl (int S) ρ-prox-regular, it results that for any 0 ̸= ζ ∈ N P S (x) ⊂ N P cl (int S)(x), we have B � x + ρ ζ ∥ζ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' ρ � ∩ cl (int S) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' NOUR AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' TAKCHE This gives that for any 0 ̸= ζ ∈ N P S (x), we have B � x + r ζ ∥ζ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r � ∩ cl (int S) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' (13) On the other hand, since r < rS 4 and using Claim 1, we have for any 0 ̸= ζ ∈ N P S (x) that B � x + r ζ ∥ζ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r � ⊂ B � x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' rS 2 � ⊂ [(bdry S) \\ bdry (int S)]c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Combining this latter with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='1(ii) and (13), we get for any 0 ̸= ζ ∈ N P S (x) that B � x + r ζ ∥ζ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' r � ∩ Sc = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' This terminates the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' □ REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Canino, On p-convex sets and geodesics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Equations 75/1 (1988) 118–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Cannarsa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Frankowska: Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 12 (2006) 350–370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Cannarsa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Sinestrari: Convexity properties of the minimum time function, Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 3 (1995) 273–298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Cannarsa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Sinestrari: Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Con- trol, Birkh¨aser, Boston (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clarke: Optimization and Nonsmooth Analysis, John Wiley, New York (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clarke, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Ledyaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Wolenski: Nonsmooth Analysis and Con trol Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Clarke, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Wolenski: Proximal smoothness and the lower-C2 property, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Convex Analysis 2/1-2 (1995) 117–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Colombo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Marigonda: Differentiability properties for a class of non-convex functions, Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 25 (2005) 1–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Federer: Curvature measures, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 93 (1959) 418–491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Mordukhovich: Variational Analysis and Generalized Differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' I: Basic Theory, Springer, Berlin (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nacry, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Thibault: Distance Function Associated to a Prox-regular set, Set-Valued Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 30 (2022) 731–750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nour, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern: Semiconcavity of the bilateral minimal time function: the linear case, Systems & Control Letters 57 (2008) 863–866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nour, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Takche: Proximal smoothness and the exterior sphere condition, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Convex Analysis 16/2 (2009) 501–514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nour, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Takche: The union of uniform closed bal ls conjecture, Control and Cybernetics 38/4B (2009) 1525–1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nour, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Takche: The θ-exterior sphere condition, ϕ-convexity and local semicon- cavity, Nonlinear Analysis 73/2 (2010) 573–589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nour, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Takche: Validity of the union of uniform closed bal ls conjecture, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Convex Analysis 18/2 (2011) 589–600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nour, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Stern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Takche: Generalized exterior sphere conditions and ϕ-convexity, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Systems 29/2 (2011) 615–622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Nour, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Takche: On the union of closed bal ls property, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 155 (2012) 376–389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' THE EXTENDED EXTERIOR SPHERE CONDITION 13 [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='Penot: Calculus Without Derivatives, Graduate Texts in Mathematics 266, Springer, New York (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Poliquin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Rockafellar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Thibault: Local differentiability of distance functions, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 352 (2000) 5231–5249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Rockafellar: Clarke’s tangent cones and the boundaries of closed sets in Rn, Nonlinear Analysis Theory Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 3 (1979) 145–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Rockafellar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Wets: Variational Analysis, Grundlehren der Mathematischen Wis- senschaften 317, Springer, Berlin (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Shapiro: Existence and differentiability of metric projections in Hilbert spaces, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} +page_content=' 4 (1994) 231–259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf'} diff 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Networks Inc. Otemachi Bldg., 1-6-1 Otemachi, Chiyoda-ku, Tokyo 100-0004 Japan +(Dated: February 1, 2023) +Programmable unitary photonic devices are emerging as promising tools to implement unitary +transformation for quantum information processing, machine learning, and optical communication. +These devices typically use a rectangular mesh of Mach-Zehnder interferometers (MZIs), which has +a clear mathematical structure and can be configured deterministically. However, this mesh archi- +tecture is sensitive to fabrication errors, and the correction techniques are still under investigation. +In contrast, the multi-plane light conversion (MPLC) architecture is more robust against fabrication +errors, but a deterministic method for configuring the converter has not yet been developed due to +its complex mathematical structure. In this work, we propose a fast iterative configuration method +for MPLC, following the mathematical review of the matrix distance and proposal of a new norm. +We show through numerical simulations that adding a few redundant layers significantly improves +the convergence of the MPLC architecture, making it a practical and attractive option. We also +consider the effects of finite resolution and crosstalk in phase shifters in our simulations. In addi- +tion, we propose a phase-insensitive distance suited for applications using only intensity detections. +Our method demonstrates orders of magnitude better accuracy and a 20-fold speed-up compared +to previous approaches. +I. +INTRODUCTION +Programmable unitary transformations implemented +on integrated photonic platforms are becoming a pow- +erful tool for a variety of applications, including quan- +tum photonics [1–7], machine learning [8–14], and op- +tical communication [15–19]. Accurate realization of a +given unitary transformation is critical, as the fidelity of +computational results and the error of optical communi- +cation can be significantly affected by the precision of the +realized transformation. A common approach to synthe- +sizing unitary transformations is to use a mesh of Mach- +Zehnder interferometers (MZIs) known as the Clements +architecture [20], which consists of phase shifters and +beam splitters (BSs). This architecture is attractive be- +cause its mathematical structure is decomposable, allow- +ing the required phase shift in each MZI to be explic- +itly determined from the given unitary transformation. +However, physical implementation artifacts such as devi- +ation in the splitting ratio of BSs can result in errors in +the synthesized transformation. These errors can become +significant as the number of optical modes increases [21]. +Several design proposals have been made in an effort to +reduce or eliminate this error, with the goal of achieving a +precise, customizable, and fabrication-error-tolerant uni- +tary transformation that can be applied to scalable and +reliable applications. +To address the challenge of implementation artifacts in +the Clements architecture, several approaches have been +proposed. One approach is local error correction, which +∗ ytaguchi@ginjo.t.u-tokyo.ac.jp +involves fixing each MZI and can be applied to any MZI- +based architecture, but requires prior knowledge of pas- +sive and active components [22]. Another approach is the +measurement of components with on-chip power moni- +tors, which allows for the calibration of each MZI but +also increases the size of the chip and the complexity of +wiring [23, 24]. Self-configuration and 3-MZI approaches +utilize an additional BS to achieve partially perfect lin- +ear operation and employ a feedback loop to adjust each +phase shift using only output signals [25, 26]. While this +method allows for infinite scalability, it also increases the +size of the circuit and may have issues with stability [27]. +It’s worth noting that these approaches primarily con- +sider the artifacts of passive BSs in the circuit, and do +not sufficiently consider the artifacts of phase shifters, +such as crosstalk. +Another architecture employs a series connection of +phase shifter arrays and unitary transformations to +achieve a highly robust universal synthesis of unitary ma- +trices that is resistant to fabrication errors. This archi- +tecture, also known as the multi-plane light conversion +(MPLC) architecture [28–31], is particularly robust be- +cause each unitary transformation can be selected from +a wide range of possible unitaries [32–35]. The unitary +transformation can be almost any well-known N-mode +mixer, which can significantly increase the flexibility and +tolerance to fabrication errors. However, configuring the +phase shifters in this architecture is challenging, and +no explicit configuration method has been known due +to its complex mathematical structure. The optimiza- +tion of this architecture must deal with the many local +minima present in its high-dimensional parameter space +[34]. As a result, previous reports have relied on heuris- +tic global searches, such as basin-hopping and simulated +arXiv:2301.13658v1 [cs.ET] 31 Jan 2023 + +2 +annealing, to configure the phase shifters [34–36]. How- +ever, these methods are time-consuming and suffer from +exponentially increasing search times as the parameter +space dimension increases. To address this issue, a ma- +chine learning-based configuration algorithm has been +proposed [37]. While this algorithm may offer a solution, +it requires an accurate initial estimation of the structure +and may result in decreased matrix fidelity if the initial +estimation contains errors. +In this research, we present a new, fast and iteratively +configurable MPLC architecture that does not require +prior knowledge and relies only on output signals. This +approach involves adding a few redundant layers to the +existing MPLC architecture and using derivative-based +optimization with gradient approximation. +This addi- +tional layer redundancy significantly improves the opti- +mization performance of the MPLC architecture, in con- +trast to the similar approach used for the Clements archi- +tecture [21, 38], which adds a large number of redundant +layers. When compared to numerical optimization of the +Clements architecture without redundancy [38], our pro- +posed method achieved 5 orders of magnitude better ac- +curacy with 1/20 fewer iterations for N = 128 modes of +transformation. Additionally, our proposed method was +able to achieve 5 orders of magnitude better accuracy and +was 23 times faster in configuration compared to the pre- +vious report that used a heuristic algorithm to optimize +the MPLC architecture [39]. +This paper is structured as follows. Before discussing +the main results, we begin by discussing important gen- +eral properties of unitary matrix optimization and in- +troducing a new distance in Section II. One key prop- +erty we cover is that unitary matrix optimization essen- +tially has no local minima. Additionally, we propose a +new distance, a phase-insensitive variant of the Frobe- +nius norm, which is invariant under phase shifts at the +output modes. Previously, the standard Frobenius norm +has even been used in phase-insensitive applications. In +Section III, we investigate the optimization properties +of the MPLC architecture with a few redundant layers +of parametrization. While the parametrization of a uni- +tary matrix can cause optimization to fall into local min- +ima, we demonstrate through numerical simulations that +these can be effectively avoided by adding a few redun- +dant layers. Our results show that this architecture can +be efficiently optimized using well-known local minimiza- +tion algorithms, such as the gradient descent algorithm, +while the Clements architecture cannot. We also study +the statistical properties of convergence. In Section IV, +we examine practical scenarios, such as when the gradi- +ent of the system is not available, only intensity detec- +tion is used, and crosstalk between phase shifters exists. +We evaluate the impact of gradient approximation and +crosstalk on the proposed method, and show that it still +performs well, albeit with a reduction in achieved matrix +accuracy after optimization or an increase in the number +of iteration until convergence. The phase-insensitive dis- +tance exhibits similar optimization properties. In Section +V, we conclude the paper. +II. +MATRIX DISTANCE USING THE +FROBENIUS NORM +This section presents some general mathematical prop- +erties of the Frobenius norm and proposes a new dis- +tance. We begin by defining the concept of unimodality +for functions on the unitary group U(N) and show that +the matrix distance using the Frobenius norm exhibits +this unimodality. We then clarify the range and expected +value of the norm. Additionally, we introduce the phase- +insensitive matrix distance for applications that only use +intensity detection. +A. +Unimodality on U(N) +Here, the concept of unimodality for a function on +U(N) is introduced. Unimodality is typically defined for +probability distributions [40]. For a multivariable func- +tion f : RN → R, unimodality is defined through the +level set L(f, α) = {x|f(x) ≤ α, x ∈ RN} and the con- +vexity of L(f, α) [41, 42]. In this paper, we extend this +definition to functions on U(N) by considering the path- +connectedness of L(f, α), as U(N) is not a convex set. +Definition A function f : U(N) → R is called uni- +modal if the level set L(f, α) = {X|f(X) ≤ α, X ∈ +U(N)} is path-connected for any α ∈ R. +In other words, any local minimum of a unimodal func- +tion on U(N) is also a global minimum. +B. +Unimodality of the Frobenius norm +We prove that the unitary matrix distance using the +Frobenius norm is unimodal. The distance between two +unitary matrices d(X, U) is defined as ∥X − U∥F , where +∥A∥F = +� +Tr [A†A] is the Frobenius norm. It is worth +noting that the matrix distance using mean square error +(MSE) � +i,j |Xij −Uij|2 is equivalent to d(X, U)2. Given +a unitary matrix U ∈ U(N), we show that the function +fU : U(N) → R, defined as fU(X) = d(X, U) is uni- +modal. First, from the definition of the Frobenius norm, +fU(X)2 is simplified as +fU(X)2 = Tr +� +(X − U)†(X − U) +� += 2N − 2 Re +� +Tr +� +U †X +�� +. +(1) +We write the eigenvalues of U †X as λi(1 ≤ k ≤ N). Since +both U and X are unitary, all the eigenvalues λk satisfy +|λk| = 1. Therefore, the eigenvalues can be written as +λk = eiθk, where −π ≤ θk ≤ π. Using these eigenvalues, +Eq. 1 can be simplified further as +fU(X)2 = 2N − 2 +N +� +k=1 +cos θk +(2) + +3 +Eq. 2 implies that fU(X)2 is unimodal because cos θk is +unimodal over the range −π ≤ θ ≤ π and their sum is +also unimodal. An algebraic proof of this unimodality is +provided in the Appendix A. +C. +Range and expected value +We derive the range and expected value of fU(X)2 +from Eq. 2. The maximum of fU(X)2 is 4N if and only +if θk = ±π for all k, and the minimum is 0 if and only +if θk = 0 for all k. In previous studies, fU(X)2 has been +normalized by N [25], 2N [38], or N 2 [35]. Normalizing +by 4N yields 0 ≤ fU(X)2/4N ≤ 1, which suggests a good +normalization of the norm with a range from 0 to 1 that +is independent of N. +To calculate the expected value +E +� +fU(X)2� +, the distribution of θk is considered. If X is +sampled from the Haar measure, then U †X is also Haar- +random due to the invariance of the Haar measure. As a +result, the eigenvalues of U †X are uniformly distributed +on the unit circle |c| = 1, and we have θk ∼ U(−π, π). +Because +E[cos θk] = +� π +−π +1 +2π cos θ dθ = 0, +(3) +the expected value of the second term in the Eq. 2 is 0. +We now conclude that E +� +fU(X)2� += 2N. This fact is ob- +served numerically in the initial value of the convergence +plots in Sec. III. +D. +Phase-insensitive distance +Here, we introduce a phase-insensitive variant of the +matrix distance using the Frobenius norm. This variant +is suitable for applications that only detect the intensity +of the output modes, as the distance should not be af- +fected by the output phases from the unitary converter. +Applications that benefit from this phase-insensitive dis- +tance include machine learning and quantum photonics, +where photodiodes or photon number counters are placed +at the output ports. Fig. 1 shows a scenario where a +complex vector (s1, s2, . . . , sn)⊤ is input into two unitary +conversion devices. The transfer matrix for these devices +is represented by P and Q, and their complex outputs +are in polar form as teiθ. The only difference in the out- +put vectors from these two devices is in their phase, with +θi ̸= θ′ +i. In applications that only detect the intensity of +output modes, these two matrices P and Q are treated +the same and a suitable matrix distance is introduced for +this purpose. In the following discussion, the matrix U +represents the given target unitary matrix, and the ma- +trix X represents the actual conversion achieved by the +unitary converter device. +To investigate the effect of output phases from the uni- +tary converter, we represent the unitary matrices U † and +… +Intensity detector +… +FIG. 1: A complex vector (s1, s2, . . . , sn)⊤ is input into +two unitary conversion devices, whose transfer matrices +are represented as P and Q. The outputs from these +devices are identical, with the exception of the phase +degrees of freedom at the outputs, when evaluated using +a phase-insensitive distance. +X as +U † = +�u1 u2 · · · un +� +X = +� +������ +x⊤ +1 +x⊤ +2 +... +x⊤ +n +� +������ +, +(4) +where ui are column vectors of U † and x⊤ +i are row vectors +of X. Since U † and X are unitary matrices, the norms of +ui and xi are all equal to 1. In this context, the output +phases of the unitary converter correspond to the phases +of xi. +The original Frobenius norm changes its value +under the global phase change of xi, which is defined as +replacing xi with eiθixi. +We analyze the dependence of the Frobenius norm on +the global phase of each xi. The Frobenius norm is ex- +panded using the column vectors ui as +∥X − U∥2 +F = +��XU † − I +��2 +F += ∥Xu1 − e1∥2 + ∥Xu2 − e2∥2 + · · · + ∥Xun − en∥2, +(5) +where ei is a unit column vector whose i-th element is 1 +and the others are 0. Expanding the term ∥Xu1 − e1∥2, +we obtain +∥Xu1 − e1∥2 += |x⊤ +1 u1 − 1|2 + |x⊤ +2 u2|2 + · · · + |x⊤ +n un|2. +(6) +Except for the first term |x⊤ +1 u1 − 1|2, the other terms +|x⊤ +i ui|2(i ≥ 2) are invariant under the global phase +change in xi. Only the first term depends on the global + +P,Q EU(N4 +phase of x1. Therefore, the term ∥Xu1 − e1∥2 is invari- +ant under the global phase change in the row vectors +x2, x3, · · · xn. Similarly, ∥Xui − ei∥2 is independent of +the global phase of xj where j ̸= i. When the global +phase of xi is changed, the term |x⊤ +i ui−1|2 takes its min- +imum value if and only if x⊤ +1 u1 is a positive real number, +because +|x⊤ +i ui − 1|2 = |x⊤ +i ui|2 − 2 Re +� +x⊤ +i ui +� ++ 1 +(7) +and only the second term 2 Re +� +x⊤ +1 u1 +� +is dependent on +the global phase of xi. This motivates the idea of replac- +ing all instances of |x⊤ +i ui − 1|2 with +� +|x⊤ +i ui| − 1 +�2 in the +Frobenius norm ∥X − U∥2 +F . As discussed in Eq. 6 and +Eq. 7, the minimum value of the Frobenius norm under +this substitution is the same as the minimum value of the +original Frobenius norm, due to the phase-dependence +property. +Based on the aforementioned consideration, we pro- +pose a phase-insensitive matrix distance. Given unitary +matrix U, we define the distance function hU : U(N) → +R using only terms whose form is |x⊤ +i uj|, which can be +obtained using intensity measurements at the outputs +from the unitary converter. We define positive real num- +bers aij = |x⊤ +i uj|2 ≥ 0. All the aij can be obtained by +multiplying column vector ui with matrix X through the +unitary converter, because +Xuj = +� +������ +x⊤ +1 uj +x⊤ +2 uj +... +x⊤ +n uj +� +������ +(8) +and |x⊤ +i uj|2 can be obtained through the intensity mea- +surement of x⊤ +i uj. By expanding all the terms in Eq. 5 +with Eq. 6 and applying the substitution discussed with +respect to Eq. 7, we define hU(X) as +hU(X) = +(√a11 − 1)2 + a12 ++ · · · ++ a1n ++ a21 ++ (√a22 − 1)2 + · · · ++ a2n +... +... ++ (√aii − 1)2 · · · ++ an1 ++ an2 ++ · · · ++ (√ann − 1)2 += +� +ij +� +δij − +��� +� +XU †� +ij +��� +�2 +. +(9) +This function is independent of the global phase of xi, +that is, the phases of each output from the unitary con- +verter, and has a minimum value identical to the original +Frobenius norm distance function fU(X). +The function hU(X) is also unimodal. +This can be +proven by reductio ad absurdum. Suppose hU(X) is not +a unimodal function and has multiple disconnected local +minima. Then, for all matrices X′ such that hU(X′) is a +local minimum, there must exist at least one set of phases +(θ1, θ2, · · · θn) that defines a matrix +Y = +� +������ +eiθ1x′⊤ +1 +eiθ2x′⊤ +2 +... +eiθnx′⊤ +n +� +������ +(10) +such that fU(Y ) is also a local minimum, where x′⊤ +i +is +the row vector of X′. +The existence of such a matrix +Y follows from the definition of hU(X). However, this +would mean that fU(X) also has multiple disconnected +local minima, which is a contradiction. Therefore, hU(X) +must be unimodal. +III. +FEW-LAYER REDUNDANT +PARAMETERIZATION +The global optimization property guaranteed by uni- +modality discussed in the previous section is derived +without any assumptions about the matrix being opti- +mized; however, the matrix synthesized by the device +is parametrized by physical parameters and unimodality +may not always hold in this parameter space. We ex- +amine this issue and propose a solution to mitigate this +difficulty. Let lU(X) ≥ 0 be a distance function from a +desired unitary matrix U, and X(p) be a unitary ma- +trix realized by a physical converter that is parametrized +by a real parameter vector p. Each element in p corre- +sponds to the amount of phase shift in the actual device. +If lU(X) is unimodal, its gradient becomes a zero vector +only when X = ±U. However, when optimizing an actual +unitary conversion device, we need to consider the scalar +optimization of lU(X(p)). +If the Jacobian of X(p) is +full-rank at any p, meaning there exists infinitesimal pa- +rameter changes ∆p for any infinitesimal matrix changes +∆X, then the function lU(X(p)) also has a single min- +imum due to the aforementioned unimodal property of +lU(X). By increasing the number of layers in the unitary +converter device, the degree of freedom in the parameter +space increases, which may make the Jacobian of X(p) +more likely to be full-rank. In this section, we demon- +strate that increasing the number of layers in unitary +converter devices by a few from its minimum requirement +significantly improves the optimization of MPLC archi- +tecture using a gradient-based optimization algorithm. +A. +Device definition and redundancy +We present the mathematical definition of the unitary +converters and the few-layer redundant parameterization. +Fig. 2a shows the architecture of the MZI-based unitary +converter, which is commonly referred to as Clements ar- +chitecture [20]. The MZI consists of two 50 : 50 BSs and +two phase shifters, which can realize an arbitrary U(2) + +5 +Layer +… +… +… +… +(a) +Layer +… +… +… +(b) +FIG. 2: Schematics of the N × m Clements architecture +(a) and N × m MPLC architecture (b). The left ports +are inputs, and the right port are outputs. The number +of layers in each architecture is specified by m. In the +Clements architecture, each layer contains either N/2 or +(N − 1)/2 MZI nodes, represented by white circles in +the figure. Each MZI node consists of two phase +shifters, represented by the variables φ and θ. In the +MPLC-based unitary converter (b), the architecture +consists of an N-port fixed unitary converter +represented by A, followed by an array of N +single-mode phase shifters. +transformation. In this paper, we do not consider any +imperfections of the MZI. Fig. 2b shows the structure of +the MPLC architecture. Each layer consists of an N-port +fixed unitary converter Ai and an array of N single-mode +phase shifters. After m layers, another array of phase +shifters is placed in a similar manner to the Clements +architecture. The overall transformation of this device, +denoted as X, is given by +X = Lm+1AmLm · · · A2L2A1L1, +(11) +where Ai is the transfer matrix of a N-port unitary con- +verter and Li is expressed as +Li = +� +���� +eiθi1 +eiθi2 +... +eiθin +� +���� . +(12) +For any i ̸= j, the matrices Ai and Aj are different. +The total number of degrees of freedom in this matrix is +(m + 1)(N − 1) + 1. This is because each phase shifter +array has N − 1 degree of freedom due to the loss of one +degree of freedom from the global phase, and the entire +device has an additional degree of freedom, the global +phase. +The N-port fixed unitary converter Ai can be +implemented using a multiport directional coupler [33], +multimode interference coupler [30], or other multiport +unitary transform devices. The device should be care- +fully chosen to ensure that the overall transformation X +is universal. The mixing entropy of a device can be used +as a measure of universality [43, 44]. To realize an arbi- +trary U(N) transformation, the total number of degrees +of freedom must exceed N 2 [45]. For the Clements ar- +chitecture, the number of layers m must satisfy m ≥ N +[20]. Similarly, the number of layers m for the MPLC ar- +chitecture must also satisfy m ≥ N, which follows from +(m + 1)(N − 1) + 1 ≥ N 2. In this context, a few-layer re- +dundant parameterized architecture is defined as having +m = N + 1, N + 2 layers for both architectures. +B. +Optimization problem setting and algorithm +We formulate the matrix optimization problem as fol- +lows. We have real parameter variables expressed as a +vector p. The number of parameters depends on m and +N. We define the normalized cost function L between +two matrices as +L(p) = +1 +4N ∥X(p) − U∥2 +F , +(13) +where X(p) is the unitary matrix realized physically by +the parameter vector p, U is the target matrix to be +achieved, and ∥·∥F is the Frombenius norm. The cost +function L is divided by 4N as discussed in the Section +II C, and 0 ≤ L ≤ 1 is always satisfied. At the beginning +of the optimization, parameters are initialized using uni- +form distribution ranging from 0 to 2π, and the target +unitary matrix U is sampled from Haar measure using +the stats module of SciPy [46]. For the MPLC architec- +ture, the matrix Ai for 1 ≤ i ≤ m is also sampled from +Haar measure. After initializing the parameters and ma- +trices, the cost function L is optimized using the quasi- +Newton optimization method L-BFGS [47] implemented +in optimize module of SciPy [46]. The derivative of the +cost function L required for L-BFGS is calculated us- +ing automatic differentiation with the JAX framework +[48]. This method starts from the initial parameters and +modifies them at each step until convergence to the lo- +cal minimum, where dL/dp = 0. In each layer, we have +N real parameters to represent N phase shifts in both +Clements and MPLC architectures. The optimization is +run 64 times while changing the initial parameters to in- +vestigate the statistical behavior. Cases with N = 8 and +N = 32 are investigated. + +cOs 0 +-sinθ +sin θ +cos 0Ai E +UN6 +(a) +(b) +(c) +(d) +FIG. 3: Convergence plots for MPLC and Clements architectures. The vertical axis shows the value of the cost +function L defined by Eq. 13, and the horizontal axis shows the number of iterations. The shaded area represents +the minimum and maximum values, the solid line represents the median, and the dotted line represents the 25% and +75% quantiles over 64 optimization trials. MPLC architecture with (a) N = 8, (b) N = 32, Clements architecture +with (c) N = 8, and (d) N = 32. +C. +Results +Fig. +3 shows the convergence plots when the num- +ber of layers is changed. +The convergence plot of the +cost function is recorded for 64 optimization trials. The +shaded area shows the range of minimum and maximum +values, the dotted line shows the 25% and 75% quantiles, +and the solid line shows the median of the trials. For +both Clements and MPLC architecture, the insufficient +parameterized layer setting results in a large amount of +errors. For MPLC architecture, the non-redundant case +of m = N results in a large variance of error, especially +for N = 8. +This suggests the presence of many local +minima in the parameter space of the MPLC architec- +ture, as previously reported in Ref. [34]. Although the +variance of non-redundant setting m = N of N = 32 +is smaller than that of N = 8, the error still remains +for N = 32, indicating the presence of inevitable local +minima for this condition as well. When we increase the +number of layers and add redundant degrees of freedom, +the variance and error become small, as shown in the +cases of m = N + 1, N + 2. In contrast, the Clements +architecture results in a large variance of error for all +conditions, even though it has the sufficient number of +degrees of freedom. +For the cases with a large number of ports, N = 128, +Fig. 4 shows the performance comparison with the pre- +vious study [38] of the Clements architecture with redun- +dancy. When compared with no redundancy, the previ- +ous study converges at L = 1.4 × 10−2 after 20000 itera- +tions, while MPLC architecture yields a result that is 6 +orders of magnitude better than the Clements architec- +ture with 1/20 fewer iterations. The MPLC architecture +with a redundant layer m = N + 1 still outperforms in +terms of both convergence speed and accuracy, even com- +pared with the Clements architecture with 128 redundant +layers. +We visualized the optimization trajectory and loss +function in the parameter space using the method re- +ported in [49, 50] and in the supplementary material of +[51]. The optimization trajectory is the path of the pa- +rameters in a high-dimensional space created by the opti- +mization. We stored the parameter history at each step of +the optimization and applied principal component anal- + +MPLC, N=8 +100 +m=N-1 +m=N+1 +m=N +m=N+2 +10-2 +Cost function L +10-4 +10-6 +10-8 +10-10 +10-12 +0 +50 +100 +150 +200 +250 +Number of iterationMPLC, N=32 +100 +m=N-1 +m=N+1 +m=N +m=N+2 +10-2 +Cost function L +10-4 +10-6 +10-8 +10-10 +0 +500 +1000 +1500 +2000 +Number of iterationClements, N=8 +100 +m=N-1 +m=N+1 +m=N +m=N+2 +10-2 +Cost function L +10-4 +10-6 +10-8 +10-10. +10-12 +0 +50 +100 +150 +200 +250 +300 +350 +400 +Number of iterationClements, N=32 +100 +m=N-1 +m=N+1 +m=N +m=N+2 +10-2 +Cost function L +10-4 +10-6 +10-8 +10-10 +0 +5000 +10000 +15000 +20000 +Number of iteration7 +FIG. 4: Comparison of the large case of N = 128 with +the previous study [38]. +ysis (PCA) to that history. The first and second PCA +components were used to project the high-dimensional +path onto a two-dimensional space. +The visualization +was performed for N = 8, m = N + 1. Fig. 5 shows +the projected trajectories and contour plots of the log of +the loss function in the projected subspace. The contour +plot for the MPLC architecture is like a simple elliptic +unimodal function, while that of the Clements architec- +ture is more complex. The difference in the contour plots +between these architectures suggests the reason for the +convergence plot difference shown in Fig. 3. +(a) +(b) +FIG. 5: The PCA projections of the optimization +trajectories and contour plot of the log of the loss +function for the MPLC architecture (a) and the +Clements architecture (b) for N = 8, m = N + 1. Each +of the four figures shows the trajectory of the +optimization when the initial parameters and the target +matrix are randomly changed. +IV. +OPTIMIZATION UNDER PRACTICAL +SETTINGS +We discuss three challenges that must be addressed +when applying the proposed method in Section III to real +device optimization. First, the optimization method used +in this section requires the gradient of the target func- +tion. While the gradient can be taken physically [52], it +requires additional external equipment, which makes the +system bulky and not scalable. Second, the optimization +method uses the complex amplitudes at the output for +optimization. Reading the complex amplitudes in a real +device requires coherent detectors at the output, which +complicates the device. While some applications require +coherent detection, many photonics-based optical com- +puting platforms and quantum computing with photonic +chips use intensity detection. Third, real devices have +crosstalk between phase shifters, which is not considered +in the optimization method. Crosstalk is especially prob- +lematic in thermo-optic phase shifters [53, 54], although +they are attractive due to their small footprints. +In this section, we report the results of derivative- +free optimization with the original and phase-insensitive +norm, using only the output signal, and examine the ef- +fect of crosstalk. +We first introduce the mathematical +formulation and then present the numerical results. The +optimization method used is the same as in Section III B. +These results pave the way for the design of optical uni- +tary converters without the need for additional compo- +nents, making the platform more scalable and versatile. +A. +Gradient approximation of multivariate +function +We use numerical gradient approximation, which uses +only function values to approximate the analytical gra- +dient, and investigate the effect of this approximation on +the optimization behavior. The gradient of multivariate +function ∇f(x1, x2, . . . , xn) is approximated by +∇f(x1, x2, . . . , xn) ≈ +� +����� +f(x1+∆,x2,...,xn)−f(x1,...,xn) +∆ +f(x1,x2+∆,...,xn)−f(x1,...,xn) +∆... +f(x1,x2,...,xn+∆)−f(x1,...,xn) +∆ +� +����� +. +(14) +where ∆ ≪ 1 represents a finite difference. +Calculat- +ing gradient approximation requires the same number of +function evaluations as the number of parameters. We +show that a derivative-based algorithm using such gra- +dient approximation can still be effective for optimizing +unitary matrices. +B. +Definition of device with intensity detection +We evaluate the optimization behavior of the phase- +insensitive distance introduced in Section II D, which +only uses intensity detectors at the outputs. We expect +the phase-insensitive distance to behave similarly to the +phase-sensitive distance during optimization because it +also has a unimodal property, as shown in Section II D. +In order to test this, we removed the last phase shifter + +2nd PCA component: 23.9% +2nd PCA component: 12.5% +4 +0 +-5 +0 +5 +5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +1st PCA component: 63.5% +lst PcA component: +80.2% +2nd PCA component: 9.4% +2nd PCA component: 15.0% +6 +2 +2 +. +-5 +0 +5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +1st PcA component: 82.9% +1st PCA component: 71.0%MPLC, N=128 +100 +m=N-1 +X +VX +m=N +X ++ +X +10-2. +m=N+1 ++ +Clements with +m=N+2 +no redundancy +Cost function ++ +10-4 +Clements with +128 redundant layers +10-6 ++++++++++++ +10-8. +10-10 +0 +500 +1000 +1500 +2000 +2500 +3000 +Number of iteration2nd PCA component: 13.7% +2nd PCA component: 13.4% +T-J +20 +2 +-2 +-2.5 +0.0 +2.5 +5.0 +7.5 +-2.5 +0.0 +2.5 +5.0 +7.5 +1stPCA component: +74.0% +1st PCA +component: +74.7% +2nd PCA component: 15.0% +2nd PCA component: 20.0% +2 +2 +-2.5 +0.0 +2.5 +5.0 +7.5 +-2 +0 +2 +4 +6 +1st PCA component: 69.0% +1st PCA component: 69.9%8 +array from the MPLC architecture, as shown in Figure 6. +Although this removes the N degree of freedom from the +architecture, we still expect the optimization behavior to +be similar to that of a standard phase-sensitive norm. +Layer +… +… +… +Removed +FIG. 6: Schematics of the MPLC architecture with the +last phase shifter array removed. +C. +Model of crosstalk +We model crosstalk by considering the interaction be- +tween adjacent phase shifters. +The crosstalk is repre- +sented by a linear combination of phase shifts, which can +be expressed as +θi = +� +j +αijθj. +(15) +The coupling model and coupling coefficients αij are +shown schematically in Fig. +7. +The coupling in the +following simulations is formulated as θ′ +i = 0.1θi−2 + +0.5θi−1 + θi + 0.5θi+1 + 0.1θi+2. If the coupled param- +eter p′ = w(p) is reversible, the unitary matrix X(p′) +realized by the coupled parameter will have a full-rank +Jacobian if the original X(p) has a full-rank Jacobian. +We use the gradient approximation for optimization. +… +… +… +FIG. 7: Crosstalk model of the MPLC architecture. +D. +Results +Fig. 8 shows the convergence plots for gradient approx- +imation using the standard Frobenius-norm-based dis- +tance. The approximation is calculated using a ∆ value +of 2−10, which corresponds to phase shifts with 10-bit res- +olution. When a redundant layer is added, the MPLC ar- +chitecture shows numerical-accuracy limited performance +(for m = N +1, N +2) with a small variance in the error. +Each iteration of the optimization requires the evaluation +of the distance the same number of times as the number +of parameters due to the gradient approximation. For +example, when N = 8 and m = N +1, each optimization +requires 8 × (8 + 2) = 80 evaluations of the distance. Us- +ing the MPLC architecture, the optimization converges +at 100 iterations for this case, so the total number of +evaluations is approximately 8000. +FIG. 8: Convergence plots for the MPLC architecture +with gradient approximation, where ∆ = 2−10. The 64 +optimization trials are shown in the same manner as in +Fig. 3. +We examined the ∆ dependence of the small error vari- +ance observed in Fig. 8, which arises from the gradient +approximation. Fig. 9 shows the optimization results for +each gradient approximation accuracy using a redundant +layer setting of m = N + 1. As the finite difference ∆ +becomes smaller, the final error also becomes smaller. If +the accuracy of the gradient approximation is not suf- +ficient, meaning ∆ is not small enough, the variance +of the optimization result is very small. +For example, +the error is in the range +� +1.5 × 10−5, 1.8 × 10−5� +for the +∆ = 2−6 case, +� +2.3 × 10−7, 2.5 × 10−7� +for the ∆ = 2−9 +case, and +� +3.5 × 10−9, 4.5 × 10−9� +for the ∆ = 2−12 case. +If the accuracy of the gradient approximation is sufficient +(∆ ≥ 2−15), the result has a non-negligible variance sim- +ilar to the one shown in Fig. +3a. +The error is in the +range +� +7.9 × 10−10, 6.8 × 10−11� +for the ∆ = 2−15 case +and +� +3.7 × 10−10, 4.6 × 10−12� +for the ∆ = 2−18 case. +These result provide criteria for designing the DAC res- +olution of a unitary converter system. +We studied the optimization property using the phase- +insensitive distance. +Fig. +10 shows the optimization +result obtained with an analytical gradient. +The con- +vergence plot is similar to the one shown in Fig. 3, in +spite of the reduced degree of freedom. However, when +using gradient approximation and comparing the accu- +racy dependence, the phase-insensitive distance shows a + +MPLC, N=8 +100 +m=N-1 +m=N+1 +m=N +m=N+2 +10-1 +10-2 +Cost function L +10-3 +10-4 +10-5 +10-6 +10-7 +50 +100 +150 +0 +200 +Number of iterationAi E +UN9 +FIG. 9: Comparison of final error when changing the +accuracy of gradient approximation for the MPLC +architecture. The 64 optimization trials are shown in +the same manner as in Fig. 3. +different result from the standard Frobenius-norm-based +distance, as shown in Fig. 11. All the optimization re- +sults have a large variance, as opposed to the cases where +∆ = 2−6, 2−9, 2−12 in Fig. 9. The finite difference ∆ +must be smaller than 2−18 to achieve an optimization re- +sult comparable to the one obtained using an analytical +gradient. The black dashed line shows the optimization +result by simulated annealing in a previous study [35]. +They achieved an error of fMSE = −50 dB, which cor- +responds to L = 2 × 10−5. This error can be achieved +using our method with a finite difference of ∆ ≤ 2−9, and +further improvement by orders of magnitude is possible +with more accurate gradient approximation. When the +accuracy is sufficient, our optimization method converges +after about 100 iterations. To compare the speed of our +method with a previous experimental report of MPLC +architecture optimization using intensity detection [39], +we also conducted optimization for a case with N = 4 +and m = N + 1. The optimization converged after about +45 iterations with ∆ = 2−18. As each iteration requires +4×(4+2) = 24 evaluations, the total number of iterations +required for convergence is 1080, representing a 23-fold +speedup compared to the previous report (∼ 25200 eval- +uations). +The effect of crosstalk is shown in Fig. 12. The gra- +dient approximation was calculated using ∆ = 2−12. Al- +though crosstalk caused a larger error and increased the +number of iterations until convergence, the performance +degradation can be mitigated by adding a few layers of +additional redundancy. This result suggests that it may +be possible to optimize the device end-to-end, including +both matrix optimization and phase shifter calibration. +V. +CONCLUSION +We proposed a fast and iteratively configurable MPLC +architecture for realizing precise and fabrication-error- +FIG. 10: Convergence plots for the MPLC architecture +using phase-insensitive distance with N = 8. The 64 +optimization trials are shown in the same manner as in +Fig. 3. +FIG. 11: Comparison of final error when changing the +accuracy of gradient approximation for the MPLC +architecture using phase-insenstive distance. The black +dashed line represents the optimization result in a +previous study [35], where L = 2.0 × 10−5 was achieved. +tolerant unitary transformation. Our numerical results +show that adding a few redundant layers to the MPLC +architecture significantly improves optimization behav- +ior. +We also examined the effect of artifacts, such as +crosstalk, and found that the proposed architecture can +be optimized end-to-end. In addition to proposing a new +architecture and optimization method, we analyzed the +distance between unitary matrices using the Frobenius +norm. +We introduced the concept of unimodality for +functions on the unitary group and proved that the ma- +trix distance using the Frobenius norm has this prop- +erty. We also calculated the expected value and range +of the matrix distance. We also introduced the phase- +insensitive norm, which is useful for applications that +only use intensity detections. We believe that this ap- +proach will enable the scalable and robust implementa- +tion of optical unitary converters and expand the use of +photonic integrated circuits in various fields. + +MPLC, N=8, m=N+1 +100 +△ = 2-6 +△ = 2-15 +△= 2-9 +△ = 2-18 +10-2. +△= 2-12 +Cost function L +10 +10-6 +10-8 +10-10 +0 +20 +40 +60 +80 +100 +120 +Number of iterationMPLC, N=8 +100 +m=N-1 +m=N+1 +m=N +m=N+2 +10-2 +Cost function L +10-4 +10-6 +10-8 +10-10. +10-12 +0 +25 +50 +75 +100 +125 +150 +175 +Number of iterationMPLC, N=8, m=N+1 +100 +△= 2-6 +△= 2-15 +△= 2-9 +△= 2-18 +10-2 +△ = 2-12 +△ = 2-21 +Cost function L +10 +10-6 +10-8 +10-10 +0 +200 +400 +600 +800 +Number of iteration10 +FIG. 12: Comparison of final error under crosstalk +using an approximated gradient with ∆ = 2−12 and +phase-insenstive distance with N = 8. +ACKNOWLEDGMENTS +We wish to acknowledge Sho Yasui for the fruitful dis- +cussion. This work is supported by JST CREST Grant +Number JPMJCR1872, Japan. +Appendix A: Unimodality of the Frobenius norm +Here, we present an algebraic proof of the unimodality +of the Frobenius norm. Let X, U ∈ U(N). +Theorem Given U, if fU(X) = ∥X − U∥F has a local +minimum at some X, then it is the global minimum. +Proof. The squared Frobenius norm can be expressed +as ∥X − U∥2 +F = 2N − 2 Re +� +Tr +� +U †X +�� +, and since this +is a local minimum, the term Re +� +Tr +� +U †X +�� +is a local +maximum. Consider Re +� +Tr +� +U †X +�� +as a function on the +manifold U(N). +We can investigate its critical points +by examining the directional derivative of the function +with respect to the tangent vector X′ at X. The tan- +gent vector X′ can be represented as X′ = ZX, where +Z is a skew-Hermitian matrix and X is any matrix on +the unitary group U(N). This is because the Lie algebra +u(N) of the unitary group U(N) is composed of skew- +Hermitian matrices 1. When X is at the critical point, +then Re +� +Tr +� +U †X′�� += 0 is satisfied for any X′. Substi- +tuting X′ = ZX, we obtain +Re +� +Tr +� +U †X′�� += Re +� +Tr +� +ZXU †�� += 0. +(A1) +To further expand this equation, we consider two sets of +special matrices Z1 and Z2, whose matrix Z1 +ij ∈ Z1, Z2 +ij ∈ +Z2 is indexed by 1 ≤ i, j ≤ N with i ̸= j. The (k, l)-th +element of these matrices [·]kl is defined as follows: +[Z1 +ij]kl = δikδjl − δilδjk +(A2) +[Z2 +ij]kl = i(δikδjl + δilδjk). +(A3) +For example, each matrix set includes the following ma- +trices: +Z1 +12 = +� +����� +0 +1 0 · · · 0 +−1 0 +0... +... +0 +· · · 0 +� +����� +, Z2 +23 = +� +������� +0 0 0 0 · · · 0 +0 0 i 0 +0 i 0 +0 0 +... +... +0 +· · · 0 +� +������� +. +(A4) +After substituting Z1 +ij, Z2 +ij, 1 ≤ i, j ≤ N for Z in Eq. A1, +we obtain +� +� +� +� +� +Re +�� +XU †� +ij +� +− Re +�� +XU †� +ji +� += 0 +Im +�� +XU †� +ij +� ++ Im +�� +XU †� +ji +� += 0 +, +(A5) +which leads to XU † = (XU †)†. Therefore, +(XU †)2 = I. +(A6) +The unitary matrix XU † can be diagonalized using a +regular matrix V , and a diagonal matrix D whose diag- +onal elements di ∈ C satisfies |di| = 1 because XU † is a +unitary matrix. We can express this diagonalization as +XU † = V DV −1. Substituting XU † in Eq. A6, we obtain +D = D†. Since |di| = 1, we have di = ±1. Now Consider +the original local maximum term Re +� +Tr +� +U †X +�� +. +We +can rewrite it as Re +� +Tr +� +U †X +�� += Re +� +Tr +� +V DV −1�� += +Re [Tr [D]] = Re [� +i di]. This value is obviously maxi- +mized when di = +1 for all i. If di = −1 for some i, +we can rotate this value to di = +1 along the unit cir- +cle |c| = 1 in the complex plane and still achieve the +maximum value. Therefore, if Re +� +Tr +� +U †X +�� +is a local +maximum, it is also a global maximum. As a result, we +now conclude that if ∥X − U∥F ≥ 0 is a local minimum, +then it must also be a global minimum. Q.E.D. +1 Another proof for X′ = ZX. Consider an identity XX† = I. +Taking the derivative of both sides, we get X′X† +X(X†)′ = O. +Then we can rewrite it as X′X† = −X(X†)′ = −(X′X†)†. Let +Z = X′X†. Then, we have Z = −Z† which indicates Z is a skew- +Hermitian matrix. Since Z = X′X†, we conclude that X′ = ZX. +□ +[1] J. +Carolan, +C. +Harrold, +C. +Sparrow, +E. +Mart´ın- +L´opez, +N. +J. +Russell, +J. +W. +Silverstone, +P. +J. +Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. +Marshall, +M. +G. +Thompson, +J. +C. +F. +Matthews, +T. Hashimoto, +J. L. O’Brien, and A. Laing, Uni- +versal +linear +optics, +Science +349, +711 +(2015), +https://www.science.org/doi/pdf/10.1126/science.aab3642. + +Effect of crosstalk. MPLC, N=8 +100 +100 +w crosstalk, m=N+1 +w crosstalk, m=N+1 +10-1 +w/o crosstalk, m=N+1 +10-1 +w crosstalk, m=N+2 +w crosstalk, m=N+3 +10-2 +10-2 +L +Cost function 1 +10-3 +10-3 +10-4 +10-4 +10-5 +10-5 +10-6 +10-6 +10-7 +10-7. +0 +100 +200 +0 +100 +20011 +[2] J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, +Integrated photonic quantum technologies, Nature Pho- +tonics 14, 273 (2020). +[3] A. W. Elshaari, W. Pernice, K. Srinivasan, O. Benson, +and V. Zwiller, Hybrid integrated quantum photonic cir- +cuits, Nature Photonics 14, 285 (2020). +[4] J. Carolan, +M. Mohseni, +J. P. Olson, +M. Prabhu, +C. Chen, D. Bunandar, M. Y. Niu, N. C. Harris, F. N. C. +Wong, M. Hochberg, S. Lloyd, and D. Englund, Varia- +tional quantum unsampling on a quantum photonic pro- +cessor, Nature Physics 16, 322 (2020), arXiv:1904.10463 +[quant-ph]. +[5] E. Pelucchi, G. Fagas, I. Aharonovich, D. Englund, +E. Figueroa, Q. Gong, H. Hannes, J. Liu, C.-Y. Lu, +N. Matsuda, J.-W. Pan, F. Schreck, F. Sciarrino, C. Sil- +berhorn, J. Wang, and K. D. J¨ons, The potential and +global outlook of integrated photonics for quantum tech- +nologies, Nature Reviews Physics 4, 194 (2022). +[6] Y. Chi, J. Huang, Z. Zhang, J. Mao, Z. Zhou, X. Chen, +C. Zhai, J. Bao, T. Dai, H. Yuan, M. Zhang, D. Dai, +B. Tang, Y. Yang, Z. Li, Y. Ding, L. K. Oxenløwe, M. G. +Thompson, J. L. O’Brien, Y. Li, Q. Gong, and J. Wang, +A programmable qudit-based quantum processor, Nature +Communications 13, 1166 (2022). +[7] L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, +T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, +L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. +Nam, V. D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, +N. Quesada, and J. Lavoie, Quantum computational ad- +vantage with a programmable photonic processor, Nature +606, 75 (2022). +[8] Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr- +Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, +D. Englund, and M. Soljaˇci´c, Deep learning with coherent +nanophotonic circuits, Nature Photonics 11, 441 (2017). +[9] M. Prabhu, +C. Roques-Carmes, +Y. Shen, +N. Har- +ris, L. Jing, J. Carolan, R. Hamerly, T. Baehr-Jones, +M. Hochberg, V. ˇCeperi´c, J. D. Joannopoulos, D. R. +Englund, and M. Soljaˇci´c, Accelerating recurrent ising +machines in photonic integrated circuits, Optica 7, 551 +(2020). +[10] H. Zhang, M. Gu, X. D. Jiang, J. Thompson, H. Cai, +S. Paesani, R. Santagati, A. Laing, Y. Zhang, M. H. +Yung, Y. Z. Shi, F. K. Muhammad, G. Q. Lo, X. S. +Luo, B. Dong, D. L. Kwong, L. C. Kwek, and A. Q. Liu, +An optical neural chip for implementing complex-valued +neural network, Nature Communications 12, 457 (2021). +[11] S. Pai, Z. Sun, T. W. Hughes, T. Park, B. Bartlett, +I. A. D. Williamson, +M. Minkov, +M. Milanizadeh, +N. Abebe, F. Morichetti, A. Melloni, S. Fan, O. Solgaard, +and D. A. B. Miller, Experimentally realized in situ +backpropagation for deep learning in nanophotonic neu- +ral networks, arXiv e-prints , arXiv:2205.08501 (2022), +arXiv:2205.08501 [cs.ET]. +[12] F. Ashtiani, A. J. Geers, and F. Aflatouni, An on-chip +photonic deep neural network for image classification, +Nature 606, 501 (2022). +[13] S. Ohno, R. Tang, K. Toprasertpong, S. Takagi, and +M. Takenaka, Si microring resonator crossbar array for +on-chip inference and training of the optical neural net- +work, ACS Photonics 9, 2614 (2022). +[14] S. Bandyopadhyay, A. Sludds, S. Krastanov, R. Hamerly, +N. Harris, D. Bunandar, M. Streshinsky, M. Hochberg, +and D. Englund, Single chip photonic deep neural +network with accelerated training, arXiv e-prints , +arXiv:2208.01623 (2022), arXiv:2208.01623 [cs.ET]. +[15] N. K. Fontaine, C. R. Doerr, M. A. Mestre, R. R. Ryf, +P. J. Winzer, L. L. Buhl, Y. Sun, X. Jiang, and R. Lin- +gle, Space-division multiplexing and all-optical mimo +demultiplexing using a photonic integrated circuit, in +OFC/NFOEC (2012) pp. 1–3. +[16] A. +Annoni, +E. +Guglielmi, +M. +Carminati, +G. +Fer- +rari, M. Sampietro, D. A. Miller, A. Melloni, and +F. Morichetti, Unscrambling light—automatically undo- +ing strong mixing between modes, Light: Science & Ap- +plications 6, e17110 (2017). +[17] D. Melati, A. Alippi, A. Annoni, N. Peserico, and A. Mel- +loni, Integrated all-optical mimo demultiplexer for mode- +and wavelength-division-multiplexed transmission, Opt. +Lett. 42, 342 (2017). +[18] K. Choutagunta, I. Roberts, D. A. B. Miller, and J. M. +Kahn, Adapting mach–zehnder mesh equalizers in direct- +detection mode-division-multiplexed links, Journal of +Lightwave Technology 38, 723 (2020). +[19] R. Tanomura, R. Tang, G. Soma, S. Ishimura, T. Tane- +mura, and Y. Nakano, All-optical dual-polarization mimo +processor based on integrated optical unitary converter, +in 2022 European Conference on Optical Communication +(ECOC) (2022) pp. 1–4. +[20] W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. +Kolthammer, and I. A. Walmsley, Optimal design for uni- +versal multiport interferometers, Optica 3, 1460 (2016). +[21] R. Burgwal, W. R. Clements, D. H. Smith, J. C. Gates, +W. S. Kolthammer, J. J. Renema, and I. A. Walmsley, +Using an imperfect photonic network to implement ran- +dom unitaries, Opt. Express 25, 28236 (2017). +[22] S. Bandyopadhyay, R. Hamerly, and D. Englund, Hard- +ware error correction for programmable photonics, Op- +tica 8, 1247 (2021). +[23] D. A. B. Miller, Self-configuring universal linear optical +component [Invited], Photon. Res. 1, 1 (2013). +[24] D. A. B. Miller, Setting up meshes of interferometers — +reversed local light interference method, Opt. Express 25, +29233 (2017). +[25] R. Hamerly, S. Bandyopadhyay, and D. Englund, Asymp- +totically fault-tolerant programmable photonics, Nature +Communications 13, 6831 (2022). +[26] R. Hamerly, S. Bandyopadhyay, and D. Englund, Accu- +rate self-configuration of rectangular multiport interfer- +ometers, Phys. Rev. Applied 18, 024019 (2022). +[27] R. Hamerly, S. Bandyopadhyay, and D. Englund, Sta- +bility of self-configuring large multiport interferometers, +Phys. Rev. Applied 18, 024018 (2022). +[28] J.-F. Morizur, +L. Nicholls, +P. Jian, +S. Armstrong, +N. Treps, B. Hage, M. Hsu, W. Bowen, J. Janousek, and +H.-A. Bachor, Programmable unitary spatial mode ma- +nipulation, J. Opt. Soc. Am. A 27, 2524 (2010). +[29] G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, +and J.-F. Morizur, Efficient and mode selective spatial +mode multiplexer based on multi-plane light conversion, +Opt. Express 22, 15599 (2014). +[30] R. Tang, T. Tanemura, and Y. Nakano, Integrated recon- +figurable unitary optical mode converter using mmi cou- +plers, IEEE Photonics Technology Letters 29, 971 (2017). +[31] R. +Tang, +T. +Tanemura, +S. +Ghosh, +K. +Suzuki, +K. Tanizawa, K. Ikeda, H. Kawashima, and Y. Nakano, +Reconfigurable all-optical on-chip mimo three-mode de- +multiplexing based on multi-plane light conversion, Opt. + +12 +Lett. 43, 1798 (2018). +[32] R. Tang, T. Tanemura, and Y. Nakano, Robust recon- +figurable optical mode mux/demux using multiport di- +rectional couplers, in 2017 Opto-Electronics and Com- +munications Conference (OECC) and Photonics Global +Conference (PGC) (2017) pp. 1–3. +[33] R. Tanomura, R. Tang, S. Ghosh, T. Tanemura, and +Y. Nakano, Robust integrated optical unitary converter +using multiport directional couplers, Journal of Light- +wave Technology 38, 60 (2020). +[34] M. Y. Saygin, I. V. Kondratyev, I. V. Dyakonov, S. A. +Mironov, S. S. Straupe, and S. P. Kulik, Robust archi- +tecture for programmable universal unitaries, Phys. Rev. +Lett. 124, 010501 (2020). +[35] R. Tanomura, R. Tang, T. Umezaki, G. Soma, T. Tane- +mura, and Y. Nakano, Scalable and robust photonic inte- +grated unitary converter based on multiplane light con- +version, Phys. Rev. Applied 17, 024071 (2022). +[36] R. Tanomura, R. Tang, T. Tanemura, and Y. Nakano, +Integrated inp optical unitary converter with compact +half-integer multimode interferometers, Opt. Express 29, +43414 (2021). +[37] S. Kuzmin, I. Dyakonov, and S. Kulik, Architecture ag- +nostic algorithm for reconfigurable optical interferometer +programming, Opt. Express 29, 38429 (2021). +[38] S. Pai, B. Bartlett, O. Solgaard, and D. A. B. Miller, Ma- +trix optimization on universal unitary photonic devices, +Phys. Rev. Applied 11, 064044 (2019). +[39] R. Tanomura, +R. Tang, +T. Suganuma, +K. Okawa, +E. Kato, T. Tanemura, and Y. Nakano, Monolithic inp +optical unitary converter based on multi-plane light con- +version, Opt. Express 28, 25392 (2020). +[40] E. W. Weisstein, Unimodal, From Mathworld – A Wol- +fram Web Resource https://mathworld.wolfram.com/ +Unimodal.html. +[41] T. W. Anderson, The integral of a symmetric unimodal +function over a symmetric convex set and some probabil- +ity inequalities, Proceedings of the American Mathemat- +ical Society 6, 170 (1955). +[42] G. Anescu, A heuristic fast gradient descent method for +unimodal optimization, Journal of Advances in Mathe- +matics and Computer Science 26, 1 (2018). +[43] R. Tang, R. Tanomura, T. Tanemura, and Y. Nakano, +Ten-port +unitary +optical +processor +on +a +silicon +photonic +chip, +ACS +Photonics +8, +2074 +(2021), +https://doi.org/10.1021/acsphotonics.1c00419. +[44] R. Tanomura, Y. Taguchi, R. Tang, T. Tanemura, and +Y. Nakano, Entropy of mode mixers for optical uni- +tary converter based on multi-plane light conversion, in +Conference on Lasers and Electro-Optics/Pacific Rim +(CLEOPR) 2022 (Optica Publishing Group, 2022) pp. +CWP13A–02. +[45] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, +Experimental realization of any discrete unitary opera- +tor, Phys. Rev. Lett. 73, 58 (1994). +[46] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- +land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- +terson, W. Weckesser, J. Bright, S. J. van der Walt, +M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. +Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Po- +lat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, +J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, +C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pe- +dregosa, P. van Mulbregt, and SciPy 1.0 Contributors, +SciPy 1.0: Fundamental algorithms for scientific com- +puting in Python, Nature Methods 17, 261 (2020). +[47] R. Fletcher, Practical Methods of Optimization, 2nd ed. +(John Wiley & Sons, New York, NY, USA, 1987). +[48] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, +C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. Van- +derPlas, S. Wanderman-Milne, and Q. Zhang, JAX: com- +posable transformations of Python+NumPy programs +(2018). +[49] M. Gallagher and T. Downs, Visualization of learning in +multilayer perceptron networks using principal compo- +nent analysis, IEEE Transactions on Systems, Man, and +Cybernetics, Part B (Cybernetics) 33, 28 (2003). +[50] H. Li, Z. Xu, G. Taylor, C. Studer, and T. Goldstein, +Visualizing the Loss Landscape of Neural Nets, arXiv +e-prints , arXiv:1712.09913 (2017), arXiv:1712.09913 +[cs.LG]. +[51] H. Li, Z. Xu, G. Taylor, C. Studer, and T. Goldstein, Vi- +sualizing the loss landscape of neural nets, in Advances in +Neural Information Processing Systems, Vol. 31, edited +by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, +N. Cesa-Bianchi, and R. Garnett (Curran Associates, +Inc., 2018). +[52] T. W. Hughes, M. Minkov, Y. Shi, and S. Fan, Training +of photonic neural networks through in situ backpropa- +gation and gradient measurement, Optica 5, 864 (2018). +[53] M. Jacques, A. Samani, E. El-Fiky, D. Patel, Z. Xing, +and D. V. Plant, Optimization of thermo-optic phase- +shifter design and mitigation of thermal crosstalk on the +soi platform, Opt. Express 27, 10456 (2019). +[54] B. +V. +Gurses, +R. +Fatemi, +A. +Khachaturian, +and +A. Hajimiri, Large-scale crosstalk-corrected thermo-optic +phase shifter arrays in silicon photonics, IEEE Journal +of Selected Topics in Quantum Electronics 28, 6101009 +(2022). + diff --git a/LdFRT4oBgHgl3EQf1zij/content/tmp_files/load_file.txt b/LdFRT4oBgHgl3EQf1zij/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b49f562085fba56a5264d54b5f4e5a393c0e4604 --- /dev/null +++ b/LdFRT4oBgHgl3EQf1zij/content/tmp_files/load_file.txt @@ -0,0 +1,1090 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf,len=1089 +page_content='Iterative configuration of programmable unitary converter based on few layer redundant multi-plane light conversion Yoshitaka Taguchi,1, ∗ Yunzhuo Wang,2 Ryota Tanomura,1 Takuo Tanemura,1 and Yasuyuki Ozeki1 1Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656 Japan 2Preferred Networks Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Otemachi Bldg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=', 1-6-1 Otemachi, Chiyoda-ku, Tokyo 100-0004 Japan (Dated: February 1, 2023) Programmable unitary photonic devices are emerging as promising tools to implement unitary transformation for quantum information processing, machine learning, and optical communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' These devices typically use a rectangular mesh of Mach-Zehnder interferometers (MZIs), which has a clear mathematical structure and can be configured deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' However, this mesh archi- tecture is sensitive to fabrication errors, and the correction techniques are still under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In contrast, the multi-plane light conversion (MPLC) architecture is more robust against fabrication errors, but a deterministic method for configuring the converter has not yet been developed due to its complex mathematical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this work, we propose a fast iterative configuration method for MPLC, following the mathematical review of the matrix distance and proposal of a new norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We show through numerical simulations that adding a few redundant layers significantly improves the convergence of the MPLC architecture, making it a practical and attractive option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We also consider the effects of finite resolution and crosstalk in phase shifters in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In addi- tion, we propose a phase-insensitive distance suited for applications using only intensity detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Our method demonstrates orders of magnitude better accuracy and a 20-fold speed-up compared to previous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' INTRODUCTION Programmable unitary transformations implemented on integrated photonic platforms are becoming a pow- erful tool for a variety of applications, including quan- tum photonics [1–7], machine learning [8–14], and op- tical communication [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Accurate realization of a given unitary transformation is critical, as the fidelity of computational results and the error of optical communi- cation can be significantly affected by the precision of the realized transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A common approach to synthe- sizing unitary transformations is to use a mesh of Mach- Zehnder interferometers (MZIs) known as the Clements architecture [20], which consists of phase shifters and beam splitters (BSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This architecture is attractive be- cause its mathematical structure is decomposable, allow- ing the required phase shift in each MZI to be explic- itly determined from the given unitary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' However, physical implementation artifacts such as devi- ation in the splitting ratio of BSs can result in errors in the synthesized transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' These errors can become significant as the number of optical modes increases [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Several design proposals have been made in an effort to reduce or eliminate this error, with the goal of achieving a precise, customizable, and fabrication-error-tolerant uni- tary transformation that can be applied to scalable and reliable applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' To address the challenge of implementation artifacts in the Clements architecture, several approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' One approach is local error correction, which ∗ ytaguchi@ginjo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='jp involves fixing each MZI and can be applied to any MZI- based architecture, but requires prior knowledge of pas- sive and active components [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Another approach is the measurement of components with on-chip power moni- tors, which allows for the calibration of each MZI but also increases the size of the chip and the complexity of wiring [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Self-configuration and 3-MZI approaches utilize an additional BS to achieve partially perfect lin- ear operation and employ a feedback loop to adjust each phase shift using only output signals [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' While this method allows for infinite scalability, it also increases the size of the circuit and may have issues with stability [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' It’s worth noting that these approaches primarily con- sider the artifacts of passive BSs in the circuit, and do not sufficiently consider the artifacts of phase shifters, such as crosstalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Another architecture employs a series connection of phase shifter arrays and unitary transformations to achieve a highly robust universal synthesis of unitary ma- trices that is resistant to fabrication errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This archi- tecture, also known as the multi-plane light conversion (MPLC) architecture [28–31], is particularly robust be- cause each unitary transformation can be selected from a wide range of possible unitaries [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The unitary transformation can be almost any well-known N-mode mixer, which can significantly increase the flexibility and tolerance to fabrication errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' However, configuring the phase shifters in this architecture is challenging, and no explicit configuration method has been known due to its complex mathematical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The optimiza- tion of this architecture must deal with the many local minima present in its high-dimensional parameter space [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' As a result, previous reports have relied on heuris- tic global searches, such as basin-hopping and simulated arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='13658v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='ET] 31 Jan 2023 2 annealing, to configure the phase shifters [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' How- ever, these methods are time-consuming and suffer from exponentially increasing search times as the parameter space dimension increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' To address this issue, a ma- chine learning-based configuration algorithm has been proposed [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' While this algorithm may offer a solution, it requires an accurate initial estimation of the structure and may result in decreased matrix fidelity if the initial estimation contains errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this research, we present a new, fast and iteratively configurable MPLC architecture that does not require prior knowledge and relies only on output signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This approach involves adding a few redundant layers to the existing MPLC architecture and using derivative-based optimization with gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This addi- tional layer redundancy significantly improves the opti- mization performance of the MPLC architecture, in con- trast to the similar approach used for the Clements archi- tecture [21, 38], which adds a large number of redundant layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' When compared to numerical optimization of the Clements architecture without redundancy [38], our pro- posed method achieved 5 orders of magnitude better ac- curacy with 1/20 fewer iterations for N = 128 modes of transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Additionally, our proposed method was able to achieve 5 orders of magnitude better accuracy and was 23 times faster in configuration compared to the pre- vious report that used a heuristic algorithm to optimize the MPLC architecture [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Before discussing the main results, we begin by discussing important gen- eral properties of unitary matrix optimization and in- troducing a new distance in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' One key prop- erty we cover is that unitary matrix optimization essen- tially has no local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Additionally, we propose a new distance, a phase-insensitive variant of the Frobe- nius norm, which is invariant under phase shifts at the output modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Previously, the standard Frobenius norm has even been used in phase-insensitive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In Section III, we investigate the optimization properties of the MPLC architecture with a few redundant layers of parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' While the parametrization of a uni- tary matrix can cause optimization to fall into local min- ima, we demonstrate through numerical simulations that these can be effectively avoided by adding a few redun- dant layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Our results show that this architecture can be efficiently optimized using well-known local minimiza- tion algorithms, such as the gradient descent algorithm, while the Clements architecture cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We also study the statistical properties of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In Section IV, we examine practical scenarios, such as when the gradi- ent of the system is not available, only intensity detec- tion is used, and crosstalk between phase shifters exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We evaluate the impact of gradient approximation and crosstalk on the proposed method, and show that it still performs well, albeit with a reduction in achieved matrix accuracy after optimization or an increase in the number of iteration until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The phase-insensitive dis- tance exhibits similar optimization properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In Section V, we conclude the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' MATRIX DISTANCE USING THE FROBENIUS NORM This section presents some general mathematical prop- erties of the Frobenius norm and proposes a new dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We begin by defining the concept of unimodality for functions on the unitary group U(N) and show that the matrix distance using the Frobenius norm exhibits this unimodality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We then clarify the range and expected value of the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Additionally, we introduce the phase- insensitive matrix distance for applications that only use intensity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Unimodality on U(N) Here, the concept of unimodality for a function on U(N) is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Unimodality is typically defined for probability distributions [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For a multivariable func- tion f : RN → R, unimodality is defined through the level set L(f, α) = {x|f(x) ≤ α, x ∈ RN} and the con- vexity of L(f, α) [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this paper, we extend this definition to functions on U(N) by considering the path- connectedness of L(f, α), as U(N) is not a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Definition A function f : U(N) → R is called uni- modal if the level set L(f, α) = {X|f(X) ≤ α, X ∈ U(N)} is path-connected for any α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In other words, any local minimum of a unimodal func- tion on U(N) is also a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Unimodality of the Frobenius norm We prove that the unitary matrix distance using the Frobenius norm is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The distance between two unitary matrices d(X, U) is defined as ∥X − U∥F , where ∥A∥F = � Tr [A†A] is the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' It is worth noting that the matrix distance using mean square error (MSE) � i,j |Xij −Uij|2 is equivalent to d(X, U)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Given a unitary matrix U ∈ U(N), we show that the function fU : U(N) → R, defined as fU(X) = d(X, U) is uni- modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' First, from the definition of the Frobenius norm, fU(X)2 is simplified as fU(X)2 = Tr � (X − U)†(X − U) � = 2N − 2 Re � Tr � U †X �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (1) We write the eigenvalues of U †X as λi(1 ≤ k ≤ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Since both U and X are unitary, all the eigenvalues λk satisfy |λk| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Therefore, the eigenvalues can be written as λk = eiθk, where −π ≤ θk ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Using these eigenvalues, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1 can be simplified further as fU(X)2 = 2N − 2 N � k=1 cos θk (2) 3 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 2 implies that fU(X)2 is unimodal because cos θk is unimodal over the range −π ≤ θ ≤ π and their sum is also unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' An algebraic proof of this unimodality is provided in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Range and expected value We derive the range and expected value of fU(X)2 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The maximum of fU(X)2 is 4N if and only if θk = ±π for all k, and the minimum is 0 if and only if θk = 0 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In previous studies, fU(X)2 has been normalized by N [25], 2N [38], or N 2 [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Normalizing by 4N yields 0 ≤ fU(X)2/4N ≤ 1, which suggests a good normalization of the norm with a range from 0 to 1 that is independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' To calculate the expected value E � fU(X)2� , the distribution of θk is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' If X is sampled from the Haar measure, then U †X is also Haar- random due to the invariance of the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' As a result, the eigenvalues of U †X are uniformly distributed on the unit circle |c| = 1, and we have θk ∼ U(−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Because E[cos θk] = � π −π 1 2π cos θ dθ = 0, (3) the expected value of the second term in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We now conclude that E � fU(X)2� = 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This fact is ob- served numerically in the initial value of the convergence plots in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Phase-insensitive distance Here, we introduce a phase-insensitive variant of the matrix distance using the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This variant is suitable for applications that only detect the intensity of the output modes, as the distance should not be af- fected by the output phases from the unitary converter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Applications that benefit from this phase-insensitive dis- tance include machine learning and quantum photonics, where photodiodes or photon number counters are placed at the output ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1 shows a scenario where a complex vector (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' , sn)⊤ is input into two unitary conversion devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The transfer matrix for these devices is represented by P and Q, and their complex outputs are in polar form as teiθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The only difference in the out- put vectors from these two devices is in their phase, with θi ̸= θ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In applications that only detect the intensity of output modes, these two matrices P and Q are treated the same and a suitable matrix distance is introduced for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In the following discussion, the matrix U represents the given target unitary matrix, and the ma- trix X represents the actual conversion achieved by the unitary converter device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' To investigate the effect of output phases from the uni- tary converter, we represent the unitary matrices U † and … Intensity detector … FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1: A complex vector (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' , sn)⊤ is input into two unitary conversion devices, whose transfer matrices are represented as P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The outputs from these devices are identical, with the exception of the phase degrees of freedom at the outputs, when evaluated using a phase-insensitive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' X as U † = �u1 u2 · · · un � X = � ������ x⊤ 1 x⊤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' x⊤ n � ������ , (4) where ui are column vectors of U † and x⊤ i are row vectors of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Since U † and X are unitary matrices, the norms of ui and xi are all equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this context, the output phases of the unitary converter correspond to the phases of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The original Frobenius norm changes its value under the global phase change of xi, which is defined as replacing xi with eiθixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We analyze the dependence of the Frobenius norm on the global phase of each xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The Frobenius norm is ex- panded using the column vectors ui as ∥X − U∥2 F = ��XU † − I ��2 F = ∥Xu1 − e1∥2 + ∥Xu2 − e2∥2 + · · · + ∥Xun − en∥2, (5) where ei is a unit column vector whose i-th element is 1 and the others are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Expanding the term ∥Xu1 − e1∥2, we obtain ∥Xu1 − e1∥2 = |x⊤ 1 u1 − 1|2 + |x⊤ 2 u2|2 + · · · + |x⊤ n un|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (6) Except for the first term |x⊤ 1 u1 − 1|2, the other terms |x⊤ i ui|2(i ≥ 2) are invariant under the global phase change in xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Only the first term depends on the global P,Q EU(N4 phase of x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Therefore, the term ∥Xu1 − e1∥2 is invari- ant under the global phase change in the row vectors x2, x3, · · · xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Similarly, ∥Xui − ei∥2 is independent of the global phase of xj where j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' When the global phase of xi is changed, the term |x⊤ i ui−1|2 takes its min- imum value if and only if x⊤ 1 u1 is a positive real number, because |x⊤ i ui − 1|2 = |x⊤ i ui|2 − 2 Re � x⊤ i ui � + 1 (7) and only the second term 2 Re � x⊤ 1 u1 � is dependent on the global phase of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This motivates the idea of replac- ing all instances of |x⊤ i ui − 1|2 with � |x⊤ i ui| − 1 �2 in the Frobenius norm ∥X − U∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' As discussed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 6 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 7, the minimum value of the Frobenius norm under this substitution is the same as the minimum value of the original Frobenius norm, due to the phase-dependence property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Based on the aforementioned consideration, we pro- pose a phase-insensitive matrix distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Given unitary matrix U, we define the distance function hU : U(N) → R using only terms whose form is |x⊤ i uj|, which can be obtained using intensity measurements at the outputs from the unitary converter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We define positive real num- bers aij = |x⊤ i uj|2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' All the aij can be obtained by multiplying column vector ui with matrix X through the unitary converter, because Xuj = � ������ x⊤ 1 uj x⊤ 2 uj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' x⊤ n uj � ������ (8) and |x⊤ i uj|2 can be obtained through the intensity mea- surement of x⊤ i uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' By expanding all the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 5 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 6 and applying the substitution discussed with respect to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 7, we define hU(X) as hU(X) = (√a11 − 1)2 + a12 + · · · + a1n + a21 + (√a22 − 1)2 + · · · + a2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' + (√aii − 1)2 · · · + an1 + an2 + · · · + (√ann − 1)2 = � ij � δij − ��� � XU †� ij ��� �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (9) This function is independent of the global phase of xi, that is, the phases of each output from the unitary con- verter, and has a minimum value identical to the original Frobenius norm distance function fU(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The function hU(X) is also unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This can be proven by reductio ad absurdum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Suppose hU(X) is not a unimodal function and has multiple disconnected local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Then, for all matrices X′ such that hU(X′) is a local minimum, there must exist at least one set of phases (θ1, θ2, · · · θn) that defines a matrix Y = � ������ eiθ1x′⊤ 1 eiθ2x′⊤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' eiθnx′⊤ n � ������ (10) such that fU(Y ) is also a local minimum, where x′⊤ i is the row vector of X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The existence of such a matrix Y follows from the definition of hU(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' However, this would mean that fU(X) also has multiple disconnected local minima, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Therefore, hU(X) must be unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' FEW-LAYER REDUNDANT PARAMETERIZATION The global optimization property guaranteed by uni- modality discussed in the previous section is derived without any assumptions about the matrix being opti- mized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' however, the matrix synthesized by the device is parametrized by physical parameters and unimodality may not always hold in this parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We ex- amine this issue and propose a solution to mitigate this difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Let lU(X) ≥ 0 be a distance function from a desired unitary matrix U, and X(p) be a unitary ma- trix realized by a physical converter that is parametrized by a real parameter vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Each element in p corre- sponds to the amount of phase shift in the actual device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' If lU(X) is unimodal, its gradient becomes a zero vector only when X = ±U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' However, when optimizing an actual unitary conversion device, we need to consider the scalar optimization of lU(X(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' If the Jacobian of X(p) is full-rank at any p, meaning there exists infinitesimal pa- rameter changes ∆p for any infinitesimal matrix changes ∆X, then the function lU(X(p)) also has a single min- imum due to the aforementioned unimodal property of lU(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' By increasing the number of layers in the unitary converter device, the degree of freedom in the parameter space increases, which may make the Jacobian of X(p) more likely to be full-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this section, we demon- strate that increasing the number of layers in unitary converter devices by a few from its minimum requirement significantly improves the optimization of MPLC archi- tecture using a gradient-based optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Device definition and redundancy We present the mathematical definition of the unitary converters and the few-layer redundant parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 2a shows the architecture of the MZI-based unitary converter, which is commonly referred to as Clements ar- chitecture [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The MZI consists of two 50 : 50 BSs and two phase shifters, which can realize an arbitrary U(2) 5 Layer … … … … (a) Layer … … … (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 2: Schematics of the N × m Clements architecture (a) and N × m MPLC architecture (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The left ports are inputs, and the right port are outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The number of layers in each architecture is specified by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In the Clements architecture, each layer contains either N/2 or (N − 1)/2 MZI nodes, represented by white circles in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Each MZI node consists of two phase shifters, represented by the variables φ and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In the MPLC-based unitary converter (b), the architecture consists of an N-port fixed unitary converter represented by A, followed by an array of N single-mode phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this paper, we do not consider any imperfections of the MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 2b shows the structure of the MPLC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Each layer consists of an N-port fixed unitary converter Ai and an array of N single-mode phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' After m layers, another array of phase shifters is placed in a similar manner to the Clements architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The overall transformation of this device, denoted as X, is given by X = Lm+1AmLm · · · A2L2A1L1, (11) where Ai is the transfer matrix of a N-port unitary con- verter and Li is expressed as Li = � ���� eiθi1 eiθi2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' eiθin � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (12) For any i ̸= j, the matrices Ai and Aj are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The total number of degrees of freedom in this matrix is (m + 1)(N − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This is because each phase shifter array has N − 1 degree of freedom due to the loss of one degree of freedom from the global phase, and the entire device has an additional degree of freedom, the global phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The N-port fixed unitary converter Ai can be implemented using a multiport directional coupler [33], multimode interference coupler [30], or other multiport unitary transform devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The device should be care- fully chosen to ensure that the overall transformation X is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The mixing entropy of a device can be used as a measure of universality [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' To realize an arbi- trary U(N) transformation, the total number of degrees of freedom must exceed N 2 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For the Clements ar- chitecture, the number of layers m must satisfy m ≥ N [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Similarly, the number of layers m for the MPLC ar- chitecture must also satisfy m ≥ N, which follows from (m + 1)(N − 1) + 1 ≥ N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this context, a few-layer re- dundant parameterized architecture is defined as having m = N + 1, N + 2 layers for both architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Optimization problem setting and algorithm We formulate the matrix optimization problem as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We have real parameter variables expressed as a vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The number of parameters depends on m and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We define the normalized cost function L between two matrices as L(p) = 1 4N ∥X(p) − U∥2 F , (13) where X(p) is the unitary matrix realized physically by the parameter vector p, U is the target matrix to be achieved, and ∥·∥F is the Frombenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The cost function L is divided by 4N as discussed in the Section II C, and 0 ≤ L ≤ 1 is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' At the beginning of the optimization, parameters are initialized using uni- form distribution ranging from 0 to 2π, and the target unitary matrix U is sampled from Haar measure using the stats module of SciPy [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For the MPLC architec- ture, the matrix Ai for 1 ≤ i ≤ m is also sampled from Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' After initializing the parameters and ma- trices, the cost function L is optimized using the quasi- Newton optimization method L-BFGS [47] implemented in optimize module of SciPy [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The derivative of the cost function L required for L-BFGS is calculated us- ing automatic differentiation with the JAX framework [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This method starts from the initial parameters and modifies them at each step until convergence to the lo- cal minimum, where dL/dp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In each layer, we have N real parameters to represent N phase shifts in both Clements and MPLC architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The optimization is run 64 times while changing the initial parameters to in- vestigate the statistical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Cases with N = 8 and N = 32 are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' cOs 0 sinθ sin θ cos 0Ai E UN6 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3: Convergence plots for MPLC and Clements architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The vertical axis shows the value of the cost function L defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 13, and the horizontal axis shows the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The shaded area represents the minimum and maximum values, the solid line represents the median, and the dotted line represents the 25% and 75% quantiles over 64 optimization trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' MPLC architecture with (a) N = 8, (b) N = 32, Clements architecture with (c) N = 8, and (d) N = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3 shows the convergence plots when the num- ber of layers is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The convergence plot of the cost function is recorded for 64 optimization trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The shaded area shows the range of minimum and maximum values, the dotted line shows the 25% and 75% quantiles, and the solid line shows the median of the trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For both Clements and MPLC architecture, the insufficient parameterized layer setting results in a large amount of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For MPLC architecture, the non-redundant case of m = N results in a large variance of error, especially for N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This suggests the presence of many local minima in the parameter space of the MPLC architec- ture, as previously reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Although the variance of non-redundant setting m = N of N = 32 is smaller than that of N = 8, the error still remains for N = 32, indicating the presence of inevitable local minima for this condition as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' When we increase the number of layers and add redundant degrees of freedom, the variance and error become small, as shown in the cases of m = N + 1, N + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In contrast, the Clements architecture results in a large variance of error for all conditions, even though it has the sufficient number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For the cases with a large number of ports, N = 128, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 4 shows the performance comparison with the pre- vious study [38] of the Clements architecture with redun- dancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' When compared with no redundancy, the previ- ous study converges at L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='4 × 10−2 after 20000 itera- tions, while MPLC architecture yields a result that is 6 orders of magnitude better than the Clements architec- ture with 1/20 fewer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The MPLC architecture with a redundant layer m = N + 1 still outperforms in terms of both convergence speed and accuracy, even com- pared with the Clements architecture with 128 redundant layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We visualized the optimization trajectory and loss function in the parameter space using the method re- ported in [49, 50] and in the supplementary material of [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The optimization trajectory is the path of the pa- rameters in a high-dimensional space created by the opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We stored the parameter history at each step of the optimization and applied principal component anal- MPLC, N=8 100 m=N-1 m=N+1 m=N m=N+2 10-2 Cost function L 10-4 10-6 10-8 10-10 10-12 0 50 100 150 200 250 Number of iterationMPLC, N=32 100 m=N-1 m=N+1 m=N m=N+2 10-2 Cost function L 10-4 10-6 10-8 10-10 0 500 1000 1500 2000 Number of iterationClements, N=8 100 m=N-1 m=N+1 m=N m=N+2 10-2 Cost function L 10-4 10-6 10-8 10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 10-12 0 50 100 150 200 250 300 350 400 Number of iterationClements, N=32 100 m=N-1 m=N+1 m=N m=N+2 10-2 Cost function L 10-4 10-6 10-8 10-10 0 5000 10000 15000 20000 Number of iteration7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 4: Comparison of the large case of N = 128 with the previous study [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' ysis (PCA) to that history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The first and second PCA components were used to project the high-dimensional path onto a two-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The visualization was performed for N = 8, m = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 5 shows the projected trajectories and contour plots of the log of the loss function in the projected subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The contour plot for the MPLC architecture is like a simple elliptic unimodal function, while that of the Clements architec- ture is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The difference in the contour plots between these architectures suggests the reason for the convergence plot difference shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 5: The PCA projections of the optimization trajectories and contour plot of the log of the loss function for the MPLC architecture (a) and the Clements architecture (b) for N = 8, m = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Each of the four figures shows the trajectory of the optimization when the initial parameters and the target matrix are randomly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' OPTIMIZATION UNDER PRACTICAL SETTINGS We discuss three challenges that must be addressed when applying the proposed method in Section III to real device optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' First, the optimization method used in this section requires the gradient of the target func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' While the gradient can be taken physically [52], it requires additional external equipment, which makes the system bulky and not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Second, the optimization method uses the complex amplitudes at the output for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Reading the complex amplitudes in a real device requires coherent detectors at the output, which complicates the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' While some applications require coherent detection, many photonics-based optical com- puting platforms and quantum computing with photonic chips use intensity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Third, real devices have crosstalk between phase shifters, which is not considered in the optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Crosstalk is especially prob- lematic in thermo-optic phase shifters [53, 54], although they are attractive due to their small footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In this section, we report the results of derivative- free optimization with the original and phase-insensitive norm, using only the output signal, and examine the ef- fect of crosstalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We first introduce the mathematical formulation and then present the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The optimization method used is the same as in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' These results pave the way for the design of optical uni- tary converters without the need for additional compo- nents, making the platform more scalable and versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gradient approximation of multivariate function We use numerical gradient approximation, which uses only function values to approximate the analytical gra- dient, and investigate the effect of this approximation on the optimization behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The gradient of multivariate function ∇f(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' , xn) is approximated by ∇f(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' , xn) ≈ � ����� f(x1+∆,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=',xn)−f(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=',xn) ∆ f(x1,x2+∆,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=',xn)−f(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=',xn) ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' f(x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=',xn+∆)−f(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=',xn) ∆ � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (14) where ∆ ≪ 1 represents a finite difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Calculat- ing gradient approximation requires the same number of function evaluations as the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We show that a derivative-based algorithm using such gra- dient approximation can still be effective for optimizing unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Definition of device with intensity detection We evaluate the optimization behavior of the phase- insensitive distance introduced in Section II D, which only uses intensity detectors at the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We expect the phase-insensitive distance to behave similarly to the phase-sensitive distance during optimization because it also has a unimodal property, as shown in Section II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In order to test this, we removed the last phase shifter 2nd PCA component: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='9% 2nd PCA component: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5% 4 0 5 0 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 1st PCA component: 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5% lst PcA component: 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='2% 2nd PCA component: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='4% 2nd PCA component: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0% 6 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 5 0 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 1st PcA component: 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='9% 1st PCA component: 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0%MPLC, N=128 100 m=N-1 X VX m=N X + X 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' m=N+1 + Clements with m=N+2 no redundancy Cost function + 10-4 Clements with 128 redundant layers 10-6 +++++++++++ 10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 10-10 0 500 1000 1500 2000 2500 3000 Number of iteration2nd PCA component: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='7% 2nd PCA component: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='4% T-J 20 2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 1stPCA component: 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0% 1st PCA component: 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='7% 2nd PCA component: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0% 2nd PCA component: 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0% 2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 2 0 2 4 6 1st PCA component: 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0% 1st PCA component: 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='9%8 array from the MPLC architecture, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Although this removes the N degree of freedom from the architecture, we still expect the optimization behavior to be similar to that of a standard phase-sensitive norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Layer … … … Removed FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 6: Schematics of the MPLC architecture with the last phase shifter array removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Model of crosstalk We model crosstalk by considering the interaction be- tween adjacent phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The crosstalk is repre- sented by a linear combination of phase shifts, which can be expressed as θi = � j αijθj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (15) The coupling model and coupling coefficients αij are shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The coupling in the following simulations is formulated as θ′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='1θi−2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5θi−1 + θi + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5θi+1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='1θi+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' If the coupled param- eter p′ = w(p) is reversible, the unitary matrix X(p′) realized by the coupled parameter will have a full-rank Jacobian if the original X(p) has a full-rank Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We use the gradient approximation for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' … … … FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 7: Crosstalk model of the MPLC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 8 shows the convergence plots for gradient approx- imation using the standard Frobenius-norm-based dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The approximation is calculated using a ∆ value of 2−10, which corresponds to phase shifts with 10-bit res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' When a redundant layer is added, the MPLC ar- chitecture shows numerical-accuracy limited performance (for m = N +1, N +2) with a small variance in the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Each iteration of the optimization requires the evaluation of the distance the same number of times as the number of parameters due to the gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For example, when N = 8 and m = N +1, each optimization requires 8 × (8 + 2) = 80 evaluations of the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Us- ing the MPLC architecture, the optimization converges at 100 iterations for this case, so the total number of evaluations is approximately 8000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 8: Convergence plots for the MPLC architecture with gradient approximation, where ∆ = 2−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The 64 optimization trials are shown in the same manner as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We examined the ∆ dependence of the small error vari- ance observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 8, which arises from the gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 9 shows the optimization results for each gradient approximation accuracy using a redundant layer setting of m = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' As the finite difference ∆ becomes smaller, the final error also becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' If the accuracy of the gradient approximation is not suf- ficient, meaning ∆ is not small enough, the variance of the optimization result is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' For example, the error is in the range � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 × 10−5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='8 × 10−5� for the ∆ = 2−6 case, � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='3 × 10−7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 × 10−7� for the ∆ = 2−9 case, and � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 × 10−9, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='5 × 10−9� for the ∆ = 2−12 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' If the accuracy of the gradient approximation is sufficient (∆ ≥ 2−15), the result has a non-negligible variance sim- ilar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The error is in the range � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='9 × 10−10, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='8 × 10−11� for the ∆ = 2−15 case and � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='7 × 10−10, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='6 × 10−12� for the ∆ = 2−18 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' These result provide criteria for designing the DAC res- olution of a unitary converter system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We studied the optimization property using the phase- insensitive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 10 shows the optimization result obtained with an analytical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The con- vergence plot is similar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3, in spite of the reduced degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' However, when using gradient approximation and comparing the accu- racy dependence, the phase-insensitive distance shows a MPLC, N=8 100 m=N-1 m=N+1 m=N m=N+2 10-1 10-2 Cost function L 10-3 10-4 10-5 10-6 10-7 50 100 150 0 200 Number of iterationAi E UN9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 9: Comparison of final error when changing the accuracy of gradient approximation for the MPLC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The 64 optimization trials are shown in the same manner as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' different result from the standard Frobenius-norm-based distance, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' All the optimization re- sults have a large variance, as opposed to the cases where ∆ = 2−6, 2−9, 2−12 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The finite difference ∆ must be smaller than 2−18 to achieve an optimization re- sult comparable to the one obtained using an analytical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The black dashed line shows the optimization result by simulated annealing in a previous study [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' They achieved an error of fMSE = −50 dB, which cor- responds to L = 2 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This error can be achieved using our method with a finite difference of ∆ ≤ 2−9, and further improvement by orders of magnitude is possible with more accurate gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' When the accuracy is sufficient, our optimization method converges after about 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' To compare the speed of our method with a previous experimental report of MPLC architecture optimization using intensity detection [39], we also conducted optimization for a case with N = 4 and m = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The optimization converged after about 45 iterations with ∆ = 2−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' As each iteration requires 4×(4+2) = 24 evaluations, the total number of iterations required for convergence is 1080, representing a 23-fold speedup compared to the previous report (∼ 25200 eval- uations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The effect of crosstalk is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The gra- dient approximation was calculated using ∆ = 2−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Al- though crosstalk caused a larger error and increased the number of iterations until convergence, the performance degradation can be mitigated by adding a few layers of additional redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This result suggests that it may be possible to optimize the device end-to-end, including both matrix optimization and phase shifter calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' CONCLUSION We proposed a fast and iteratively configurable MPLC architecture for realizing precise and fabrication-error- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 10: Convergence plots for the MPLC architecture using phase-insensitive distance with N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The 64 optimization trials are shown in the same manner as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 11: Comparison of final error when changing the accuracy of gradient approximation for the MPLC architecture using phase-insenstive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The black dashed line represents the optimization result in a previous study [35], where L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 × 10−5 was achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' tolerant unitary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Our numerical results show that adding a few redundant layers to the MPLC architecture significantly improves optimization behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We also examined the effect of artifacts, such as crosstalk, and found that the proposed architecture can be optimized end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' In addition to proposing a new architecture and optimization method, we analyzed the distance between unitary matrices using the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We introduced the concept of unimodality for functions on the unitary group and proved that the ma- trix distance using the Frobenius norm has this prop- erty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We also calculated the expected value and range of the matrix distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We also introduced the phase- insensitive norm, which is useful for applications that only use intensity detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We believe that this ap- proach will enable the scalable and robust implementa- tion of optical unitary converters and expand the use of photonic integrated circuits in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' MPLC, N=8, m=N+1 100 △ = 2-6 △ = 2-15 △= 2-9 △ = 2-18 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' △= 2-12 Cost function L 10 10-6 10-8 10-10 0 20 40 60 80 100 120 Number of iterationMPLC, N=8 100 m=N-1 m=N+1 m=N m=N+2 10-2 Cost function L 10-4 10-6 10-8 10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 10-12 0 25 50 75 100 125 150 175 Number of iterationMPLC, N=8, m=N+1 100 △= 2-6 △= 2-15 △= 2-9 △= 2-18 10-2 △ = 2-12 △ = 2-21 Cost function L 10 10-6 10-8 10-10 0 200 400 600 800 Number of iteration10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 12: Comparison of final error under crosstalk using an approximated gradient with ∆ = 2−12 and phase-insenstive distance with N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' ACKNOWLEDGMENTS We wish to acknowledge Sho Yasui for the fruitful dis- cussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This work is supported by JST CREST Grant Number JPMJCR1872, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Appendix A: Unimodality of the Frobenius norm Here, we present an algebraic proof of the unimodality of the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Let X, U ∈ U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Theorem Given U, if fU(X) = ∥X − U∥F has a local minimum at some X, then it is the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The squared Frobenius norm can be expressed as ∥X − U∥2 F = 2N − 2 Re � Tr � U †X �� , and since this is a local minimum, the term Re � Tr � U †X �� is a local maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Consider Re � Tr � U †X �� as a function on the manifold U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We can investigate its critical points by examining the directional derivative of the function with respect to the tangent vector X′ at X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The tan- gent vector X′ can be represented as X′ = ZX, where Z is a skew-Hermitian matrix and X is any matrix on the unitary group U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This is because the Lie algebra u(N) of the unitary group U(N) is composed of skew- Hermitian matrices 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' When X is at the critical point, then Re � Tr � U †X′�� = 0 is satisfied for any X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Substi- tuting X′ = ZX, we obtain Re � Tr � U †X′�� = Re � Tr � ZXU †�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (A1) To further expand this equation, we consider two sets of special matrices Z1 and Z2, whose matrix Z1 ij ∈ Z1, Z2 ij ∈ Z2 is indexed by 1 ≤ i, j ≤ N with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' The (k, l)-th element of these matrices [·]kl is defined as follows: [Z1 ij]kl = δikδjl − δilδjk (A2) [Z2 ij]kl = i(δikδjl + δilδjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (A3) For example, each matrix set includes the following ma- trices: Z1 12 = � ����� 0 1 0 · · · 0 −1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 0 · · 0 � ����� , Z2 23 = � ������� 0 0 0 0 · · · 0 0 0 i 0 0 i 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 0 · · 0 � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (A4) After substituting Z1 ij, Z2 ij, 1 ≤ i, j ≤ N for Z in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A1, we obtain � � � � � Re �� XU †� ij � − Re �� XU †� ji � = 0 Im �� XU †� ij � + Im �� XU †� ji � = 0 , (A5) which leads to XU † = (XU †)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Therefore, (XU †)2 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (A6) The unitary matrix XU † can be diagonalized using a regular matrix V , and a diagonal matrix D whose diag- onal elements di ∈ C satisfies |di| = 1 because XU † is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We can express this diagonalization as XU † = V DV −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Substituting XU † in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A6, we obtain D = D†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Since |di| = 1, we have di = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Now Consider the original local maximum term Re � Tr � U †X �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' We can rewrite it as Re � Tr � U †X �� = Re � Tr � V DV −1�� = Re [Tr [D]] = Re [� i di].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' This value is obviously maxi- mized when di = +1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' If di = −1 for some i, we can rotate this value to di = +1 along the unit cir- cle |c| = 1 in the complex plane and still achieve the maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Therefore, if Re � Tr � U †X �� is a local maximum, it is also a global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' As a result, we now conclude that if ∥X − U∥F ≥ 0 is a local minimum, then it must also be a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1 Another proof for X′ = ZX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Consider an identity XX† = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Taking the derivative of both sides, we get X′X† +X(X†)′ = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Then we can rewrite it as X′X† = −X(X†)′ = −(X′X†)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Let Z = X′X†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Then, we have Z = −Z† which indicates Z is a skew- Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Since Z = X′X†, we conclude that X′ = ZX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' □ [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Carolan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Harrold, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sparrow, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Mart´ın- L´opez, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Russell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Silverstone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Shadbolt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Matsuda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Oguma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Itoh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Marshall, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Thompson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Matthews, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hashimoto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' O’Brien, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Laing, Uni- versal linear optics, Science 349, 711 (2015), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='aab3642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Effect of crosstalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' MPLC, N=8 100 100 w crosstalk, m=N+1 w crosstalk, m=N+1 10-1 w/o crosstalk, m=N+1 10-1 w crosstalk, m=N+2 w crosstalk, m=N+3 10-2 10-2 L Cost function 1 10-3 10-3 10-4 10-4 10-5 10-5 10-6 10-6 10-7 10-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 0 100 200 0 100 20011 [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sciarrino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Laing, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Thompson, Integrated photonic quantum technologies, Nature Pho- tonics 14, 273 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Elshaari, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Pernice, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Srinivasan, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Benson, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zwiller, Hybrid integrated quantum photonic cir- cuits, Nature Photonics 14, 285 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Carolan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Mohseni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Olson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Prabhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bunandar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Niu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Harris, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Wong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hochberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lloyd, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, Varia- tional quantum unsampling on a quantum photonic pro- cessor, Nature Physics 16, 322 (2020), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='10463 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Pelucchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fagas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Aharonovich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Figueroa, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hannes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Matsuda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Pan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Schreck, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sciarrino, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sil- berhorn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Wang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J¨ons, The potential and global outlook of integrated photonics for quantum tech- nologies, Nature Reviews Physics 4, 194 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Chi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Mao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Dai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Yuan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Dai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ding, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Oxenløwe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Thompson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' O’Brien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gong, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Wang, A programmable qudit-based quantum processor, Nature Communications 13, 1166 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Madsen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Laudenbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Askarani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Rortais, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Vincent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bulmer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Miatto, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Neuhaus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Helt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Collins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gerrits, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nam, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Vaidya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Menotti, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Dhand, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Vernon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Quesada, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lavoie, Quantum computational ad- vantage with a programmable photonic processor, Nature 606, 75 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Shen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Harris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Skirlo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Prabhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Baehr- Jones, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hochberg, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Larochelle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Soljaˇci´c, Deep learning with coherent nanophotonic circuits, Nature Photonics 11, 441 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Prabhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Roques-Carmes, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Shen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Har- ris, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Jing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Carolan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hamerly, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Baehr-Jones, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hochberg, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' ˇCeperi´c, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Joannopoulos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Soljaˇci´c, Accelerating recurrent ising machines in photonic integrated circuits, Optica 7, 551 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Thompson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Cai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Paesani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Santagati, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Laing, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Yung, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Shi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Muhammad, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Luo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Dong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kwong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kwek, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Liu, An optical neural chip for implementing complex-valued neural network, Nature Communications 12, 457 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Pai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hughes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Park, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bartlett, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Williamson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Minkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Milanizadeh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Abebe, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Morichetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Melloni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fan, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Solgaard, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Miller, Experimentally realized in situ backpropagation for deep learning in nanophotonic neu- ral networks, arXiv e-prints , arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='08501 (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='08501 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='ET].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ashtiani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Geers, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Aflatouni, An on-chip photonic deep neural network for image classification, Nature 606, 501 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ohno, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Toprasertpong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Takagi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Takenaka, Si microring resonator crossbar array for on-chip inference and training of the optical neural net- work, ACS Photonics 9, 2614 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bandyopadhyay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sludds, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Krastanov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hamerly, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Harris, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bunandar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Streshinsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hochberg, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, Single chip photonic deep neural network with accelerated training, arXiv e-prints , arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='01623 (2022), arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='01623 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='ET].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fontaine, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Doerr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Mestre, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ryf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Winzer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Buhl, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Jiang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lin- gle, Space-division multiplexing and all-optical mimo demultiplexing using a photonic integrated circuit, in OFC/NFOEC (2012) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Annoni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Guglielmi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Carminati, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fer- rari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Sampietro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Miller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Melloni, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Morichetti, Unscrambling light—automatically undo- ing strong mixing between modes, Light: Science & Ap- plications 6, e17110 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Melati, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Alippi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Annoni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Peserico, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Mel- loni, Integrated all-optical mimo demultiplexer for mode- and wavelength-division-multiplexed transmission, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 42, 342 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Choutagunta, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Roberts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Miller, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kahn, Adapting mach–zehnder mesh equalizers in direct- detection mode-division-multiplexed links, Journal of Lightwave Technology 38, 723 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanomura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Soma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ishimura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tane- mura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, All-optical dual-polarization mimo processor based on integrated optical unitary converter, in 2022 European Conference on Optical Communication (ECOC) (2022) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Clements, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Humphreys, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Metcalf, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kolthammer, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Walmsley, Optimal design for uni- versal multiport interferometers, Optica 3, 1460 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Burgwal, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Clements, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gates, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kolthammer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Renema, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Walmsley, Using an imperfect photonic network to implement ran- dom unitaries, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Express 25, 28236 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bandyopadhyay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hamerly, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, Hard- ware error correction for programmable photonics, Op- tica 8, 1247 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Miller, Self-configuring universal linear optical component [Invited], Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1, 1 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Miller, Setting up meshes of interferometers — reversed local light interference method, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Express 25, 29233 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hamerly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bandyopadhyay, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, Asymp- totically fault-tolerant programmable photonics, Nature Communications 13, 6831 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hamerly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bandyopadhyay, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, Accu- rate self-configuration of rectangular multiport interfer- ometers, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Applied 18, 024019 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [27] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hamerly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bandyopadhyay, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Englund, Sta- bility of self-configuring large multiport interferometers, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Applied 18, 024018 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Morizur, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nicholls, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Jian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Armstrong, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Treps, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hage, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hsu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bowen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Janousek, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bachor, Programmable unitary spatial mode ma- nipulation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A 27, 2524 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Labroille, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Denolle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Jian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Genevaux, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Treps, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Morizur, Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Express 22, 15599 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Integrated recon- figurable unitary optical mode converter using mmi cou- plers, IEEE Photonics Technology Letters 29, 971 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [31] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ghosh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Suzuki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanizawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ikeda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kawashima, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Reconfigurable all-optical on-chip mimo three-mode de- multiplexing based on multi-plane light conversion, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 12 Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 43, 1798 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Robust recon- figurable optical mode mux/demux using multiport di- rectional couplers, in 2017 Opto-Electronics and Com- munications Conference (OECC) and Photonics Global Conference (PGC) (2017) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanomura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ghosh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Robust integrated optical unitary converter using multiport directional couplers, Journal of Light- wave Technology 38, 60 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Saygin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kondratyev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Dyakonov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Mironov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Straupe, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kulik, Robust archi- tecture for programmable universal unitaries, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 124, 010501 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanomura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Umezaki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Soma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tane- mura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Scalable and robust photonic inte- grated unitary converter based on multiplane light con- version, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Applied 17, 024071 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanomura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Integrated inp optical unitary converter with compact half-integer multimode interferometers, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Express 29, 43414 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kuzmin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Dyakonov, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kulik, Architecture ag- nostic algorithm for reconfigurable optical interferometer programming, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Express 29, 38429 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Pai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bartlett, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Solgaard, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Miller, Ma- trix optimization on universal unitary photonic devices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Applied 11, 064044 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanomura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Suganuma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Okawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kato, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Monolithic inp optical unitary converter based on multi-plane light con- version, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Express 28, 25392 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [40] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Weisstein, Unimodal, From Mathworld – A Wol- fram Web Resource https://mathworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='com/ Unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [41] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Anderson, The integral of a symmetric unimodal function over a symmetric convex set and some probabil- ity inequalities, Proceedings of the American Mathemat- ical Society 6, 170 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [42] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Anescu, A heuristic fast gradient descent method for unimodal optimization, Journal of Advances in Mathe- matics and Computer Science 26, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [43] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanomura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Ten-port unitary optical processor on a silicon photonic chip, ACS Photonics 8, 2074 (2021), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='1021/acsphotonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='1c00419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanomura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Taguchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Tanemura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nakano, Entropy of mode mixers for optical uni- tary converter based on multi-plane light conversion, in Conference on Lasers and Electro-Optics/Pacific Rim (CLEOPR) 2022 (Optica Publishing Group, 2022) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' CWP13A–02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Reck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zeilinger, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bernstein, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bertani, Experimental realization of any discrete unitary opera- tor, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 73, 58 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [46] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Virtanen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gommers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Oliphant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Haber- land, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Reddy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Cournapeau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Burovski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Pe- terson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Weckesser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bright, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' van der Walt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Brett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Wilson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Millman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Mayorov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Nelson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Kern, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Larson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Carey, ˙I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Po- lat, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Feng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Moore, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' VanderPlas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Laxalde, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Perktold, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Cimrman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Henriksen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Quintero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Archibald, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Ribeiro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Pe- dregosa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' van Mulbregt, and SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0 Contributors, SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='0: Fundamental algorithms for scientific com- puting in Python, Nature Methods 17, 261 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [47] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fletcher, Practical Methods of Optimization, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' (John Wiley & Sons, New York, NY, USA, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bradbury, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Frostig, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hawkins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Johnson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Leary, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Maclaurin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Necula, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Paszke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Van- derPlas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Wanderman-Milne, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Zhang, JAX: com- posable transformations of Python+NumPy programs (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gallagher and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Downs, Visualization of learning in multilayer perceptron networks using principal compo- nent analysis, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 33, 28 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Taylor, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Studer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Goldstein, Visualizing the Loss Landscape of Neural Nets, arXiv e-prints , arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='09913 (2017), arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='09913 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content='LG].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Taylor, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Studer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Goldstein, Vi- sualizing the loss landscape of neural nets, in Advances in Neural Information Processing Systems, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' 31, edited by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Larochelle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Grauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Cesa-Bianchi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Garnett (Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [52] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hughes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Minkov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Shi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fan, Training of photonic neural networks through in situ backpropa- gation and gradient measurement, Optica 5, 864 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Jacques, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Samani, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' El-Fiky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Patel, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Xing, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Plant, Optimization of thermo-optic phase- shifter design and mitigation of thermal crosstalk on the soi platform, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Express 27, 10456 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' [54] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Gurses, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Fatemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Khachaturian, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} +page_content=' Hajimiri, Large-scale crosstalk-corrected thermo-optic phase shifter arrays in silicon photonics, IEEE Journal of Selected Topics in Quantum Electronics 28, 6101009 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQf1zij/content/2301.13658v1.pdf'} diff --git a/MdFRT4oBgHgl3EQf2zi9/content/2301.13662v1.pdf b/MdFRT4oBgHgl3EQf2zi9/content/2301.13662v1.pdf new file mode 100644 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sha256:fe3411e7f0a4c96f836a63467cff67910904942293336b37e744eddd3d01195c +size 7209005 diff --git a/PNFPT4oBgHgl3EQfnjVw/content/tmp_files/2301.13130v1.pdf.txt b/PNFPT4oBgHgl3EQfnjVw/content/tmp_files/2301.13130v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..04223be771a8d8443fc5f17779f034f754b530bf --- /dev/null +++ b/PNFPT4oBgHgl3EQfnjVw/content/tmp_files/2301.13130v1.pdf.txt @@ -0,0 +1,1281 @@ +arXiv:2301.13130v1 [math.DG] 30 Jan 2023 +STABILITY OF CONVEX DISKS +HUNTER STUFFLEBEAM +Abstract. We prove that topological disks with positive curvature and strictly +convex boundary of large length are close to round spherical caps of constant +boundary curvature in the Gromov-Hausdorff and Sormani-Wenger Intrinsic +Flat senses. This proves stability for a theorem of F. Hang and X. Wang in +[HW09]. As an intermediate step we obtain a result concerning gauge fixing +and compactness for solutions of a Liouville type PDE. +1. Introduction +Inequalities in geometric analysis, such as the isoperimetric and systolic, Faber- +Krahn and Penrose, relate given geometric objects to understood model cases, +taking as input data bounds on curvatures, volumes, eigenvalues, energies, etc. Via +such relationships, much work has been done to understand the structure of spaces +with natural geometric conditions phrased in terms of such quantities. +Given an inequality for which one has some understanding of extremizers (the +geometric objects which realize equality), one might ask if an object nearly re- +alizing equality must somehow share characteristics with the extremizer(s). The +first problem of understanding the extremizers might be called a rigidity problem. +The second problem of understanding near extremizers might be called a stability +problem. A classical example is the isoperimetric inequality in Euclidean space. Its +extremizers are balls, and the stability problem has received much attention–for +example consider the work of Fusco-Maggi-Pratelli in [FMP08]. +The aim of this paper is to establish the following stability theorem for convex +disks. +Throughout, given a 2-manifold (M, g) we use the notation Kg for the +Gaussian curvature, κg for the geodesic curvature of the boundary, and Lg for the +length functional of g. +The Gromov-Hausdorff metric on the space of compact +metric spaces is denoted by dGH, and the Sormani-Wenger Intrinsic Flat metric on +the space of integral current spaces is denoted by dIF . +Theorem 1.1 (Stability of the Convex Spherical Cap). Fix c > 0 and let δ > 0. +Then there exists an ε = ε(c, δ) > 0 such that if (M, g) is a two dimensional +manifold with Kg ⩾ 1, κg ⩾ c > 0, and Lg(∂M) ⩾ 2π(1 − ε)/ +√ +1 + c2, then +dGH((M, g), Bcot−1(c)) < δ, +where Bcot−1(c) is a closed geodesic disk of radius cot−1(c) in the round sphere S2. +The same conclusion holds, with potentially different ε, with dIF in place of dGH. +This result was motivated by the desire for a stability theorem corresponding to +an old result of V. Toponogov: +Key words and phrases. Gromov-Hausdorff, Sormani-Wenger Instrinsic Flat, Liouville Equa- +tion, Convexity, Stability. +1 + +2 +HUNTER STUFFLEBEAM +Theorem 1.2 (V. Toponogov). Let (M, g) be a closed surface with Kg ⩾ 1. Then +any simple closed geodesic has length bounded from above by 2π, and this length is +attained if and only if (M, g) is isometric to the round sphere. +There are at least two proofs of this result–the original due to Toponogov using +the celebrated triangle comparison theorem, and a modern PDE proof due to F. +Hang and X. Wang (see [Kli11] and [HW09], respectively). In both arguments, +one cuts open the closed manifold along the largest closed geodesic to obtain two +disks with geodesic boundary. The length inequality, and a corresponding rigidity +theorem, is then proven for these disks. +The PDE proof of the result applies +immediately to the case of convex disks in general: +Theorem 1.3 (F. Hang and X. Wang [HW09]). Let (M, g) be a smooth, compact +surface with boundary. +Suppose Kg ⩾ 1 and κg ⩾ c ⩾ 0. +Then Lg(∂M) ⩽ +2π/ +√ +1 + c2. Moreover, equality holds if and only if (M, g) is isometric to Bcot−1(c). +To attempt a proof of stability for Toponogov’s theorem, it is thus natural to +begin by trying to prove stability in the disk rigidity theorem of Hang-Wang. How- +ever, while stability in the convex case does hold as is shown by our following work, +this does not extend to the case of disks with weakly convex boundary. And indeed, +stability does not generally hold in Toponogov’s theorem: +Remark 1.1 (Failure of Stability in Toponogov’s Theorem). Fix any small ε > 0 +and δ > 0. There exists a rotationally symmetric metric g = dr2 + φ2(r)dθ2 on S2 +which has Kg ⩾ 1, a simple closed geodesic of length 2π − ε, and +d((S2, g), (S2, grd)) ⩾ δ, +where d = dGH, dIF . Here grd denotes the round metric on S2. +This follows an idea of [CC96] [CM97], and was explored in some detail in [WZ21]. +One takes a good football metric on S2/Zk and carefully smoothes the tips by gluing +in spherical caps and rescaling. The interested reader is encouraged to look to the +latter source for the specifics of the construction, and it is not hard to deduce from +it the claimed properties in the remark above pertaining to the case d = dGH. +In addressing the dIF case, one can apply work of [LP18] to see that stability for +Toponogov’s Theorem still fails, since in the setting of closed Alexandrov spaces of +non-negative curvature and uniform upper diameter bounds, convergence in dIF to +a nonzero space would imply dGH convergence to the same limit. +Let us now remark on the main ideas of the proof of Theorem 1.1. +A more +detailed description will be given shortly, after the requisite notation and setup +has been properly introduced. Proceeding by way of contradiction, we obtain a se- +quence of convex topological disks with c-convex boundary whose boundary lengths +converge to the extremal value while remaining bounded away from the model disk +in the metric d. +By the Gauss-Bonnet and Uniformization Theorems, studying +this sequence of metrics on topological disks amounts to studying a corresponding +sequence of conformal factors for metrics on the unit disk of R2. New conformal +factors for constant curvature ≡ 1 disks with isometric boundaries to the original +sequence are produced, to be compared to the original sequence. A major hurdle +in handling the compactness of these constant curvature conformal factors is the +action of the conformal group of the disk. By fixing the gauge and applying some +results from conformal mapping and elliptic PDE theory, we obtain converging + +STABILITY OF CONVEX DISKS +3 +subsequences of both the comparison conformal factors and the differences between +them and those of the original sequence. This gives subconvergence for the original +sequence of conformal factors. We then upgrade this analytic convergence of confor- +mal factors to geometric convergence of the manifolds to the model disk, obtaining +the desired contradiction. +1.1. A Comment on Notation. Throughout this paper, Riemannian metrics will +often be written as being conformally equivalent to geuc, the standard Euclidean +metric on R2. We will often reference geometrical quantities defined with respect +to such a metric g = gu = e2ugeuc by the conformal factor u. For example, we may +write du for the distance function dgu deriving from gu, and κu for the geodesic +curvature of the boundary for the metric gu. ∆ will denote the Euclidean Laplace +operator defined by ∆ ..= div ◦ ∇. +We will also have to refer to various geometrical quantities, such as metric balls, +which depend upon a choice of metric and/or distance function. In general, we +will denote by Br(x, d) the open metric ball of radius r about x with respect to the +distance d. In case d = dg derives from a Riemannian metric g, we will interchange- +ably use the notation Br(x, g) and Br(x, dg) as is most convenient for exposition. In +the special case x = 0 and g = geuc, we will simply write Br for Br(0, geuc). In case +a measure is omitted from an integral, it is understood that the implied measure +is the standard volume measure on the underlying space. In all cases, it should be +quite clear what is implied. +Finally, we will follow tradition in letting Ψ = Ψ(x) = Ψ(x|a1, a2, . . .) denote a +non-negative function, which may change from line to line, depending on a variable +x and any number of parameters ai with the property that if the ai are all held +fixed, Ψ ց 0 as x → 0. +2. Preliminaries +2.1. Our Setup and a Review of Hang-Wang’s Argument. For completeness +of exposition, and to set some notation which we will use throughout the rest of +the paper, we briefly recall the proof of Toponogov’s Theorem in [HW09], which +we restate for convenience: +Theorem 2.1 (F. Hang and X. Wang). Let (M 2, g) be a compact surface with +boundary γ, Kg ⩾ 1, and κg(γ) ⩾ c ⩾ 0. Then Lg(γ) ⩽ 2π/ +√ +1 + c2, with equality +iff (M 2, g) is isometric to Bcot−1(c). +Proof. By Gauss-Bonnet and Uniformization, there is an isometry of (M 2, g) with +(B1, e2ugeuc) for some smooth u: B1 → R, and the curvature conditions translate +to +� +−∆u = Kge2u ⩾ e2u +on B1 +∂nu + 1 = κg(γ)eu ⩾ ceu +on S1. +By the sub-super solution method, we can produce a constant curvature comparison +factor v: B1 → R to u, which satisfies the following: + + + + + + + + + +−∆v = e2v +on B1 +∂nv + 1 ⩾ cev +on S1 +u ⩾ v +on B1 +v = u +on S1. + +4 +HUNTER STUFFLEBEAM +The constant curvature comparison manifold (B1, e2vgeuc) can therefore be realized +isometrically as a smooth domain Ω in the standard S2 with boundary that is +uniformly c-convex. Some elementary geometry tells us that the smallest geodesic +disk D ⊂ S2 containing Ω is of radius at most cot−1(c), and this disk has boundary +length Lgrd(∂D) ⩽ 2π/ +√ +1 + c2. Since u = v on S1, +Lg(γ) = Le2vgeuc(S1) = Lgrd(∂Ω) ⩽ 2π/ +� +1 + c2. +Now suppose equality is obtained. The construction above forces ∂Ω = ∂D, +which forces the geodesic curvature of ∂Ω to be identically c. Thus, our comparison +factor v must satisfy + + + + + +−∆v = e2v +on B1 +∂nv + 1 = cev +on S1 +v = u +on S1. +Since on S1 we have cev = ∂nv + 1 ⩾ ∂nu + 1 ⩾ ceu = cev we conclude that γ also +has constant geodesic curvature c. Setting w = u − v, we have that + + + + + +−∆w ⩾ 0 +on B1 +∂nw = 0 +on S1 +w = 0 +on B1. +It then follows easily that u ≡ v on B1, which proves that (M 2, g) is isometric to a +geodesic disk with constant boundary curvature c in the standard unit S2. +□ +In this paper, we are interested in the consequences of the assumption that +Lg(γ) is nearly equal to 2π/ +√ +1 + c2. Let then ε > 0 be small, and consider a +compact surface (M 2, g) with boundary γ, Kg ⩾ 1, κg(γ) ⩾ c > 0, and Lg(γ) ⩾ +2π(1 − ε)/ +√ +1 + c2. We set out to prove that (M 2, g) is Gromov-Hausdorff and +Sormani-Wenger Intrinsic Flat close to the spherical cap characterizing the equality +case. Construct exactly as above the manifolds (B1, e2ugeuc) and (B1, e2vgeuc), with +the latter corresponding isometrically to some domain Ω in the standard S2. The +jumping off point is the following estimate for the inradius of a strictly convex +domain in a sphere in terms of the length of the boundary, and is a direct adaptation +of the more general Theorem 1.2 in [Dra18]: +Theorem 2.2 (Inradius Estimate for Convex Domains). Let Ω be a smooth convex +domain in the standard S2, with boundary of length L and κg(∂Ω) ⩾ c > 0. Let rin +denote the inradius of Ω. Then +rin ⩾ cot−1(c) − cot−1 +� +c sec +� +L +√ +1 + c2 +4 +�� +. +Remark 2.1. Notice the importance of the positivity of c in this estimate. Indeed, +the failure of an inradius lower bound for domains with boundary having segments +of zero geodesic curvature allows for collapsing, and is exactly what underlies the +failure of stability in the weakly convex case without further assumptions. +In particular, our constant curvature comparison manifold (B1, e2vgeuc), when +realized isometrically as a domain Ω in S2, has a large inball of radius +rin ⩾ cot−1(c) − cot−1 � +c sec +�π +2 (1 − ε) +�� +. + +STABILITY OF CONVEX DISKS +5 +Additionally, as proven above, Ω has the geodesic disk D as an outball of radius +rout ⩽ cot−1(c). We thus have observed the following, which will be a fundamental +lemma for us: +Lemma 2.1. Let 0 < ε < 1 and fix a compact surface (M 2, g) with boundary γ, +Kg ⩾ 1, κg(γ) ⩾ c > 0, and Lg(γ) ⩾ 2π(1 − ε)/ +√ +1 + c2. Let (B1, e2vgeuc) ↔ Ω ⊂ +S2 be the constant curvature comparison disk in S2, which has inradius rin and +outradius rout. Then +• cot−1(c) − Ψ(ε|c) ⩽ rin ⩽ rout ⩽ cot−1(c) +• c ⩽ cot(rout) ⩽ cot(rin) ⩽ c + Ψ(ε|c). +2.2. Outline of the Proof. Here we explain the broad-strokes idea of the argu- +ment for proving our main theorem, which we restate for convenience: +Theorem 2.3 (Stability of the Convex Spherical Cap). Fix c > 0 and let δ > 0. +Then there exists an ε = ε(c, δ) > 0 such that if (M, g) is a two dimensional +manifold with Kg ⩾ 1, κg ⩾ c > 0, and Lg(∂M) ⩾ 2π(1 − ε)/ +√ +1 + c2, then +dGH((M, g), Bcot−1(c)) < δ, +where Bcot−1(c) is a closed geodesic disk of radius cot−1(c) in the round sphere S2. +The same conclusion holds with dIF in place of dGH. +We will prove this by way of contradiction, supposing that there exists a δ0 > 0 +such that, for every sequence εk ց 0, we can find an (Mk, gk) as in the statement +with Lk(∂Mk) ⩾ 2π(1 − εk)/ +√ +1 + c2 but +d((Mk, gk), Bcot−1(c)) ⩾ δ0 > 0, +where d = dGH or dIF . +First, let’s fix the notation for the model spaces that we will be comparing our +given manifolds to. Given c ⩾ 0, define the function ρc : B1 → R by the formula +ρc(x) ..= log +� +2Rc +1 + |Rcx|2 +� +where +Rc ..= +� +1 + c2 − c. +Then (B1, e2ρcgeuc) is isometric, via the dilation Rc·Id: B1 → BRc, to (BRc, e2ρ0geuc), +which under stereographic projection Φ: S2 \ N → R2 from the north pole N = e3 +is isometric to the geodesic disk of radius cot−1(c) in S2 centered at the south pole +S = −e3. In other words, our model extremizer Bcot−1(c) can be isometrically real- +ized as (B1, e2ρcgeuc). Here, and throughout when convenient, we may identify Br +with Br × {0} ⊂ R3, in particular when considering the stereographic projection +from the standard embedding of round S2. +Our goal is to estimate the sequence of distances d((B1, e2ukgeuc), (B1, e2ρcgeuc)), +and extract a subsequence which converges to 0 to derive a contradiction. +At +the outset, we remark that by the compactness theorems of M. Gromov and S. +Wenger, any sequence of manifolds satisfying our assumptions will subconverge +in the Gromov-Hausdorff (GH) and Sormani-Wenger Intrinsic Flat (SWIF) senses, +respectively, to compact metric spaces. In general, given our assumptions the SWIF +limit will embed isometrically as a subset of the GH limit, but outright we do not +know much more about what they are, or whether they agree. + +6 +HUNTER STUFFLEBEAM +To identify both limits as being the round spherical cap, we will first control the +differences of conformal factors wk = uk − vk and vk − ρc. Writing for λ ⩾ 1 +eλuk − eλρc = +� +eλvk − eλρc� +eλwk + +� +eλwk − 1 +� +eλρc, +we will obtain W 1,p +loc convergence by showing that eλwk → 1 in W 1,p and that +eλvk → eλρc in Cm +loc for any m ⩾ 0. +The control on wk follows from standard +elliptic PDE techniques and a result of H. Brezis-F. Merle in [BM91]. The control +on vk − ρc is more subtle, and involves some results from the theory of conformal +mappings of convex domains. +With this control established, we can show that our sequence has a local Gromov- +Lipschitz (GL) sublimit on each Br ⊂ B1. Using variants of the Sobolev Trace +Theorem and the Bishop-Gromov Theorem, we can identify the local GL limits +as spherical caps. We then glue these local limits together to identify the model +disk (B1, e2ρcgeuc) as the SWIF limit. To identify the GH limit, we apply a recent +theorem due to R. Perales [Per18] showing that the SWIF and GH limits must +agree. In either case, we will obtain a convergent subsequence to the model disk, +forcing a contradiction and establishing the main theorem. +2.3. A Remark on the Brezis-Merle and other Blow-Up Theories. Evi- +dently, the study of sequences of Riemannian surfaces is linked, via the uniformiza- +tion process described above, to the study of sequences of solutions to the Liouville +equation on a two dimensional domain: +−∆u = K(x)e2u +on +Ω ⊂ R2. +The geometric interpretation is that the metric e2ugeuc on Ω has Gaussian curvature +K. In their seminal 1991 paper [BM91], H. Brezis and F. Merle studied the blow +up behavior of solutions to this equation. Their analysis, which includes a study of +uniform a-priori estimates for sequences of such solutions u, require Lp bounds on +the potentials K for p > 1. We only have L1 bounds on K, however, rendering their +conclusions unavailable to us. Nonetheless, we will find great use in the following +fundamental estimate from [BM91]: +Theorem 2.4 (H. Brezis and F. Merle). Assume Ω ⊂ R2 is a bounded domain and +let u be a solution of +� +−∆u = f +on Ω +u = 0 +on ∂Ω +with f ∈ L1(Ω). Then for every δ ∈ (0, 4π), we have the estimate +� +Ω +e +4π−δ +∥f∥L1(Ω) |u(x)|dx ⩽ 4π2 +δ (diamΩ)2. +Several authors have recently investigated possible extensions of the blow-up +analysis, in particular with attention to geometric applications. For example [LST21], +[LT17], and [CL22] have studied, among other things, the compactness of sequences +of Riemannian surfaces with curvature bounds via an analysis of this equation. We +remark that their results seem to be largely unavailable to us here, given the more +general nature of our curvature bounds and the desire for identifying exact limits +to converging sequences. + +STABILITY OF CONVEX DISKS +7 +3. Stability of Convex Disks +3.1. Controlling the Difference of Conformal Factors. The goal of this sec- +tion, largely a rapid-fire sequence of lemmas, is to prove the following proposition: +Proposition 3.1. Let c > 0, λ ⩾ 1, p ∈ [1, 2), and ε > 0 small. Given (B1, e2ugeuc) +and (B1, e2vgeuc) as in the proof of Hang-Wang’s Theorem1.3 with κu ⩾ c and +Lu(S1) ⩾ 2π(1 − ε)/ +√ +1 + c2, set w = u − v. Then +∥eλw − 1∥W 1,p(B1) ⩽ Ψ(ε|c, p, λ). +We begin by collecting together some basic facts about w: +Lemma 3.1. The difference of conformal factors w = u − v satisfies the following: +(i) w ⩾ 0 on B1 +(ii) −∆w ⩾ 0 on B1 +(iii) w = 0 on S1 +(iv) ∂nw ⩽ 0 on S1 +(v) | +� +S1 ∂nw| ⩽ Ψ(ε|c). +Proof. Items (i), (ii), (iii), and (iv) are rather immediate from the construction of +v via the sup-super solution method, so we focus on item (v). This relies upon +the inradius estimate of Lemma 2.1 in a crucial way, and is in a sense the most +‘geometrically informed’ result of the Lemma. +We seek to estimate +0 ⩾ +� +S1 ∂nw = +� +S1(∂nu + 1) − (∂nv + 1) = +� +S1(κu − κv)eu. +By the inradius estimate 2.1, we may consider the new comparison disk (B1, e2fgeuc) +where +� +−∆f = e2f +on B1 +∂nf + 1 = cot(rin)ef +on S1, +which is isometric to a geodesic disk in S2 of constant boundary curvature cot(rin), +serving as an inball for (B1, e2vgeuc). +By the Gauss-Bonnet Theorem, we have +(A) 2π = +� +B1 +e2f + +� +S1 cot(rin)ef = Area(B1, e2fgeuc) + +� +S1 cot(rin)ef +(B) 2π = +� +B1 +e2v + +� +S1 κvev = Area(B1, e2vgeuc) + +� +S1 κvev +and by direct comparison +(C) Area(B1, e2fgeuc) ⩽ Area(B1, e2vgeuc). +(A), (B), and (C) together imply that +� +S1 κvev ⩽ +� +S1 cot(rin)ef. +Finally, observe that +� +S1 ef = Lf(S1) ⩽ +� +S1 ev = Lv(S1) ⩽ +2π +√ +1 + c2 . + +8 +HUNTER STUFFLEBEAM +Using parts (i)-(iii) of the Lemma, the inradius estimate 2.1, together with the prior +two observations, we get +� +S1 cev = +� +S1 ceu ⩽ +� +S1 κueu ⩽ +� +S1 κvev +⩽ +� +S1 cot(rin)ef ⩽ +� +S1(c + Ψ)ef ⩽ +� +S1 cev + Ψ. +Seeing as though +� +S1 cev = cLv(S1) ∈ +�2πc(1 − ε) +√ +1 + c2 , +2πc +√ +1 + c2 +� +it follows that, as desired, +0 ⩾ +� +S1 ∂nw = +� +S1(κu − κv)eu ⩾ −Ψ. +□ +Lemma 3.2. ∥∆w∥L1(B1) ⩽ Ψ(ε|c). +Proof. By the divergence theorem and Lemma 3.1, we have +0 ⩽ − +� +B1 +∆w = − +� +S1 ∂nw ⩽ Ψ. +□ +Lemma 3.3. For all λ ⩾ 1, and for all ε > 0 small enough (depending on λ), we +have +π ⩽ +� +B1 +eλw ⩽ 4π + Ψ(ε|c, λ). +Proof. In Theorem 2.4, take Ω = B1, Ψ as above, and δ = 4π − λΨ. For sufficiently +small ε > 0 (depending on λ), we have δ = δ(λ) ∈ (0, 4π). Then (since also w ⩾ 0) +π ⩽ +� +B1 +eλw ⩽ +� +B1 +e +Ψ +∥∆w∥L1(Ω) |λw(x)| ⩽ +4π2 +4π − λΨ · 4 = 4π + Ψ. +□ +Lemma 3.4. For all p ∈ [1, 2), ∥∇w∥Lp(B1) ⩽ Ψ(ε|c, p). +Proof. This is immediate from the following, concerning solutions of the Poisson +equation on the unit ball: +Proposition 3.2. 1 Fix p ∈ [1, 2). Then there exists a C = C(p) > 0 such that if +w is a smooth solution of +� +−∆w = f +on B1 +w = 0 +on S1, +then +∥∇w∥Lp(B1) ⩽ C(p)∥f∥L1(B1). +□ +1To prove this, one can compute directly from the Green’s Representation of the solution w +and apply the Minkowski integral inequality. + +STABILITY OF CONVEX DISKS +9 +Lemma 3.5. For all p ∈ [1, 2), ∥w∥W 1,p(B1) ⩽ Ψ(ε|c, p). Moreover, for all p ∈ +[1, ∞), ∥w∥Lp(B1) ⩽ Ψ(ε|c, p). +Proof. Since w = 0 along S1, the Poincar´e Inequality implies the first claim in the +Lemma, after which the second part follows from the Sobolev Embedding Theorem. +□ +Lemma 3.6. For every λ ⩾ 1 and p ∈ [1, 2), ∥eλw∥p +W 1,p(B1) ⩽ 4π + Ψ(ε|c, p, λ). In +particular, ∥∇eλw∥Lp(B1) ⩽ Ψ(ε|c, p, λ). +Proof. By Lemma 3.3, +∥eλw∥p +Lp(B1) ⩽ 4π + Ψ +for any p ∈ [1, ∞), provided ε is small enough depending upon the choice of p. +Now, observe that +∇(eλw) = λeλw∇w +so that if p ∈ [1, 2), and we choose q ∈ (1, 2/p), +∥∇(eλw)∥p +Lp(B1) = λp +� +B1 +eλpw|∇w|p ⩽ λp +�� +B1 +eλpq′w +� 1 +q′ �� +B1 +|∇w|pq +� 1 +q +⩽ λp (4π + Ψ) +1 +q′ +�� +B1 +|∇w|pq +� 1 +q +⩽ Ψ. +Indeed, q ∈ (1, 2/p) implies that λpq′ ∈ (1, ∞) and pq ∈ (1, 2), so that Lemma 3.3 +applies to the first integral factor and Lemma 3.5 applies in the second. As such, +∥eλw∥p +W 1,p(B1) = ∥eλw∥p +Lp(B1) + ∥∇(eλw)∥p +Lp(B1) ⩽ 4π + Ψ. +□ +At last, we can establish the main proposition of the section: +Proof of Proposition 3.1. Suppose not. +Then there exist c0 > 0, λ0 ⩾ 1, p0 ∈ +[1, 2), and η0 > 0 such that for every k ⩾ 1, there is a function wk satisfying the +conclusions of the all the previous lemmas with some Ψ = Ψ(k−1|c0, p0, λ0) ց 0, +and +∥eλ0wk − 1∥W 1,p0(B1) ⩾ η0 > 0. +By the Poincar´e Inequality, if +ck = 1 +π +� +B1 +eλ0wk +then by Lemma 3.6, +∥eλ0wk − ck∥Lp0(B1) ⩽ ∥∇eλ0wk∥Lp0(B1) ⩽ Ψ(k−1). +By Lemma 3.3, ck ∈ [1, 4 + Ψ(k−1)], so up to extracting a subsequence, ck → c ∈ +[1, 4]. In fact, it must be that c = 1, which can be seen from the Sobolev Trace +Theorem and the fact that wk = 0 on S1: +∥1−c∥Lp0(S1) ⩽ ∥eλ0wk −c∥W 1,p0(B1) ⩽ ∥eλ0wk −ck∥W 1,p0(B1)+∥ck−c∥W 1,p0(B1) → 0. +But then we find that +∥eλ0wk − 1∥p0 +W 1,p0(B1) = ∥eλ0wk − 1∥p0 +Lp0(B1) + ∥∇eλ0wk∥p0 +Lp0(B1) ⩽ Ψ(k−1), + +10 +HUNTER STUFFLEBEAM +contradicting the existence of η0. The first claim thus follows, and by the Sobolev +Embedding Theorem the second claim follows as well. +□ +3.2. Estimates for the Constant Curvature Comparison Metrics. In this +section we study a sequence of c-convex constant curvature 1 comparison disks +(B1, e2vkgeuc) whose boundary lengths go to the extremal length 2π/ +√ +1 + c2. This +is complicated by the fact that the process of representing such disks via the Uni- +formization Theorem has a large gauge invariance and consequent loss of compact- +ness. +To illustrate this, recall that we isometrically represented (B1, e2vgeuc) as a +smooth domain Ω in the round sphere, and that this choice is unique only up +to action of the symmetry group O(3) ⟳ S2. Under stereographic projection, such +a domain is isometric to a disk of the form (D, e2ρ0geuc) for some smooth domain +D ⊂ R2, where e2ρ0geuc is the round metric in stereographic coordinates. Since we +seek to relate v to ρ0 on B1 via pullback of (D, e2ρ0geuc) by an isometry F : B1 → D, +the ambiguity in the choice of Ω, and thus D, matters a great deal insofar as esti- +mates are concerned. +In particular, the gauge invariance that we seek to control manifests in the +following form. Given a function f : B1 → R and a diffeomorphism φ ∈ Diff(B1), +we employ the notation (f)φ ..= f ◦ φ + log |φ′|. Observe that if φ ∈ Conf(B1) is +a Mobius transformation of B1, then (B1, e2(f)φgeuc) is isometric to (B1, e2fgeuc) +under the action of pullback by φ. Thus, the conformal factors f and (f)φ represent +the same geometric object, and therefore the corresponding curvature data in PDE +form is invariant under this action of Conf(B1). For example, if v is any solution +of −∆v = e2v on B1 and φ ∈ Conf(B1), we have (using an apostrophe to denote +complex differentiation) +−∆(v)φ = −∆(v ◦ φ) = −(∆u) ◦ φ|φ′|2 = e2v◦φ|φ′|2 = e2(v)φ. +Here, of course, we use the fact that the Mobius transformations are holomorphic +on B1. +To combat this gauge invariance, we will first choose the Ωk in a way which +takes uniform advantage of the quantitative inradius estimate from Lemma 2.1. +The resulting isometries relating the (Dk, e2ρ0geuc) to the (B1, e2vkgeuc) then form +a sequence of conformal diffeomorphisms whose images converge in a nice way. +After normalizing these maps we obtain a limit mapping, and these normalizations +provide us with our final choice of gauge for looking at the sequence of constant +curvature comparison disks. Before proceeding, we quickly recall for the reader’s +convenience the necessary complex-analytic framework. +3.2.1. Review of Conformal Mappings. In this section we introduce the concepts +that we will need from the theory of conformal mappings in the plane. Primarily, +we are concerned with the behavior of a sequence of conformal (i. e. biholomorphic) +mappings Fk : B1 → Dk, where the domains Dk in R2 are smooth and uniformly +convex. +2 In this section it is natural to identify R2 with C, and to view a map +between regions in R2 as a complex valued mapping in the standard way. +2The subject of conformal, and more broadly harmonic, mappings in the plane is wonderfully +rich, and the interested reader should especially look to the books by Pommerenke [Pom11] and +Duren [Dur04]. + +STABILITY OF CONVEX DISKS +11 +In our applications, our domains Dk will converge to a limiting domain D. The +relevant form of convergence is the following notion, due to C. Carath´eodory: 3 +Definition 3.1 (Kernel Convergence). Let Dk ⊂ C be simply connected domains +with 0 ∈ Dk. The kernel of the sequence {Dk} is defined to be {0} if 0 /∈ int(� Dk), +and otherwise is defined to be the largest domain D ⊂ C containing 0 with the +property that each point of D possesses a neighborhood in D which lies in cofinitely +many Dk. We then say that Dk → D (with respect to 0) in the sense of kernel +convergence if every subsequence of the collection {Dk} has the same kernel D. +There are a few examples where kernel convergence is easily verified. For exam- +ple, a sequence of increasing open sets containing 0 has their union as their kernel. +In our case, each of our sets will be of the form Brk ⊂ Dk ⊂ BR, where rk ր R, so +again the kernel is simply the union of the Dk. Another way to phrase the definition +of the kernel, which makes the last example clear, is as follows: for each n ⩾ 1 let +Cn denote the connected component of int (Dn ∩ Dn+1 ∩ · · · ) containing 0. If they +exist for all n, the union of the Cn is defined to be the kernel, and otherwise it is +defined to be {0}. +The raison d’ˆetre for this notion of convergence is the following famous theorem: +Theorem 3.1 (Carath´eodory’s Convergence Theorem4). Let {Dk ⊂ C} be a se- +quence of simply connected domains containing 0, and {Fk : B1 → Dk} a sequence +of bijective conformal mappings with Fk(0) = 0 and F ′ +k(0) > 0. +Then the Fk +converge to a limit function F uniformly on compact subsets of B1 if and only if +Dk → D ̸= C in the kernel sense. +In the event of convergence with D = {0}, F ≡ 0. If we have convergence with +D ̸= {0}, then D is simply connected, F is a bijective conformal mapping of B1 and +D, and the F −1 +k +converge locally uniformly on D to F −1. +Lastly, it is worth recalling the Kellogg-Warschawski Theorem, which ensures +that the limit maps we get will be smooth up to and on the boundary of our +regions of interest: +Theorem 3.2 (Kellogg-Warschawski5). Suppose F : B1 → D is a conformal bijec- +tion, where the boundary of the domain D is a Jordan curve of regularity Cm,α for +some m ⩾ 1 and α ∈ (0, 1). Then F (m) has an α-H¨older continuous extension to +B1. In particular, if D is smooth then all derivatives of F extend continuously to +B1. +3.2.2. Fixing the Gauge. The goal of this section is to prove the following proposi- +tion. Recall that we use the notation (f)φ ..= f ◦φ+log |φ′|, and that if φ ∈ Mob(B1) +is a Mobius transformation of B1, then (B1, e2(f)φgeuc) is isometric to (B1, e2fgeuc) +under the action of pullback by φ. +3This definition often takes slightly different forms. The current one is convenient for us and +is found in [Dur04] +4Many different conceptions of this result exist, and the one we are using here, as stated +in [Dur04], is convenient for our purposes. For other conceptions, the article [IK18] is a great +reference. +5As stated in [Pom11]. + +12 +HUNTER STUFFLEBEAM +Proposition 3.3. Given εk ց 0 and smooth functions vk : B1 → R with + + + + + +−∆vk = e2vk +on B1 +∂nvk + 1 ⩾ ceuk +on S1 +� +S1 evk ⩾ 2π − εk, +we can find a sequence of Mobius transformations φk ∈ Conf(B1) such that (vk)φk → +ρc in Cm +loc(B1) for any m ⩾ 0. +Geometrically, of course, we are showing that the constant curvature comparison +disks (B1, e2vkgeuc) have isometric representations (B1, e2(vk)φk geuc) which converge +locally in the Cm +loc Cheeger-Gromov sense to the model space (B1, e2ρcgeuc) for every +m ⩾ 0. +To begin proving this, we make an initial choice of gauge by conveniently choosing +the domains Ωk ⊂ S2 which isometrically realize our constant curvature comparison +disks (B1, e2vkgeuc). We consider round S2 to be isometrically embedded in R3 in +the standard way, and we let Φ: S2 \ N → R2 denote stereographic projection +from the north pole N. Now, choose Ωk to lie in the southern hemisphere of S2, +containing the south pole S. Recalling that Ωk has an outball of radius cot−1(c) +by Lemma 2.1, we can choose Ωk so that its outball is centered at S. By Lemma +2.1, Ωk also has an inball of radius cot−1(c + Ψ(εk)). It thus follows that with Ωk +chosen in this way, the geodesic disk of radius 2cot−1(c+Ψ(εk))−cot−1(c) centered +at S is contained in Ωk. As k → ∞ and εk ց 0, this disk expands up to the fixed +outball of radius cot−1(c) for all the Ωk. +Under Φ, Ωk is isometric to a disk of the form (Dk, e2ρ0geuc), where Dk is a +smooth convex domain in R2 containing 0 and contained in B1. More precisely, the +Dk all lie within BRc, and each contains the ball Bℓk, where +ℓk ..= +sin[2cot−1(c + Ψ(εk)) − cot−1(c)] +1 + cos[2cot−1(c + Ψ(εk)) − cot−1(c)] ր Rc. +Consequently, we can easily see that Dk → BRc in the kernel sense: we have +BℓN ⊂ CN ..= int(DN ∩ DN+1 ∩ · · · ) ⊂ BRc, and so the kernel, which we recall as +being the union of CN, is exactly BRc. Denote by Fk : B1 → Dk the smooth map +which provides an isometry between our two disks (Dk, e2ρ0geuc) and (B1, e2vkgeuc). +Claim 3.1. F is a conformal diffeomorphism of B1 and Dk. +Proof. Indeed, Fk an isometry means that e2vkgeuc = F ∗ +k (e2ρ0geuc) = e2ρ0◦FkF ∗ +k geuc. +Expanding out, we find that +(e2vkgeuc)ij = + + + + + +e2ρ0◦Fk|∂1F|2 +i = j = 1 +e2ρ0◦Fk⟨∂iFk, ∂jFk⟩ +i ̸= j +e2ρ0◦Fk|∂2F|2 +i = j = 2. +Thus, it follows that Fk is a conformal diffeomorphism of B1 and Dk. +□ +One easily sees from the Cauchy-Riemann Equations that Fk is a biholomor- +phic mapping and, if we use the complex analytic notation (−)′ for differentiation, +|∂iFk|2 = |F ′ +k|2 for i = 1, 2. Thus, +Corollary 3.1. Under the isometry Fk, we have vk = ρ0 ◦ Fk + log |F ′ +k|. + +STABILITY OF CONVEX DISKS +13 +Our task now is to investigate the convergence of the maps Fk : B1 → Dk, and it +is here where our specific choices concerning the domains Ωk help. The main result +of this section boils down to the following: +Proposition 3.4. Given εk ց 0, vk, and Fk all as above, we can find Mobius +transformations φk ∈ Conf(B1) such that the maps ˆFk ..= Fk ◦ φk → Rc · Id in +Cm +loc(B1) for any m ⩾ 0. In particular, (vk)φk → ρc in Ck +loc(B1) for any k ⩾ 1. +Proof. Choose ηk ∈ Conf(B1) such that ˜Fk ..= Fk ◦ η−1 +k +satisfies the normalization +conditions ˜Fk(0) = 0 and ˜F ′ +k(0) > 0. Then ˜Fk : B1 → Dk satisfies all the conditions +of the Carath´eodory Convergence Theorem 3.1, so we have a conformal diffeomor- +phism ˜F : B1 → BRc to which the ˜Fk limit in C0 +loc(B1). It follows that ˜F is simply +a Mobius transformation ψ ∈ Conf(B1) scaled by the factor Rc, so ˜Fk → Rc · ψ. +Now, let ˆFk ..= ˜Fk ◦ ψ−1 = Fk ◦ η−1 +k +◦ ψ−1 ..= Fk ◦ φk. We see that ˆFk → Rc ·Id in +C0 +loc(B1). To see that the φk provide the right change of gauge, we simply compute +the pullback of (Dk, e2ρ0geuc) by ˆFk: +(B1, e2(ρ0) ˆ +Fk geuc) = ˆF ∗ +k (Dk, e2ρ0geuc) = φ∗ +kF ∗ +k (Dk, e2ρ0geuc) += φ∗ +k(B1, e2vkgeuc) = (B1, e2(vk)φk geuc). +By the Cauchy Integral Theorem, for any m ⩾ 0 we have that ˆFk → Rc · Id in +Cm +loc(B1). Therefore, +(ρ0) ˆ +Fk(x) = ρ0 ◦ ˆFk(x) + log | ˆF ′ +k(x)| → ρ0(Rc · x) + log Rc = ρc(x) +and so +(vk)φk = (ρ0) ˆ +Fk → ρc +in Cm +loc(B1) for any m ⩾ 0, since ρ0 is smooth on B1. +□ +Now, recall that since φk ∈ Conf(B1) the disk (B1, e2(vk)φk geuc) is isometric to +(B1, e2vkgeuc). Therefore, ˜vk ..= (vk)φk satisfies all of the same curvature conditions +that vk does, namely, +� +−∆˜vk = e2˜vk +on B1 +∂n˜vk + 1 ⩾ ce˜vk +on S1. +Similarly, our original disk (B1, e2ukgeuc) is isometric to (B1, e2(uk)φk geuc) so the +functions ˜uk ..= (uk)φk similarly satisfy all the same curvature conditions: +� +−∆˜uk = K˜uke2˜uk ⩾ e2˜uk +on B1 +∂n˜uk + 1 ⩾ ce˜uk +on S1. +Of course, the relations ˜uk ⩾ ˜vk on B1, ˜uk = ˜vk on S1, and +L˜uk(S1) = +� +S1 e˜uk = +� +S1 e˜vk ⩾ 2π(1 − εk)/ +� +1 + c2 +continue to hold. Therefore, if ˜wk ..= ˜uk − ˜vk, all of the estimates of the previous +section for wk apply also to ˜wk. Our gauge fixing process thus consists of replacing +uk, vk, and wk with their counterparts ˜uk, ˜vk, and ˜wk. These conformal factors +(and their difference, respectively) all represent the same geometric objects up to +isometry with the same estimates, but with the property that the new constant cur- +vature comparison factors ˜vk enjoy good local convergence to the model conformal +factor ρc. + +14 +HUNTER STUFFLEBEAM +From here on out, for clarity of notation we will denote our gauge-corrected +conformal factors and their difference with the original unaccented quantities uk, +vk, and wk, implicitly assuming a-priori correction by the maps φk. +3.3. Producing a Metric Space Limit. We now have the estimates that we +need to show that the disks (B1, e2ukgeuc) converge to a limit. First, let us collect +what we have shown so far. For λ ⩾ 1 we have +eλuk − eλρc = +� +eλvk − eλρc� +eλwk + +� +eλwk − 1 +� +eλρc. +By the results of Sections 3.1 and 3.2, the pair of product terms on the right +hand side tends to zero in W 1,p +loc (B1) for every p ∈ [1, 2) and in Lq +loc(B1) for every +q ∈ [1, ∞), as can be readily seen by H¨older’s Inequality and the Sobolev Embedding +Theorem. In particular, we get strong local Sobolev convergence of the metrics +e2ukgeuc to the model metric e2ρcgeuc. To bootstrap this analytic convergence up +to geometric convergence, we begin by showing that the distance functions of the +e2ukgeuc subconverge to a semi-definite distance function on B1, which we will +eventually show is the distance function of the model metric. +Proposition 3.5. There exists a continuous function d∞ on B1 × B1 such that, +upto subsequence, duk → d∞ in C0 +loc(B1 × B1). +Proof. Fix 0 < r < 1. For any (x, y) ∈ B1 × B1 with x ̸= y, we have that (away +from the measure zero cut locuses of y and x, respectively) +|∇xduk(x, y)| = e2uk(x) +and +|∇yduk(x, y)| = e2uk(y). +Here, the norms and gradients are Euclidean. So, with 1 ⩽ p < ∞ and the Lp(Br) +bound on euk, we obtain +� +Br×Br +|∇xduk(x, y)|p+|∇yduk(x, y)|pdxdy = +� +Br×Br +e2puk(x)+e2puk(y)dxdy ⩽ C(p, r). +Thus, {duk} is bounded in W 1,p(Br × Br) for any 1 ⩽ p < ∞. By the Morrey- +Sobolev embedding theorem this sequence is also bounded in Cα(Br × Br) for any +α = 1−4/p ∈ (0, 1). By the compact embedding of H¨older spaces, we see that there +is a continuous function d∞ on Br × Br such that, upto subsequence, duk → d∞ +in Cα(Br × Br) for each α ∈ (0, 1). By letting r ր 1 we thus obtain pointwise +convergence of a subsequence to d∞ on B1 × B1 which is uniform on any compact +subset. +□ +We remark that d∞ defines a semi-metric on B1 to which the duk limit. Sym- +metry and the triangle inequality follow directly from the origin of d∞ as a limit of +metrics, but a priori we do not know that d∞ is positive definite. We next seek to +identify d∞ with dρc, the distance function associated to the round metric on B1 +with constant boundary curvature c. +3.4. Identifying the Local Gromov-Lipschitz Limits. In this section we show +that, for any r < 1, the sequence of metric spaces underlying the +� +Br, e2ukgeuc +� +con- +verges in the Gromov-Lipschitz sense to the underlying metric space of the spherical +domain +� +Br, e2ρcgeuc +� +. Recall that Gromov-Lipschitz convergence of metric spaces +means the following: + +STABILITY OF CONVEX DISKS +15 +Definition 3.2. Let (X, dX) and (Y, dY ) be metric spaces, and L the class of bi- +Lipschitz homeomorphisms between them. Then the Gromov-Lipschitz distance is +defined as +dGL((X, dX), (Y, dY )) ..= inf +F ∈L log +� +Lip(F), Lip(F −1) +� +provided such homeomorphisms exist, and is defined as +∞ otherwise. +We thus want to apply this notion of distance, for each r ∈ (0, 1), to the sequence +of metric spaces (Br, duk). Based off of the work in the previous section our candi- +date GL limit is (Br, d∞). However, d∞ is only known to be a semi-metric at this +point, so we instead first prove directly that d∞ = dρc. The uniform convergence +from Proposition 3.5 then implies the desired GL convergence (see [BBI01]). The +following argument identifying d∞ is inspired by one in [LST21]. +We’ll need the following Sobolev trace-type inequality to identify the limit. The +proof follows the exact same argument as in the standard case (see, for example, +[EG15]) where the curve γ is the boundary of a Lipschitz domain. +Proposition 3.6 (A Sobolev Trace Theorem). Let Ω be a precompact domain in +R2. Let γ ⊂ Ω be a curve of finite length which as a set is Lipschitz, in the sense +that it is locally the graph of a Lipschitz function over its tangent line. +Then, for any p ⩾ 1, there exists a C = C(Ω, γ, p) > 0 such that: if u ∈ W 1,p(Ω), +then the trace operator T : W 1,p(Ω) → Lp(γ) is a bounded linear operator with +operator norm ∥T ∥ ⩽ C. +We also need the following proposition, which is a version of the standard Bishop- +Gromov Theorem6 where we allow for the distance balls for both metrics in con- +sideration to make contact with their respective convex boundaries: +Proposition 3.7. Let Ω ⊂ R2 be a precompact, smooth set. Suppose g and gc are +two smooth Riemannian metrics on Ω, with the properties that secg ⩾ secgc ≡ 1, +and κg(∂Ω) ⩾ κgc(∂Ω) ≡ c ⩾ 0. Let x ∈ Ω and s ∈ (0, π). Then +volg (Bs(x, g) ⊂ Ω) ⩽ volgc (Bs(x, gc) ⊂ Ω) +Proof. By convexity of the boundaries, Ω can be covered by a global normal coor- +dinate chart in each metric, and the volume comparison follows from the standard +proof. +More specifically, the assumption on curvatures tells us that g ⩽ gc on +Ω, and expressing the volumes as integrals of √det g and √det gc in the normal +coordinates proves the claim. See for example [Lee18] Chapter 11. +□ +Now we are ready to prove that d∞ = dρc. +Claim 3.2 (d∞ ⩽ dρc). +Proof. Fix any x, y ∈ B1 and r < 1 so that x, y ∈ Br. Let γ be the ρc geodesic in +B1 from x to y, which by convexity lives inside of Br. Since each ρc geodesic in B1 +is certainly an admissible curve in the above Trace Theorem, +euk → eρc +in +W 1,p(Br) +for all +p ∈ [1, 2) +implies that +euk → eρc +in +Lp(γ) +for all +p ∈ [1, 2). +6Technically, since we call for sectional curvature bounds, this might be more accurately called +a boundary version of the G¨unther volume comparison. + +16 +HUNTER STUFFLEBEAM +Therefore, +dρc(x, y) = +� +γ +eρc = lim +� +γ +euk = lim Luk(γ) ⩾ lim inf duk(x, y) = d∞(x, y). +□ +Claim 3.3 (d∞ ⩾ dρc). +Proof. For the sake of contradiction, suppose that there were to exist some x, y ∈ +Br ⊂ B1 with +R∞ ..= d∞(x, y) < R ..= dρc(x, y). +Since y ∈ BR(x, d∞) but y /∈ BR(x, dρc), by continuity of d∞ we have that +volρc (BR(x, dρc)) < volρc (BR(x, d∞)) . +Next, choose an η > 0 small and recall that duk → d∞ uniformly on Br. For all +large enough k we then have that +Br ∩ BR−η(x, d∞) ⊂ BR(x, duk). +Putting these two facts together with convergence of e2uk to e2ρc in L1(Br) and +applying our version of Bishop-Gromov with boundaries, we obtain +volρc +� +Br ∩ BR−η(x, d∞) +� += lim voluk +� +Br ∩ BR−η(x, d∞) +� +⩽ lim inf voluk (BR(x, duk)) +⩽ volρc (BR(x, dρc)) . +We now send r ր 1 and η ց 0 (equivalently, take the union of the sets Brj ∩ +BR−ηj(x, d∞) for a sequence rj ր 1 and ηj ց 0) to obtain a contradiction with +our first strict volume estimate above. +□ +Therefore, we have identified the limit semi-metric d∞ as the bonafide dis- +tance function dρc deriving from the round metric. +Consequently, by our work +in the previous section, we can say with proof that for every r < 1, the manifolds +(Br, e2ukgeuc) converge in the Gromov-Lipschitz sense to the round spherical cap +(Br, e2ρcgeuc). +3.5. Identifying the Intrinsic Flat Limit. In this section, we prove that the se- +quence (B1, e2ukgeuc) converges in the Sormani-Wenger Intrinsic Flat (SWIF) sense +to the spherical cap (B1, e2ρcgeuc). By Wenger’s Compactness Theorem (discussed +shortly), the sequence (B1, e2ukgeuc) is sure to subconverge to some limit. +Our +task will be identifying it via direct geometric constructions and estimates, taking +advantage of the local GL convergence from the previous section. +We first recall the general idea of SWIF convergence. For an earnest introduction +to these ideas, the original papers of C. Sormani and S. Wenger [SW11], and L. +Ambrosio and B. Kircheim [AK00], are excellent. +An integral current space is, +loosely speaking, a metric space (X, d) paired with an integral current structure +T on (X, d) in the sense of Ambrosio-Kircheim.7 In a nutshell, the intrinsic flat +distance is the Federer-Fleming flat distance between two integral current spaces +isometrically embedded in the most efficient way into a common complete metric +7For our purposes, (X, d) will be the metric space underlying a smooth compact Riemannian n- +manifold, and the natural integral current structure T is simply integration of differential n-forms +over the manifold. + +STABILITY OF CONVEX DISKS +17 +space. It is thus analogous to the Gromov-Hausdorff distance8, where our objects +are now integral current spaces instead of compact metric spaces, the ambient +embedding space is now a complete metric space instead of a compact one, and +the Hausdorff metric of the ambient space is replaced by the Federer-Fleming flat +distance between integral currents. As proved in [SW11], two integral current spaces +have SWIF distance 0 iff they are isometric via an orientation preserving map. +What is most important for us is the following method for estimating the SWIF +distance: +Definition 3.3 (SWIF Convergence). Let (M1, g1) and (M2, g2) be compact, ori- +ented Riemannian n-manifolds, and suppose that there exist distance preserving9 +inclusions φi : Mi → Z of the Mi into a common, metrically complete Riemann- +ian manifold Z. Suppose also that we can find oriented submanifolds An ⊂ Z and +Bn+1 ⊂ Z such that, in the language of currents, �(φ1)#M1�−�(φ2)#M2� = A+∂B. +That is, for every ω ∈ Ωn(Z), +� +M1 +φ∗ +1ω − +� +M2 +φ∗ +2ω = +� +A +ω + +� +∂B +ω. +Then we have that dIF ((M1, g1), (M2, g2)) ⩽ voln(A) + voln+1(B). +In [Wen10], S. Wenger proved the following compactness theorem for sequences +of integral current spaces, which directly applies to our sequence (B1, e2ukgeuc): +Theorem 3.3 (Wenger’s Compactness Theorem, for Riemannian Manifolds). Sup- +pose that (Mj, gj) is a sequence of oriented Riemannian n-manifolds with uniformly +bounded diameters, n-volumes, and boundary (n−1)-volumes. Then there exists an +integral current space (possible the zero space) to which the (Mj, gj) subconverge in +the SWIF sense. +We reiterate that even when both GH and SWIF limits for a sequence of integral +current spaces exist, they may not agree. However, what is known in the presence +of uniform upper volume and boundary volume bounds is that the SWIF limit +isometrically embeds into the GH limit. Lastly, we remark that GL convergence +of the underlying metric spaces implies SWIF convergence. All of this is proven in +[SW11]. +Now let us return to our problem, fixing r < 1. We estimate the SWIF distance +directly via the triangle inequality: +dIF ((B1, e2ukgeuc), (B1, e2ρcgeuc)) ⩽dIF ((B1, e2ukgeuc), (Br, e2ukgeuc)) ++ dIF ((Br, e2ukgeuc), (Br, e2ρcgeuc)) ++ dIF ((Br, e2ρcgeuc), (B1, e2ρcgeuc)). +The middle term can be estimated by the fact that GL Convergence implies SWIF +Convergence. The first and third terms can then be estimated directly, by construct- +ing embedding spaces to estimate the flat distances. To be precise, to estimate the +first term we define our ambient embedding spaces to be the manifolds +Bk ..= (B1, e2ukgeuc) × [0, εk] +8We assume some familiarity with the GH distance, but remark that the book [BBI01] provides +a wonderful introduction to this and many other topics in metric geometry. +9Namely, dZ(φi(x), φi(y)) = di(x, y) for every x, y ∈ Mi. + +18 +HUNTER STUFFLEBEAM +equipped with the standard Riemannian product measures eukgeuc ⊕ dt2. +Our +distance preserving embeddings φk : (Br, e2ukgeuc) → Bk and ψk : (B1, e2ukgeuc) → +Bk are simply +φk(x) ..= (x, 0) +and +ψk(x) ..= (x, εk). +Thus, (expressing the condition on differential forms in Definition 3.3) we have +�(ψk)∗(B1, e2ukgeuc)� − �(φk)∗(Br, e2ukgeuc)� = Ak + ∂Bk +where Ak ..= S1 × [0, εk] + (B1 \ Br, e2ukgeuc) × {εk}. So, we may estimate +dIF ((B1, e2ukgeuc), (Br, e2ukgeuc)) ⩽ Area(Ak) + Vol(Bk) +⩽ 2πεk + Area(B1 \ Br, e2ukgeuc) ++ Area(B1, e2ukgeuc)εk +⩽ 4πεk + Area(B1 \ Br, e2ukgeuc). +Consider the last term +Area(B1 \ Br, e2ukgeuc) = Area(B1, e2ukgeuc) − Area(Br, e2ukgeuc) +⩽ Area(B1, e2vkgeuc) − Area(Br, e2ukgeuc). +By the inradius estimate 2.1, Area(B1, e2vkgeuc) → Area(B1, e2ρcgeuc), and since +GL convergence implies volume convergence, the results of the last section imply +that Area(Br, e2ukgeuc) → Area(Br, e2ρcgeuc). +Therefore, we conclude that, for +large k, +dIF ((B1, e2ukgeuc), (Br, e2ukgeuc)) ⩽ 4πεk + Ψ(1 − r|c, k). +Arguing similarly (and in fact more easily) for the third term, we also find +dIF ((Br, e2ρcgeuc), (B1, e2ρcgeuc)) ⩽ Ψ(1 − r|c). +Since the middle term is small by the aforementioned GL convergence, we conclude +by arbitrariness of r < 1 that (B1, e2ukgeuc) → (B1, e2ρcgeuc) in the SWIF sense. +3.6. Showing the Intrinsic Flat Limit ≡ Gromov-Hausdorff Limit. We are +now ready to show that (B1, e2ukgeuc) → (B1, e2ρcgeuc) in the GH sense. By Gro- +mov’s Compactness Theorem, the (B1, e2ukgeuc) subconverge to some metric space +(X, d), but the relationship between (B1, e2ρcgeuc) and (X, d) is a priori unclear. +Utilizing recent work of R. Perales, we will show that the SWIF limit agrees with +the GH limit. To state it, we introduce the following notation: +Let (M, g) be a Riemannian manifold with boundary ∂M, let d denote the +distance function of g, and for δ > 0 define the δ-inner region of M by +M δ ..= {x ∈ M : d(x, ∂M) > δ}. +On M δ there are two distance functions, the restriction d|Mδ and the induced length +metric dMδ. Given this notation, we can state the following expedient adaptation +of Theorem 1.2 in [Per18]: +Theorem 3.4 (R. Perales). Let δ, Di, L, V, θ > 0 and δi ց 0. Let (Mk, gk) be a +sequence of compact, oriented 2-manifolds with boundary such that +(A) Kgk ⩾ 0 +(B) AreagkMk ⩽ A +(C) Lengthgk∂Mk ⩽ L. +(D) diam(M δi +k , dM +δi +k ) ⩽ Di + +STABILITY OF CONVEX DISKS +19 +(E) There exists some qk ∈ M δ +k such that AreagkBδ(qk, gk) ⩾ θδ2 +(F) There exists a compact metric space (X∂, d∂) such that (∂Mk, dk) → (X∂, d∂) +in GH +Then up to a subsequence (not relabeled), there is a compact metric space (X, dX) +such that +(Mk, dk) → (X, dX) +in the GH sense +and an integral current space (Y ⊂ X, dX, T ) such that +(Mk, dk, Tk) → (Y, dX, T ) +in the SWIF sense +where we also have that X \ X∂ ⊂ Y . Moreover, +(G) If X∂ ⊂ Y , then Y = X. +Let’s carefully check the conditions in this theorem when we take (Mk, gk) = +(B1, e2ukgeuc). Items (A), (B), (C) all follow immediately from the hypotheses, so +we focus on the the last four conditions: +(D) By convexity, each δi-inner region of every (B1, e2ukgeuc) is convex10, and +hence the induced length metric is exactly the restriction of dk to the δi- +inner region. The distance functions dk are bounded uniformly by π, so we +may take Di = π for all i. +(E) Take qk = 0 for every k, δ = π/4, and θ = π/8, for instance. Then +Bδ(0, ρc) = {x ∈ B1 : dρc(0, x) < δ} = B1/(1+ +√ +2) ⊂ B2/3. +Since duk → dρc uniformly on B2/3, for all large k we have that +Bδ(0, gk) = {x ∈ B1 : duk(0, x) < δ} ⊃ {x ∈ B1 : dρc(0, x) < δ} = Bδ(0, ρc). +Therefore, +AreagkBδ(0, gk) ⩾ AreagkBδ(0, ρc) → AreaρcBδ(0, ρc) +and for all large k we have +AreagkBδ(0, gk) ⩾ 1 +2AreaρcBδ(0, ρc) = π(1 − cos(δ)) ⩾ π +4 δ2. +To check that 0 ∈ (B1, e2ukgeuc)δ for all k, first let ω ∈ S1. Then we have +lim duk(0, ω/ +√ +3) = dρc(0, ω/ +√ +3) = π/3. +and thus for all large k and every ω ∈ S1, +duk(0, ω/ +√ +3) > π/4. +Consequently, given any η ∈ S1, +duk(0, η) ⩾ inf +ω∈S1 duk(0, ω/ +√ +3) > π/4 = δ. +(F), (G) By our setup, ∂(B1, e2ukgeuc) is isometric to ∂(B1, e2vkgeuc). The latter +boundary is isometric to ∂Ωk ⊂ S2, a convex simple closed curve which con- +verges in the Hausdorff, and therefore Gromov-Hausdorff, sense to the cir- +cle of length 2π/ +√ +1 + c2 which bounds the geodesic disk of radius cot−1(c) +about the south pole–an isometric realization of the SWIF limit of the +(B1, e2ukgeuc). Since the SWIF limit is the entire closed disk, the last con- +dition is also satisfied, and we can at last conclude that the SWIF limit is +isometric to the GH limit. +10For instance, see Theorem 8.9 in [CE75]. + +20 +HUNTER STUFFLEBEAM +Remark 3.1. It would be interesting to know if one can directly estimate the GH +distance between (B1, e2ρcgeuc) and the (B1, e2ukgeuc). A natural approach would +be to consider the identity maps φk : Br → Br ⊂ B1, and try to prove that for +large k they are ε-Hausdorff approximations of (Br, e2ρcgeuc) and (B1, e2ukgeuc). +While the local GL convergence easily shows that these maps have small distortion, +it seems more challenging to show directly that the ε-neighborhood of their images +in (B1, e2ukgeuc) cover all of B1, due to a priori non-uniformity of the convergence +of the duk to dρc. +3.7. Proof of the Main Theorem. To summarize, we supposed that there were +a δ0 > 0 such that, for every sequence εk ց 0, we could find an (Mk, gk) as in the +statement with Lk(∂Mk) ⩾ 2π(1 − εk)/ +√ +1 + c2 but +d((Mk, gk), Bcot−1(c)) ⩾ δ0 > 0, +where d = dGH or dIF . As shown in the previous sections, for either choice of d any +such sequence of disks will subconverge to Bcot−1(c), contradicting the existence of +δ0. Hence, the main result follows. +4. Acknowledgments +The author would like to thank Renato Bettiol and their adviser Davi M´aximo +for being calming voices when an error in an early draft was found, as well as +the Fields Institute for its hospitality during the Thematic Program on Nonsmooth +Riemannian and Lorentzian Geometry. Much of the final work was completed there +during the author’s visit. +References +[AK00] Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Mathematica +185 (2000), no. 1, 1–80. +[BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, American +mathematical Society, 2001. +[BM91] Ham Brezis and Frank Merle, Uniform estimates and blowup behavior for solutions of +δ(u) = v(x)eu in two dimensions, Communications in Partial Differential Equations 16 +(1991), no. 8-9, 1223–1253. +[CC96] Jeff Cheeger and Tobias H. Colding, Lower bounds on ricci curvature and the almost +rigidity of warped products, Annals of Mathematics 144 (1996), no. 1, 189–237. +[CE75] Jeff Cheeger and David G. Ebin, Comparison theorems in riemannian geometry, 1975. +[CL22] Jingyi Chen and Yuxiang Li, Uniform convergence of metrics on alexandrov surfaces +with bounded integral curvature, arXiv, 2022. +[CM97] Tobias H. Colding and William P. Minicozzi II, Harmonic functions with polynomial +growth, Journal of Differential Geometry 46 (1997), no. 1, 1 –77. +[Dra18] Kostiantyn Drach, Inradius estimates for convex domains in 2-dimensional alexandrov +spaces, Analysis and Geometry in Metric Spaces 6 (2018), no. 1, 165–173. +[Dur04] Peter Larkin Duren, Harmonic mappings in the plane, Cambridge University Press, +2004. +[EG15] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of func- +tions, CRC Press, Taylor and Francis Group, 2015. +[FMP08] N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative isoperimetric inequality, +Annals of Mathematics 168 (2008), no. 3, 941–980. +[HW09] Fengbo Hang and Xiaodong Wang, Rigidity theorems for compact manifolds with bound- +ary and positive ricci curvature, Journal of Geometric Analysis 19 (2009), no. 3, 628– +642. +[IK18] M. Izuki and T. Koyama, An elementary proof of the carath´eodory kernel convergence +theorem, Azerbaijan Journal of Mathematics 8 (January 2018), no. 1, 69–85 (English). + +STABILITY OF CONVEX DISKS +21 +[Kli11] Wilhelm P.A. Klingenberg, Riemannian geometry, De Gruyter, Berlin, New York, 2011. +[Lee18] John M. Lee, Introduction to riemannian manifolds, Springer, 2018. +[LP18] Nan Li and Raquel Perales, The sormani-wenger intrinsic flat convergence of alexandrov +spaces, Journal of Topology and Analysis (2018). +[LST21] Yuxiang Li, Jianxin Sun, and Hongyan Tang, Metrics on a surface with bounded total +curvature, International Mathematics Research Notices 2022 (2021), no. 17, 13212– +13245. +[LT17] Yuxiang Li and Hongyan Tang, Metrics on s2 with bounded ∥kg∥L1 log L1 and small +∥kg − 1∥L1, arXiv, 2017. +[Per18] Raquel Perales, Convergence of manifolds and metric spaces with boundary, Journal of +Topology and Analysis 12 (2018), no. 03, 735–774. +[Pom11] Christian Pommerenke, Boundary behaviour of conformal maps, Springer, 2011. +[SW11] Christina Sormani and Stefan Wenger, The intrinsic flat distance between riemannian +manifolds and other integral current spaces, Journal of Differential Geometry 87 (2011), +no. 1. +[Wen10] Stefan Wenger, Compactness for manifolds and integral currents with bounded diameter +and volume, Calculus of Variations and Partial Differential Equations 40 (2010), no. 3-4, +423–448. +[WZ21] Bing Wang and Xinrui Zhao, Canonical diffeomorphisms of manifolds near spheres, +arXiv, 2021. +The University of Pennsylvania, Department of Mathematics, David Rittenhouse +Lab., 209 South 33rd Street, Philadelphia, PA 19104. +Email address: hstuff(at)sas(dot)upenn(dot)edu + diff --git a/PNFPT4oBgHgl3EQfnjVw/content/tmp_files/load_file.txt b/PNFPT4oBgHgl3EQfnjVw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec324c74c17e844d455f8b4257bec73ec841a1cd --- /dev/null +++ b/PNFPT4oBgHgl3EQfnjVw/content/tmp_files/load_file.txt @@ -0,0 +1,614 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf,len=613 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='13130v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='DG] 30 Jan 2023 STABILITY OF CONVEX DISKS HUNTER STUFFLEBEAM Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We prove that topological disks with positive curvature and strictly convex boundary of large length are close to round spherical caps of constant boundary curvature in the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' This proves stability for a theorem of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Hang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Wang in [HW09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' As an intermediate step we obtain a result concerning gauge fixing and compactness for solutions of a Liouville type PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Introduction Inequalities in geometric analysis, such as the isoperimetric and systolic, Faber- Krahn and Penrose, relate given geometric objects to understood model cases, taking as input data bounds on curvatures, volumes, eigenvalues, energies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Via such relationships, much work has been done to understand the structure of spaces with natural geometric conditions phrased in terms of such quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Given an inequality for which one has some understanding of extremizers (the geometric objects which realize equality), one might ask if an object nearly re- alizing equality must somehow share characteristics with the extremizer(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The first problem of understanding the extremizers might be called a rigidity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The second problem of understanding near extremizers might be called a stability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' A classical example is the isoperimetric inequality in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Its extremizers are balls, and the stability problem has received much attention–for example consider the work of Fusco-Maggi-Pratelli in [FMP08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The aim of this paper is to establish the following stability theorem for convex disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Throughout, given a 2-manifold (M, g) we use the notation Kg for the Gaussian curvature, κg for the geodesic curvature of the boundary, and Lg for the length functional of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The Gromov-Hausdorff metric on the space of compact metric spaces is denoted by dGH, and the Sormani-Wenger Intrinsic Flat metric on the space of integral current spaces is denoted by dIF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1 (Stability of the Convex Spherical Cap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Fix c > 0 and let δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then there exists an ε = ε(c, δ) > 0 such that if (M, g) is a two dimensional manifold with Kg ⩾ 1, κg ⩾ c > 0, and Lg(∂M) ⩾ 2π(1 − ε)/ √ 1 + c2, then dGH((M, g), Bcot−1(c)) < δ, where Bcot−1(c) is a closed geodesic disk of radius cot−1(c) in the round sphere S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The same conclusion holds, with potentially different ε, with dIF in place of dGH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' This result was motivated by the desire for a stability theorem corresponding to an old result of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Toponogov: Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Gromov-Hausdorff, Sormani-Wenger Instrinsic Flat, Liouville Equa- tion, Convexity, Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1 2 HUNTER STUFFLEBEAM Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2 (V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Toponogov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let (M, g) be a closed surface with Kg ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then any simple closed geodesic has length bounded from above by 2π, and this length is attained if and only if (M, g) is isometric to the round sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' There are at least two proofs of this result–the original due to Toponogov using the celebrated triangle comparison theorem, and a modern PDE proof due to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Hang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Wang (see [Kli11] and [HW09], respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In both arguments, one cuts open the closed manifold along the largest closed geodesic to obtain two disks with geodesic boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The length inequality, and a corresponding rigidity theorem, is then proven for these disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The PDE proof of the result applies immediately to the case of convex disks in general: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Hang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Wang [HW09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let (M, g) be a smooth, compact surface with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Suppose Kg ⩾ 1 and κg ⩾ c ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then Lg(∂M) ⩽ 2π/ √ 1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Moreover, equality holds if and only if (M, g) is isometric to Bcot−1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To attempt a proof of stability for Toponogov’s theorem, it is thus natural to begin by trying to prove stability in the disk rigidity theorem of Hang-Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' How- ever, while stability in the convex case does hold as is shown by our following work, this does not extend to the case of disks with weakly convex boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' And indeed, stability does not generally hold in Toponogov’s theorem: Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1 (Failure of Stability in Toponogov’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Fix any small ε > 0 and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' There exists a rotationally symmetric metric g = dr2 + φ2(r)dθ2 on S2 which has Kg ⩾ 1, a simple closed geodesic of length 2π − ε, and d((S2, g), (S2, grd)) ⩾ δ, where d = dGH, dIF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Here grd denotes the round metric on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' This follows an idea of [CC96] [CM97], and was explored in some detail in [WZ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' One takes a good football metric on S2/Zk and carefully smoothes the tips by gluing in spherical caps and rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The interested reader is encouraged to look to the latter source for the specifics of the construction, and it is not hard to deduce from it the claimed properties in the remark above pertaining to the case d = dGH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In addressing the dIF case, one can apply work of [LP18] to see that stability for Toponogov’s Theorem still fails, since in the setting of closed Alexandrov spaces of non-negative curvature and uniform upper diameter bounds, convergence in dIF to a nonzero space would imply dGH convergence to the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let us now remark on the main ideas of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' A more detailed description will be given shortly, after the requisite notation and setup has been properly introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proceeding by way of contradiction, we obtain a se- quence of convex topological disks with c-convex boundary whose boundary lengths converge to the extremal value while remaining bounded away from the model disk in the metric d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the Gauss-Bonnet and Uniformization Theorems, studying this sequence of metrics on topological disks amounts to studying a corresponding sequence of conformal factors for metrics on the unit disk of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' New conformal factors for constant curvature ≡ 1 disks with isometric boundaries to the original sequence are produced, to be compared to the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' A major hurdle in handling the compactness of these constant curvature conformal factors is the action of the conformal group of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By fixing the gauge and applying some results from conformal mapping and elliptic PDE theory, we obtain converging STABILITY OF CONVEX DISKS 3 subsequences of both the comparison conformal factors and the differences between them and those of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' This gives subconvergence for the original sequence of conformal factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We then upgrade this analytic convergence of confor- mal factors to geometric convergence of the manifolds to the model disk, obtaining the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' A Comment on Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Throughout this paper, Riemannian metrics will often be written as being conformally equivalent to geuc, the standard Euclidean metric on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We will often reference geometrical quantities defined with respect to such a metric g = gu = e2ugeuc by the conformal factor u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For example, we may write du for the distance function dgu deriving from gu, and κu for the geodesic curvature of the boundary for the metric gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' ∆ will denote the Euclidean Laplace operator defined by ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= div ◦ ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We will also have to refer to various geometrical quantities, such as metric balls, which depend upon a choice of metric and/or distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In general, we will denote by Br(x, d) the open metric ball of radius r about x with respect to the distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In case d = dg derives from a Riemannian metric g, we will interchange- ably use the notation Br(x, g) and Br(x, dg) as is most convenient for exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In the special case x = 0 and g = geuc, we will simply write Br for Br(0, geuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In case a measure is omitted from an integral, it is understood that the implied measure is the standard volume measure on the underlying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In all cases, it should be quite clear what is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Finally, we will follow tradition in letting Ψ = Ψ(x) = Ψ(x|a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=') denote a non-negative function, which may change from line to line, depending on a variable x and any number of parameters ai with the property that if the ai are all held fixed, Ψ ց 0 as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Our Setup and a Review of Hang-Wang’s Argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For completeness of exposition, and to set some notation which we will use throughout the rest of the paper, we briefly recall the proof of Toponogov’s Theorem in [HW09], which we restate for convenience: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Hang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Wang).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let (M 2, g) be a compact surface with boundary γ, Kg ⩾ 1, and κg(γ) ⩾ c ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then Lg(γ) ⩽ 2π/ √ 1 + c2, with equality iff (M 2, g) is isometric to Bcot−1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By Gauss-Bonnet and Uniformization, there is an isometry of (M 2, g) with (B1, e2ugeuc) for some smooth u: B1 → R, and the curvature conditions translate to � −∆u = Kge2u ⩾ e2u on B1 ∂nu + 1 = κg(γ)eu ⩾ ceu on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the sub-super solution method, we can produce a constant curvature comparison factor v: B1 → R to u, which satisfies the following: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −∆v = e2v on B1 ∂nv + 1 ⩾ cev on S1 u ⩾ v on B1 v = u on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 4 HUNTER STUFFLEBEAM The constant curvature comparison manifold (B1, e2vgeuc) can therefore be realized isometrically as a smooth domain Ω in the standard S2 with boundary that is uniformly c-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Some elementary geometry tells us that the smallest geodesic disk D ⊂ S2 containing Ω is of radius at most cot−1(c), and this disk has boundary length Lgrd(∂D) ⩽ 2π/ √ 1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since u = v on S1, Lg(γ) = Le2vgeuc(S1) = Lgrd(∂Ω) ⩽ 2π/ � 1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Now suppose equality is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The construction above forces ∂Ω = ∂D, which forces the geodesic curvature of ∂Ω to be identically c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Thus, our comparison factor v must satisfy \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆v = e2v on B1 ∂nv + 1 = cev on S1 v = u on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since on S1 we have cev = ∂nv + 1 ⩾ ∂nu + 1 ⩾ ceu = cev we conclude that γ also has constant geodesic curvature c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Setting w = u − v, we have that \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆w ⩾ 0 on B1 ∂nw = 0 on S1 w = 0 on B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' It then follows easily that u ≡ v on B1, which proves that (M 2, g) is isometric to a geodesic disk with constant boundary curvature c in the standard unit S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ In this paper, we are interested in the consequences of the assumption that Lg(γ) is nearly equal to 2π/ √ 1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let then ε > 0 be small, and consider a compact surface (M 2, g) with boundary γ, Kg ⩾ 1, κg(γ) ⩾ c > 0, and Lg(γ) ⩾ 2π(1 − ε)/ √ 1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We set out to prove that (M 2, g) is Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat close to the spherical cap characterizing the equality case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Construct exactly as above the manifolds (B1, e2ugeuc) and (B1, e2vgeuc), with the latter corresponding isometrically to some domain Ω in the standard S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The jumping off point is the following estimate for the inradius of a strictly convex domain in a sphere in terms of the length of the boundary, and is a direct adaptation of the more general Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2 in [Dra18]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2 (Inradius Estimate for Convex Domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let Ω be a smooth convex domain in the standard S2, with boundary of length L and κg(∂Ω) ⩾ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let rin denote the inradius of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then rin ⩾ cot−1(c) − cot−1 � c sec � L √ 1 + c2 4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Notice the importance of the positivity of c in this estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Indeed, the failure of an inradius lower bound for domains with boundary having segments of zero geodesic curvature allows for collapsing, and is exactly what underlies the failure of stability in the weakly convex case without further assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In particular, our constant curvature comparison manifold (B1, e2vgeuc), when realized isometrically as a domain Ω in S2, has a large inball of radius rin ⩾ cot−1(c) − cot−1 � c sec �π 2 (1 − ε) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' STABILITY OF CONVEX DISKS 5 Additionally, as proven above, Ω has the geodesic disk D as an outball of radius rout ⩽ cot−1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We thus have observed the following, which will be a fundamental lemma for us: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let 0 < ε < 1 and fix a compact surface (M 2, g) with boundary γ, Kg ⩾ 1, κg(γ) ⩾ c > 0, and Lg(γ) ⩾ 2π(1 − ε)/ √ 1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let (B1, e2vgeuc) ↔ Ω ⊂ S2 be the constant curvature comparison disk in S2, which has inradius rin and outradius rout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then cot−1(c) − Ψ(ε|c) ⩽ rin ⩽ rout ⩽ cot−1(c) c ⩽ cot(rout) ⩽ cot(rin) ⩽ c + Ψ(ε|c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Outline of the Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Here we explain the broad-strokes idea of the argu- ment for proving our main theorem, which we restate for convenience: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3 (Stability of the Convex Spherical Cap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Fix c > 0 and let δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then there exists an ε = ε(c, δ) > 0 such that if (M, g) is a two dimensional manifold with Kg ⩾ 1, κg ⩾ c > 0, and Lg(∂M) ⩾ 2π(1 − ε)/ √ 1 + c2, then dGH((M, g), Bcot−1(c)) < δ, where Bcot−1(c) is a closed geodesic disk of radius cot−1(c) in the round sphere S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The same conclusion holds with dIF in place of dGH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We will prove this by way of contradiction, supposing that there exists a δ0 > 0 such that, for every sequence εk ց 0, we can find an (Mk, gk) as in the statement with Lk(∂Mk) ⩾ 2π(1 − εk)/ √ 1 + c2 but d((Mk, gk), Bcot−1(c)) ⩾ δ0 > 0, where d = dGH or dIF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' First, let’s fix the notation for the model spaces that we will be comparing our given manifolds to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Given c ⩾ 0, define the function ρc : B1 → R by the formula ρc(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= log � 2Rc 1 + |Rcx|2 � where Rc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= � 1 + c2 − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then (B1, e2ρcgeuc) is isometric, via the dilation Rc·Id: B1 → BRc, to (BRc, e2ρ0geuc), which under stereographic projection Φ: S2 \\ N → R2 from the north pole N = e3 is isometric to the geodesic disk of radius cot−1(c) in S2 centered at the south pole S = −e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In other words, our model extremizer Bcot−1(c) can be isometrically real- ized as (B1, e2ρcgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Here, and throughout when convenient, we may identify Br with Br × {0} ⊂ R3, in particular when considering the stereographic projection from the standard embedding of round S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Our goal is to estimate the sequence of distances d((B1, e2ukgeuc), (B1, e2ρcgeuc)), and extract a subsequence which converges to 0 to derive a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' At the outset, we remark that by the compactness theorems of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Gromov and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Wenger, any sequence of manifolds satisfying our assumptions will subconverge in the Gromov-Hausdorff (GH) and Sormani-Wenger Intrinsic Flat (SWIF) senses, respectively, to compact metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In general, given our assumptions the SWIF limit will embed isometrically as a subset of the GH limit, but outright we do not know much more about what they are, or whether they agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 6 HUNTER STUFFLEBEAM To identify both limits as being the round spherical cap, we will first control the differences of conformal factors wk = uk − vk and vk − ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Writing for λ ⩾ 1 eλuk − eλρc = � eλvk − eλρc� eλwk + � eλwk − 1 � eλρc, we will obtain W 1,p loc convergence by showing that eλwk → 1 in W 1,p and that eλvk → eλρc in Cm loc for any m ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The control on wk follows from standard elliptic PDE techniques and a result of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Brezis-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Merle in [BM91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The control on vk − ρc is more subtle, and involves some results from the theory of conformal mappings of convex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' With this control established, we can show that our sequence has a local Gromov- Lipschitz (GL) sublimit on each Br ⊂ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Using variants of the Sobolev Trace Theorem and the Bishop-Gromov Theorem, we can identify the local GL limits as spherical caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We then glue these local limits together to identify the model disk (B1, e2ρcgeuc) as the SWIF limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To identify the GH limit, we apply a recent theorem due to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Perales [Per18] showing that the SWIF and GH limits must agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In either case, we will obtain a convergent subsequence to the model disk, forcing a contradiction and establishing the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' A Remark on the Brezis-Merle and other Blow-Up Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Evi- dently, the study of sequences of Riemannian surfaces is linked, via the uniformiza- tion process described above, to the study of sequences of solutions to the Liouville equation on a two dimensional domain: −∆u = K(x)e2u on Ω ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The geometric interpretation is that the metric e2ugeuc on Ω has Gaussian curvature K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In their seminal 1991 paper [BM91], H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Brezis and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Merle studied the blow up behavior of solutions to this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Their analysis, which includes a study of uniform a-priori estimates for sequences of such solutions u, require Lp bounds on the potentials K for p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We only have L1 bounds on K, however, rendering their conclusions unavailable to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Nonetheless, we will find great use in the following fundamental estimate from [BM91]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='4 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Brezis and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Merle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Assume Ω ⊂ R2 is a bounded domain and let u be a solution of � −∆u = f on Ω u = 0 on ∂Ω with f ∈ L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then for every δ ∈ (0, 4π), we have the estimate � Ω e 4π−δ ∥f∥L1(Ω) |u(x)|dx ⩽ 4π2 δ (diamΩ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Several authors have recently investigated possible extensions of the blow-up analysis, in particular with attention to geometric applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For example [LST21], [LT17], and [CL22] have studied, among other things, the compactness of sequences of Riemannian surfaces with curvature bounds via an analysis of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We remark that their results seem to be largely unavailable to us here, given the more general nature of our curvature bounds and the desire for identifying exact limits to converging sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' STABILITY OF CONVEX DISKS 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Stability of Convex Disks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Controlling the Difference of Conformal Factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The goal of this sec- tion, largely a rapid-fire sequence of lemmas, is to prove the following proposition: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let c > 0, λ ⩾ 1, p ∈ [1, 2), and ε > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Given (B1, e2ugeuc) and (B1, e2vgeuc) as in the proof of Hang-Wang’s Theorem1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3 with κu ⩾ c and Lu(S1) ⩾ 2π(1 − ε)/ √ 1 + c2, set w = u − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then ∥eλw − 1∥W 1,p(B1) ⩽ Ψ(ε|c, p, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We begin by collecting together some basic facts about w: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The difference of conformal factors w = u − v satisfies the following: (i) w ⩾ 0 on B1 (ii) −∆w ⩾ 0 on B1 (iii) w = 0 on S1 (iv) ∂nw ⩽ 0 on S1 (v) | � S1 ∂nw| ⩽ Ψ(ε|c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Items (i), (ii), (iii), and (iv) are rather immediate from the construction of v via the sup-super solution method, so we focus on item (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' This relies upon the inradius estimate of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1 in a crucial way, and is in a sense the most ‘geometrically informed’ result of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We seek to estimate 0 ⩾ � S1 ∂nw = � S1(∂nu + 1) − (∂nv + 1) = � S1(κu − κv)eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the inradius estimate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1, we may consider the new comparison disk (B1, e2fgeuc) where � −∆f = e2f on B1 ∂nf + 1 = cot(rin)ef on S1, which is isometric to a geodesic disk in S2 of constant boundary curvature cot(rin), serving as an inball for (B1, e2vgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the Gauss-Bonnet Theorem, we have (A) 2π = � B1 e2f + � S1 cot(rin)ef = Area(B1, e2fgeuc) + � S1 cot(rin)ef (B) 2π = � B1 e2v + � S1 κvev = Area(B1, e2vgeuc) + � S1 κvev and by direct comparison (C) Area(B1, e2fgeuc) ⩽ Area(B1, e2vgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' (A), (B), and (C) together imply that � S1 κvev ⩽ � S1 cot(rin)ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Finally, observe that � S1 ef = Lf(S1) ⩽ � S1 ev = Lv(S1) ⩽ 2π √ 1 + c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 8 HUNTER STUFFLEBEAM Using parts (i)-(iii) of the Lemma, the inradius estimate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1, together with the prior two observations, we get � S1 cev = � S1 ceu ⩽ � S1 κueu ⩽ � S1 κvev ⩽ � S1 cot(rin)ef ⩽ � S1(c + Ψ)ef ⩽ � S1 cev + Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Seeing as though � S1 cev = cLv(S1) ∈ �2πc(1 − ε) √ 1 + c2 , 2πc √ 1 + c2 � it follows that, as desired, 0 ⩾ � S1 ∂nw = � S1(κu − κv)eu ⩾ −Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' ∥∆w∥L1(B1) ⩽ Ψ(ε|c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the divergence theorem and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1, we have 0 ⩽ − � B1 ∆w = − � S1 ∂nw ⩽ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For all λ ⩾ 1, and for all ε > 0 small enough (depending on λ), we have π ⩽ � B1 eλw ⩽ 4π + Ψ(ε|c, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='4, take Ω = B1, Ψ as above, and δ = 4π − λΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For sufficiently small ε > 0 (depending on λ), we have δ = δ(λ) ∈ (0, 4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then (since also w ⩾ 0) π ⩽ � B1 eλw ⩽ � B1 e Ψ ∥∆w∥L1(Ω) |λw(x)| ⩽ 4π2 4π − λΨ · 4 = 4π + Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For all p ∈ [1, 2), ∥∇w∥Lp(B1) ⩽ Ψ(ε|c, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' This is immediate from the following, concerning solutions of the Poisson equation on the unit ball: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1 Fix p ∈ [1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then there exists a C = C(p) > 0 such that if w is a smooth solution of � −∆w = f on B1 w = 0 on S1, then ∥∇w∥Lp(B1) ⩽ C(p)∥f∥L1(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ 1To prove this, one can compute directly from the Green’s Representation of the solution w and apply the Minkowski integral inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' STABILITY OF CONVEX DISKS 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For all p ∈ [1, 2), ∥w∥W 1,p(B1) ⩽ Ψ(ε|c, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Moreover, for all p ∈ [1, ∞), ∥w∥Lp(B1) ⩽ Ψ(ε|c, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since w = 0 along S1, the Poincar´e Inequality implies the first claim in the Lemma, after which the second part follows from the Sobolev Embedding Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For every λ ⩾ 1 and p ∈ [1, 2), ∥eλw∥p W 1,p(B1) ⩽ 4π + Ψ(ε|c, p, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In particular, ∥∇eλw∥Lp(B1) ⩽ Ψ(ε|c, p, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3, ∥eλw∥p Lp(B1) ⩽ 4π + Ψ for any p ∈ [1, ∞), provided ε is small enough depending upon the choice of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Now, observe that ∇(eλw) = λeλw∇w so that if p ∈ [1, 2), and we choose q ∈ (1, 2/p), ∥∇(eλw)∥p Lp(B1) = λp � B1 eλpw|∇w|p ⩽ λp �� B1 eλpq′w � 1 q′ �� B1 |∇w|pq � 1 q ⩽ λp (4π + Ψ) 1 q′ �� B1 |∇w|pq � 1 q ⩽ Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Indeed, q ∈ (1, 2/p) implies that λpq′ ∈ (1, ∞) and pq ∈ (1, 2), so that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3 applies to the first integral factor and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='5 applies in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' As such, ∥eλw∥p W 1,p(B1) = ∥eλw∥p Lp(B1) + ∥∇(eλw)∥p Lp(B1) ⩽ 4π + Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ At last, we can establish the main proposition of the section: Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then there exist c0 > 0, λ0 ⩾ 1, p0 ∈ [1, 2), and η0 > 0 such that for every k ⩾ 1, there is a function wk satisfying the conclusions of the all the previous lemmas with some Ψ = Ψ(k−1|c0, p0, λ0) ց 0, and ∥eλ0wk − 1∥W 1,p0(B1) ⩾ η0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the Poincar´e Inequality, if ck = 1 π � B1 eλ0wk then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='6, ∥eλ0wk − ck∥Lp0(B1) ⩽ ∥∇eλ0wk∥Lp0(B1) ⩽ Ψ(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3, ck ∈ [1, 4 + Ψ(k−1)], so up to extracting a subsequence, ck → c ∈ [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In fact, it must be that c = 1, which can be seen from the Sobolev Trace Theorem and the fact that wk = 0 on S1: ∥1−c∥Lp0(S1) ⩽ ∥eλ0wk −c∥W 1,p0(B1) ⩽ ∥eλ0wk −ck∥W 1,p0(B1)+∥ck−c∥W 1,p0(B1) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' But then we find that ∥eλ0wk − 1∥p0 W 1,p0(B1) = ∥eλ0wk − 1∥p0 Lp0(B1) + ∥∇eλ0wk∥p0 Lp0(B1) ⩽ Ψ(k−1), 10 HUNTER STUFFLEBEAM contradicting the existence of η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The first claim thus follows, and by the Sobolev Embedding Theorem the second claim follows as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Estimates for the Constant Curvature Comparison Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In this section we study a sequence of c-convex constant curvature 1 comparison disks (B1, e2vkgeuc) whose boundary lengths go to the extremal length 2π/ √ 1 + c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' This is complicated by the fact that the process of representing such disks via the Uni- formization Theorem has a large gauge invariance and consequent loss of compact- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To illustrate this, recall that we isometrically represented (B1, e2vgeuc) as a smooth domain Ω in the round sphere, and that this choice is unique only up to action of the symmetry group O(3) ⟳ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Under stereographic projection, such a domain is isometric to a disk of the form (D, e2ρ0geuc) for some smooth domain D ⊂ R2, where e2ρ0geuc is the round metric in stereographic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since we seek to relate v to ρ0 on B1 via pullback of (D, e2ρ0geuc) by an isometry F : B1 → D, the ambiguity in the choice of Ω, and thus D, matters a great deal insofar as esti- mates are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In particular, the gauge invariance that we seek to control manifests in the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Given a function f : B1 → R and a diffeomorphism φ ∈ Diff(B1), we employ the notation (f)φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= f ◦ φ + log |φ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Observe that if φ ∈ Conf(B1) is a Mobius transformation of B1, then (B1, e2(f)φgeuc) is isometric to (B1, e2fgeuc) under the action of pullback by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Thus, the conformal factors f and (f)φ represent the same geometric object, and therefore the corresponding curvature data in PDE form is invariant under this action of Conf(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For example, if v is any solution of −∆v = e2v on B1 and φ ∈ Conf(B1), we have (using an apostrophe to denote complex differentiation) −∆(v)φ = −∆(v ◦ φ) = −(∆u) ◦ φ|φ′|2 = e2v◦φ|φ′|2 = e2(v)φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Here, of course, we use the fact that the Mobius transformations are holomorphic on B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To combat this gauge invariance, we will first choose the Ωk in a way which takes uniform advantage of the quantitative inradius estimate from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The resulting isometries relating the (Dk, e2ρ0geuc) to the (B1, e2vkgeuc) then form a sequence of conformal diffeomorphisms whose images converge in a nice way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' After normalizing these maps we obtain a limit mapping, and these normalizations provide us with our final choice of gauge for looking at the sequence of constant curvature comparison disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Before proceeding, we quickly recall for the reader’s convenience the necessary complex-analytic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Review of Conformal Mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In this section we introduce the concepts that we will need from the theory of conformal mappings in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Primarily, we are concerned with the behavior of a sequence of conformal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' biholomorphic) mappings Fk : B1 → Dk, where the domains Dk in R2 are smooth and uniformly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 2 In this section it is natural to identify R2 with C, and to view a map between regions in R2 as a complex valued mapping in the standard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 2The subject of conformal, and more broadly harmonic, mappings in the plane is wonderfully rich, and the interested reader should especially look to the books by Pommerenke [Pom11] and Duren [Dur04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' STABILITY OF CONVEX DISKS 11 In our applications, our domains Dk will converge to a limiting domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The relevant form of convergence is the following notion, due to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Carath´eodory: 3 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1 (Kernel Convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let Dk ⊂ C be simply connected domains with 0 ∈ Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The kernel of the sequence {Dk} is defined to be {0} if 0 /∈ int(� Dk), and otherwise is defined to be the largest domain D ⊂ C containing 0 with the property that each point of D possesses a neighborhood in D which lies in cofinitely many Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We then say that Dk → D (with respect to 0) in the sense of kernel convergence if every subsequence of the collection {Dk} has the same kernel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' There are a few examples where kernel convergence is easily verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For exam- ple, a sequence of increasing open sets containing 0 has their union as their kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In our case, each of our sets will be of the form Brk ⊂ Dk ⊂ BR, where rk ր R, so again the kernel is simply the union of the Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Another way to phrase the definition of the kernel, which makes the last example clear, is as follows: for each n ⩾ 1 let Cn denote the connected component of int (Dn ∩ Dn+1 ∩ · · · ) containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' If they exist for all n, the union of the Cn is defined to be the kernel, and otherwise it is defined to be {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The raison d’ˆetre for this notion of convergence is the following famous theorem: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1 (Carath´eodory’s Convergence Theorem4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let {Dk ⊂ C} be a se- quence of simply connected domains containing 0, and {Fk : B1 → Dk} a sequence of bijective conformal mappings with Fk(0) = 0 and F ′ k(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then the Fk converge to a limit function F uniformly on compact subsets of B1 if and only if Dk → D ̸= C in the kernel sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In the event of convergence with D = {0}, F ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' If we have convergence with D ̸= {0}, then D is simply connected, F is a bijective conformal mapping of B1 and D, and the F −1 k converge locally uniformly on D to F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Lastly, it is worth recalling the Kellogg-Warschawski Theorem, which ensures that the limit maps we get will be smooth up to and on the boundary of our regions of interest: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2 (Kellogg-Warschawski5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Suppose F : B1 → D is a conformal bijec- tion, where the boundary of the domain D is a Jordan curve of regularity Cm,α for some m ⩾ 1 and α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then F (m) has an α-H¨older continuous extension to B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In particular, if D is smooth then all derivatives of F extend continuously to B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Fixing the Gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The goal of this section is to prove the following proposi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Recall that we use the notation (f)φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= f ◦φ+log |φ′|, and that if φ ∈ Mob(B1) is a Mobius transformation of B1, then (B1, e2(f)φgeuc) is isometric to (B1, e2fgeuc) under the action of pullback by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3This definition often takes slightly different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The current one is convenient for us and is found in [Dur04] 4Many different conceptions of this result exist, and the one we are using here, as stated in [Dur04], is convenient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For other conceptions, the article [IK18] is a great reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 5As stated in [Pom11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 12 HUNTER STUFFLEBEAM Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Given εk ց 0 and smooth functions vk : B1 → R with \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆vk = e2vk on B1 ∂nvk + 1 ⩾ ceuk on S1 � S1 evk ⩾ 2π − εk, we can find a sequence of Mobius transformations φk ∈ Conf(B1) such that (vk)φk → ρc in Cm loc(B1) for any m ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Geometrically, of course, we are showing that the constant curvature comparison disks (B1, e2vkgeuc) have isometric representations (B1, e2(vk)φk geuc) which converge locally in the Cm loc Cheeger-Gromov sense to the model space (B1, e2ρcgeuc) for every m ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To begin proving this, we make an initial choice of gauge by conveniently choosing the domains Ωk ⊂ S2 which isometrically realize our constant curvature comparison disks (B1, e2vkgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We consider round S2 to be isometrically embedded in R3 in the standard way, and we let Φ: S2 \\ N → R2 denote stereographic projection from the north pole N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Now, choose Ωk to lie in the southern hemisphere of S2, containing the south pole S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Recalling that Ωk has an outball of radius cot−1(c) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1, we can choose Ωk so that its outball is centered at S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1, Ωk also has an inball of radius cot−1(c + Ψ(εk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' It thus follows that with Ωk chosen in this way, the geodesic disk of radius 2cot−1(c+Ψ(εk))−cot−1(c) centered at S is contained in Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' As k → ∞ and εk ց 0, this disk expands up to the fixed outball of radius cot−1(c) for all the Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Under Φ, Ωk is isometric to a disk of the form (Dk, e2ρ0geuc), where Dk is a smooth convex domain in R2 containing 0 and contained in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' More precisely, the Dk all lie within BRc, and each contains the ball Bℓk, where ℓk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= sin[2cot−1(c + Ψ(εk)) − cot−1(c)] 1 + cos[2cot−1(c + Ψ(εk)) − cot−1(c)] ր Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Consequently, we can easily see that Dk → BRc in the kernel sense: we have BℓN ⊂ CN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= int(DN ∩ DN+1 ∩ · · · ) ⊂ BRc, and so the kernel, which we recall as being the union of CN, is exactly BRc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Denote by Fk : B1 → Dk the smooth map which provides an isometry between our two disks (Dk, e2ρ0geuc) and (B1, e2vkgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' F is a conformal diffeomorphism of B1 and Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Indeed, Fk an isometry means that e2vkgeuc = F ∗ k (e2ρ0geuc) = e2ρ0◦FkF ∗ k geuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Expanding out, we find that (e2vkgeuc)ij = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 e2ρ0◦Fk|∂1F|2 i = j = 1 e2ρ0◦Fk⟨∂iFk, ∂jFk⟩ i ̸= j e2ρ0◦Fk|∂2F|2 i = j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Thus, it follows that Fk is a conformal diffeomorphism of B1 and Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ One easily sees from the Cauchy-Riemann Equations that Fk is a biholomor- phic mapping and, if we use the complex analytic notation (−)′ for differentiation, |∂iFk|2 = |F ′ k|2 for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Thus, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Under the isometry Fk, we have vk = ρ0 ◦ Fk + log |F ′ k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' STABILITY OF CONVEX DISKS 13 Our task now is to investigate the convergence of the maps Fk : B1 → Dk, and it is here where our specific choices concerning the domains Ωk help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The main result of this section boils down to the following: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Given εk ց 0, vk, and Fk all as above, we can find Mobius transformations φk ∈ Conf(B1) such that the maps ˆFk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= Fk ◦ φk → Rc · Id in Cm loc(B1) for any m ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In particular, (vk)φk → ρc in Ck loc(B1) for any k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Choose ηk ∈ Conf(B1) such that ˜Fk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= Fk ◦ η−1 k satisfies the normalization conditions ˜Fk(0) = 0 and ˜F ′ k(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then ˜Fk : B1 → Dk satisfies all the conditions of the Carath´eodory Convergence Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1, so we have a conformal diffeomor- phism ˜F : B1 → BRc to which the ˜Fk limit in C0 loc(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' It follows that ˜F is simply a Mobius transformation ψ ∈ Conf(B1) scaled by the factor Rc, so ˜Fk → Rc · ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Now, let ˆFk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= ˜Fk ◦ ψ−1 = Fk ◦ η−1 k ψ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= Fk ◦ φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We see that ˆFk → Rc ·Id in C0 loc(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To see that the φk provide the right change of gauge, we simply compute the pullback of (Dk, e2ρ0geuc) by ˆFk: (B1, e2(ρ0) ˆ Fk geuc) = ˆF ∗ k (Dk, e2ρ0geuc) = φ∗ kF ∗ k (Dk, e2ρ0geuc) = φ∗ k(B1, e2vkgeuc) = (B1, e2(vk)φk geuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the Cauchy Integral Theorem, for any m ⩾ 0 we have that ˆFk → Rc · Id in Cm loc(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Therefore, (ρ0) ˆ Fk(x) = ρ0 ◦ ˆFk(x) + log | ˆF ′ k(x)| → ρ0(Rc · x) + log Rc = ρc(x) and so (vk)φk = (ρ0) ˆ Fk → ρc in Cm loc(B1) for any m ⩾ 0, since ρ0 is smooth on B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Now, recall that since φk ∈ Conf(B1) the disk (B1, e2(vk)φk geuc) is isometric to (B1, e2vkgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Therefore, ˜vk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= (vk)φk satisfies all of the same curvature conditions that vk does, namely, � −∆˜vk = e2˜vk on B1 ∂n˜vk + 1 ⩾ ce˜vk on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Similarly, our original disk (B1, e2ukgeuc) is isometric to (B1, e2(uk)φk geuc) so the functions ˜uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= (uk)φk similarly satisfy all the same curvature conditions: � −∆˜uk = K˜uke2˜uk ⩾ e2˜uk on B1 ∂n˜uk + 1 ⩾ ce˜uk on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Of course, the relations ˜uk ⩾ ˜vk on B1, ˜uk = ˜vk on S1, and L˜uk(S1) = � S1 e˜uk = � S1 e˜vk ⩾ 2π(1 − εk)/ � 1 + c2 continue to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Therefore, if ˜wk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= ˜uk − ˜vk, all of the estimates of the previous section for wk apply also to ˜wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Our gauge fixing process thus consists of replacing uk, vk, and wk with their counterparts ˜uk, ˜vk, and ˜wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' These conformal factors (and their difference, respectively) all represent the same geometric objects up to isometry with the same estimates, but with the property that the new constant cur- vature comparison factors ˜vk enjoy good local convergence to the model conformal factor ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 14 HUNTER STUFFLEBEAM From here on out, for clarity of notation we will denote our gauge-corrected conformal factors and their difference with the original unaccented quantities uk, vk, and wk, implicitly assuming a-priori correction by the maps φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Producing a Metric Space Limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We now have the estimates that we need to show that the disks (B1, e2ukgeuc) converge to a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' First, let us collect what we have shown so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For λ ⩾ 1 we have eλuk − eλρc = � eλvk − eλρc� eλwk + � eλwk − 1 � eλρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the results of Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2, the pair of product terms on the right hand side tends to zero in W 1,p loc (B1) for every p ∈ [1, 2) and in Lq loc(B1) for every q ∈ [1, ∞), as can be readily seen by H¨older’s Inequality and the Sobolev Embedding Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In particular, we get strong local Sobolev convergence of the metrics e2ukgeuc to the model metric e2ρcgeuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To bootstrap this analytic convergence up to geometric convergence, we begin by showing that the distance functions of the e2ukgeuc subconverge to a semi-definite distance function on B1, which we will eventually show is the distance function of the model metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' There exists a continuous function d∞ on B1 × B1 such that, upto subsequence, duk → d∞ in C0 loc(B1 × B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Fix 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For any (x, y) ∈ B1 × B1 with x ̸= y, we have that (away from the measure zero cut locuses of y and x, respectively) |∇xduk(x, y)| = e2uk(x) and |∇yduk(x, y)| = e2uk(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Here, the norms and gradients are Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' So, with 1 ⩽ p < ∞ and the Lp(Br) bound on euk, we obtain � Br×Br |∇xduk(x, y)|p+|∇yduk(x, y)|pdxdy = � Br×Br e2puk(x)+e2puk(y)dxdy ⩽ C(p, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Thus, {duk} is bounded in W 1,p(Br × Br) for any 1 ⩽ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the Morrey- Sobolev embedding theorem this sequence is also bounded in Cα(Br × Br) for any α = 1−4/p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the compact embedding of H¨older spaces, we see that there is a continuous function d∞ on Br × Br such that, upto subsequence, duk → d∞ in Cα(Br × Br) for each α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By letting r ր 1 we thus obtain pointwise convergence of a subsequence to d∞ on B1 × B1 which is uniform on any compact subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ We remark that d∞ defines a semi-metric on B1 to which the duk limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Sym- metry and the triangle inequality follow directly from the origin of d∞ as a limit of metrics, but a priori we do not know that d∞ is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We next seek to identify d∞ with dρc, the distance function associated to the round metric on B1 with constant boundary curvature c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Identifying the Local Gromov-Lipschitz Limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In this section we show that, for any r < 1, the sequence of metric spaces underlying the � Br, e2ukgeuc � con- verges in the Gromov-Lipschitz sense to the underlying metric space of the spherical domain � Br, e2ρcgeuc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Recall that Gromov-Lipschitz convergence of metric spaces means the following: STABILITY OF CONVEX DISKS 15 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let (X, dX) and (Y, dY ) be metric spaces, and L the class of bi- Lipschitz homeomorphisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then the Gromov-Lipschitz distance is defined as dGL((X, dX), (Y, dY )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= inf F ∈L log � Lip(F), Lip(F −1) � provided such homeomorphisms exist, and is defined as +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We thus want to apply this notion of distance, for each r ∈ (0, 1), to the sequence of metric spaces (Br, duk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Based off of the work in the previous section our candi- date GL limit is (Br, d∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' However, d∞ is only known to be a semi-metric at this point, so we instead first prove directly that d∞ = dρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The uniform convergence from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='5 then implies the desired GL convergence (see [BBI01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The following argument identifying d∞ is inspired by one in [LST21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We’ll need the following Sobolev trace-type inequality to identify the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The proof follows the exact same argument as in the standard case (see, for example, [EG15]) where the curve γ is the boundary of a Lipschitz domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='6 (A Sobolev Trace Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let Ω be a precompact domain in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let γ ⊂ Ω be a curve of finite length which as a set is Lipschitz, in the sense that it is locally the graph of a Lipschitz function over its tangent line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then, for any p ⩾ 1, there exists a C = C(Ω, γ, p) > 0 such that: if u ∈ W 1,p(Ω), then the trace operator T : W 1,p(Ω) → Lp(γ) is a bounded linear operator with operator norm ∥T ∥ ⩽ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We also need the following proposition, which is a version of the standard Bishop- Gromov Theorem6 where we allow for the distance balls for both metrics in con- sideration to make contact with their respective convex boundaries: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let Ω ⊂ R2 be a precompact, smooth set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Suppose g and gc are two smooth Riemannian metrics on Ω, with the properties that secg ⩾ secgc ≡ 1, and κg(∂Ω) ⩾ κgc(∂Ω) ≡ c ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let x ∈ Ω and s ∈ (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then volg (Bs(x, g) ⊂ Ω) ⩽ volgc (Bs(x, gc) ⊂ Ω) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By convexity of the boundaries, Ω can be covered by a global normal coor- dinate chart in each metric, and the volume comparison follows from the standard proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' More specifically, the assumption on curvatures tells us that g ⩽ gc on Ω, and expressing the volumes as integrals of √det g and √det gc in the normal coordinates proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' See for example [Lee18] Chapter 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Now we are ready to prove that d∞ = dρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2 (d∞ ⩽ dρc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Fix any x, y ∈ B1 and r < 1 so that x, y ∈ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let γ be the ρc geodesic in B1 from x to y, which by convexity lives inside of Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since each ρc geodesic in B1 is certainly an admissible curve in the above Trace Theorem, euk → eρc in W 1,p(Br) for all p ∈ [1, 2) implies that euk → eρc in Lp(γ) for all p ∈ [1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 6Technically, since we call for sectional curvature bounds, this might be more accurately called a boundary version of the G¨unther volume comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 16 HUNTER STUFFLEBEAM Therefore, dρc(x, y) = � γ eρc = lim � γ euk = lim Luk(γ) ⩾ lim inf duk(x, y) = d∞(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3 (d∞ ⩾ dρc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For the sake of contradiction, suppose that there were to exist some x, y ∈ Br ⊂ B1 with R∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= d∞(x, y) < R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= dρc(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since y ∈ BR(x, d∞) but y /∈ BR(x, dρc), by continuity of d∞ we have that volρc (BR(x, dρc)) < volρc (BR(x, d∞)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Next, choose an η > 0 small and recall that duk → d∞ uniformly on Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For all large enough k we then have that Br ∩ BR−η(x, d∞) ⊂ BR(x, duk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Putting these two facts together with convergence of e2uk to e2ρc in L1(Br) and applying our version of Bishop-Gromov with boundaries, we obtain volρc � Br ∩ BR−η(x, d∞) � = lim voluk � Br ∩ BR−η(x, d∞) � ⩽ lim inf voluk (BR(x, duk)) ⩽ volρc (BR(x, dρc)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We now send r ր 1 and η ց 0 (equivalently, take the union of the sets Brj ∩ BR−ηj(x, d∞) for a sequence rj ր 1 and ηj ց 0) to obtain a contradiction with our first strict volume estimate above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' □ Therefore, we have identified the limit semi-metric d∞ as the bonafide dis- tance function dρc deriving from the round metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Consequently, by our work in the previous section, we can say with proof that for every r < 1, the manifolds (Br, e2ukgeuc) converge in the Gromov-Lipschitz sense to the round spherical cap (Br, e2ρcgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Identifying the Intrinsic Flat Limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In this section, we prove that the se- quence (B1, e2ukgeuc) converges in the Sormani-Wenger Intrinsic Flat (SWIF) sense to the spherical cap (B1, e2ρcgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By Wenger’s Compactness Theorem (discussed shortly), the sequence (B1, e2ukgeuc) is sure to subconverge to some limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Our task will be identifying it via direct geometric constructions and estimates, taking advantage of the local GL convergence from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We first recall the general idea of SWIF convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' For an earnest introduction to these ideas, the original papers of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Sormani and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Wenger [SW11], and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Ambrosio and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Kircheim [AK00], are excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' An integral current space is, loosely speaking, a metric space (X, d) paired with an integral current structure T on (X, d) in the sense of Ambrosio-Kircheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='7 In a nutshell, the intrinsic flat distance is the Federer-Fleming flat distance between two integral current spaces isometrically embedded in the most efficient way into a common complete metric 7For our purposes, (X, d) will be the metric space underlying a smooth compact Riemannian n- manifold, and the natural integral current structure T is simply integration of differential n-forms over the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' STABILITY OF CONVEX DISKS 17 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' It is thus analogous to the Gromov-Hausdorff distance8, where our objects are now integral current spaces instead of compact metric spaces, the ambient embedding space is now a complete metric space instead of a compact one, and the Hausdorff metric of the ambient space is replaced by the Federer-Fleming flat distance between integral currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' As proved in [SW11], two integral current spaces have SWIF distance 0 iff they are isometric via an orientation preserving map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' What is most important for us is the following method for estimating the SWIF distance: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3 (SWIF Convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let (M1, g1) and (M2, g2) be compact, ori- ented Riemannian n-manifolds, and suppose that there exist distance preserving9 inclusions φi : Mi → Z of the Mi into a common, metrically complete Riemann- ian manifold Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Suppose also that we can find oriented submanifolds An ⊂ Z and Bn+1 ⊂ Z such that, in the language of currents, �(φ1)#M1�−�(φ2)#M2� = A+∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' That is, for every ω ∈ Ωn(Z), � M1 φ∗ 1ω − � M2 φ∗ 2ω = � A ω + � ∂B ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then we have that dIF ((M1, g1), (M2, g2)) ⩽ voln(A) + voln+1(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' In [Wen10], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Wenger proved the following compactness theorem for sequences of integral current spaces, which directly applies to our sequence (B1, e2ukgeuc): Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3 (Wenger’s Compactness Theorem, for Riemannian Manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Sup- pose that (Mj, gj) is a sequence of oriented Riemannian n-manifolds with uniformly bounded diameters, n-volumes, and boundary (n−1)-volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then there exists an integral current space (possible the zero space) to which the (Mj, gj) subconverge in the SWIF sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We reiterate that even when both GH and SWIF limits for a sequence of integral current spaces exist, they may not agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' However, what is known in the presence of uniform upper volume and boundary volume bounds is that the SWIF limit isometrically embeds into the GH limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Lastly, we remark that GL convergence of the underlying metric spaces implies SWIF convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' All of this is proven in [SW11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Now let us return to our problem, fixing r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We estimate the SWIF distance directly via the triangle inequality: dIF ((B1, e2ukgeuc), (B1, e2ρcgeuc)) ⩽dIF ((B1, e2ukgeuc), (Br, e2ukgeuc)) + dIF ((Br, e2ukgeuc), (Br, e2ρcgeuc)) + dIF ((Br, e2ρcgeuc), (B1, e2ρcgeuc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The middle term can be estimated by the fact that GL Convergence implies SWIF Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The first and third terms can then be estimated directly, by construct- ing embedding spaces to estimate the flat distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To be precise, to estimate the first term we define our ambient embedding spaces to be the manifolds Bk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= (B1, e2ukgeuc) × [0, εk] 8We assume some familiarity with the GH distance, but remark that the book [BBI01] provides a wonderful introduction to this and many other topics in metric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 9Namely, dZ(φi(x), φi(y)) = di(x, y) for every x, y ∈ Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 18 HUNTER STUFFLEBEAM equipped with the standard Riemannian product measures eukgeuc ⊕ dt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Our distance preserving embeddings φk : (Br, e2ukgeuc) → Bk and ψk : (B1, e2ukgeuc) → Bk are simply φk(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= (x, 0) and ψk(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= (x, εk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Thus, (expressing the condition on differential forms in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='3) we have �(ψk)∗(B1, e2ukgeuc)� − �(φk)∗(Br, e2ukgeuc)� = Ak + ∂Bk where Ak .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= S1 × [0, εk] + (B1 \\ Br, e2ukgeuc) × {εk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' So, we may estimate dIF ((B1, e2ukgeuc), (Br, e2ukgeuc)) ⩽ Area(Ak) + Vol(Bk) ⩽ 2πεk + Area(B1 \\ Br, e2ukgeuc) + Area(B1, e2ukgeuc)εk ⩽ 4πεk + Area(B1 \\ Br, e2ukgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Consider the last term Area(B1 \\ Br, e2ukgeuc) = Area(B1, e2ukgeuc) − Area(Br, e2ukgeuc) ⩽ Area(B1, e2vkgeuc) − Area(Br, e2ukgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By the inradius estimate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1, Area(B1, e2vkgeuc) → Area(B1, e2ρcgeuc), and since GL convergence implies volume convergence, the results of the last section imply that Area(Br, e2ukgeuc) → Area(Br, e2ρcgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Therefore, we conclude that, for large k, dIF ((B1, e2ukgeuc), (Br, e2ukgeuc)) ⩽ 4πεk + Ψ(1 − r|c, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Arguing similarly (and in fact more easily) for the third term, we also find dIF ((Br, e2ρcgeuc), (B1, e2ρcgeuc)) ⩽ Ψ(1 − r|c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since the middle term is small by the aforementioned GL convergence, we conclude by arbitrariness of r < 1 that (B1, e2ukgeuc) → (B1, e2ρcgeuc) in the SWIF sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Showing the Intrinsic Flat Limit ≡ Gromov-Hausdorff Limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' We are now ready to show that (B1, e2ukgeuc) → (B1, e2ρcgeuc) in the GH sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' By Gro- mov’s Compactness Theorem, the (B1, e2ukgeuc) subconverge to some metric space (X, d), but the relationship between (B1, e2ρcgeuc) and (X, d) is a priori unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Utilizing recent work of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Perales, we will show that the SWIF limit agrees with the GH limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To state it, we introduce the following notation: Let (M, g) be a Riemannian manifold with boundary ∂M, let d denote the distance function of g, and for δ > 0 define the δ-inner region of M by M δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='.= {x ∈ M : d(x, ∂M) > δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' On M δ there are two distance functions, the restriction d|Mδ and the induced length metric dMδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Given this notation, we can state the following expedient adaptation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='2 in [Per18]: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='4 (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Perales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let δ, Di, L, V, θ > 0 and δi ց 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let (Mk, gk) be a sequence of compact, oriented 2-manifolds with boundary such that (A) Kgk ⩾ 0 (B) AreagkMk ⩽ A (C) Lengthgk∂Mk ⩽ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' (D) diam(M δi k , dM δi k ) ⩽ Di STABILITY OF CONVEX DISKS 19 (E) There exists some qk ∈ M δ k such that AreagkBδ(qk, gk) ⩾ θδ2 (F) There exists a compact metric space (X∂, d∂) such that (∂Mk, dk) → (X∂, d∂) in GH Then up to a subsequence (not relabeled), there is a compact metric space (X, dX) such that (Mk, dk) → (X, dX) in the GH sense and an integral current space (Y ⊂ X, dX, T ) such that (Mk, dk, Tk) → (Y, dX, T ) in the SWIF sense where we also have that X \\ X∂ ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Moreover, (G) If X∂ ⊂ Y , then Y = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Let’s carefully check the conditions in this theorem when we take (Mk, gk) = (B1, e2ukgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Items (A), (B), (C) all follow immediately from the hypotheses, so we focus on the the last four conditions: (D) By convexity, each δi-inner region of every (B1, e2ukgeuc) is convex10, and hence the induced length metric is exactly the restriction of dk to the δi- inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The distance functions dk are bounded uniformly by π, so we may take Di = π for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' (E) Take qk = 0 for every k, δ = π/4, and θ = π/8, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then Bδ(0, ρc) = {x ∈ B1 : dρc(0, x) < δ} = B1/(1+ √ 2) ⊂ B2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since duk → dρc uniformly on B2/3, for all large k we have that Bδ(0, gk) = {x ∈ B1 : duk(0, x) < δ} ⊃ {x ∈ B1 : dρc(0, x) < δ} = Bδ(0, ρc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Therefore, AreagkBδ(0, gk) ⩾ AreagkBδ(0, ρc) → AreaρcBδ(0, ρc) and for all large k we have AreagkBδ(0, gk) ⩾ 1 2AreaρcBδ(0, ρc) = π(1 − cos(δ)) ⩾ π 4 δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To check that 0 ∈ (B1, e2ukgeuc)δ for all k, first let ω ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Then we have lim duk(0, ω/ √ 3) = dρc(0, ω/ √ 3) = π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' and thus for all large k and every ω ∈ S1, duk(0, ω/ √ 3) > π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Consequently, given any η ∈ S1, duk(0, η) ⩾ inf ω∈S1 duk(0, ω/ √ 3) > π/4 = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' (F), (G) By our setup, ∂(B1, e2ukgeuc) is isometric to ∂(B1, e2vkgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The latter boundary is isometric to ∂Ωk ⊂ S2, a convex simple closed curve which con- verges in the Hausdorff, and therefore Gromov-Hausdorff, sense to the cir- cle of length 2π/ √ 1 + c2 which bounds the geodesic disk of radius cot−1(c) about the south pole–an isometric realization of the SWIF limit of the (B1, e2ukgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Since the SWIF limit is the entire closed disk, the last con- dition is also satisfied, and we can at last conclude that the SWIF limit is isometric to the GH limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 10For instance, see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='9 in [CE75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 20 HUNTER STUFFLEBEAM Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' It would be interesting to know if one can directly estimate the GH distance between (B1, e2ρcgeuc) and the (B1, e2ukgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' A natural approach would be to consider the identity maps φk : Br → Br ⊂ B1, and try to prove that for large k they are ε-Hausdorff approximations of (Br, e2ρcgeuc) and (B1, e2ukgeuc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' While the local GL convergence easily shows that these maps have small distortion, it seems more challenging to show directly that the ε-neighborhood of their images in (B1, e2ukgeuc) cover all of B1, due to a priori non-uniformity of the convergence of the duk to dρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Proof of the Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' To summarize, we supposed that there were a δ0 > 0 such that, for every sequence εk ց 0, we could find an (Mk, gk) as in the statement with Lk(∂Mk) ⩾ 2π(1 − εk)/ √ 1 + c2 but d((Mk, gk), Bcot−1(c)) ⩾ δ0 > 0, where d = dGH or dIF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' As shown in the previous sections, for either choice of d any such sequence of disks will subconverge to Bcot−1(c), contradicting the existence of δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Hence, the main result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Acknowledgments The author would like to thank Renato Bettiol and their adviser Davi M´aximo for being calming voices when an error in an early draft was found, as well as the Fields Institute for its hospitality during the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Much of the final work was completed there during the author’s visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' References [AK00] Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Mathematica 185 (2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1, 1–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, American mathematical Society, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [BM91] Ham Brezis and Frank Merle, Uniform estimates and blowup behavior for solutions of δ(u) = v(x)eu in two dimensions, Communications in Partial Differential Equations 16 (1991), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 8-9, 1223–1253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [CC96] Jeff Cheeger and Tobias H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Colding, Lower bounds on ricci curvature and the almost rigidity of warped products, Annals of Mathematics 144 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1, 189–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [CE75] Jeff Cheeger and David G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Ebin, Comparison theorems in riemannian geometry, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [CL22] Jingyi Chen and Yuxiang Li, Uniform convergence of metrics on alexandrov surfaces with bounded integral curvature, arXiv, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [CM97] Tobias H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Colding and William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Minicozzi II, Harmonic functions with polynomial growth, Journal of Differential Geometry 46 (1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1, 1 –77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [Dra18] Kostiantyn Drach, Inradius estimates for convex domains in 2-dimensional alexandrov spaces, Analysis and Geometry in Metric Spaces 6 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1, 165–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [Dur04] Peter Larkin Duren, Harmonic mappings in the plane, Cambridge University Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [EG15] Lawrence C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Evans and Ronald F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Gariepy, Measure theory and fine properties of func- tions, CRC Press, Taylor and Francis Group, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [FMP08] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Fusco, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Maggi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Pratelli, The sharp quantitative isoperimetric inequality, Annals of Mathematics 168 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3, 941–980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [HW09] Fengbo Hang and Xiaodong Wang, Rigidity theorems for compact manifolds with bound- ary and positive ricci curvature, Journal of Geometric Analysis 19 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3, 628– 642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [IK18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Izuki and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Koyama, An elementary proof of the carath´eodory kernel convergence theorem, Azerbaijan Journal of Mathematics 8 (January 2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1, 69–85 (English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' STABILITY OF CONVEX DISKS 21 [Kli11] Wilhelm P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Klingenberg, Riemannian geometry, De Gruyter, Berlin, New York, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [Lee18] John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Lee, Introduction to riemannian manifolds, Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [LP18] Nan Li and Raquel Perales, The sormani-wenger intrinsic flat convergence of alexandrov spaces, Journal of Topology and Analysis (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [LST21] Yuxiang Li, Jianxin Sun, and Hongyan Tang, Metrics on a surface with bounded total curvature, International Mathematics Research Notices 2022 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 17, 13212– 13245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [LT17] Yuxiang Li and Hongyan Tang, Metrics on s2 with bounded ∥kg∥L1 log L1 and small ∥kg − 1∥L1, arXiv, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [Per18] Raquel Perales, Convergence of manifolds and metric spaces with boundary, Journal of Topology and Analysis 12 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 03, 735–774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [Pom11] Christian Pommerenke, Boundary behaviour of conformal maps, Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [SW11] Christina Sormani and Stefan Wenger, The intrinsic flat distance between riemannian manifolds and other integral current spaces, Journal of Differential Geometry 87 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [Wen10] Stefan Wenger, Compactness for manifolds and integral currents with bounded diameter and volume, Calculus of Variations and Partial Differential Equations 40 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' 3-4, 423–448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' [WZ21] Bing Wang and Xinrui Zhao, Canonical diffeomorphisms of manifolds near spheres, arXiv, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' The University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=', 209 South 33rd Street, Philadelphia, PA 19104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} +page_content=' Email address: hstuff(at)sas(dot)upenn(dot)edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf'} diff --git a/PtFRT4oBgHgl3EQf6Djm/content/2301.13675v1.pdf 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switching in an antiferromagnet + +M. A. Weiss1, A. Herbst1, J. Schlegel1, T. Dannegger1, M. Evers1, A. Donges1, M. Nakajima2, A. +Leitenstorfer1, S. T. B. Goennenwein1, U. Nowak1 & T. Kurihara1,3 +1Department of Physics, University of Konstanz, D-78457 Konstanz, Germany. +2Institute of Laser Engineering, Osaka University, Japan. +3Institute for Solid State Physics, The University of Tokyo, Japan. + +Owing to their high magnon frequencies, antiferromagnets are key materials for future high-speed +spintronics. Picosecond switching of antiferromagnetic order has been viewed a milestone for +decades and pursued only by using ultrafast external perturbations. Here, we show that picosecond +spin switching occurs spontaneously due to thermal fluctuations in the antiferromagnetic orthoferrite +Sm0.7Er0.3FeO3. By analysing the correlation between the pulse-to-pulse polarisation fluctuations of +two femtosecond optical probes, we extract the autocorrelation of incoherent magnon fluctuations. +We observe a strong enhancement of the magnon fluctuation amplitude and the coherence time +around the critical temperature of the spin reorientation transition. The spectrum shows two distinct +modes, one corresponding to the quasi-ferromagnetic mode and another one which has not been +previously reported in pump-probe experiments. Comparison to a stochastic spin dynamics +simulation reveals this new mode as smoking gun of ultrafast spontaneous spin switching within the +double-well anisotropy potential. + +Some of the most intriguing effects in physics rest on fluctuations. Incoherent thermal fluctuations of spins +critically determine the magnetic properties of correlated systems. Thermally excited magnons are a driving force +of a rich variety of fundamental physical phenomena such as phase transitions and spin caloritronic effects1–3 +whereas their non-deterministic properties promise unique applications such as probabilistic computing4. +Incoherent spin fluctuations have traditionally been studied either indirectly through the temperature dependence +of macroscopic properties such as heat capacity, conductivity, and magnetic susceptibility5,6, or in the frequency +domain through optical probes relying on e.g. the Raman effect7,8 or diffraction9. For measuring relatively slow +spin fluctuation dynamics in paramagnets in the range of MHz to GHz frequencies, spin noise spectroscopy +(SNS)10–15 has been employed. In contrast to paramagnets, correlated spin systems exhibit collective excitations, +i.e., magnons. Antiferromagnets (AFM) possess especially high frequency magnons reaching into the THz range16, +and are thereby attracting enormous attention from the viewpoint of accelerating the conventional ferromagnet- +based spintronics devices17,18. However, due to their ultrafast dynamics that go beyond state-of-the-art electronics, +conventional SNS cannot detect antiferromagnetic spin fluctuations. The rich spin physics in AFMs arising from +their complicated spin textures have only been resolved with ultrafast pump-probe spectroscopy, where time +resolutions down to femto- or even attoseconds19 are available. Still, such pump-probe measurements are of +perturbative nature and therefore, they cannot detect the incoherent dynamics that spontaneously exist due to +thermal or quantum mechanisms. A femtosecond dynamical access to the spin fluctuations of AFMs has not been +reported to the best of our knowledge. +In this work, we experimentally demonstrate the spontaneous incoherent sub-THz magnon fluctuation dynamics +in the AFM for the first time. This is achieved by a novel experimental principle, inspired by the emerging field of +sub-cycle quantum optics20–22. Here, we analyse magnon fluctuation dynamics via their temporal autocorrelation +function, by measuring the statistical correlations of polarisation noise imprinted on two subsequent femtosecond +probe pulses [Figure 1a]. The two linearly polarised, spectrally separate pulses with a variable time delay Δ𝑡 are +focused on the sample. Upon transmission of the pulses at times t and 𝑡 + ∆t, transient magnetisation fluctuations +δ𝑀�(𝑡) and δ𝑀�(𝑡 + Δ𝑡) parallel to the propagation direction introduce polarisation changes δα(t) and δα(𝑡 + Δ𝑡) +to each probe, respectively, via the Faraday effect. The polarisation states of the transmitted probe pulses is +individually analysed with independent polarimetric detectors. The pulse-to-pulse fluctuations of the detector +output is extracted by sub-harmonic lock-in amplification21, multiplied in real time and averaged over ~108 pulses +at each delay position. By this method, the time correlation trace of the out-of-plane sub-THz magnetisation +dynamics 〈δ𝑀c(𝑡)δ𝑀c(𝑡 + Δ𝑡)〉 is precisely unravelled22,23 [See Methods section for further details]. + +Our sample is a single crystal of the orthoferrite Sm0.7Er0.3FeO324. In this material, the electron spins of the Fe3+ +ions are antiferromagnetically coupled. A Dzyaloshinskii-Moriya interaction25,26 (DMI) results in a slight spin canting +and a weak net ferromagnetic moment (net magnetisation 𝑴). Orthoferrites have two exchange modes with +resonance at multi-THz frequencies and two magnon modes in the sub-THz regime27,28. The latter include a +quasiferromagnetic mode (F mode) and a quasiantiferromagnetic mode (AF mode). The F mode is characterised +by a precession of the weak ferromagnetic moment around its equilibrium axis, whereas the AF mode results in +its longitudinal oscillation. Sm0.7Er0.3FeO3 shows a temperature-induced second-order spin reorientation transition +(SRT) close to room temperature29. In the SRT region (𝑇� < 𝑇 < 𝑇U), 𝑴 continuously rotates from along the +crystallographic a-axis at 𝑇L = 310 K to the c-axis at 𝑇U = 330 K24. +The SRT is expressed by a change in the free +energy potential (see Fig. 1b). 𝑇� marks the +temperature where the anisotropy difference +between a- and c-axis disappears, causing an +enhanced magnetic susceptibility in the a-c-plane +and strong softening of the F mode resonance +frequency24,30. When such a system is thermally +populated, one expects a strong enhancement of +the angular distribution width at 𝑇�. +First, we investigate the magnon noise dynamics +as a function of the time delay ∆t and its scaling +with the probed volume Ω at a temperature of +294.15 K (Fig. 2a). The confocal position of the +sample is identified by the maximum signal +amplitude (z = 0 µm, green graph). A systematic +decrease of the amplitude is observed with +extending z up to ±20 µm (blue, yellow, magenta, +and red graphs in Fig. 2a). The signals are +symmetric around ∆t = 0, consistent with the fact +that we probe an autocorrelation of the temporal +dynamics of the system. All waveforms exhibit a +distinct peak at ∆t = 0 followed by a gradual +decrease and a slow oscillation around the zero +level that lasts for several tens of picoseconds. +Figure 2b shows the noise amplitude at ∆t = 0 as +a function of the longitudinal sample position with +respect to the optical focus (blue circles). As +illustrated by the black graph, the correlated +Faraday noise amplitude is fitted well by a function +which is inversely proportional to volume (z) +probed by the fundamental Gaussian mode +〈δ𝛼(𝑡)�〉 ∝ +� +�(�). Note that in paramagnets, the +Figure 1 | Schematic illustration of the experimental setup +and spin system. a, Due to the Faraday effect, two spectrally +separated fs pulses (orange and red) of variable time delay ∆t +experience a polarisation rotation proportional to out-of-plane spin +fluctuations 𝛿𝑀� . Corresponding rotation angles 𝛿𝛼�,� are +measured in separate elipsometers. After extraction of the pulse- +to-pulse fluctuations from each branch, the cross-correlation +function 〈𝛿𝛼�𝛿𝛼�〉 is calculated in real time as a function of the +delay time ∆t. L1,L2: Lenses; DM: Dichroic mirror; HWP: Half- +wave plate; WP: Wollaston prism; BPD: Balanced photo detector; +BP: electronic band-pass filter; Amp: Transimpedence amplifier; +Subh: Sub-harmonic demodulation scheme. b, Temperature +evolution of the free energy F and its influence on the spin noise +dynamics (red cones) close to the spin reorientation in +Sm0.7Er0.3FeO3. The weak ferromagnetic moment 𝑴 (black arrow) +gradually rotates from 𝜃 = 0° at 𝑇 ≤ 𝑇L to 𝜃 = ±90° at 𝑇 ≥ 𝑇U. +Here, 𝜃 is the angle between 𝑴 and the a-axis of the sample. +Figure 2 | Dependence of magnon noise dynamics on probing +volume. a, Spin noise waveforms recorded at a constant temperature +of 294.15 K for multiple longitudinal sample positions relative to the +laser focus. The confocal position 𝑧 = 0 was determined by maximising +the amplitude at 𝛥𝑡 = 0 (green graph). Amplitudes decrease +monotonically with increasing distance from focus (blue, yellow, +magenta and red graphs). b, Correlated noise amplitude at 𝛥𝑡 = 0 as +a function of lateral sample position relative to the focus (blue open +circles). The longitudinal position dependence was fitted with the +function given in Methods. + +a +α,(t) +OM0 +BPD +a +L2 +DM +图 +HAmp +BP +L1 +Sα,(t+△t) +Subh +C +WP +Smo.Ero3Feo +Dataanalysis +Subh +BPD +Cross-correlation +BP +(Sα,(t) Sα2(t+△t) +b +H +0 +0 +T=Tu +0 +I>>I +>>a +0.20 +20μm +-10 μm +o μm +0.15 ++7.5 μm ++20 μm +0.10 +0.05 +0.00 +-0.05 +-40 +-20 +0 +20 +40 +timedelay△t(ps) +Experiment +Fit +0.20 +0.15 +0.10 +0.05 +. +0.00 +-40 +-20 +0 +20 +40 +sampleposition (um)amplitude δ𝛼 of the statistical fluctuations of 𝑁 independent spins within the probing volume Ω is known to follow +the scaling law δ𝛼 ∝ +� +√� ∝ +� +√� +10,13. Here, we find the same volume scaling of Faraday noise as in conventional SNS. +Note that this dependence is not trivially understood for correlated spin systems where individual spins are coupled +to form collective magnons. Nevertheless, in the following we fix our sample position to z = 0 µm to maximise the +signal based on this feature. +Figure 3a depicts the striking variation of spin noise autocorrelation waveforms found around the SRT. The +amplitudes, oscillation periods and lifetimes depend strongly on temperature. To focus on the temperature +evolution of the signal amplitude, the amplitude at ∆t = 0 is depicted in Fig. 3b. A sharp amplitude increase is +observed in the region around 312.15 K. This point is slightly higher but close to the estimated lower threshold +temperature 𝑇L ~ 305 K of the SRT in our sample, around which temperature the anisotropy softening results in +an enhanced magnetic susceptibility27. This close coincidence indicates that the amplitude of the observed +magnon noise can be naively understood as the angle distribution of spins due to thermal occupation of the +anisotropy potential well by magnons, consistent with the model described in Fig. 1b. Beyond this temperature, +the noise amplitude decreases continuously, almost disappearing around the upper threshold 𝑇U = 320 K. The +sharp decrease observed towards the higher temperature side is explained by the equilibrium rotation of the spin +system within the SRT. In this temperature region, the net magnetisation 𝑴 continuously rotates from 𝑴 // a (in- +plane) to 𝑴 // c (out-of-plane). Our Faraday probe is sensitive only to the c-axis magnetisation fluctuation δ𝑀� of +the F mode [Supplementary Materials], which is expected to reduce as δ𝑀� ∝ cos(θ) towards higher temperature. +Therefore, the noise amplitude is expected to decrease continuously. +These findings are analysed exploiting simulations of the spin noise around the SRT in a generic orthoferrite with +parameters fitting the equilibrium properties observed experimentally [Methods]. The theory is based on an +atomistic spin model and the stochastic Landau-Lifshitz-Gilbert equation31,32. Figure 3c depicts the simulated c- +axis magnon noise autocorrelation trace in a 192x192x192 orthoferrite spin lattice. The simulation reproduces the +temporal shape of the noise waveforms in Fig. 3a, including the symmetry around ∆t = 0 and the temperature +evolution of both the time-zero peak amplitude and the subsequent slow oscillation. The peak amplitude of the +calculated waveforms is shown as a function of temperature in Fig. 3d. In the simulations, the SRT manifests as +a strong noise enhancement around 302 K followed by a decrease at higher temperatures. The nearly quantitative +agreement between the theoretical calculation and the experimental data allows us to investigate the stochastic +nature of the spin noise dynamics from a microscopic viewpoint in following discussions. +We now analyse the dynamics in the frequency domain. The Fourier spectra of the detected noise waveforms +are shown in Fig. 4a. Interestingly, two distinct spectral peaks (purple and green arrows) are observed for most +temperatures. The frequencies and amplitudes of both modes are strongly dependent on temperature. While the +two peaks are clearly distinguishable for 𝑇 ≤ 304 K, they exhibit similar frequencies when approaching the noise +enhancement region around 𝑇L and eventually become indistinguishable due to the strong broadening. At +temperatures 𝑇 ≫ 𝑇L, the spectral amplitude is significantly reduced because of the SRT. Figure 4b shows the +Figure 3 | Ultrafast magnon noise dynamics near spin reorientation in Sm0.7Er0.3FeO3. a, Correlated noise as a function of time +delay 𝛥𝑡 between probing pulses for multiple temperatures near the spin reorientation in Sm0.7Er0.3FeO3. b, Experimentally +determined time-zero amplitude as a function of temperature. In the region of noise enhancement around 312 K. Error bars are added +considering the uncertainty associated with background subtraction procedure [Methods]. c, Magnon noise simulation based on +atomistic spin models and the stochastic Landau-Lifshitz-Gilbert equation. d, Temperature evolution of the simulated time-zero +amplitude. + +noise +elel +a +c +b +1.2 +Experiment +Simulation +294.15 K +301.37 K +3.0 +0.8 +0.6 +303.15K +301.72 K +0.4 +correl +2.0 +ise +0.4 +307.15K +301.89K +ted +300 +310 +320 +correlated noi +noise +temperature (K) +T +1.2 +noise +(0.5x)313.15K +(0.5x) 302.41 K +1.0 +E +0.2 +amplitude +319.15 K +303.45 K +0.8 +D +0.4 +323.65 K +311.00 K +? +(a.u.) +0.0 +G +0 +1 +1 +1 +-80 +-40 +0 +40 +80 +-80 +-40 +0 +40 +80 +300 +305 +310 +315 +time delay △t (ps) +time delay △t (ps) +temperature (K)peak frequencies of each mode as a function of temperature. Both modes experience a strong frequency reduction +around 𝑇L ≈ 305 K, which closely resembles the softening behaviour of the F mode around the SRT. The +temperature dependence of the high-frequency (HF) mode (green full circles in Fig. 4b) is shown to quantitatively +match with pump-probe data24 (pink crosses), clearly identifying this HF mode as the F mode in Sm0.7Er0.3FeO3. +On the other hand, the low-frequency mode (LF mode) has no correspondence in pump-probe measurements. To +the best of our knowledge, the appearance of such a mode in the SRT region of orthoferrites is reported here for +the first time. +The Fourier spectra of the stochastic LLG simulations are depicted in Fig. 4c. The appearance of the two peaks +and their softening around 𝑇L,sim ≈ 301.5 K (Fig. 4d) matches the experimental results in Fig. 4b. This agreement +between the temperature dependence of the simulated F mode fluctuation and the HF mode seen in the +experiment further supports our assignment to the original F mode. Conversely, the LF feature appears in the +experimental data from the spectrum recorded at a temperature of 𝑇 = 294.15 K to temperatures well beyond 𝑇L ≈ + 305 K, whereas it is observed in a narrower temperature region 𝑇 ⪆ 𝑇L,sim in the simulation (Fig. 4c). Both the +experimental and simulated temperature dependence of the LF spectral amplitude (Suppl. Figs. 1,4) follow a +similar trend as the time-zero amplitude as a function of temperature shown in Fig. 3b. This finding suggests the +underlying LF dynamics to contribute significantly to the total noise amplitude (Figs. 3b,d). +To gain insights into the physical origin of the LF feature, we now investigate the simulation data in more detail. +Figure 5a shows results for the temporal evolution of the c-axis projection of the normalised magnetisation 𝑚�/𝑚� +(𝑚� is the magnetisation at saturation). For 𝑇 < 𝑇L,sim, the equilibrium axis of the normalised magnetisation is +parallel to the a-axis. Consequently, fluctuations of 𝑚�/𝑚� centred around the origin are observed. When +approaching 𝑇L,sim, the fluctuations increase in amplitude and oscillation period in agreement with our previous +discussion. For 𝑇 ⪆ 𝑇L,sim, 𝑚�/𝑚� switching between two discrete states with similar amplitude but different sign +(up- and down-state) become prominent, resembling random telegraph noise (RTN33–35) on a picosecond time +scale. With increasing temperature, the number of observed switches gradually decreases until no more switching +events are recorded for 𝑇 ≫ 𝑇L,sim. Here, 𝑚�/𝑚� always fluctuates around a preferred state. When comparing the +temperatures at which the LF peak is observed in the simulated spectra (Fig. 4c) with the temperatures at which +RTN is recorded in the magnetisation time traces (Fig. 5a), it becomes clear that the emergence of the LF peak is +inherently linked to the observation of picosecond RTN. It should be noted that Fourier transform of a RTN signal +should result in an exponentially decaying autocorrelation function and thus in a Lorentzian spectrum centred +around zero36. In contrast, in our observations the LF mode exhibits a peak at finite frequencies. We attribute this +finding to the limited time window over which our traces are analysed. +Figure 4 | Magnon noise spectra near SRT in Sm0.7Er0.3FeO3. a, Fourier spectra of the measured magnon noise waveforms from +Fig. 2a. Two maxima are resolved for 𝑇 < 307.15 K (purple and green arrows), while only the low-frequency (purple arrows) mode +prevails for higher temperatures. The spectra were fitted with a double lorentzian function (dashed lines) where distinction of the two +separate peaks was possible. b, Temperature evolution of the high-frequency mode (dark green full circles) and the low-frequency +mode (purple full circles) from the spectra shown in a. The values were obtained from the fits and evaluation of the 2nd derivative zero +crossing points in a [Methods]. For comparison, quasiferromagnetic magnon mode (F mode) frequency data obtained by THz pump- +near IR probe30 are shown as pink crosses. The pump-probe data is shifted by -7.5 K to compensate for the different amount of +stationary heating in our experiment. c, Fourier spectra of the simulated waveforms (Fig. 2c). Green and purple arrows indicate the +center frequencies of the F mode and the low-frequency feature, respectively. The spectra were fitted with a triple Lorentzian function +(dashed lines). d, Temperature evolution of the simulated c-axis projection of the F mode frequency. The values were obtained from +the fits in c. + +5.0 +a +2.0 +Experiment +Simulation +个 +50 +b +40 +294.15 K +301.37 K +个 +4.0 +30 +303.15K +301.72 K +spectral +HFmode +20 +(GHz) +oLFmode +PPdata +10 +litude +3.0 +I amplitude (a.u.) +295 +300 +305 +310 +315 +ampl +1.0 +temperature (K) +307.15K +301.89K +2.0 +d +(0.2x) 313.15 K +(0.2x) 302.41 K- +1.0 +319.15 K +303.46 K +(GHz) +50 +323.65 K +311.00 K +0 +0 +1 +1 +1 +1 +1 +1 +do +20 +40 +60 +80 +0 +20 +40 +60 +80 +298 +300 +302 +304 +306 +frequency (GHz) +frequency (GHz) +temperature (K)To investigate this RTN behaviour in further detail, we plot the trajectory of the LLG-simulated magnetisation for +different temperatures around the SRT (Figs. 5b-f). At 𝑇 = 298.24 K < 𝑇L,sim (Fig. 5b), the equilibrium +magnetisation points along the a-axis (θ = 0) and F mode noise can be observed in the b- and c-projections (see +Suppl. +Fig. +5). +At 𝑇 = 𝑇L,sim ≈ 301.54 K (Fig. 5c), an enhancement of the F mode noise is observed. The spin system then rotates +towards finite angle θ for 𝑇 > 𝑇L,sim. The ±θ states are energetically degenerate and switching between them +occurs for temperatures slightly above 𝑇L,sim, i.e. for 𝑇 = 302.88 K ⪆ 𝑇L,sim (Fig. 5d). This results in an additional +magnetisation noise contribution (Suppl. Fig 5c) and accounts for the strongly enhanced noise amplitude in the +𝑇 = 302.41 K ⪆ 𝑇L,sim waveform in Fig. 4c. The configuration of the sublattice magnetisation vectors and the net +magnetisation for the switching ±θ states is shown in Fig 5f. As the temperature is elevated even further (Fig. 5e), +the switching probability becomes lower until no more switching occurs. Here, the up-state is always preferred +because of the initial conditions of the simulation. When approaching the upper threshold temperature 𝑇�,���, +switching is observed in the a-projection due to anisotropy softening along this direction. Beyond 𝑇�,���, the +equilibrium magnetisation becomes parallel to the c-axis (not shown) and the F mode contribution to the noise +vanishes (see Supplementary Material Figs 2,4). +The physical picture of the RTN dynamics can be clearly understood by a model considering the stochastic +switching between two energetically degenerate quasi-equilibrium states, which manifest as ±θ rotation states of +the equilibrium magnetisation due to the SRT (see Fig. 5g-j). In Sm0.7Er0.3FeO3, the free energy density exhibits a +periodic shape for 𝑇 < 𝑇L, whereas for 𝑇 > 𝑇L it evolves into a double-potential well with minima at ±θ24. The +rotation angle θ and the height of the potential barrier separating the two minima gradually increase until 𝑇 = 𝑇U +37. +Random switching between ±θ states occurs when the height of the potential barrier is low compared to the +thermal energy of the system (Fig. 5i, 𝑇 ⪆ 𝑇L). Further increase of the temperature in the order of 1 K significantly +changes the barrier height, while the thermal energy only changes marginally. As a result, the average lifetime 𝜏 +on each quasi-equilibrium state increases, and the switching probability declines. Note that this model even +Figure 5 | Picosecond random switching in Sm0.7Er0.3FeO3. a, Simulated time traces of the c-axis component 𝑚� of the +magnetisation normalised to the saturation magnetisation 𝑚� for multiple temperatures near spin reorientation in Sm0.7Er0.3FeO3. b- +e, Simulated trajectories (purple lines) of the normalised magnetisation (red arrow) for multiple temperatures near spin reorientation +in Sm0.7Er0.3FeO3. At temperature 𝑇 ⪆ 𝑇L,sim (d) switching events are recorded in the c-direction. The precession cones of the F mode +magnon are indicated in yellow. f, Illustration of the sublattice magnetisation vectors in Sm0.7Er0.3FeO3 magnetisation (blue, mint, +green, yellow) and the net magnetisation (red) for the energetically degenerate ±θ states. The canting angle of the sublattice +magnetisation vectors and the thus resulting net magnetisation is highly exaggerated for visibility. g-j, Orthoferrite potential landscape +for different temperatures across the SRT. The red-dotted line indicates the thermal energy Etherm of the system. At temperatures 𝑇 < +𝑇� (g), the potential exhibits a parabolic shape and the fluctuations are restricted to a single minimum around θ = 0. Slightly above 𝑇� +(h,i), a double-well potential develops and the particle randomly switches between the minima located at ±θ. For 𝑇� > 𝑇 ≫ 𝑇� (j), the +energy barrier between the minima is larger than the thermal energy of the system and no more switching events occur. + +298.24K +g +Mwwwwwwwwwwwwwwwww +T> TL +0 +0 +VE +200 +400 +600 +800 +1000 +1200 +time (ps) +T= 298.24 K +T= 301.54 K +T= 302.88 K +b +T= 311.00 K +C +c +e +103 +2 +2 +2 +2 +20 +20 +20 +3.1 +3.1 +3.1 +3.1 +2reproduces our observation that the temperature at which the time-zero amplitude of the autocorrelation becomes +maximal (Figs. 3b,d) is slightly higher than 𝑇L (Fig. 4b,d). +The evident connection between RTN in the simulated time traces and the LF peak in the spectra firmly +establishes picosecond RTN as the physical origin of the experimental LF feature. It should be noted that +stochastic physical systems exhibiting RTN have lately gathered prominence as a key enabler for probabilistic +computing. For this purpose, systems showing high-frequency RTN are desired to implement faster computing +times and higher precision33–35. While RTN on electronic timescales was intensively studied for decades in systems +exhibiting charge carrier traps e.g. in commercial semiconductor structures38,39, the fastest RTN device reported +so far remained in the nanosecond regime exploiting in-plane magnetic tunnel junctions40,41. To the best of our +knowledge, the picosecond RTN reported here marks the record switching speed to date. We attribute the high +rate in Sm0.7Er0.3FeO3 to the magnon frequency in the sub-THz region which results in shorter intervals between +switching attempts as compared to conventional ferromagnets41,42. This result further highlights the capability of +ultrafast SNS as a unique tool for probing the stochastic dynamics near a magnetic phase transition. +In summary, we demonstrate the first time-domain observation of sub-THz magnon fluctuations in the +antiferromagnetic orthoferrite Sm0.7Er0.3FeO3 near the spin reorientation transition. The drastic increase of +amplitude and coherence time within the SRT together with the low-frequency peak observed in the spectrum is +direct evidence of ultrafast random spin switching between two equilibrium states of the magnetic free energy. The +random spin switching speed reported here is the fastest ever marked and may provide a key ingredient for +ultrafast probabilistic computing, operating at THz frequencies. Our experimental concept is not only limited to +orthoferrites but also applicable to a wide range of correlated magnetic systems. Furthermore, our results shed +new light on THz magnonics in AFMs, where the influence of incoherent spin dynamics has largely been dismissed. + +Methods: + +Experiment and data post-processing: + This study exploits a modelocked Er:fibre laser system emitting pulses of a width of 150 fs at a central wavelength +of 1.55 µm, repetition rate of 40 MHz and with total energy of 5 nJ. This output is frequency doubled in a +periodically-poled lithium niobate (PPLN) device, reaching a transparency window of the orthoferrites43. +Subsequently, the pulses are spectrally split by a dichroic mirror, resulting in two linearly polarised, spectrally +distinct femtosecond pulse trains with a time delay Δ𝑡 provided by an optical delay stage in one branch. After +spatial recombination by another dichroic mirror, the pulses are focused to a spot diameter of <2 µm on the +orthoferrite sample with a transmissive objective lens with a numerical aperture of 0.4. Central wavelengths are +set to 770 nm and 780 nm, respectively, with polarisation along the a-axis of the sample. The sample is a 𝑑 = +10 µm thick, c-cut plate of single-crystal Sm0.7Er0.3FeO3. Upon transmission of the pulses through the sample of +thickness 𝑑 at times t and 𝑡 + ∆t, transient magnetisation fluctuations δ𝑀�(𝑡) parallel to the propagation direction +of the pulse trains introduce polarisation noise δα(t)∝ 𝑉(λ) ⋅ 𝑑 ⋅ μ� ⋅ δ𝑀�(𝑡) via the magneto-optic Faraday effect, +where 𝑉(λ) is the Verdet constant and μ� is the vacuum permeability. After collimating them with an additional lens, +the two probe beams are spatially separated with a dichroic mirror and sent into individual analysers to monitor +their polarisation. Each detector consists of a half-wave plate (HWP), a Wollaston prism and a pair of balanced +photo diodes. The angles of the HWPs are set to compensate for stationary components of the polarisation rotation +induced by e.g. biaxial birefringence of the sample44. The Wollaston prisms (WP) separate p- and s-polarised +components of the probes. The intensity difference of the polarisation components is then detected in balanced +photodetectors (BPD) and amplified with a transimpedance amplifier (Amp), respectively. Subsequently, the +signals pass a 20 MHz bandpass filter (BP) and they are demodulated at the first subharmonic of the laser +repetition rate (20 MHz) with a radio-frequency lock-in amplifier20. The outputs of these demodulation channels +now comprise the polarisation noise amplitudes δα(𝑡) and δα(𝑡 + Δ𝑡), respectively, as well as uncorrelated +components dominated by the shot noise of the photons in the probes. In a last step, both demodulation signals +are multiplied in real time inside the lock-in amplifier. This product is averaged over approximately 108 pulse pairs +per time delay Δ𝑡 to effectively yield the cross-correlation of Faraday noise 〈δ𝛼(𝑡)δ𝛼(𝑡 + Δ𝑡)〉. In this way, we +extract the tiny portion of correlated fluctuations originating from the sample response out of a much larger +uncorrelated background. At the same time, the two-colour scheme avoids detrimental interference effects at short +delay times, providing sensitive access to high frequencies. Furthermore, it enables the two probe beams to +collinearly overlap before the objective lens, allowing for the beam spots on the sample surface to maximally mode +matched. This is crucial to obtain magnon correlation signals with sufficiently strong amplitude at measurable +levels. + +The raw waveforms are post-processed with a third-order Savitzky-Golay filter for smoothing, as well as a linear +baseline subtraction, where the average values of the first and last 5 ps serve as a reference. Close to the SRT, +the combined effects of magnon softening and thermally induced random switching result in the magnetisation +noise not fully decaying to zero within the observation window. In this case, the linear baseline subtraction as +described above cannot be used without distorting the waveform. From 307.15 K upwards, we therefore use the +average baseline of all waveforms recorded below 307.15 K as the reference for our linear baseline subtraction. +Furthermore, in the SRT region above 307.15 K, a strong baseline increase is observed, the amplitude of which +is strongly temperature dependent and most prominent at around 311 K. The baseline increase is asymmetric +around ∆t = 0 and therefore must result from external factors, such as a slight misalignment of the delay stage. +In the temperature region between 310.35 K and 311.95 K no signature of correlated noise was observed because +of the large asymmetric background. Hence no meaningful evaluation could be carried out and we consequently +neglect the data in this region in our discussion. In all other reliable datasets recorded for temperatures larger than +307.15 K where slight asymmetric background is superimposed with the correlated spin noise, a slope correction +is employed to remove the asymmetry. To account for potential artefacts in the correlated spin noise introduced +by this procedure, we assign a relative uncertainty of 10% of the amplitude determined at each time delay. + +Atomistic spin model simulations: +Sm0.7Er0.3FeO3 is modelled as a generic orthoferrite with magnetic moments on the Fe sites only. These are treated +as classical vectors 𝑺𝒊 on a simple cubic lattice with four sublattices. The rare earth moments order only at very +low temperatures of typically 𝑇 < 6 K and are, hence, neglected. In orthoferrites, the nearest neighbour exchange +constant 𝐽� is of the order of −20 meV leading to an antiferromagnetic order with a Néel temperature in the range +of 630 K, whereas the next nearest neighbour exchange constant 𝐽� is much smaller and of the order of −1 meV +45–47. A reorientation transition can be modelled by different thermally stable anisotropies, as it is done in Ref. 48. +Here the reorientation transition is due to a competition of second-order on-site anisotropy, favouring the [001] +direction, and a second-order two-site anisotropy, favouring the [001] plane. The low-temperature state is +dominated by the on-site anisotropy whereas the high-temperature state is dominated by the more thermally stable +two-site anisotropy. With these two anisotropies one would obtain a first-order reorientation transition. By adding +a small cubic anisotropy preferring the [111] direction, one obtains a second-order reorientation transition in +agreement with experiments. The strength of the cubic anisotropy determines the width of the reorientation +transition. The weak ferromagnetism caused by the canting of the antiparallelly aligned sublattice magnetisations +originates from the oxygen-mediated Dzyaloshinskii-Moriya interaction (DMI). +Consequently, the Hamiltonian of our model reads +𝐻{𝑺𝒊} = − � 𝐽�𝑆� +�𝑆�� +⟨�,�⟩ + − � 𝐽�𝑆� +�𝑆�� +�⟨�,�⟩� +− � 𝜀���𝐷�� +� 𝑆� +�𝑆� +� +⟨�,�⟩ +− � 𝜅�� +⟨�,�⟩ +𝑆��𝑆�� − � κ� +��𝑆��𝑆�� +� +− � 𝜅� +����𝑆��𝑆��𝑆��𝑆�� +� + , +using the Einstein notation where 𝑖 and 𝑗 denote the site indices and 𝜐,𝜂 and 𝜆 the Cartesian directions. The first +two double sums correspond to the nearest neighbour interaction with 𝐽� = −22.32 meV and the next nearest +neighbour interaction with 𝐽� = −1.4 meV. The DMI is included in the nearest neighbour shell with DMI vectors +having a length of 0.18 meV for in-plane interactions and 0.25 meV for out-of-plane interactions. The directions of +the DMI vectors can be obtained using the symmetry rules of Ref 26. In the Hamiltonian 𝐷�� +� corresponds to the 𝜐- +component of the DMI vector describing the interaction between the spins on lattice site 𝑖 and 𝑗. The two-site +anisotropy is also included in the first shell with 𝜅�� = −0.1255 meV, leading to an easy 𝑥-𝑦-plane. The second +order on-site anisotropy is 𝜅� +�� = 0.905 meV, which results in an easy 𝑧-axis. There is also a small in-plane +contribution with 𝜅� +�� = 𝜅� +�� = 0.015 meV and 𝜅� +�� = 𝜅� +�� = −0.015 meV. The latter contribution is not necessary +for the reorientation transition but lifts the degeneracy of the spins in the 𝑥𝑦-plane. The fourth-order on-site +anisotropy is 𝜅� +���� = 𝜅� +���� = 𝜅� +���� = 0.036 meV. With these parameters, the model undergoes a reorientation +transition between 302 K and 322 K, where the Néel vector rotates continuously from the 𝑧-direction to the 𝑥𝑦�- +direction and the magnetisation from the 𝑥𝑦�-direction to the 𝑧-direction [Supplementary]. Note that the 𝑥, 𝑦 and +𝑧 coordinate axes of the Hamiltonian are parallel or antiparallel to the connection lines of the iron atoms, but not +parallel to the crystallographic axes of an orthoferrite 𝑎, 𝑏, 𝑐. The crystallographic unit vectors are given by +𝒆� = 1 +√2 +�𝒆� − 𝒆��, 𝒆� = 1 +√2 +�𝒆� + 𝒆��, 𝒆� = 𝒆�, + +so that the 𝑎-direction corresponds to the 𝑥𝑦�-direction, the 𝑏-direction to the 𝑥𝑦-direction and the 𝑐-direction +equals the 𝑧-direction. +To simulate the dynamics of the magnetic moments, the stochastic Landau-Lifshitz-Gilbert equation31,32 is used +which reads +d +d𝑡 𝑺� = − +𝛾 +𝜇�(1 + 𝛼�) 𝑺� × (𝑯� + 𝛼𝑺� × 𝑯�), +with +𝑯� = − 𝜕𝐻 +𝜕𝑺� ++ 𝝃�, +and a thermal Gaussian white noise term with +⟨𝝃�(𝑡)⟩ = 0, �𝜉��(𝑡)𝜉��(𝑡�)� = 2𝜇�𝛼𝑘�𝑇 +𝛾� +𝛿��𝛿��𝛿(𝑡 − 𝑡′) +with the dimensionless damping parameter 𝛼 = 0.0002. The gyromagnetic ratio is set to the value of a free electron +𝛾� = 1.76086 × 1011 �� +� , the magnitude of the magnetic moments is 𝜇� = 3.66𝜇�. On this basis, the time evolution +of a system of 1923 spins is numerically calculated via the stochastic Heun method. Our main output is the +magnetisation 𝒎(𝑡), which reads +𝒎(𝑡) = 1 +𝑁 � 𝑺� +� + . +𝑁 is the number of spins and 𝑺� denotes the spin at lattice site 𝑖. Thus, 𝒎(𝑡) is dimensionless and normalised to +unity in the following. The spectral noise amplitude 𝑃� (spectral power density) is calculated via +𝑃�(𝑓) = ��𝑚��(𝑓)� +� +𝑇 +� +with the Fourier transform +𝑚��(𝑓) = ∫ 𝑚�(𝑡)𝑒����� +� +� +d𝑡. +Furthermore, the spectral amplitude is averaged over 20 to 30 simulation runs. The time dependent correlated +noise amplitude (autocovariance) is determined by inverse Fourier transform of the spectral noise amplitude, +taking advantage of the Wiener-Khinchin theorem. + +Z-scan: +When a sample of thickness 𝑑 is placed into the focus of a Gaussian laser beam, the illuminated volume forms a +position-dependent hyperboloid +Ω(z) = 𝜋𝑤� +� �𝑑 + +��/12���� +�� +� +�, +where 𝑧 is the position of the sample along the axis of light propagation relative to the laser focus, 𝑤� is the beam +radius at its narrowest point and 𝑧� = +���� +� is the wavelength λ dependent Rayleigh length. We estimate 𝑧� to be 7 +µm. In conventional SNS, the statistical fluctuation of 𝑁 spins in the probing volume Ω results in Faraday noise of +the order δ𝛼 ∝ +� +√� . For a correlated noise amplitude, we hence expect 〈δ𝛼�〉 ∝ +� +� at ∆t = 0. +Inserting the formula for Ω(z) then yields the following fitting function for the correlated noise amplitude at ∆t = 0 +as a function of lateral sample position (Fig 2b): +𝐴 +𝜋𝑤� +� �𝑑 + 𝑑�/12 + 𝑑𝑧� +𝑧� +� +� + +where 𝐴 is a proportionality constant. + +Spectra: +The experimental noise waveforms are interpolated, zero-padded and smoothed. Subsequently, a FFT algorithm +is harnessed to obtain high-resolution spectra. The spectral features are analysed using a double Lorentzian peak +fit. In spectra where the F mode and the LF feature (see Fig. 4b) overlap, the centre frequencies are obtained by +estimating the zero-crossings of the second derivative. The simulated spectra are interpolated and smoothed, as +well. The spectral features are then analysed using a triple Lorentzian peak fit where the peaks correspond to the +LF feature, the F mode and the AF mode, respectively (see Suppl. Fig 4a-c). + +Acknowledgements: +This research was supported by the Overseas Research Fellowship of the Japan Society for the Promotion of +Science (JSPS), Zukunftskolleg Fellowship from the University of Konstanz, JSPS KAKENHI (JP21K14550, +JP20K22478, JP20H02206) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) +– Project-ID 425217212 -SFB 1432. + +Author Contributions: +T.K. and A.L. conceived the experiment. M.A.W., A.H. and T.K. developed the experimental system, performed +the measurements, and analysed the data. J.S., T.D., M.E. and A.D. performed the numerical simulations under +supervision of U.N. M.N. produced the specimen. T.K., A.L. and S.T.B.G. co-supervised the project. M.A.W. and +T.K. wrote the manuscript with help of all co-authors. + +Additional Information: +Correspondence and requests for materials should be addressed to T. Kurihara. Supplementary Information is +available for this paper. + +References +1. Weller, D. & Moser, A. Thermal effect limits in ultrahigh-density magnetic recording. IEEE Trans. Magn. 35, 4423–4439; +10.1109/20.809134 (1999). +2. Moriya, T. Spin Fluctuations in Itinerant Electron Magnetism (Springer International Publishing, Cham, 20). +3. Bauer, G. E. W., Saitoh, E. & van Wees, B. J. Spin caloritronics. Nature materials 11, 391–399; 10.1038/nmat3301 (2012). +4. Camsari, K. Y., Salahuddin, S. & Datta, S. Implementing p-bits With Embedded MTJ. IEEE Electron Device Lett. 38, 1767– +1770; 10.1109/LED.2017.2768321 (2017). +5. Falk, H. & Bruch, L. W. Susceptibility and Fluctuation. Phys. Rev. 180, 442–444; 10.1103/physrev.180.442 (1969). +6. Eckel, S., Sushkov, A. O. & Lamoreaux, S. K. Magnetic susceptibility and magnetization fluctuation measurements of +mixed gadolinium-yttrium iron garnets. Phys. Rev. B 79; 10.1103/physrevb.79.014422 (2009). +7. van Loosdrecht, P. H. M., Boucher, J. P., Martinez, G., Dhalenne, G. & Revcolevschi, A. Magnetic fluctuation spectrum of +CuGeO3: Raman scattering. J. Appl. Phys. 79, 5395; 10.1063/1.362316 (1996). +8. Weber, M. C. et al. Raman spectroscopy of rare-earth orthoferrites RFeO3 ( R =La, Sm, Eu, Gd, Tb, Dy). Phys. Rev. B 94; +10.1103/PhysRevB.94.214103 (2016). +9. Yakhou, F., Létoublon, A., Livet, F., Boissieu, M. de & Bley, F. Magnetic domain fluctuations observed by coherent X-ray +scattering. Journal of Magnetism and Magnetic Materials 233, 119–122; 10.1016/S0304-8853(01)00241-4 (2001). +10. Crooker, S. A., Rickel, D. G., Balatsky, A. V. & Smith, D. L. Spectroscopy of spontaneous spin noise as a probe of spin +dynamics and magnetic resonance. Nature 431, 49–52; 10.1038/nature02804 (2004). +11. Römer, M., Hübner, J. & Oestreich, M. Spin noise spectroscopy in semiconductors. The Review of scientific instruments +78, 103903; 10.1063/1.2794059 (2007). + +12. Zapasskii, V. S. et al. Optical spectroscopy of spin noise. Physical review letters 110, 176601; +10.1103/PhysRevLett.110.176601 (2013). +13. Zapasskii, V. S. Spin-noise spectroscopy: from proof of principle to applications. Adv. Opt. Photon. 5, 131; +10.1364/AOP.5.000131 (2013). +14. Berski, F., Kuhn, H., Lonnemann, J. G., Hübner, J. & Oestreich, M. Ultrahigh bandwidth spin noise spectroscopy: detection +of +large +g-factor +fluctuations +in +highly-n-doped +GaAs. +Physical +review +letters +111, +186602; +10.1103/PhysRevLett.111.186602 (2013). +15. Hübner, J. et al. Rapid scanning of spin noise with two free running ultrafast oscillators. Optics express 21, 5872–5878; +10.1364/OE.21.005872 (2013). +16. Bloch, F. Nuclear Induction. Phys. Rev. 70, 460–474; 10.1103/physrev.70.460 (1946). +17. Keffer, F. & Kittel, C. Theory of Antiferromagnetic Resonance. Phys. Rev. 85, 329–337; 10.1103/physrev.85.329 (1952). +18. Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. Nature Nanotech 11, 231–241; +10.1038/nnano.2016.18 (2016). +19. Siegrist, F. et al. Light-wave dynamic control of magnetism. Nature 571, 240–244; 10.1038/s41586-019-1333-x (2019). +20. Riek, C. et al. Subcycle quantum electrodynamics. Nature 541, 376–379; 10.1038/nature21024 (2017). +21. Riek, C. et al. Direct sampling of electric-field vacuum fluctuations. Science 350, 420–423; 10.1126/science.aac9788 +(2015). +22. Benea-Chelmus, I.-C., Settembrini, F. F., Scalari, G. & Faist, J. Electric field correlation measurements on the +electromagnetic vacuum state. Nature 568, 202–206; 10.1038/s41586-019-1083-9 (2019). +23. Yang, L. et al. Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within +quantum-dot ensembles. Nat Commun 5, 4949; 10.1038/ncomms5949 (2014). +24. Fitzky, G., Nakajima, M., Koike, Y., Leitenstorfer, A. & Kurihara, T. Ultrafast Control of Magnetic Anisotropy by Resonant +Excitation of 4f Electrons and Phonons in Sm_{0.7}Er_{0.3}FeO_{3}. Physical review letters 127, 107401; +10.1103/PhysRevLett.127.107401 (2021). +25. Dzyaloshinsky, I. A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics. Journal of Physics and +Chemistry of Solids 4, 241–255; 10.1016/0022-3697(58)90076-3 (1958). +26. Moriya, T. Anisotropic Superexchange Interaction and Weak Ferromagnetism. Phys. Rev. 120, 91–98; +10.1103/PhysRev.120.91 (1960). +27. Herrmann, G. F. Resonance and high frequency susceptibility in canted antiferromagnetic substances. Journal of Physics +and Chemistry of Solids 24, 597–606; 10.1016/S0022-3697(63)80001-3 (1963). +28. Herrmann, G. F. Magnetic Resonances and Susceptibility in Orthoferrites. Phys. Rev. 133, A1334-A1344; +10.1103/physrev.133.a1334 (1964). +29. Zhao, X. et al. Crystal growth and spin reorientation transition in Sm0.4Er0.6FeO3 orthoferrite. Solid State +Communications 231-232, 43–47; 10.1016/j.ssc.2016.02.002 (2016). +30. Yamaguchi, K., Kurihara, T., Minami, Y., Nakajima, M. & Suemoto, T. Terahertz time-domain observation of spin +reorientation in orthoferrite ErFeO3 through magnetic free induction decay. Physical review letters 110, 137204; +10.1103/PhysRevLett.110.137204 (2013). +31. Gilbert, T. L. Classics in Magnetics A Phenomenological Theory of Damping in Ferromagnetic Materials. IEEE Trans. Magn. +40, 3443–3449; 10.1109/TMAG.2004.836740 (2004). +32. Nowak, U. Classical Spin Models. In Handbook of magnetism and advanced magnetic materials, edited by H. Kronmüller +& S. S. P. Parkin (John Wiley & Sons, Chichester, England, 2007). +33. Borders, W. A. et al. Integer factorization using stochastic magnetic tunnel junctions. Nature 573, 390–393; +10.1038/s41586-019-1557-9 (2019). +34. Debashis, P. et al. Hardware implementation of Bayesian network building blocks with stochastic spintronic devices. +Scientific reports 10, 16002; 10.1038/s41598-020-72842-6 (2020). + +35. Kaiser, J. & Datta, S. Probabilistic computing with p-bits. Appl. Phys. Lett. 119, 150503; 10.1063/5.0067927 (2021). +36. Machlup, S. Noise in Semiconductors: Spectrum of a Two‐Parameter Random Signal. Journal of Applied Physics 25, +341–343; 10.1063/1.1721637 (1954). +37. Suemoto, T., Nakamura, K., Kurihara, T. & Watanabe, H. Magnetization-free measurements of spin orientations in +orthoferrites using terahertz time domain spectroscopy. Appl. Phys. Lett. 107, 42404; 10.1063/1.4927431 (2015). +38. Hung, K. K., Ko, P. K., Hu, C. & Cheng, Y. C. Random telegraph noise of deep-submicrometer MOSFETs. IEEE Electron +Device Lett. 11, 90–92; 10.1109/55.46938 (1990). +39. Ralls, K. S. et al. Discrete Resistance Switching in Submicrometer Silicon Inversion Layers: Individual Interface Traps and +Low-Frequency ( 1f ?) Noise. Phys. Rev. Lett. 52, 228–231; 10.1103/PhysRevLett.52.228 (1984). +40. Safranski, C. et al. Demonstration of Nanosecond Operation in Stochastic Magnetic Tunnel Junctions. Nano letters 21, +2040–2045; 10.1021/acs.nanolett.0c04652 (2021). +41. Hayakawa, K. et al. Nanosecond Random Telegraph Noise in In-Plane Magnetic Tunnel Junctions. Physical review letters +126, 117202; 10.1103/PhysRevLett.126.117202 (2021). +42. Kanai, S., Hayakawa, K., Ohno, H. & Fukami, S. Theory of relaxation time of stochastic nanomagnets. Phys. Rev. B 103; +10.1103/PhysRevB.103.094423 (2021). +43. Wood, D. L., Remeika, J. P. & Kolb, E. D. Optical Spectra of Rare‐Earth Orthoferrites. Journal of Applied Physics 41, +5315–5322; 10.1063/1.1658669 (1970). +44. Usachev, P. A. Optical Properties of Thulium Orthoferrite TmFeO[sub 3]. Phys. Solid State 47, 2292; 10.1134/1.2142893 +(2005). +45. White, R. L. Review of Recent Work on the Magnetic and Spectroscopic Properties of the Rare‐Earth Orthoferrites. J. +Appl. Phys. 40, 1061–1069; 10.1063/1.1657530 (1969). +46. Treves, D. Studies on Orthoferrites at the Weizmann Institute of Science. J. Appl. Phys. 36, 1033–1039; +10.1063/1.1714088 (1965). +47. Shapiro, S. M., Axe, J. D. & Remeika, J. P. Neutron-scattering studies of spin waves in rare-earth orthoferrites. Phys. Rev. +B 10, 2014–2021; 10.1103/PhysRevB.10.2014 (1974). +48. Donges, A. et al. Magnetization compensation and spin reorientation transition in ferrimagnetic DyCo5 : Multiscale +modeling and element-specific measurements. Phys. Rev. B 96; 10.1103/PhysRevB.96.024412 (2017). + + +Supplementary Material +Experiment: LF feature spectral amplitude versus temperature +The spectral amplitude of the LF peak as a function of temperature is depicted in Suppl. Fig.1. As in the +temperature evolution of the time-zero peak (Fig. 3b), a strong noise enhancement is observed at 𝑇  ≈ 312 K  > +𝑇�. In this temperature range, the spectral amplitude is an order of magnitude larger than the F-mode noise (see +Suppl. Fig. 2). This finding suggests that random switching between two quasi-equilibrium states (see main text) +is the major contributor to the observed total noise amplitude (Fig. 3b) for temperatures 𝑇  ⪆ 𝑇�. + +Theory: Equilibrium properties of the simulated orthoferrite +Supplementary Figure 2 shows the equilibrium properties of the simulated orthoferrite as a function of temperature. +The equilibrium Néel vector 𝑛�/𝑛� and magnetisation vector components 𝑚�/𝑚� (β = 𝑎, 𝑏, 𝑐), normalised to the +Suppl. Figure 1 | Spectral amplitude as a function of temperature of measured LF peak in Sm0.7Er0.3FeO3. +Suppl. Figure 2 | Equilibrium properties of the simulated orthoferrite. a,b, Néel vector 𝑛�/𝑛� (a) and magnetisation vector +components 𝑚�/𝑚� (b) in a temperature range where the systems exhibits magnetic ordering. The dashed line indicates the +tempearute 𝑇� ≈ 630 K at which the magnetic ordering is lost and the system transitions into a paramagnetic state. c,d, Néel vector +𝑛�/𝑛� (c) and magnetisation vector components 𝑚�/𝑚� (d) for multiple temperatures across the spin reorientation transition, where +the Néel and magnetisation vector rotate at 𝑇�,��� ≈ 301.5 K from the c-axis and a-axis by 90° to the a- and c-axis at 𝑇�,��� ≈ 322 +K, respectively. + +3.0 +1 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +295 +300 +305 +310 +315 +320 +325 +temperature (K)6 +β= c +a +β=c +X +X +2.5 +β=a +0.8 +q=g +I(×10-3) +&XX +X +q= +X +2 +0.6 +TN ++ +X +1.5 +[Sw/ +X +- +X +1 +0.4 +X +1 +Imp/ +- +1 +xx +×I ++1 +0.2 +X +0.5 +1 +0 +100 +200 +300 +400 +500 +600 +0 +100 +200 +300 +400 +500 +600 +temperature (K) +temperature (K) +1 +3 +T +C +d +2.5× +0.8* +XXxX +XXXXX +X +2Su +0.6 +? +3 +=C +1 +1 +*=β=a +- +1.5 +np/ +[s/dul +*-β=b +β=b +0.4 +X +X +X +0.2 +以 +I* +* +0.5 +0x +0× +X +280 +290 +300 +310 +320 +330 +280 +290 +300 +310 +320 +330 +temperature (K)saturation values 𝑛� and 𝑚�, are obtained by taking the average of multiple simulation runs. The simulated system +exhibits a transition to the paramagnetic state at Néel temperature at 𝑇�,��� ≈ 630 K and furthermore shows a +temperature-induced 2nd order reorientation transition in a finite temperature window, where the Néel and +magnetisation vector rotate at 𝑇�,��� ≈ 301.5 K from the c-axis and a-axis by 90° to the a- and c-axis at 𝑇�,��� ≈ +322 K, respectively. + +Theory: Resonance modes in Sm0.7Er0.3FeO3 +The resonance modes of Sm0.7Er0.3FeO₃ at temperatures below the SRT are shown in Suppl. Fig 3. They were +determined from the eigenfunctions of the linearised LLG equation. Below the SRT (low temperature regime, LT), +the Néel vector is oriented along the c-axis, while is parallel to the a-axis above the SRT (high temperature regime, +HT). For both temperature regimes, we see two sub-THz modes emerge with pairwise collinear sublattice +magnetisations, as well as two multi-THz exchange modes with non-collinear sublattice magnetisation vectors. In +terms of the net magnetisation vector, the modes can be grouped into F modes, characterised by an elliptical +trajectory of the magnetisation vector, and AF modes, where the norm of the magnetisation oscillates along a fixed +direction. + + +Theory: Extended data +Suppl. Fig. 4a-c shows the spectra of simulated spin noise projected along the crystalline axes for multiple +temperatures across the SRT. In both, a- and c-axis projections up to three noise peaks can be distinguished, +namely LF peak (<100 GHz), F mode peak (100-200 GHz) and AF mode peak (~600 GHz), whereas only the F +mode is observed in the b-axis projection. The spectral amplitude of the LF peak as a function of temperature is +plotted in Suppl. Fig. 4d. Along the a-axis, the LF peak is observed for 𝑇 ⪆ 𝑇�,��� ≈ 301.5 K and along the c-axis +close to the upper threshold temperature 𝑇�,��� ≈ 320 K. The temperature ranges in which the LF peak is recorded +precisely coincides with the observation of RTN in the magnetisation time traces (Suppl. Fig.5). This further +supports our claim that random switching between two quasi-equilibrium states, resulting in picosecond RTN is +the physical mechanism behind the emergence of the LF peak in the experiment. Furthermore, no LF peak and +no RTN is observed in the b-axis projection of simulated spin noise. This is expected because the b-axis is the +hard axis in Sm0.7Er0.3FeO3. Consequently no quasi-equilibrium states arise in this direction at any temperature. +Suppl. Fig. 4e shows the temperature evolution of the F mode magnon’s spectral amplitude for different crystalline +Suppl. Figure 3 | Resonance modes of Sm0.7Er0.3FeO₃ across the spin reorientation transition. a-d, Resonance modes in the +low temperature regime (below the lower threshold temperature 𝑇�). e-f, Resonance modes in the high temperature regime (above +the upper threshold temperature 𝑇�). The blue, yellow, mint, and green arrows symbolise the sublattice magnetisation vectors, while +the purple arrow represents the Néel vector. The red arrow indicates the net magnetisation vector, for visibility magnified by a factor +of 75 relative to the sublattice magnetisation vectors. The lines of the same colours represent the respective trajectories of the +magnetisation vectors in time. + +Suppl. Figure 3 I Resonance modes of Smo.7Ero.3FeO3 across the spin reorientation transition. a-d, Resonance modes in th +otemneratiireaxes. Most notably, a strong enhancement along the c-axis is observed close to 𝑇�,��� in accord with the +Fluctuation-Dissipation Theorem. For 𝑇�,��� < 𝑇 < 𝑇�,��� , the SRT takes place and consequently the c-axis +projection of the F mode noise reduces as ∝ cos(θ). At 𝑇�,���, the equilibrium magnetisation 𝑴 is alinged, with the +c-axis and no more transversal F mode noise can be observed in the c-axis direction. At the same time, the F +mode noise in the a-axis direction is maximal at 𝑇�,���. For all temperatures, the b-axis component of the +magnetisation 𝑚� = 0. Consequently, a transversal F mode noise component is always observed in this direction, +although at smaller amplitude compared to a- and c-axis, because the b-axis is the hard axis in our system. The +latter is furthermore expressed as the ellipticity of noise cone in the trajectory plots (Fig. 5b-e).The longitudinal AF +mode noise can only be observed along a specific axis, if the equilibrium magnetisation component along the said +axis is non-zero. As a consequence, no AF mode noise is observed along the b-axis in Suppl. Fig. 4f. Furthermore, +the AF mode spectral amplitude in a-direction is maximal for 𝑇 < 𝑇�,��� (𝑴 // a), then reduces in the SRT region +(𝑇�,��� < 𝑇 < 𝑇�,���) until 𝑴 // c for 𝑇 ≥ 𝑇�,���. In contrast, the AF noise along the c-axis is first zero, then increases +during the SRT and is maximal for 𝑇 ≥ 𝑇�,���. We plot the centre frequency of the F mode and AF mode as a +function of temperature in Suppl. Fig. 4g. In accord with previous observations30, the F mode experiences strong +softening close to the SRT region, whereas only slight frequency changes of the AF mode are observed. The +above result evidences that our stochastic spin model can correctly reproduce our experimental observations and +is consistent with physical expectations. +Suppl. Figure 4 | Spectra and evaluation of simulated spin noise in Sm0.7Er0.3FeO3. for different projections. a-c, Raw spectra +of simulated spin noise for different temperatures near spin reorientation. The peak at ~600 GHz is attributed to the +quasiantiferromagnetic magnon mode (AF mode), the middle peak, which shows a strong temperature dependence is attributed to +the quasiferromagnetic mode (F mode), and the low-frequency feature (LF) to random switching between two energetically +degenerate quasi-equilibrium states. d-f, temperature evolution of the LF peak, F mode, and AF mode spectral amplitude for different +axes. g, central frequency of F mode and AF mode as a function of temperature. + +a-axisprojection +b-axisprojection +c-axisprojection +0.08 +0.04 +HH +0.2 +a +298.24K +b +298.24 K +c +298.24 K +301.54K +301.54K +301.54K +0.06 +302.41K +302.41K +302.41K +(a. +305.2K +305.2K +305.2K +litude +321.45K +321.45K +321.45K +330.73K +330.73K +330.73K +0.00 +0.00 +0.0 +0 +100 +200 +500 +600 +0 +100 +200 +500 +600 +0 +100 +200 +600 +frequency (GHz) +frequency (GHz) +frequency (GHz) +5 +0.6 +0.06 +LF, a axis +e +d +Fmode, a-axis +-AF mode, a-axis +4 +-LF,b-axis +F mode, b-axis +AFmode,b-axis +LF, c-axis +Fmode, c-axis +AFmode,c-axis +3 +2spectral +spectral +0.0 +0.00 +295300 +305 +310315320325 +5330335 +295300 +305 +310315320 +325330335 +295300 +305310315320 +325330335 +temperature (K) +temperature (K) +temperature (K) +600 +g +-Fmode,a-axis +-F mode, c-axis +frequency +400 +-AF mode, a-axis +-AF mode, c-axis +300 +200 +100 +295300305310315320325330335 +temperature (K) +Supplementary Figure 5 shows different projections of the simulated magnetisation time traces. At temperatures +𝑻 ⪆ 𝑻L,sim and 𝑻 ⪅ 𝑻U,sim switching events are recorded in the c- (Suppl. Fig 5c) and a-direction (Suppl. Fig 5a), +respectively, due to anisotropy softening. For all temperatures, the fluctuations are centred around 0 in the b-axis +projection. +Suppl. Figure 5 | Simulated time traces of different components of the normalised magnetisation near spin reorientation +in Sm0.7Er0.3FeO3. a-c, normalised a-axis (a), b-axis (b) and c-axis (c) component of the magnetisation. For every temperature, +only the first time trace from a total number of 20 simulated traces is shown. The first 50 ps are not shown, because here the +system still equilibrates from the initial conditions. + +2 +a +0 +2 +298.24K +2 +0 +-2 +302.07 K +2 +(x10-3) +0 +2 +305.2 K +2 +319.12 K +2 +-2 +320.28 K +2 +0 +-2 +—330.73 K +200 +400 +600 +800 +1000 +1200 +time (ps) +b +0 +298.24 K +-1 +1 +302.07 K +1 +ms (x10-3) +0 +WAA +wwww +305.2 K0www-wwwwww +319.12K +-1 +320.28K +1 +Owwww +-1 +330.73K +200 +400 +600 +800 +1000 +1200 +time (ps) +2 +0 +-2 +298.24 K +2 +-2 +302.07 K +2 +2 +302.41 K +2 +0 +2 +305.2 K +2 +0 +-2 +319.12 K +2 +-2 +330.73 K +200 +400 +600 +800 +1000 +1200 +time (ps) \ No newline at end of file diff --git a/RNA0T4oBgHgl3EQfDv8H/content/tmp_files/load_file.txt b/RNA0T4oBgHgl3EQfDv8H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2617ac7e64eed251c215dc1fda500f8fd5d163c3 --- /dev/null +++ b/RNA0T4oBgHgl3EQfDv8H/content/tmp_files/load_file.txt @@ -0,0 +1,1189 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf,len=1188 +page_content='Ultrafast spontaneous spin switching in an antiferromagnet M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Weiss1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Herbst1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Schlegel1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Dannegger1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Evers1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Donges1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Nakajima2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Leitenstorfer1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Goennenwein1, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Nowak1 & T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Kurihara1,3 1Department of Physics, University of Konstanz, D-78457 Konstanz, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 2Institute of Laser Engineering, Osaka University, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 3Institute for Solid State Physics, The University of Tokyo, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Owing to their high magnon frequencies, antiferromagnets are key materials for future high-speed spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Picosecond switching of antiferromagnetic order has been viewed a milestone for decades and pursued only by using ultrafast external perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Here, we show that picosecond spin switching occurs spontaneously due to thermal fluctuations in the antiferromagnetic orthoferrite Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' By analysing the correlation between the pulse-to-pulse polarisation fluctuations of two femtosecond optical probes, we extract the autocorrelation of incoherent magnon fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' We observe a strong enhancement of the magnon fluctuation amplitude and the coherence time around the critical temperature of the spin reorientation transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The spectrum shows two distinct modes, one corresponding to the quasi-ferromagnetic mode and another one which has not been previously reported in pump-probe experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Comparison to a stochastic spin dynamics simulation reveals this new mode as smoking gun of ultrafast spontaneous spin switching within the double-well anisotropy potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Some of the most intriguing effects in physics rest on fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Incoherent thermal fluctuations of spins critically determine the magnetic properties of correlated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Thermally excited magnons are a driving force of a rich variety of fundamental physical phenomena such as phase transitions and spin caloritronic effects1–3 whereas their non-deterministic properties promise unique applications such as probabilistic computing4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Incoherent spin fluctuations have traditionally been studied either indirectly through the temperature dependence of macroscopic properties such as heat capacity, conductivity, and magnetic susceptibility5,6, or in the frequency domain through optical probes relying on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' the Raman effect7,8 or diffraction9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' For measuring relatively slow spin fluctuation dynamics in paramagnets in the range of MHz to GHz frequencies, spin noise spectroscopy (SNS)10–15 has been employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In contrast to paramagnets, correlated spin systems exhibit collective excitations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=', magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Antiferromagnets (AFM) possess especially high frequency magnons reaching into the THz range16, and are thereby attracting enormous attention from the viewpoint of accelerating the conventional ferromagnet- based spintronics devices17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' However, due to their ultrafast dynamics that go beyond state-of-the-art electronics, conventional SNS cannot detect antiferromagnetic spin fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The rich spin physics in AFMs arising from their complicated spin textures have only been resolved with ultrafast pump-probe spectroscopy, where time resolutions down to femto- or even attoseconds19 are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Still, such pump-probe measurements are of perturbative nature and therefore, they cannot detect the incoherent dynamics that spontaneously exist due to thermal or quantum mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' A femtosecond dynamical access to the spin fluctuations of AFMs has not been reported to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In this work, we experimentally demonstrate the spontaneous incoherent sub-THz magnon fluctuation dynamics in the AFM for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' This is achieved by a novel experimental principle, inspired by the emerging field of sub-cycle quantum optics20–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Here, we analyse magnon fluctuation dynamics via their temporal autocorrelation function, by measuring the statistical correlations of polarisation noise imprinted on two subsequent femtosecond probe pulses [Figure 1a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The two linearly polarised, spectrally separate pulses with a variable time delay Δ𝑡 are focused on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Upon transmission of the pulses at times t and 𝑡 + ∆t, transient magnetisation fluctuations δ𝑀�(𝑡) and δ𝑀�(𝑡 + Δ𝑡) parallel to the propagation direction introduce polarisation changes δα(t) and δα(𝑡 + Δ𝑡) to each probe, respectively, via the Faraday effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The polarisation states of the transmitted probe pulses is individually analysed with independent polarimetric detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The pulse-to-pulse fluctuations of the detector output is extracted by sub-harmonic lock-in amplification21, multiplied in real time and averaged over ~108 pulses at each delay position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' By this method, the time correlation trace of the out-of-plane sub-THz magnetisation dynamics 〈δ𝑀c(𝑡)δ𝑀c(𝑡 + Δ𝑡)〉 is precisely unravelled22,23 [See Methods section for further details].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Our sample is a single crystal of the orthoferrite Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In this material, the electron spins of the Fe3+ ions are antiferromagnetically coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' A Dzyaloshinskii-Moriya interaction25,26 (DMI) results in a slight spin canting and a weak net ferromagnetic moment (net magnetisation 𝑴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Orthoferrites have two exchange modes with resonance at multi-THz frequencies and two magnon modes in the sub-THz regime27,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The latter include a quasiferromagnetic mode (F mode) and a quasiantiferromagnetic mode (AF mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The F mode is characterised by a precession of the weak ferromagnetic moment around its equilibrium axis, whereas the AF mode results in its longitudinal oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3 shows a temperature-induced second-order spin reorientation transition (SRT) close to room temperature29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In the SRT region (𝑇� < 𝑇 < 𝑇U), 𝑴 continuously rotates from along the crystallographic a-axis at 𝑇L = 310 K to the c-axis at 𝑇U = 330 K24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The SRT is expressed by a change in the free energy potential (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 𝑇� marks the temperature where the anisotropy difference between a- and c-axis disappears, causing an enhanced magnetic susceptibility in the a-c-plane and strong softening of the F mode resonance frequency24,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' When such a system is thermally populated, one expects a strong enhancement of the angular distribution width at 𝑇�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' First, we investigate the magnon noise dynamics as a function of the time delay ∆t and its scaling with the probed volume Ω at a temperature of 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The confocal position of the sample is identified by the maximum signal amplitude (z = 0 µm, green graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' A systematic decrease of the amplitude is observed with extending z up to ±20 µm (blue, yellow, magenta, and red graphs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The signals are symmetric around ∆t = 0, consistent with the fact that we probe an autocorrelation of the temporal dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' All waveforms exhibit a distinct peak at ∆t = 0 followed by a gradual decrease and a slow oscillation around the zero level that lasts for several tens of picoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figure 2b shows the noise amplitude at ∆t = 0 as a function of the longitudinal sample position with respect to the optical focus (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' As illustrated by the black graph, the correlated Faraday noise amplitude is fitted well by a function which is inversely proportional to volume \uf057(z) probed by the fundamental Gaussian mode 〈δ𝛼(𝑡)�〉 ∝ � �(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Note that in paramagnets, the Figure 1 | Schematic illustration of the experimental setup and spin system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' a, Due to the Faraday effect, two spectrally separated fs pulses (orange and red) of variable time delay ∆t experience a polarisation rotation proportional to out-of-plane spin fluctuations 𝛿𝑀� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Corresponding rotation angles 𝛿𝛼�,� are measured in separate elipsometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' After extraction of the pulse- to-pulse fluctuations from each branch, the cross-correlation function 〈𝛿𝛼�𝛿𝛼�〉 is calculated in real time as a function of the delay time ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' L1,L2: Lenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' DM: Dichroic mirror;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' HWP: Half- wave plate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' WP: Wollaston prism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' BPD: Balanced photo detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' BP: electronic band-pass filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Amp: Transimpedence amplifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Subh: Sub-harmonic demodulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' b, Temperature evolution of the free energy F and its influence on the spin noise dynamics (red cones) close to the spin reorientation in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The weak ferromagnetic moment 𝑴 (black arrow) gradually rotates from 𝜃 = 0° at 𝑇 ≤ 𝑇L to 𝜃 = ±90° at 𝑇 ≥ 𝑇U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Here, 𝜃 is the angle between 𝑴 and the a-axis of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figure 2 | Dependence of magnon noise dynamics on probing volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' a, Spin noise waveforms recorded at a constant temperature of 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K for multiple longitudinal sample positions relative to the laser focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The confocal position 𝑧 = 0 was determined by maximising the amplitude at 𝛥𝑡 = 0 (green graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Amplitudes decrease monotonically with increasing distance from focus (blue, yellow, magenta and red graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' b, Correlated noise amplitude at 𝛥𝑡 = 0 as a function of lateral sample position relative to the focus (blue open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The longitudinal position dependence was fitted with the function given in Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' a α,(t) OM0 BPD a L2 DM 图 HAmp BP L1 Sα,(t+△t) Subh C WP Smo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='Ero3Feo Dataanalysis Subh BPD Cross-correlation BP (Sα,(t) Sα2(t+△t) b H 0 0 T=Tu 0 I>>I >>a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='20 20μm 10 μm o μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='5 μm +20 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='05 40 20 0 20 40 timedelay△t(ps) Experiment Fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='00 40 20 0 20 40 sampleposition (um)amplitude δ𝛼 of the statistical fluctuations of 𝑁 independent spins within the probing volume Ω is known to follow the scaling law δ𝛼 ∝ � √� ∝ � √� 10,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Here, we find the same volume scaling of Faraday noise as in conventional SNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Note that this dependence is not trivially understood for correlated spin systems where individual spins are coupled to form collective magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Nevertheless, in the following we fix our sample position to z = 0 µm to maximise the signal based on this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figure 3a depicts the striking variation of spin noise autocorrelation waveforms found around the SRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The amplitudes, oscillation periods and lifetimes depend strongly on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' To focus on the temperature evolution of the signal amplitude, the amplitude at ∆t = 0 is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' A sharp amplitude increase is observed in the region around 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' This point is slightly higher but close to the estimated lower threshold temperature 𝑇L ~ 305 K of the SRT in our sample, around which temperature the anisotropy softening results in an enhanced magnetic susceptibility27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' This close coincidence indicates that the amplitude of the observed magnon noise can be naively understood as the angle distribution of spins due to thermal occupation of the anisotropy potential well by magnons, consistent with the model described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Beyond this temperature, the noise amplitude decreases continuously, almost disappearing around the upper threshold 𝑇U = 320 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The sharp decrease observed towards the higher temperature side is explained by the equilibrium rotation of the spin system within the SRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In this temperature region, the net magnetisation 𝑴 continuously rotates from 𝑴 // a (in- plane) to 𝑴 // c (out-of-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Our Faraday probe is sensitive only to the c-axis magnetisation fluctuation δ𝑀� of the F mode [Supplementary Materials], which is expected to reduce as δ𝑀� ∝ cos(θ) towards higher temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Therefore, the noise amplitude is expected to decrease continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' These findings are analysed exploiting simulations of the spin noise around the SRT in a generic orthoferrite with parameters fitting the equilibrium properties observed experimentally [Methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The theory is based on an atomistic spin model and the stochastic Landau-Lifshitz-Gilbert equation31,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figure 3c depicts the simulated c- axis magnon noise autocorrelation trace in a 192x192x192 orthoferrite spin lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The simulation reproduces the temporal shape of the noise waveforms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 3a, including the symmetry around ∆t = 0 and the temperature evolution of both the time-zero peak amplitude and the subsequent slow oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The peak amplitude of the calculated waveforms is shown as a function of temperature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In the simulations, the SRT manifests as a strong noise enhancement around 302 K followed by a decrease at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The nearly quantitative agreement between the theoretical calculation and the experimental data allows us to investigate the stochastic nature of the spin noise dynamics from a microscopic viewpoint in following discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' We now analyse the dynamics in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The Fourier spectra of the detected noise waveforms are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Interestingly, two distinct spectral peaks (purple and green arrows) are observed for most temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The frequencies and amplitudes of both modes are strongly dependent on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' While the two peaks are clearly distinguishable for 𝑇 ≤ 304 K, they exhibit similar frequencies when approaching the noise enhancement region around 𝑇L and eventually become indistinguishable due to the strong broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' At temperatures 𝑇 ≫ 𝑇L, the spectral amplitude is significantly reduced because of the SRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figure 4b shows the Figure 3 | Ultrafast magnon noise dynamics near spin reorientation in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' a, Correlated noise as a function of time delay 𝛥𝑡 between probing pulses for multiple temperatures near the spin reorientation in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' b, Experimentally determined time-zero amplitude as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In the region of noise enhancement around 312 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Error bars are added considering the uncertainty associated with background subtraction procedure [Methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' c, Magnon noise simulation based on atomistic spin models and the stochastic Landau-Lifshitz-Gilbert equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' d, Temperature evolution of the simulated time-zero amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' noise elel a c b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='2 Experiment Simulation 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='37 K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='6 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15K 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='72 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='4 correl 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 ise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='4 307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15K 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='89K ted 300 310 320 correlated noi noise temperature (K) T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='2 noise (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='5x)313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15K (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='5x) 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='41 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='2 amplitude 319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='45 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='8 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='4 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='65 K 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='00 K ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 G 0 1 1 1 80 40 0 40 80 80 40 0 40 80 300 305 310 315 time delay △t (ps) time delay △t (ps) temperature (K)peak frequencies of each mode as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Both modes experience a strong frequency reduction around 𝑇L ≈ 305 K, which closely resembles the softening behaviour of the F mode around the SRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The temperature dependence of the high-frequency (HF) mode (green full circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4b) is shown to quantitatively match with pump-probe data24 (pink crosses), clearly identifying this HF mode as the F mode in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' On the other hand, the low-frequency mode (LF mode) has no correspondence in pump-probe measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' To the best of our knowledge, the appearance of such a mode in the SRT region of orthoferrites is reported here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The Fourier spectra of the stochastic LLG simulations are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The appearance of the two peaks and their softening around 𝑇L,sim ≈ 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='5 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4d) matches the experimental results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' This agreement between the temperature dependence of the simulated F mode fluctuation and the HF mode seen in the experiment further supports our assignment to the original F mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Conversely, the LF feature appears in the experimental data from the spectrum recorded at a temperature of 𝑇 = 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K to temperatures well beyond 𝑇L ≈ 305 K, whereas it is observed in a narrower temperature region 𝑇 ⪆ 𝑇L,sim in the simulation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Both the experimental and simulated temperature dependence of the LF spectral amplitude (Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 1,4) follow a similar trend as the time-zero amplitude as a function of temperature shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' This finding suggests the underlying LF dynamics to contribute significantly to the total noise amplitude (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 3b,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' To gain insights into the physical origin of the LF feature, we now investigate the simulation data in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figure 5a shows results for the temporal evolution of the c-axis projection of the normalised magnetisation 𝑚�/𝑚� (𝑚� is the magnetisation at saturation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' For 𝑇 < 𝑇L,sim, the equilibrium axis of the normalised magnetisation is parallel to the a-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Consequently, fluctuations of 𝑚�/𝑚� centred around the origin are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' When approaching 𝑇L,sim, the fluctuations increase in amplitude and oscillation period in agreement with our previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' For 𝑇 ⪆ 𝑇L,sim, 𝑚�/𝑚� switching between two discrete states with similar amplitude but different sign (up- and down-state) become prominent, resembling random telegraph noise (RTN33–35) on a picosecond time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' With increasing temperature, the number of observed switches gradually decreases until no more switching events are recorded for 𝑇 ≫ 𝑇L,sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Here, 𝑚�/𝑚� always fluctuates around a preferred state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' When comparing the temperatures at which the LF peak is observed in the simulated spectra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4c) with the temperatures at which RTN is recorded in the magnetisation time traces (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5a), it becomes clear that the emergence of the LF peak is inherently linked to the observation of picosecond RTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' It should be noted that Fourier transform of a RTN signal should result in an exponentially decaying autocorrelation function and thus in a Lorentzian spectrum centred around zero36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In contrast, in our observations the LF mode exhibits a peak at finite frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' We attribute this finding to the limited time window over which our traces are analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Figure 4 | Magnon noise spectra near SRT in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' a, Fourier spectra of the measured magnon noise waveforms from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Two maxima are resolved for 𝑇 < 307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K (purple and green arrows), while only the low-frequency (purple arrows) mode prevails for higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The spectra were fitted with a double lorentzian function (dashed lines) where distinction of the two separate peaks was possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' b, Temperature evolution of the high-frequency mode (dark green full circles) and the low-frequency mode (purple full circles) from the spectra shown in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The values were obtained from the fits and evaluation of the 2nd derivative zero crossing points in a [Methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' For comparison, quasiferromagnetic magnon mode (F mode) frequency data obtained by THz pump- near IR probe30 are shown as pink crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The pump-probe data is shifted by -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='5 K to compensate for the different amount of stationary heating in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' c, Fourier spectra of the simulated waveforms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Green and purple arrows indicate the center frequencies of the F mode and the low-frequency feature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The spectra were fitted with a triple Lorentzian function (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' d, Temperature evolution of the simulated c-axis projection of the F mode frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The values were obtained from the fits in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 Experiment Simulation 个 50 b 40 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='37 K 个 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 30 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15K 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='72 K spectral HFmode 20 (GHz) oLFmode PPdata 10 litude 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 I amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=') 295 300 305 310 315 ampl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 temperature (K) 307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15K 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='89K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 d (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='2x) 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='2x) 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='41 K- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='0 319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='15 K 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='46 K (GHz) 50 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='65 K 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='00 K 0 0 1 1 1 1 1 1 do 20 40 60 80 0 20 40 60 80 298 300 302 304 306 frequency (GHz) frequency (GHz) temperature (K)To investigate this RTN behaviour in further detail, we plot the trajectory of the LLG-simulated magnetisation for different temperatures around the SRT (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5b-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' At 𝑇 = 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='24 K < 𝑇L,sim (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5b), the equilibrium magnetisation points along the a-axis (θ = 0) and F mode noise can be observed in the b- and c-projections (see Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' At 𝑇 = 𝑇L,sim ≈ 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='54 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5c), an enhancement of the F mode noise is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The spin system then rotates towards finite angle θ for 𝑇 > 𝑇L,sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The ±θ states are energetically degenerate and switching between them occurs for temperatures slightly above 𝑇L,sim, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' for 𝑇 = 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='88 K ⪆ 𝑇L,sim (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' This results in an additional magnetisation noise contribution (Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Fig 5c) and accounts for the strongly enhanced noise amplitude in the 𝑇 = 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='41 K ⪆ 𝑇L,sim waveform in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The configuration of the sublattice magnetisation vectors and the net magnetisation for the switching ±θ states is shown in Fig 5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' As the temperature is elevated even further (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5e), the switching probability becomes lower until no more switching occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Here, the up-state is always preferred because of the initial conditions of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' When approaching the upper threshold temperature 𝑇�,���, switching is observed in the a-projection due to anisotropy softening along this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Beyond 𝑇�,���, the equilibrium magnetisation becomes parallel to the c-axis (not shown) and the F mode contribution to the noise vanishes (see Supplementary Material Figs 2,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The physical picture of the RTN dynamics can be clearly understood by a model considering the stochastic switching between two energetically degenerate quasi-equilibrium states, which manifest as ±θ rotation states of the equilibrium magnetisation due to the SRT (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5g-j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' In Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3, the free energy density exhibits a periodic shape for 𝑇 < 𝑇L, whereas for 𝑇 > 𝑇L it evolves into a double-potential well with minima at ±θ24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The rotation angle θ and the height of the potential barrier separating the two minima gradually increase until 𝑇 = 𝑇U 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Random switching between ±θ states occurs when the height of the potential barrier is low compared to the thermal energy of the system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 5i, 𝑇 ⪆ 𝑇L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Further increase of the temperature in the order of 1 K significantly changes the barrier height, while the thermal energy only changes marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' As a result, the average lifetime 𝜏 on each quasi-equilibrium state increases, and the switching probability declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Note that this model even Figure 5 | Picosecond random switching in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' a, Simulated time traces of the c-axis component 𝑚� of the magnetisation normalised to the saturation magnetisation 𝑚� for multiple temperatures near spin reorientation in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' b- e, Simulated trajectories (purple lines) of the normalised magnetisation (red arrow) for multiple temperatures near spin reorientation in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' At temperature 𝑇 ⪆ 𝑇L,sim (d) switching events are recorded in the c-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The precession cones of the F mode magnon are indicated in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' f, Illustration of the sublattice magnetisation vectors in Sm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='7Er0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='3FeO3 magnetisation (blue, mint, green, yellow) and the net magnetisation (red) for the energetically degenerate ±θ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The canting angle of the sublattice magnetisation vectors and the thus resulting net magnetisation is highly exaggerated for visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' g-j, Orthoferrite potential landscape for different temperatures across the SRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' The red-dotted line indicates the thermal energy Etherm of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' At temperatures 𝑇 < 𝑇� (g), the potential exhibits a parabolic shape and the fluctuations are restricted to a single minimum around θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' Slightly above 𝑇� (h,i), a double-well potential develops and the particle randomly switches between the minima located at ±θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' For 𝑇� > 𝑇 ≫ 𝑇� (j), the energy barrier between the minima is larger than the thermal energy of the system and no more switching events occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content=' 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNA0T4oBgHgl3EQfDv8H/content/2301.02006v1.pdf'} +page_content='24K g Mwwwwwwwwwwwwwwwww T 1 able to pin a vortex on its own. So far, this weak-to-strong pinning transition +has been studied for isotropic defect potentials, resulting in a critical exponent µ = 2 for the onset +of the strong pinning force density Fpin ∼ npfp(ξ/a0)2(κ − 1)µ, with np denoting the density of +defects and a0 the intervortex distance. This result is owed to the special rotational symmetry of +the defect producing a finite trapping area Strap ∼ ξ2 at the strong-pinning onset. The behavior +changes dramatically when studying anisotropic defects with no special symmetries: the strong +pinning then originates out of isolated points with length scales growing as ξ(κ − 1)1/2, resulting +in a different force exponent µ = 5/2. Our analysis of the strong pinning onset for arbitrary defect +potentials ep(R), with R a planar coordinate, makes heavy use of the Hessian matrix describing its +curvature and leads us to interesting geometrical structures: the strong pinning onset is characterized +by the appearance of unstable areas of elliptical shape whose boundaries mark the locations where +vortices jump. The associated locations of asymptotic vortex positions define areas of bistable vortex +states; these bistable regions assume the shape of a crescent with boundaries that correspond to +the spinodal lines in a thermodynamic first-order transition and cusps corresponding to critical end- +points. Both, unstable and bistable areas grow with κ > 1 and join up into larger domains; for a +uniaxially anisotropic defect, two face to face crescents merge into the ring-shaped area previously +encountered for the isotropic defect. Both, onset and merger points are defined by local differential +properties of the Hessian’s determinant D(R), specifically, its minima and saddle points. Extending +our analysis to the case of a random two-dimensional pinning landscape, we discuss the topological +properties of unstable and bistable regions as expressed through the Euler characteristic, with the +latter related to the local differential properties of D(R) through Morse theory. +I. +INTRODUCTION +Vortex pinning by material defects1 determines the +phenomenological +properties +of +all +technically +rele- +vant (type II) superconducting materials, e.g., their +dissipation-free transport or magnetic response. Similar +applies to the pinning of dislocations in metals2 or do- +main walls in magnets3, with the commonalities found +in the topological defects of the ordered phase being +pinned by defects in the host material: these topolog- +ical defects are the vortices4, dislocations5, or domain +walls6,7 appearing within the respective ordered phases— +superconducting, crystalline, or magnetic. +The theory +describing the pinning of topological defects has been +furthest developed in superconductors, with the strong +pinning paradigm8,9 having been strongly pushed during +the last decade10–13. In its simplest form, it boils down to +the setup involving a single vortex subject to one defect +and the cage potential14,15 of other vortices. While still +exhibiting a remarkable complexity, it produces quanti- +tative results which benefit the comparison between the- +oretical predictions and experimental findings16. So far, +strong pinning has focused on isotropic defects, with the +implicit expectation that more general potential shapes +would produce small changes. This is not the case, as +first demonstrated by Buchacek et al.17 in their study of +correlation effects between defects that can be mapped +to the problem of a string pinned to an anisotropic pin- +ning potential. In the present work, we generalize strong +pinning theory to defect potentials of arbitrary shape. +We find that this simple generalization has pronounced +(geometric) effects near the onset of strong pinning that +even change the growth of the pinning force density +Fpin ∝ (κ − 1)µ with increasing pinning strength κ > 1 +in a qualitative manner, changing the exponent µ from +µ = 2 for isotropic defects8,10 to µ = 5/2 for general +anisotropic pinning potentials. +The pinning of topological defects poses a rather +complex problem that has been attacked within two +paradigms, weak-collective- and strong pinning. These +have been developed in several stages: originating in the +sixties of the last century, weak pinning and creep9 has +been further developed with the discovery of high tem- +perature superconductors as a subfield of vortex matter +physics18. Strong pinning was originally introduced by +Labusch8 and by Larkin and Ovchinnikov9 and has been +further developed recently with several works studying +critical currents10, current–voltage characteristics11,19, +magnetic field penetration12,20,21, and creep13,21,22; re- +sults on numerical simulations involving strong pins have +been reported in Refs. 23–25. The two theories come to- +gether at the onset of strong pinning: an individual defect +is qualified as weak if it is unable to pin a vortex, i.e., a +vortex traverses the pin smoothly. Crossing a strong pin, +arXiv:2301.02254v1 [cond-mat.supr-con] 5 Jan 2023 + +2 +however, the vortex undergoes jumps that mathemati- +cally originate in bistable distinct vortex configurations, +‘free’ and ‘pinned’. Quantitatively, the onset of strong +pinning is given by the Labusch criterion κ = 1, with +the Labusch parameter κ ≡ max[−e′′ +p]/ ¯C ∼ fp/ξ ¯C, the +dimensionless ratio of the negative curvature e′′ +p of the +isotropic pinning potential and the effective elasticity ¯C +of the vortex lattice. Strong pinning appears for κ > 1, +i.e., when the lattice is soft compared to the curvatures +in the pinning landscape. +So far, the strong pinning transition at κ = 1 has been +described for defects with isotropic pinning potentials; +it can be mapped10 to the magnetic transition in the +h-T (field–temperature) space, with the strong-pinning +phenomenology at κ > 1 corresponding to the first-order +Ising magnetic transition at T < Tc and the critical point +at T = Tc corresponding to the strong pinning transition +at κ = 1. +The role of the reduced temperature T/Tc +is then assumed by the Labusch parameter κ and the +bistabilities associated with the ferromagnetic phases at +T/Tc < 1 translate to the bistable pinned and free vor- +tex states at κ > 1, with the bistability disappearing +on approaching the critical point, T/Tc = 1 and κ = 1, +respectively. +A first attempt to account for correlations between +defects has been done in Ref. 17. +The latter analysis +takes into account the enhanced pinning force excerted +by pairs of isotropic defects that can be cast in the form +of anisotropic effective pinning centers. Besides shifting +the onset of strong pinning to κ = 1/2 (with κ defined +for one individual defect), the analysis unravelled quite +astonishing (geometric) features that appeared as a con- +sequence of the symmetry reduction in the pinning po- +tential. In the present paper, we take a step back and +study the transition to strong pinning for anisotropic de- +fect potentials ep(R), with R a planar coordinate, see +Fig. 1. Note that collective effects of many weak defects +can add up to effectively strong pins that smoothen the +transition at κ = 1, thereby turning the strong pinning +transition into a weak-to-strong pinning crossover. +We find that the onset of strong pinning proceeds quite +differently when going from the isotropic defect to the +anisotropic potential of a generic defect without spe- +cial symmetries and further on to a general random pin- +ning landscape. The simplest comparison is between an +isotropic and a uniaxially anisotropic defect, acting on +a vortex lattice that is directed along the magnetic field +B ∥ ez chosen parallel to the z-axis; for convenience, we +place the defect at the origin of our coordinate system +r = (R, z) and have it act only in the z = 0-plane. In +this setup, see Fig. 1, the pinning potential ep(R) acts on +the nearest vortex with a force fp(R) = −∇Rep|z=0 at- +tracting the vortex to the defect; the presence of the other +vortices constituting the lattice renormalizes the vortex +elasticity ¯C. With the pinning potential acting in the +z = 0 plane, the vortex is deformed with a pronounced +cusp at z = 0, see Fig. 1; we denote the tip position of the +vortex where the cusp appears by ˜R, while the asymp- +˜R +¯R +defect +tip +vortex +asymptotic +x +y +z +FIG. 1. Sketch of a vortex interacting with a defect located +at the origin. The vortex approaches the asymptotic position +¯R at z → ±∞ and is attracted to the defect residing at the +origin; the cusp at z = 0 defines the tip position ˜R and its +angle quantifies the pinning strength. +totic position of the vortex at z → ±∞ is fixed at ¯R. +With this setup the problem can be reduced to a planar +one, with the tip coordinate ˜R and the asymptotic co- +ordinate ¯R determining the location and full shape (and +hence the pinning force) of the vortex line. +In the case of an isotropic pin, e.g., produced by a +point-like defect11, strong pinning first appears on a cir- +cle of finite radius Rm ∼ ξ, typically of order of the vortex +core radius ξ, see left panel of Fig. 2(a). This is owed to +the fact that, given the radial symmetry, the Labusch cri- +terion κ = maxR[−e′′ +p(R)]/ ¯C = 1 is satisfied on a circle +R = Rm where the (negative) curvature −e′′ +p > 0 is max- +imal. Associated with the radius Rm where the tip is lo- +cated at κ = 1, ˜R(κ = 1) ≡ ˜Rm = Rm, there is an asymp- +totic vortex position ¯R(κ = 1) = ¯Rm > ˜Rm. Increasing +the Labusch parameter beyond κ = 1, the circle of ra- +dius ¯Rm transforms into a ring ¯R− < ¯R < ¯R+ of finite +width. Vortices placed inside the ring at small distances +¯R < ¯R− near the defect are qualified as ‘pinned’, while +vortices at large distances ¯R > ¯R+ away from the pin +are described as ‘free’, see right panel in Fig. 2(a); phys- +ically, we denote a vortex configuration as ‘free’ when +it is smoothly connected to the asymptotic undeformed +state, while a ‘pinned’ vortex is localized to a finite region +around the defect. Vortices placed inside the bistable ring +at ¯R− < ¯R < ¯R+ acquire two possible states, pinned and +free (colored magenta in Fig. 2, the superposition of red +(pinned state) and blue (free state) colors). +The +onset +of +strong +pinning +for +the +uniaxially + +3 +anisotropic defect proceeds in several stages. Let us con- +sider an illustrative example and assume a defect with +an anisotropy aligned with the axes and a steeper po- +tential along x. In this situation, strong pinning as de- +fined by the criterion κm = 1, with a properly gener- +alized Labusch parameter κm, appears out of two points +(±¯xm, 0) where the Labusch criterion κm = 1 is met first, +see Fig. 2(b) left. Increasing κm > 1 beyond unity, two +bistable domains spread around these points and develop +two crescent-shaped areas (with their large extent along +¯y) in asymptotic ¯R-space, see Fig. 2(b) right. Vortices +with asymptotic positions within these crescent-shaped +regions experience bistability, while outside these regions +the vortex state is unique. Classifying the bistable solu- +tions as ‘free’ and ‘pinned’ is not possible, with the sit- +uation resembling the one around the gas–liquid critical +point with a smooth crossover (from blue to white to red) +between phases. With κm increasing further, the cusps +of the crescents approach one another. As the arms of +the two crescents touch and merge at a sufficiently large +value of κm, the topology of the bistable area changes: +the two merged crescents now define a ring-like geometry +and separate ¯R-space into an inside region where vortices +are pinned, an outside region where vortices are free and +the bistable region with pinned and free states inside the +ring-like region. +As a result, the pinning geometry of +the isotropic defect is recovered, though with the perfect +ring replaced by a deformed ring with varying width. Us- +ing the language describing a thermodynamic first-order +transition, the cusps of the crescents correspond to criti- +cal points while its boundaries map to spinodal lines; the +merging of critical points changing the topology of the +bistable regions of the pinning landscape goes beyond the +standard thermodynamic analogue of phase diagrams. +The bistable area is defining the trapping area where +vortices get pinned to the defect; this trapping area is one +of the relevant quantities determining the pinning force +density Fpin, the other being the jumps in energy associ- +ated with the difference between the bistable states8,10, +see the discussion in Secs. II C, II E, and III G below. It +is the change in the bistable- and hence trapping geom- +etry that modifies the exponent µ in Fpin ∝ (κ − 1)µ, +replacing the exponent µ = 2 for isotropic defects by the +new exponent µ = 5/2 for general anisotropic pinning +potentials. +While the existence of bistable regions B ¯R in the space +of asymptotic vortex positions ¯R is an established ele- +ment of strong pinning theory by now, in the present pa- +per, we introduce the new concept of unstable domains +U ˜R in tip-space. The two coordinates ˜R and ¯R represent +dual variables in the sense of the thermodynamic analog, +with the asymptotic coordinate ¯R corresponding to the +driving field h in the Ising model and the tip position +˜R replacing the magnetic response m; from a thermody- +namic perspective it is then quite natural to change view +by going back and forth between intensive (h) and exten- +sive (m) variables. In tip space ˜R, the onset of pinning +appears at isolated points ˜Rm that grow into ellipses as +−3 +0 +−3 +0 +−3 +0 +−3 +0 +−3 +0 +−3 +0 +−3 +0 +−3 +0 +−1 +0 +2 +(a) +(b) +˜ +R/ξ − ˜ +Rm( ˜φ)/ξ +¯y/ξ +¯x/ξ +¯y/ξ +¯x/ξ +¯y/ξ +¯x/ξ +¯y/ξ +¯x/ξ +¯Rm +¯R− +¯R+ +¯R0 +FIG. 2. Illustration of bistable regions in asymptotic ¯R-space +for a vortex pinned to a defect located at the origin. (a) For an +isotropic defect (Lorentzian shape with κ = 1, 1.5), pinning +appears at κ = 1 along a ring with radius ¯Rm, with the red +area corresponding to pinned states and free states colored in +blue. With increasing pinning strength κ, see right panel at +κ = 1.5, a bistable region (in magenta) appears in a ring ge- +ometry, with vortices residing inside, ¯R < ¯R−, being pinned +and vortices outside, ¯R > ¯R+, remaining free. Vortices with +asymptotic positions inside the ring ( ¯R− < ¯R < ¯R+) exhibit +bistable states, pinned and free. The dashed circle ¯R0 marks +the crossing of pinned and free branches, see Fig. 4. (b) For +a uniaxially anisotropic defect, see Eq. (94) with ϵ = 0.3 and +largest (negative) curvature along x, pinning appears in two +points (±¯xm, 0) along the x-axis. +As the pinning strength +increases beyond unity, see right panel, bistable regions (ma- +genta) develop in a crescent-shape geometry. +Pinned- and +free-like states are smoothly connected as indicated by the +crossover of colors (see Sec. III C for the precise description +of coloring in terms of an ‘order parameter’). +As κm fur- +ther increases, the cusps of the two crescents merge on the +y-axis, changing the topology of the ¯R-plane through sepa- +ration into inner and outer regions (not shown). A ring-like +bistable region appears as in (a), with the inner (outer) region +corresponding to unique vortex states that are pinned (free), +while vortices residing inside the ring-shaped domain exhibit +bistable states, pinned and free. +κ is increased beyond unity. These ellipses describe un- +stable areas U ˜R in the ˜R-plane across which vortex tips +jump when flipping between bistable states; they relate to +the bistable crescent-shaped areas B ¯R in asymptotic space +through the force balance equation; the latter determines +the vortex shape with elastic and pinning forces compen- + +4 +sating one another. The unstable regions U ˜R in tip space +are actually more directly accessible than the bistable re- +gions B ¯R in asymptotic space and play an equally central +role in the discussion of the strong pinning landscape. +The simplification introduced by the concept of unsta- +ble domains U ˜R in tip space ˜R is particularly evident +when going from individual defects as described above to +a generic pinning landscape. Here, we focus on a model +pinning potential landscape (or short pinscape) confined +to the two-dimensional (2D) R plane at z = 0; such a pin- +scape can be produced, e.g., by defects that reside in the +z = 0 plane. The pinned vortex tip ˜R then still resides in +the z = 0 plane as well and the strong pinning problem +remains two-dimensional. +For a 2D random pinscape, +unstable ellipses appear sequentially out of different (iso- +lated) points and at different pinning strength κm; their +assembly defines the unstable area U ˜R, with each newly +appearing ellipse changing the topology of U ˜R, specif- +ically, its number of components. +Increasing κm, the +ellipses first grow in size, then deform away from their +original elliptical shapes, and finally touch and merge in a +hyperbolic geometry. Such mergers change, or more pre- +cisely reduce, the number of components in U ˜R and hence +correspond again to topological transitions as described +by a change in the Euler characteristic χ associated with +the shape of U ˜R. +Furthermore, these mergers tend to +produce U ˜R shapes that are non-simply connected, again +implying a topological transition in U ˜R with a change +in χ. Such non-simply connected parts of U ˜R separate +the tip space into ‘inner’ and ‘outer’ regions that allows +to define proper ‘pinned’ states (localized near a poten- +tial minimum) in the ‘inner’ of U ˜R, while ‘free’ states +(smoothly connected to asymptotically undeformed vor- +tices) occupy the regions outside of U ˜R. +The discussion below is dominated by three mathemat- +ical tools: for one, it is the Hessian matrix H(R) of the +pinning potential17,26 ep(R), its eigenvalues λ±(R) and +eigenvectors v±(R), its determinant det[H](R) and trace +tr[H](R). +The Hessian matrix involves the curvatures +Hij = ∂i∂jep(R), i, j ∈ {x, y}, of the pinning potential, +that in turn are the quantities determining strong pin- +ning, as can be easily conjectured from the form of the +Labusch parameter κ ∝ −e′′ +p for the isotropic defect. The +second tool is the Landau-type expansion of the total pin- +ning energy near the strong-pinning onset around ˜Rm at +κm = 1 (appearance of a critical point) as well as near +merging around ˜Rs at κ( ˜Rs) ≡ κs = 1 (disappearance +of a pair of critical points); the standard manipulations +as they are known from the description of a thermody- +namic first-order phase transition produce most of the +new results. Third, the topological structure of the un- +stable domain U ˜R associated with a generic 2D pinning +landscape, i.e., its components and their connectedness, +is conveniently described through its Euler characteristic +χ with the help of Morse theory. +The structure of the paper is as follows: In Section +II, we briefly introduce the concepts of strong pinning +theory with a focus on the isotropic defect. The onset +of strong pinning by a defect of arbitrary shape is pre- +sented in Sec. III; we start with a translation and ex- +tension of the strong pinning ideas from the isotropic +situation to a general anisotropic one, that leads us to +the Hessian analysis of the pinning potential as our ba- +sic mathematical tool. Close to onset, we find (using a +Landau-type expansion, see Sec. III A) that the unstable +(Sec. III B) and bistable (Sec. III C) domains are asso- +ciated with minima of the determinant of the Hessian +curvature matrix and assume the shape of an ellipse and +a crescent, respectively. Due to the anisotropy, the ge- +ometry of the trapping region depends non-trivially on +the Labusch parameter and the critical exponent for the +pinning force is changed from µ = 2 to µ = 5/2, see Sec. +III G. The analytic solution of the strong pinning onset +for a weakly uniaxial defect presented in Sec. IV leads +us to define new hyperbolic points associated with sad- +dle points of the determinant of the Hessian curvature +matrix. These hyperbolic points describe the merging of +unstable and bistable domains, see Sec. V A, and allow +us to relate the new results for the anisotropic defect to +our established understanding of isotropic defects. In a +final step, we extend the local perspective on the pin- +scape, as acquired through the analysis of minima and +saddles of the determinant of the Hessian curvature ma- +trix, to a global description in terms of the topological +characteristics of the unstable domain U ˜R: in Sec. VI, +we discuss strong pinning in a two-dimensional pinning +potential of arbitrary shape, e.g., as it appears when mul- +tiple pinning defects overlap (though all located in one +plane). We follow the evolution of the unstable domain +U ˜R with increasing pinning strength κm and express its +topological properties through the Euler characteristic χ; +the latter is related to the local differential properties of +the pinscape’s curvature, its minima, saddles, and max- +ima, through Morse theory. Finally, in Appendix A, we +map the two-dimensional Landau-type theories (involv- +ing two order parameters) describing onset and merging, +to effective one-dimensional Landau theories and rederive +previous results following standard statistical mechanics +calculations as they are performed in the analysis of the +critical point in the van der Waals gas. +II. +STRONG PINNING THEORY +We start with a brief introduction to strong pinning +theory, keeping a focus on the transition region at mod- +erate values of κ > 1. We consider an isotropic defect +(Sec. II A) and determine the unstable and bistable ring +domains for this situation in Sec. II B. We derive the +general expression for the pinning force density Fpin in +Sec. II C, determine the relevant scales of the strong pin- +ning characteristic near the crossover in Sec. II D, and +apply the results to derive the scaling Fpin ∝ (κ − 1)2 +for the isotropic defect (Sec. II E). In Sec. II F, we relate +the strong pinning theory for the isotropic defect to the +Landau mean-field description for the Ising model in a + +5 +magnetic field. +A. +Isotropic defect +The standard strong-pinning setup involves a vortex +lattice directed along z with a lattice constant a0 deter- +mined by the induction B = φ0/a2 +0 that is interacting +with a dilute set of randomly arranged defects of den- +sity np. This many-body problem can be reduced10,13,20 +to a much simpler effective problem involving an elastic +string with effective elasticity ¯C that is pinned by a de- +fect potential ep(R) acting in the origin, as described by +the energy function +epin( ˜R; ¯R) = +¯C +2 ( ˜R − ¯R)2 + ep( ˜R) +(1) +depending on the tip- and asymptotic coordinates ˜R and +¯R of the vortex, see Fig. 1. +The energy (or Hamilto- +nian) epin( ˜R; ¯R) of this setup involves an elastic term +and the pinning energy ep(R) evaluated at the location +˜R of the vortex tip. We denote the depth of the pin- +ning potential by ep. A specific example is the point- +like defect that produces an isotropic pinning potential +which is determined by the form of the vortex11 and as- +sumes a Lorentzian shape ep(R) = −ep/(1 + R2/2ξ2) +with R = |R|; in Sec. III below, we will consider pin- +ning potentials of arbitrary shape ep(R) but assume a +small (compared to the coherence length ξ) extension +along z. +‘Integrating out’ the vortex lattice, the re- +maining string or vortex is described by the effective +elasticity ¯C ≈ νε(a2 +0/λL) +� +c66c44(0) ∼ εε0/a0. +Here, +ε0 = (φ0/4πλL)2 is the vortex line energy, λL denotes +the London penetration depth, ε < 1 is the anisotropy +parameter for a uniaxial material18, and ν is a numerical, +see Refs. 23 and 25. +The most simple pinning geometry is for a vortex that +traverses the defect through its center. Given the rota- +tional symmetry of the isotropic defect, we choose a vor- +tex that impacts the defect in a head-on collision from the +left with asymptotic coordinate ¯R = (¯x, 0) and increase +¯x along the x-axis; finite impact parameters ¯y ̸= 0 will +be discussed later. The geometry then simplifies consid- +erably and involves the asymptotic vortex position ¯x and +the tip position ˜x of the vortex, reducing the problem to +a one-dimensional one; the full geometry of the deformed +string can be determined straightforwardly20 once the tip +position ˜x has been found. The latter follows from mini- +mizing (1) with respect to ˜x at fixed asymptotic position +¯x and leads to the non-linear equation +¯C(˜x − ¯x) = −∂xep|x=˜x = fp(˜x). +(2) +This can be solved graphically, see Fig. 3, and produces +either a single solution or multiple solutions—the appear- +ance of multiple tip solutions is the signature of strong +pinning. The relevant parameter that distinguishes the +¯C(˜x − ¯x+) +¯C(˜x − ¯x−) +˜xp− +˜xp+ +˜xf− +˜xf+ ˜x +fp(˜x) +0 +ξ +¯x− +¯x+ +0 +˜xm +¯xm +˜x +fp(˜x) +κ < 1 +¯C(˜x − ¯xm) +FIG. 3. +Graphical illustration13 of the self-consistent solu- +tion of the microscopic force-balance equation Eq. (2) for a +Lorentzian potential with κ = 2.5. The vortex coordinates ˜x +and ¯x are expressed in units of ξ. When moving the asymp- +totic vortex position ¯x across the bistable interval [¯x−, ¯x+], +we obtain three solutions describing pinned ˜xp ≲ ξ, free ˜xf +close to ¯x, and unstable ˜xus states; they define the corre- +sponding pinned (red), free (blue), and unstable (black dot- +ted) branches. The tip-positions at the edges of the bistable +interval denoted by ˜xp+ and ˜xf− denote jump points where +the vortex tip turns unstable, see Eq. (3); they are defined +by the condition f ′ +p(˜xp+) = f ′ +p(˜xf−) = ¯C (black solid dots). +The associated positions ˜xf+ and ˜xp− denote the tip landing +points after the jump (open circles); they are given by the +second solution of Eq. (2) at the same asymptotic position ¯x. +The open red/blue circles and the cross mark the positions of +metastable minima and the unstable maximum in Fig. 4. The +lower right inset shows the weak-pinning situation at κ < 1, +here implemented with a larger ¯C, where the tip solution ˜x is +unique for all ¯x. +two cases is found by taking the derivative of (2) with +respect to ¯x that leads to +∂¯x˜x = +1 +1 − f ′p(˜x)/ ¯C , +(3) +where prime denotes the derivative, f ′ +p(x) = ∂xfp(x) = +−∂2 +xep(x). Strong pinning involves vortex instabilities, +i.e., jumps in the tip coordinate ˜x, that appear when the +denominator in (3) vanishes; this leads us to the strong +pinning parameter κ first introduced by Labusch8, +κ = max +˜x +f ′ +p(˜x) +¯C += f ′ +p(˜xm) +¯C +, +(4) +with ˜xm defined as the position of maximal force deriva- +tive f ′ +p, i.e., f ′′ +p (˜xm) = 0, or maximal negative curva- +ture −e′′ +p of the defect potential. Defining the force scale +fp ≡ ep/ξ and estimating the force derivative or curva- +ture f ′ +p = −e′′ +p ∼ fp/ξ produces a Labusch parameter +κ ∼ ep/ ¯Cξ2; for the Lorentzian potential, we find that +f ′ +p(˜xm) = ep/4ξ2 at ˜xm = +√ +2 ξ and hence κ = ep/4 ¯Cξ2. +We see that strong pinning is realized for either large +pinning energy ep or small effective elasticity ¯C. + +6 +As follows from Fig. 3 (inset), for κ < 1 (large ¯C) the +solution to Eq. (2) is unique for all values of ¯x and pinning +is weak, while for κ > 1 (small ¯C), multiple solutions +appear in the vicinity of ˜xm and pinning is strong. These +multiple solutions appear in a finite interval ¯x ∈ [¯x−, ¯x+] +and we denote them by ˜x = ˜xf, ˜xp, ˜xus, see Fig. 3; they +are associated with free (weakly deformed vortex with +˜xf close to ¯x), pinned (strongly deformed vortex with +˜xp < ξ), and unstable vortex states. +Inserting the solutions ˜x(¯x) = ˜xf(¯x), ˜xp(¯x), ˜xus(¯x) of +Eq. (2) at a given vortex position ¯x back into the pinning +energy epin(˜x; ¯x), we find the energies of the correspond- +ing branches, +ei +pin(¯x) ≡ epin[˜xi(¯x); ¯x], +i = f, p, us. +(5) +The pair ep(˜x) and ei +pin(¯x) of energies in tip- and asymp- +totic spaces then has its correspondence in the force: as- +sociated with fp(˜x) in tip space are the force branches +f i +pin(¯x) in asymptotic ¯x-space defined as +f i +pin(¯x) = fp[˜xi(¯x)], +i = f, p, us. +(6) +Using Eq. (2), it turns out that the force fpin can be +written as the total derivative of epin, +fpin(¯x) = −depin[˜x(¯x); ¯x] +d¯x +. +(7) +The multiple branches ei +pin and f i +pin associated with a +strong pinning situation at κ > 1 are shown in Figs. 4 +and 5(b). +B. +Unstable and bistable domains U ˜ +R and B ¯ +R +Next, we identify the unstable (in ˜x) and bistable (in +¯x) domains of the pinning landscape that appear as sig- +natures of strong pinning when κ increases beyond unity. +Figure 5(a) shows the force profile fp(˜x) as experienced +by the tip coordinate ˜x. A vortex passing the defect on a +head-on trajectory from left to right undergoes a forward +jump in the tip from −˜xf− to −˜xp−; subsequently, the +tip follows the pinned branch until ˜xp+ and then returns +back to the free state with a forward jump from ˜xp+ to +˜xf+. The jump positions (later indexed by a subscript +‘jp’) are determined by the two solutions of the equation +f ′ +p(x) +��� +−˜xf−,˜xp+ = ¯C +(8) +that involves the curvature of the pinning potential ep(x); +the landing positions −˜xp− and ˜xf+ (later indexed by a +subscript ‘lp’), on the other hand, are given by the second +solution of the force-balance equation (2) that involves +the driving term ¯C(˜x − ¯x) and hence depends on the +asymptotic position ¯x. Finally, the positions in asymp- +totic space ¯x where the vortex tip jumps are obtained +again from the force balance equation (2), +¯x− = ˜xf− − fp(˜xf−)/ ¯C, +(9) +¯x+ = ˜xp+ − fp(˜xp+)/ ¯C. +epin +¯x +epin +0 +−¯x0 +∆efp +pin +−¯x− +ξ +∆epf +pin +¯x+ +˜x +˜xp +˜xus +˜xf +FIG. 4. +Multi-valued pinning energy landscape ei +pin(¯x) for a +defect producing a Lorentzian-shaped potential with κ = 2.5; +the branches i = p, f, us correspond to the pinned (red), free +(blue), and unstable (black dotted) vortex states. The bista- +bility extends over the intervals |¯x| ∈ [¯x−, ¯x+] where the dif- +ferent branches coexist; pinned and free vortex branches cut +at the branch crossing point ¯x = ¯x0. A vortex traversing the +defect from left to right assumes the free and pinned states +marked with thick colored lines and undergoes jumps ∆efp +pin +and ∆epf +pin in energy (vertical black solid lines) at the bound- +aries −¯x− and ¯x+. The asymmetric occupation of states pro- +duces a finite pinning force density Fpin. Inset: Total energy +epin(˜x; ¯x) versus vortex tip position ˜x for a fixed vortex po- +sition ¯x (vertical dashed line in the main figure). The points +˜xf, ˜xp, and ˜xus mark the free, pinned, and unstable solutions +of the force-balance equation (2); they correspond to local +minima and the maximum in epin(˜x; ¯x) and are marked with +corresponding symbols in Fig. 3. +Note that the two pairs of tip jump and landing posi- +tions, ˜xp+, ˜xf+ and ˜xf−, ˜xp− are associated with only +two asymptotic positions ¯x+ and ¯x−. +Let us generalize the geometry and consider a vortex +moving parallel to ¯x, impacting the defect at a finite dis- +tance ¯y. We then have to extend the above discussion to +the entire z = 0 plane, see Fig. 5. For an isotropic de- +fect, the jump- and landing points now define jump cir- +cles with radii ˜Rjp given by ˜Rf− = ˜xf− and ˜Rp+ = ˜xp+ +(solid circles in Fig. 5(c)) and landing circles with radii +˜Rlp given by ˜Rf+ = ˜xf+, ˜Rp− = ˜xp− (dashed circles in +Fig. 5(c)). Their combination defines an unstable ring +˜Rp+ < ˜R < ˜Rf− in tip space where tips cannot reside. +The existence of unstable domains U ˜R in tip space is a +signature of strong pinning. +Figures 5(b) and (d) show the corresponding results +in asymptotic coordinates ¯x and ¯R, respectively. +The +pinning force fpin(¯x) = fp[˜x(¯x)] shown in (b) is simply +an ‘outward tilted’ version of fp(˜x), with S-shaped over- +hangs that generate bistable intervals [−¯x+, −¯x−] and +[¯x−, ¯x+]. +Extending them to the asymptotic ¯R-plane +with radii ¯R− ≡ ¯x− and ¯R+ ≡ ¯x+, see Fig. 5(d), we +obtain a ring ¯R− < ¯R < ¯R+ that marks the location of +bistability. Again, the appearance of bistable domains + +7 +(a) +(c) +(b) +(d) +˜x +˜xp+ +−˜xp− +ξ +˜xf+ +−˜xf− +˜x +fp +˜y +¯x +¯x+ +−¯x− +¯x +ξ +ξ +fpin +¯y +¯R− +¯R+ +¯R0 +FIG. 5. (a) and (b): Force profiles fp(˜x) and fpin(¯x) in tip- +and asymptotic coordinates for a Lorentzian-shaped poten- +tial with κ = 2.5. +The tip of a vortex moving from left +to right along the x-axis approaches the defect on the free +branch (thick blue line) undergoes a jump (arrow) from −˜xf− +to −˜xp−, follows the pinned branch (red) until ˜xp+ and then +jumps back (arrow) to the free (blue) state at ˜xf+. Extend- +ing these jump positions to the (˜x, ˜y)-plane, see (c), defines +jump (solid) and landing (dashed) circles, with the jump cir- +cles enclosing an unstable domain U ˜ +R characteristic of strong +pinning. The force profile fpin(¯x) in (b) includes free (blue), +pinned (red), and unstable branches (black dotted). (d) Ex- +tending the bistable intervals [−¯x+, −¯x−] and [¯x−, ¯x+] to the +[¯x, ¯y]-plane defines a bistable ring B ¯ +R (magenta), again a +strong pinning characteristic. The dashed circle of radius ¯R0 +in (d) marks the branch crossing point. Vortices passing the +defect with a finite impact parameter ¯y ̸= 0 move on a straight +line in asymptotic space, see (d); the associated trajectory in +tip space is nontrivial, see (c) and undergoes jumps at pinning +(circle ˜Rf−) and depinning (circle ˜Rp+). +B ¯R in asymptotic space is a signature of strong pinning. +Both, the size of the unstable- and bistable rings depend +on the Labusch parameter κ; they appear out of circles +with radii ˜R = ˜xm and ¯R = ¯xm = ˜xm − fp(˜xm)/ ¯C at +κ = 1 and grow in radius and width when κ increases. +The unstable and bistable domains U ˜R and B ¯R (see Ref. +27) will exhibit interesting non-trivial behavior as a func- +tion of κ when generalizing the analysis to defect poten- +tials of arbitrary shape. +1. +Alternative strong pinning formulation +An alternative formulation of strong pinning physics is +centered on the local differential properties of the pinning +energy epin(˜x; ¯x), i.e., its extremal points in ˜x at different +values of the asymptotic coordinate ¯x. +We start from +equation (1) restricted to one dimension and rearrange +terms to arrive at the expression +epin(˜x; ¯x) = eeff(˜x) − ¯C¯x ˜x + ¯C¯x2/2 +(10) +with the effective pinning energy +eeff(˜x) = ep(˜x) + ¯C˜x2/2 +(11) +involving both pinning and elastic terms. Equation (10) +describes a particle at position ˜x subject to the potential +eeff(˜x) and the force term f ˜x = − ¯C¯x ˜x, see also Ref. +26. +The potential eeff(˜x) can trap two particle states +if there is a protecting maximum with negative curva- +ture ∂2 +˜xeeff = ∂2 +˜xepin < 0, preventing its escape from the +metastable state at forces f = ± ¯C¯x with ¯x ∈ [¯x+, ¯x−]; +the maximum in epin at ˜xus then separates two minima +in epin defining distinct branches with different tip coor- +dinates ˜xp and ˜xf, see the inset of Fig. 4. +As the asymptotic position ¯x approaches the bound- +aries ¯x±, one of the minima joins up with the maximum +to define an inflection point with +[∂2 +˜xeeff]˜xjp = [∂2 +˜xepin]˜xjp = 0, +(12) +that corresponds to the instability condition (8) where +the vortex tip jumps; the persistent second minimum in +epin(˜x; ¯x) defines the landing position ˜xlp and the condi- +tion for a flat inflection point [∂˜xepin]˜xjp = 0 defines the +associated asymptotic coordinate ±¯x±. +Finally, strong pinning vanishes at the Labusch point +κ = 1, with the inflection point in eeff(˜x) coalescing with +the second minimum at ˜xm, hence +[∂2 +˜xeeff]˜xm = 0 +and +(13) +[∂3 +˜xeeff]˜xm = [∂3 +˜xep]˜xm = 0. +Note the subtle use of epin versus eeff versus ep in the +above discussion; as we go to higher derivatives, first +the asymptotic coordinate ¯x turns irrelevant in the sec- +ond derivative ∂2 +˜xepin = ∂2 +˜xeeff and then all of the elas- +tic response, i.e., ¯C, drops out in the third derivative +[∂3 +˜xepin] = [∂3 +˜xep]. +The above alternative formulation of strong pinning +turns out helpful in several discussions below, e.g., the +derivation of strong pinning characteristics near the tran- +sition in Secs. II D and III A and in the generalization of +the instability condition to an anisotropic defect in Sec. +III and furthermore provides an inspiring link to the Lan- +dau theory of phase transitions discussed below in Sec. +II F. +C. +Pinning force density Fpin +Next, we determine the pinning force density Fpin at +strong pinning, assuming a random homogeneous distri- +bution of pins with a small density np, npa0ξ2 ≪ 1, see + +8 +Refs. 13 and 20. The derivation of Fpin is conveniently +done in asymptotic ¯R coordinates where vortex trajec- +tories follow simple straight lines. Vortices approach the +pin by following the free branch until its termination, +jump to the pinned branch to again follow this to its +termination, and finally jump back to the free branch. +This produces an asymmetric pinned-branch occupation +pc( ¯R) that leads to the pinning force density (we assume +vortices approaching the defect along ¯x from the left; fol- +lowing convention, we include a minus sign) +Fc = −np +� d2 ¯R +a2 +0 +� +pc( ¯R)f p +pin( ¯R) + (1 − pc( ¯R))f f +pin( ¯R) +� += −np +� d2 ¯R +a2 +0 +pc( ¯R)[∂x∆efp +pin( ¯R)] e¯x, +(14) +with the energy difference ∆efp +pin( ¯R) = ef +pin( ¯R) − ep +pin( ¯R) +and e¯x the unit vector along ¯x; the ¯y-component of the +pinning force density vanishes due to the antisymmetry +in fpin,¯y. For the isotropic defect, the jumps ∆efp +pin( ¯R) +in energy appearing upon changing branches are inde- +pendent of angle and the average in (14) separates in ¯x +and ¯y coordinates; note that the energy jumps are no +longer constant for an anisotropic defect and hence such +a separation does not occur. +Furthermore, i) all vor- +tices approaching the defect within the transverse length +|¯y| < ¯R− get pinned, see Fig. 5(d), while those passing +further away follow a smooth (weak pinning) trajectory +that does not undergo jumps and hence do not contribute +to the pinning force, and ii) all vortices that get pinned +contribute the same force that is most easily evaluated +for a head-on vortex–defect collision on the ¯x-axis with +pc(¯x) = Θ(¯x + ¯x−) − Θ(¯x − ¯x+) and +⟨fpin⟩ = − +� a0/2 +−a0/2 +d¯x +a0 +� +pc(¯x)f p +pin(¯x) + (1 − pc(¯x))f f +pin(¯x) +� += +∆efp +pin(−¯x−) + ∆epf +pin(¯x+) +a0 +, +(15) +where we have replaced −∆efp +pin(¯x+) by ∆epf +pin(¯x+) > 0. +Hence, the average pinning force ⟨fpin⟩ is given by the +jumps in the pinning energy ei +pin(¯x) associated with dif- +ferent branches i = p, f, see Fig. 4. +Finally, accounting for trajectories with finite impact +parameter |¯y| < ¯R−, we arrive at the result for the pin- +ning force density Fpin acting on the vortex system, +Fpin = np +2 ¯R− +a0 +⟨fpin⟩ = np +2 ¯R− +a0 +∆efp +pin + ∆epf +pin +a0 +, +(16) +where the factor 2 ¯R−/a0 accounts for the averaging of +the pinning force along the y-axis. As strong pins act +independently, a consequence of the small defect density +np, the pinning force density is linear in the defect den- +sity, Fpin ∝ np. If pinning is weak, i.e., κ < 1, we have no +jumps, ⟨fpin⟩ = 0, and Fpin|strong = 0. A finite pinning +force then only arises from correlations between pinning +defects and scales in density as9,10 Fpin|weak ∝ n2 +p. This +contribution to the pinning force density Fpin continues +beyond κ = 1, hence, while the strong pinning onset at +κ = 1 can be formulated in terms of a transition, weak +pinning goes to strong pinning in a smooth crossover. +Knowing the pinning force density Fpin, the motion of +the vortex lattice follows from the bulk dynamical equa- +tion +ηv = FL(j) − Fpin. +(17) +Here, η = BHc2/ρnc2 is the Bardeen-Stephen viscosity28 +(per unit volume; ρn is the normal state resistivity) and +FL = j × B/c is the Lorentz force density driving the +vortex system. The pinning force density Fpin is directed +along v, in our case along x. +Next, we determine the strong pinning characteristics +¯x−, ¯x+, ˜xf±, ˜xp±, ∆efp +pin and ∆epf +pin as a function of the +Labusch parameter κ close to the strong pinning transi- +tion, i.e., κ ≳ 1. +D. +Strong pinning characteristics near the +transition +Near the strong pinning transition at κ ≳ 1, we can +derive quantitative results for the strong pinning char- +acteristics by expanding the pinning energy epin(˜x; ¯x) in +˜x at fixed ¯x; this reminds about the Landau expansion +of the free energy f(φ, h) in the order parameter φ at a +fixed field h in a thermodynamic transition, see Sec. II F +below for a detailed discussion. +We expand epin(˜x; ¯x) in ˜x around the point of first +instability ˜xm by introducing the relative tip and asymp- +totic positions ˜u = ˜x − ˜xm and ¯u = ¯x − ¯xm and make +use of our alternative strong pinning formulation sum- +marized in Sec. II B 1. At ˜xm and close to κ = 1, we have +[∂2 +˜xepin]˜xm = [∂2 +˜xep]˜xm + ¯C = ¯C(1−κ) and [∂3 +˜xepin]˜xm = 0, +hence, +epin(˜x; ¯x) ≈ +¯C +2 (1 − κ) ˜u2 + γ +24 ˜u4 − ¯C¯u˜u, +(18) +where we have introduced the shape parameter γ = +[∂4 +xep]˜xm describing the quartic term in the expansion +and we have made use of the force balance equation (2) +to rewrite fp(˜xm) = ¯C(˜xm − ¯xm); furthermore, we have +dropped all irrelevant terms that do not depend on ˜u. +We find the jump and landing positions ˜xjp and ˜xlp ex- +ploiting the differential properties of epin(˜x) at a fixed ¯x: +As discussed above, the vortex tip jumps at the bound- +aries ¯x± of the bistable regime, where epin develops a flat +inflection point at ˜xjp with one minimum joining up with +the unstable maximum and the second minimum at the +landing position ˜xlp staying isolated. Within our fourth- +order expansion the jump positions at (de)pinning are +placed symmetrically with respect to the onset at ˜xm, +˜xp+ = ˜xm + ˜ujp, +˜xf− = ˜xm − ˜ujp +(19) + +9 +and imposing the condition [∂2 +˜uepin]˜xjp = 0 (that is equiv- +alent to the jump condition f ′ +p[˜xf−] = f ′ +p[˜xp+] = ¯C of Eq. +(8), see also Fig. 3), we find that +˜ujp ≈ − +� +2 ¯C +γ (κ − 1)1/2. +(20) +In order to find the (symmetric) landing positions, it +is convenient to shift the origin of the expansion to the +jump position, ˜u → ˜u − ˜ujp ≡ ˜u′, and define the jump +distance ∆˜u, +˜xf+ = ˜xp+ + ∆˜u, +˜xp− = ˜xf− − ∆˜u. +(21) +At the jump position, the linear and quadratic terms in +˜u′ vanish, resulting in the expansion (up to an irrelevant +constant) +epin(˜xp+ + ˜u′; ¯x+) ≈ γ +6 ˜ujp˜u′ 3 + γ +24 ˜u′ 4 +(22) +and similar at ˜xf− and ¯x− for a left moving vortex. This +expression is minimal at the landing position ˜xf+, i.e., at +˜u′ = ∆˜u, [∂˜u′epin]∆˜u = 0, and we find the jump distance +∆˜u = −3˜ujp. +(23) +Inserting this result back into (22), we obtain the jump +in energy ∆epf +pin = epin(˜xp+; ¯x+) − epin(˜xf+; ¯x+), +∆epf +pin(¯x+) ≈ γ +72(∆˜u)4 ≈ 9 ¯C2 +2γ (κ − 1)2, +(24) +and similar at ¯x−. Note that all these results have been +obtained without explicit knowledge of the asymptotic +coordinates ¯x± where these tip jumps are triggered. The +latter follow from the force equation (2) that corresponds +to the condition [∂˜xepin]˜xjp = 0 for a flat inflection point. +Using the expansion (18) of the pinning energy, we find +that +¯x± − ¯xm = ∓2 +3 ˜ujp(κ − 1) = ±2 +3 +� +2 ¯C +γ (κ − 1)3/2. +(25) +The pair ¯xm and ˜xm of asymptotic and tip positions +depends on the details of the potential; while ˜xm derives +solely from the shape ep(˜x), ¯xm as given by (2) involves +¯C and shifts ∝ (κ − 1). For a Lorentzian potential, we +find that +˜xm = +√ +2ξ, +¯xm = 2 +√ +2ξ + +√ +2ξ(κ − 1). +(26) +The shape coefficient is γ = 3ep/4ξ4 and the Labusch +parameter is given by κ = ep/4 ¯Cξ2 (hence ¯C2/γ = +ep/12κ2), providing us with the results +˜ujp ≈ −ξ [2(κ−1)/3]1/2 and ∆epf +pin ≈ 3 +8ep(κ−1)2. (27) +E. +Pinning force density for the isotropic defect +Using the results of Sec. II D in the expression (16) +for the pinning force density, we find, to leading order in +κ − 1, +Fpin = 9np +¯xm +a0 +¯C2 +γa0 +(κ − 1)2. +(28) +The scaling Fpin ∼ np(ξ/a0)2fp(κ − 1)2 (with ¯Cξ2/ep ∼ +1/κ, up to a numerical) uniquely derives from the scaling +∝ (κ − 1)2 of the energy jumps in (24), as the asymp- +totic trapping length ¯x− ∼ ξ remains finite as κ → 1 for +the isotropic defect; this will change for the anisotropic +defect. +F. +Relation to Landau’s theory of phase transitions +The expansion (18) of the pinning energy epin(˜x; ¯x) +around the inflection point ˜xm of the force takes the same +form as the Landau free energy of a phase transition10, +f(φ; h) = r0 +2 (T/Tc − 1)φ2 + uφ4 − hφ, +(29) +with the straightforward transcription ˜u ↔ φ, ¯C(1−κ) ↔ +r0(T/Tc − 1), γ/24 ↔ u and the conjugate field ¯C¯u ↔ h. +The functional (29) describes a one-component oder pa- +rameter φ driven by h, e.g., an Ising model with magne- +tization density φ in an external magnetic field h. This +model develops a mean-field transition with a first-order +line in the h–T phase diagram that terminates in a criti- +cal point at T = Tc and h = 0. The translation to strong +pinning describes a strong pinning region at large κ that +terminates (upon decreasing κ) at κ = 1. The ferromag- +netic phases with φ = ± +� +r0(1 − T/Tc)/4u correspond +to pinned and unpinned states, the paramagnetic phase +at T > Tc with φ = 0 translates to the unpinned domain +at κ < 1. The spinodals associated with the hysteresis in +the first-order magnetic transition correspond to the ter- +mination of the free and pinned branches at ¯x±; indeed, +the flat inflection points appearing in epin(˜x; ¯x) at the +boundaries of the bistable region B ¯R as discussed in Sec. +II B correspond to the disappearance of metastable mag- +netic phases in (29) at the spinodals of the first-order +transition where ∂φf(φ; h) = ∂2 +φf(φ; h) = 0. When in- +cluding correlations between defects, the unpinned phase +at κ < 1 transforms into a weakly pinned phase that +continues beyond κ = 1 into the strongly pinned phase. +Including such correlations, the strong-pinning transition +at the onset of strong pinning at κ = 1 transforms into a +weak-to-strong pinning crossover. +III. +ANISOTROPIC DEFECTS +Let us generalize the above analysis to make it fit for +the ensuing discussion of an arbitrary pinning landscape + +10 +or short, pinscape. +Central to the discussion are the +unstable and bistable domains U ˜R and B ¯R in tip- and +asymptotic space. The boundary of the unstable domain +U ˜R in tip space is determined by the jump positions of the +vortex tip. The latter follows from the local differential +properties of epin( ˜R; ¯R) at fixed asymptotic coordinate +¯R, for the isotropic defect, the appearence of an inflection +point [∂2 +˜xepin(˜x, ¯x)] = 0, see Eq. (12). In generalizing this +condition to the anisotropic situation, we have to study +the Hessian matrix of epin( ˜R; ¯R) defined in Eq. (1), +� +Hess +� +epin( ˜R; ¯R)| ¯R +�� +ij = ¯Cδij + Hij( ˜R) +(30) +with +Hij( ˜R) = ∂˜xi∂˜xjep( ˜R; ¯R) +(31) +the Hessian matrix associated with the defect potential +ep( ˜R). The vortex tip jumps when the pinning landscape +epin( ˜R; ¯R) at fixed ¯R opens up in an unstable direction, +i.e., develops an inflection point; this happens when the +lower eigenvalue λ−( ˜R) < 0 of the Hessian matrix Hij( ˜R) +matches up with ¯C, +λ−( ˜R) + ¯C = 0, +(32) +and strong pinning appears in the location where this +happens first, say in the point ˜Rm, implying that the +eigenvalue λ−( ˜R) has a minimum at ˜Rm. Furthermore, +the eigenvector v−( ˜Rm) associated with the eigenvalue +λ−( ˜Rm) provides the unstable direction in the pinscape +epin( ˜R; ¯R) along which the vortex tip escapes. +Defining the reduced curvature function +κ( ˜R) ≡ −λ−( ˜R) +¯C +, +(33) +we find the generalized Labusch parameter +κm ≡ κ( ˜Rm), +(34) +and the Labusch criterion takes the form +κm = 1. +(35) +The latter has to be read as a double condition: i) find +the location ˜Rm where the smaller eigenvalue λ−( ˜R) is +negative and largest, from which ii), one obtains the crit- +ical elasticity ¯C where strong pinning sets in. +A useful variant of the strong pinning condition (32) is +provided by the representation of the determinant of the +Hessian matrix, +D( ˜R) ≡ det +� +Hess +� +epin( ˜R; ¯R)| ¯R +�� +, +(36) +in terms of its eigenvalues λ±( ˜R), D( ˜R) += +[ ¯C + +λ−( ˜R)][ ¯C + λ+( ˜R)]; near onset, the second factor ¯C + +λ+( ˜R) stays positive and the strong pinning onset ap- +pears in the point ˜Rm where D( ˜R) has a minimum which +touches zero for the first time, i.e., the two conditions +∇D( ˜R)| ˜Rm = 0 and D( ˜Rm) = 0 are satisfied simultane- +ously. The latter conditions make sure that the minima +of λ−( ˜R) and D( ˜R) line up at ˜Rm. Note that the Hes- +sian determinant D( ˜R) does not depend on the asymp- +totic coordinate ¯R as it involves only second derivatives +of epin( ˜R; ¯R). +The Labusch criterion defines the situation where +jumps of vortex tips appear for the first time in the iso- +lated point ˜Rm. Increasing the pinning strength, e.g., +by decreasing the elasticity ¯C for a fixed pinning poten- +tial ep(R) (alternatively, the pinning scale ep could be +increased at fixed ¯C) the condition (32) is satisfied on +the boundary of a finite domain and we can define the +unstable domain U ˜R through (see also Ref. 27) +U ˜R = +� +˜R | λ−( ˜R) + ¯C ≤ 0 +� +. +(37) +Once the latter has been determined, the bistable do- +main B ¯R follows straightforwardly from the force balance +equation +¯C( ˜R − ¯R) = fp( ˜R) = fpin( ¯R), +(38) +i.e.,27 +B ¯R = +� +¯R = ˜R − fp( ˜R)/ ¯C | +˜R ∈ U ˜R +� +. +(39) +In a last step, one then evaluates the energy jumps ap- +pearing at the boundary of B ¯R and proper averaging pro- +duces the pinning force density Fpin. +Let us apply the above generalized formulation to +the isotropic situation. Choosing cylindrical coordinates +(r, ϕ), the Hessian matrix Hij is already diagonal; close to +the inflection point ˜Rm, where e′′′ +p ( ˜Rm) = 0, the eigenval- +ues are λ−( ˜R) = e′′ +p( ˜R) < 0 and λ+( ˜R) = e′ +p( ˜R)/ ˜R > 0, +producing results in line with our discussion above. +A. +Expansion near strong pinning onset +With our focus on the strong pinning transition near +κ( ˜Rm) = 1, we can obtain quantitative results using the +expansion of the pinning energy epin( ˜R; ¯R), Eq. (1), close +to ˜Rm, cf. Sec. II D. Hence, we construct the Landau-type +pinning energy corresponding to (29) for the case of an +anisotropic pinning potential, i.e., we generalize (18) to +the two-dimensional situation. +When generalizing the strong pinning problem to the +anisotropic situation, we are free to define local coor- +dinate systems (˜u, ˜v) and (¯u, ¯v) in tip- and asymptotic +space centered at ˜Rm and ¯Rm, where the latter is asso- +ciated with ˜Rm through the force balance equation (38) +in the original laboratory system. Furthermore, we fix +our axes such that the unstable direction coincides with +the u-axis, i.e., the eigenvector v−( ˜Rm) associated with +λ−( ˜Rm) points along u; as a result, the mixed term ∝ ˜u˜v +is absent from the expansion. +Keeping all potentially + +11 +relevant terms up to fourth order in ˜u and ˜v in the ex- +pansion, we then have to deal with an expression of the +form +epin( ˜R; ¯R) = +¯C + λ− +2 +˜u2 + +¯C + λ+ +2 +˜v2 − ¯C ¯u˜u − ¯C ¯v˜v ++ a +2 ˜u˜v2 + a′ +2 ˜u2˜v + b′ +6 ˜u3 + b′′ +6 ˜v3 +(40) ++ α +4 ˜u2˜v2 + β +6 ˜u3˜v + β′′ +6 ˜u˜v3 + γ +24 ˜u4 + γ′′ +24 ˜v4, +with λ± = λ±( ˜Rm), +˜R = ˜Rm + δ ˜R, +δ ˜R = (˜u, ˜v), +(41) +¯R = ¯Rm + δ ¯R, +δ ¯R = (¯u, ¯v), +and coefficients given by the corresponding derivatives of +ep(R), e.g., a ≡ ∂u∂2 +vep(R)| ˜Rm, . . . , γ′′ ≡ ∂4 +vep(R)| ˜Rm. +As we are going to see, the primed terms in this expan- +sion vanish due to the condition of a minimal Hessian +determinant at the onset of strong pinning, while double- +primed terms will turn out irrelevant to leading order in +the small distortions ˜u and ˜v. +The first term in (40) drives the strong pinning tran- +sition as it changes sign when λ− = − ¯C. Making use of +the Labusch parameter κm defined in (34), we can replace +(see also (18)) +¯C + λ− → ¯C(1 − κm). +(42) +In our further considerations below, the quantity κm − +1 ≪ 1 acts as the small parameter; it assumes the role of +the distance 1 − T/Tc to the critical point in the Landau +expansion of a thermodynamic phase transition. +The second term in (40) stabilizes the theory along +the v direction as ¯C +λ+ > 0 close to the Labusch point, +while the sign of the cubic term a ˜u˜v2/2 determines the +direction of the instability along x, i.e., to the right (a > +0) or left (a < 0). The quartic terms ∝ α, γ > 0 bound +the pinning energy at large distances, while the term ∝ β +determines the skew angle in the shape of the unstable +domain U ˜R, see below. Finally, we have used the force +balance equation (38) in the derivation of the driving +terms ¯C ¯u˜u and ¯C ¯v˜v. +The parameters in (40) are constrained by the require- +ment of a minimal determinant D( ˜R) at the strong pin- +ning onset ˜R = ˜Rm and κm = 1, i.e., its gradient has to +vanish, +∇ ˜R D( ˜R) +�� ˜Rm = 0, +(43) +and its Hessian Hess[D( ˜R)] has to satisfy the relations +det +� +Hess +� +D( ˜R) +���� ˜Rm > 0, +(44) +tr +� +Hess +� +D( ˜R) +���� ˜Rm > 0. +(45) +Making use of the expansion (40), the determinant D( ˜R) +reads +D( ˜R) = +� +[∂2 +˜uepin][∂2 +˜vepin] − [∂˜u∂˜vepin]2� +˜R +(46) +with +∂2 +˜uepin = ¯C (1−κm) + a′˜v + b′˜u + α˜v2/2 + β˜u˜v + γ˜u2/2, +∂2 +˜vepin = ¯C + λ+ + a˜u + b′′˜v + α˜u2/2 + β′′˜u˜v + γ′′˜v2/2, +∂˜u∂˜vepin = a˜v + a′˜u + α˜u˜v + β˜u2/2 + β′′˜v2/2, +and produces the gradient +∇ ˜R D( ˜R) +��� ˜Rm = ( ¯C + λ+)(b′, a′), +(47) +hence the primed parameters indeed vanish, a′ = 0 and +b′ = 0. The Hessian then takes the form +Hess +� +D( ˜R) +���� ˜Rm += ( ¯C + λ+) +� +γ +β +β +δ +� +(48) +at the Labusch point κm = 1, where we have introduced +the parameter +δ ≡ α − 2a2 +¯C +1 +1 + λ+/ ¯C . +(49) +The stability conditions (44) and (45) translate, respec- +tively, to +γδ − β2 > 0 +(50) +(implying δ > 0) and +γ + δ > 0. +(51) +The Landau-type theory (40) involves the two ‘order +parameters’ ˜u and ˜v and is driven by the dual coordinates +¯u and ¯v. This n = 2 theory involves a soft order param- +eter ˜u and the stiff ˜v, allowing us to integrate out ˜v and +reformulate the problem as an effective one-dimensional +Landau theory (A6) of the van der Waals kind—the way +of solving the strong pinning problem near onset in this +1D formulation is presented in Appendix A 1. +B. +Unstable domain U ˜ +R +Next, we determine the unstable domain U ˜R in tip +space as defined in (37). +We will find that, up to +quadratic order, the boundary of U ˜R has the shape of +an ellipse with the semiaxes lengths scaling as √κm − 1. +1. +Jump line J ˜ +R +We find the unstable domain U ˜R by determining its +boundary ∂U ˜R that is given by the set of jump positions +˜Rjp making up the jump line J ˜R. The boundary ∂U ˜R is +determined by the condition ¯C +λ− = 0 or, equivalently, +the vanishing of the determinant +D( ˜Rjp) ≡ 0. +(52) + +12 +The latter condition guarantees the existence of an un- +stable direction parallel to the eigenvector v−( ˜Rjp) as- +sociated with the eigenvalue λ−( ˜Rjp) where the energy +(40) turns flat, cf. our discussion in Sec. II B. The edges +of the unstable domain U ˜R therefore correspond to a line +of inflection points in epin( ˜R; ¯R) along which one of the +bistable tip configurations of the force balance equation +(38) coalesces with the unstable solution. Near the onset +of strong pinning, the unstable domain U ˜R is closely con- +fined around the point ˜Rm where v−( ˜Rm) ∥ ˆu. The un- +stable direction v−( ˜Rjp) is therefore approximately ho- +mogeneous within the unstable domain U ˜R and is parallel +to the u axis. This fact will be of importance later, when +determining the topological properties of the unstable +domain U ˜R. +Inspection of the condition (52) with D( ˜R) given by +Eq. (46) shows that the components of δ ˜Rjp scale as +√κm − 1: in the product [∂2 +˜uepin][∂2 +˜vepin], the first fac- +tor involves the small constant ¯C(1 − κm) plus quadratic +terms (as a′ = 0 and b′ = 0), while the second factor +comes with the large constant ¯C + λ+ plus corrections. +The leading term in [∂˜u∂˜vepin] is linear in ˜v with the re- +maining terms providing corrections. To leading order, +the condition of vanishing determinant then produces the +quadratic form +[γ ˜u2 + 2β ˜u˜v + δ ˜v2] ˜Rjp = 2 ¯C (κm − 1) . +(53) +With γ and δ positive, this form is associated with an +elliptic geometry of extent ∝ √κm − 1. For later conve- +nience, we rewrite Eq. (53) in matrix form +δ ˜RT +jpMjp δ ˜Rjp = ¯C(κm − 1) +(54) +with +Mjp = +� +γ/2 +β/2 +β/2 +δ/2 +� +(55) +and det Mjp = (γδ − β2)/4 > 0, see Eq. (50). The jump +line J ˜R can be expressed in the parametric form +˜ujp(|˜v| < ˜vc) = − 1 +γ +� +β˜v +± +� +2γ ¯C(κm − 1) − (γδ − β2)˜v2 +� +, +(56) +with +˜vc = +� +2γ ¯C(κm − 1)/(γδ − β2) +(57) +and is shown in Fig. 6 for the example of an anisotropic +potential inspired by the uniaxial defect in Sec. IV with +10 % anisotropy. The associated unstable domain U ˜R as- +sumes a compact elliptic shape, with the parameter β de- +scribing the ellipse’s skew. Comparing with the isotropic +defect, this ellipse assumes the role of the ring bounded +by solid lines in Fig. 5(c), see Sec. III E for a discussion +of its different topology. +−2 +−1 +0 +1 +−2 +0 +˜u/ξ√κm − 1 +˜v/ξ√κm − 1 +FIG. 6. +Jump line J ˜ +R (solid red/blue, see Eq. (54)) and +landing line (dashed red/blue, see Eq. (63)) L ˜ +R in tip space +˜R (in units of ξ), with the ellipse J ˜ +R representing the edge +∂U ˜ +R of the unstable domain U ˜ +R. +We choose parameters +κm − 1 = 10−2, with λ− = −0.25 ep/ξ2, λ+ = 0.05 ep/ξ2, and +a = 0.07 ep/ξ3, α = 0.1 ep/ξ4, β = 0, γ = 0.75 ep/ξ4 inspired +by the choice of the uniaxial defect with 10 % anisotropy in +Sec. IV; the dotted ellipse shows the effect of a finite skew +parameter β = 0.05 ep/ξ4 on the jump ellipse J ˜ +R. +Along +the edges of U ˜ +R, one of the stable tip configurations coalesces +with the unstable solution of (38) and the total pinning energy +epin( ˜R; ¯R) develops an inflection line in the tip coordinate ˜R. +Crosses correspond to the contact points (67) between the +two ellipses J ˜ +R and L ˜ +R. +Blue and red colors identify dif- +ferent types of vortex deformations upon jump and landing. +Pairs of solid and open circles connected via long arrows are, +respectively, examples of pairs of jumping- and landing tip po- +sitions for vortices approaching the defect from the left (top) +and right (bottom), see Fig. 5(c) for the isotropic problem’s +counterpart. The unstable direction v−( ˜Rjp), shown as short +black arrows for different points on the ellipse, always points +in the u−direction and are parallel to the tangent vector of +the unstable ellipse at the contact points (67). +An additional result of the above discussion concerns +the terms that we need to keep in the expansion of +the pinning energy (40): indeed, dropping corrections +amounts to dropping terms with double-primed coeffi- +cients and we find that the simplified expansion +epin( ˜R; ¯R) = +¯C +2 (1 − κm) ˜u2 + +¯C + λ+ +2 +˜v2 + a +2 ˜u˜v2 ++ α +4 ˜u2˜v2 + β +6 ˜u3˜v + γ +24 ˜u4 − ¯C ¯u˜u − ¯C ¯v˜v +(58) +produces all of our desired results to leading order. +2. +Landing line L ˜ +R +We find the landing positions ˜Rlp by extending the +discussion of the isotropic situation in Sec. II D to two +dimensions: we shift the origin of the expansion (58) to +the jump point ˜Rjp and find the landing point ˜Rlp = + +13 +˜Rjp + ∆ ˜R by minimizing the total energy epin(∆ ˜R) at +the landing position. Below, we use ∆ ˜R both as a vari- +able and as the jump distance to avoid introducing more +coordinates. +We exploit the differential properties of epin at the +jump and landing positions. At landing, epin( ˜Rjp + ∆ ˜R) +has a minimum, hence, the configuration is force free, in +particular along ˜v, +∂˜vepin( ˜Rjp + ∆ ˜R) ≈ [∂˜v∂˜uepin] ˜Rjp∆˜u ++ [∂2 +˜vepin] ˜Rjp∆˜v = 0, +from which we find that ∆˜u and ∆˜v are related via +∆˜v ≈ − +[∂˜v∂˜uepin] ˜Rjp +[∂2 +˜vepin] ˜Rjp +∆˜u. +(59) +Here, we have dropped higher order terms in the expan- +sion, assuming that the jump is mainly directed along the +unstable u-direction—indeed, using the expansion (58), +we find that +∆˜v ≈ − +a˜vjp +¯C + λ+ +∆˜u ∝ +√ +κm − 1 ∆˜u. +(60) +Note that we cannot interchange the roles of ˜u and ˜v in +this force analysis, as higher order terms in the expression +for the force along ˜u cannot be dropped. +At the jump position ˜Rjp, the state is force-free, +i.e., the derivatives [∂˜uepin] ˜Rjp and [∂˜vepin] ˜Rjp vanish, +and the Hessian determinant vanishes as well. +There- +fore, the expansion of epin( ˜Rjp + ∆ ˜R) has no linear +terms and the second order terms [∂2 +˜uepin] ˜Rjp∆˜u2/2 + +[∂˜u∂˜vepin] ˜Rjp∆˜u∆˜v + [∂2 +˜vepin] ˜Rjp∆˜v2/2 combined with +Eq. (59) can be expressed through the Hessian determi- +nant, {[∂2 +˜uepin][∂2 +˜vepin] − [∂˜u∂˜vepin]2} ˜Rjp∆˜u2/2 = 0, that +vanishes as well. Therefore, the expansion of epin around +˜Rjp starts at third order in ∆ ˜R ≈ (∆˜u, 0) and takes the +form (we make use of (60), dropping terms ∝ ∆˜v and a +constant) +epin( ˜Rjp + ∆ ˜R) ≈ 1 +6 +� +γ˜ujp + β˜vjp +� +∆˜u3 + γ +24∆˜u4. (61) +Minimizing this expression with respect to ∆˜u (as epin is +minimal at ˜Rlp), we obtain the result +∆˜u ≈ −3(γ˜ujp + β˜vjp)/γ. +(62) +Making use of the quadratic form (54), we can show +that the equation for the landing position ˜Rlp = ˜Rjp + +∆ ˜R can be cast into a similar quadratic form (with δ ˜Rlp +measured relative to ˜Rm) +δ ˜RT +lpMlp δ ˜Rlp = ¯C(κm − 1), +(63) +but with the landing matrix now given by +Mlp = 1 +4Mjp + +� +� +0 +0 +0 +3 +4 +�δ +2 − β2 +2γ +� +� +� . +(64) +In the following, we will refer to the solutions of Eq. (63) +as the ‘landing’ or ‘stable’ ellipse ˜Rlp and write the jump +distance in a parametric form involving the shape ˜ujp(˜v) +in Eq. (56) of the jumping ellipse, +∆˜u(˜v) = −3 [γ ˜ujp(˜v) + β ˜v] /γ, +(65) +∆˜v(˜v) = − +� +a/( ¯C + λ+) +� +˜v ∆˜u(˜v). +(66) +The landing line derived from (63) is displayed as a +dashed line in Fig. 6. Two tip jumps connected by an +arrow are shown for illustration, with solid dots marking +the jump position ˜Rjp of the tip and open dots its land- +ing position ˜Rlp; they describe tip jumps for a vortex +approaching the unstable ellipse once from the left (up- +per pair) and another time from the right (lower pair). +The different topologies associated with jumps and land- +ing showing up for the isotropic defect in Fig. 5(c) (two +concentric circles) and for the generic onset in Fig. 6 (two +touching ellipses) will be discussed later. +Inspecting the matrix equation (63), we can gain sev- +eral insights on the landing ellipse L ˜R: (i) the matrix +Mjp/4 on the right-hand side of (64) corresponds to an +ellipse with the same geometry as for J ˜R but double in +size, (ii) the remaining matrix with vanishing entries in +the off-diagonal and the Mxx elements leaves the size dou- +bling of the stable ellipse L ˜R at ˜v = 0 unchanged, and +(iii) the finite Myy component exactly counterbalances +the doubling along the v−direction encountered in (i), +cf. the definiton (55) of Mjp, up to a term proportional +to the skew parameter β accounting for deviations of the +semiaxis from the v−axis. Altogether, the stable ellipse +L ˜R extends with a double width along the u−axis and +smoothly overlaps with the unstable ellipse at the two +contact points ˜vc,±. The latter are found by imposing +the condition ∆˜u = ∆˜v = 0 in Eqs. (65) and (66); we +find them located (relative to ˜Rm) at +δ ˜Rc,± = ± (−β/γ, 1) ˜vc, +(67) +with the endpoint coordinate ˜vc given in Eq. (57), and +mark them with crosses in Fig. 6. As anticipated, the +contact points are off-set with respect to the v−axis for +a finite skew parameter β. At these points, the unstable +and the stable tip configurations coincide and the vortex +tip undergoes no jump. +Furthermore, the vector tan- +gent to the jump (or landing) ellipse is parallel to the +u−direction at the contact points. To see that, we con- +sider (56) and find that +∂˜u +∂˜v +��� +˜v→±˜vc≈ ± +� +� +� +˜v2c − 2γ ¯C(κm − 1) +γβ − δ2 +� +� +−1 +→ ±∞, +(68) +hence, the corresponding tangents ∂˜u˜v vanish. +The asymptotic positions ¯R where the vortex tips +jump and land belong to the boundary of the bistable +region B ¯R; for the isotropic case in Fig. 5(d) these cor- +respond to the circles with radii ¯R− (pinning) and ¯R+ + +14 +(depinning) with jump and landing radii ˜Rf−( ¯R−) and +˜Rp−( ¯R−) and ˜Rp+( ¯R+) and ˜Rf+( ¯R+), respectively, see +Fig. 5(c). +For the anisotropic defect, we have only a +single jump/landing event at one asymptotic position ¯R +that we are going to determine in the next section. +C. +Bistable domain B ¯ +R +The set of asymptotic positions ¯R corresponding to +the tip positions ˜Rjp along the edges of U ˜R forms the +boundary ∂B ¯R of the bistable domain B ¯R; they are re- +lated through the force-balance equation (38), with ev- +ery vortex tip position ˜Rjp ∈ ∂U ˜R defining an associated +asymptotic position ¯R( ˜Rjp) ∈ ∂B ¯R. +At the onset of strong pinning, the bistable domain cor- +responds to the isolated point ¯Rm, related to ˜Rm through +(38). Beyond the Labusch point, B ¯R expands out of ¯Rm +and its geometry is found by evaluating the force bal- +ance equation (38) at a given tip position ˜Rjp ∈ ∂U ˜R, +¯R( ˜Rjp) = ˜Rjp − fp( ˜Rjp)/ ¯C ∈ ∂B ¯R. Using the expansion +(58) for epin( ˜R; ¯R), this force equation can be expressed +as ∇Repin(R; ¯R) +�� ˜R = 0, or explicitly (we remind that we +measure ¯R = ¯Rm + (¯u, ¯v) relative to ¯Rm), +¯C¯u = ¯C(1 − κm)˜u + a +2 ˜v2 + γ +6 ˜u3 + β +2 ˜u2˜v + α +2 ˜u˜v2, +¯C¯v = ( ¯C + λ+)˜v + a ˜u˜v + β +6 ˜u3 + α +2 ˜u2˜v. +(69) +Inserting the results for the jump ellipse J ˜R, Eq. (56), +into Eqs. (69), we find the crescent-shape bistable domain +B ¯R shown in Fig. 7; let us briefly derive the origin of this +shape. +Solving (69) to leading order, ¯C¯u(0) ≈ (a/2)˜v2 and +¯C¯v(0) ≈ ( ¯C + λ+)˜v, we find the parabolic approximation +¯u +(0) ≈ a +2 ¯C +1 +(1 + λ+/ ¯C)2 ¯v +(0) 2, +(70) +telling that the extent of B ¯R scales as (κm − 1) along ¯u +and ∝ (κm − 1)1/2 along ¯v, i.e., we find a flat parabola +opening towards positive ¯u for a > 0, see Fig. 7. +In order to find the width of B ¯R, we have to solve (69) +to the next higher order, ¯u = ¯u(0) + ¯u(1); for β = 0, we +find the correction +¯u +(1) = (1 − κm)˜u + γ +6 ¯C ˜u3 + α +2 ¯C ˜u˜v2 +(71) +that produces a ¯v ↔ −¯v symmetric crescent. Inserting +the two branches (56) of the jump ellipse, we arrive at +the width of the crescent that scales as (κm − 1)3/2. The +correction to ¯v is ∝ (κm − 1) and we find the closed form +¯v ≈ [1 + (λ+ + a˜u)/ ¯C] ˜v +(72) +with a small antisymmetric (in ˜u) correction. For a finite +β ̸= 0, the correction ¯u(1) picks up an additional term +0 +0.5 +−2 +0 +−2 +0 +−2 +0 +−10 +−5 +0 +5 +π/2 − θ∗ +θ∗ +¯u/ξ(κm − 1) +¯v/ξ√κm − 1 +(a) +¯u/ξ√κm − 1 +¯v/ξ√κm − 1 +(b) +˜u/ξ√κm − 1 +FIG. 7. (a) Bistable domain B ¯ +R in asymptotic ¯R-space mea- +sured in units of ξ; the same parameters as in Fig. 6 have +been used. Note the different scaling of the axes in κm − 1; +the right panel (b) shows B ¯ +R in isotropic scales. The bistable +domain B ¯ +R is elongated along the transverse direction ¯v and +narrow/bent along the unstable direction ¯u, giving B ¯ +R its pe- +culiar crescent-like shape. The branch crossing line ¯R0, see +(77), is shown as a dashed black line. +Black crosses mark +the cusps of B ¯ +R and are associated with the contact points +of U ˜ +R through the force balance equation (38); they corre- +spond to critical end-points in the thermodynamic Ising ana- +logue, while the boundaries ∂B ¯ +R map to spinodals. Blue and +red colors identify different characters of vortex tip configu- +rations as quantified through the ‘order parameter’ ˜u of the +Landau expansion (at β = 0), see text, while magenta is as- +sociated to the bistable area B ¯ +R; the blue and red branches +extend to the far side of the crescent and terminate in the +blue and red colored boundaries ∂Bb +¯ +R and ∂Br +¯ +R, respectively. +Thin horizontal lines show vortex trajectories that proceed +smoothly in asymptotic space, see also Fig. 5(d). Blue and +red dots mark the asymptotic positions associated with vor- +tex tip jumps that happen at the exit of B ¯ +R; they correspond +to the pairs of tip positions in Fig. 6. (b) Bistable domain +B ¯ +R in isotropic scaled coordinates ¯u and ¯v showing the ‘true’ +shape of B ¯ +R. Vortices impacting on the bistable domain with +an angle |θ| ≤ θ∗ undergo a single jump on the far side of B ¯ +R, +with the pinning force density directed along u and scaling +as F ∥ +pin ∝ (κ − 1)5/2. Vortices crossing B ¯ +R at large angles +close to π/2 jump either never, once, or twice; at θ = π/2 the +pinning force density is small, F ⊥ +pin ∝ (κ − 1)3, and directed +along v. +(β/2 ¯C) ˜u2˜v that breaks the ¯v ↔ −¯v symmetry and the +crescent is distorted. +Viewing the boundary ∂B ¯R as a parametric curve in +the variable ˜v with ˜u = ˜ujp(˜v) given by Eq. (56), we +obtain the boundary ∂B ¯R in the form of two separate +arcs that define the crescent-shaped domain B ¯R in Fig. +7(a). +The two arcs merge in two cusps at ¯Rc,± that +are associated to the touching points (67) in dual space +and derive from Eqs. (69); measured with respect to ¯Rm, + +15 +these cusps are located at +δ ¯Rc,± = (¯uc, ±¯vc) +(73) +≈ +�� +a/2 ¯C +� +˜v2 +c, ±(1 + λ+/ ¯C)˜vc +� +. +The coloring in Fig. 7 indicates the characters ‘red’ and +‘blue’ of the vortex states; these are defined in terms of +the ‘order parameter’ ˜u− ˜um(¯v) of the Landau functional +(58) that changes sign at the branch crossing line Eq. +(77), with the shift +˜um(¯v) = −β +γ ˜v(¯v) ≈ −β +γ +¯v +1 + λ+/ ¯C , +(74) +˜um(¯v) = 0 for our symmetric case with β = 0 in Fig. 7. +Going beyond the cusps (or critical points) at ¯Rc,±, the +two states smoothly crossover between ‘red’ and ‘blue’ +(indicated by the smooth blue–white–red transition), as +known for the van der Waals gas (or Ising magnet) above +the critical point. Within the bistable region B ¯R, both +‘red’ and ‘blue’ states coexist and we color this region in +magenta. +The geometry of the bistable domain B ¯R is very differ- +ent from the ring-shaped geometry of the isotropic prob- +lem discussed in Sec. II A, see Fig. 5(d); in the discussion +of the uniaxial anisotropic defect below, we will learn how +these two geometries are interrelated. +Comparing the +overall dimensions of the crescent with the ring in Fig. +5(d), we find the following scaling behavior in κm − 1: +while the crescent B ¯R grows along ¯v as (κm − 1)1/2, the +isotropic ring involves the characteristic size ξ of the de- +fect, ¯R− ∼ ξ and hence its extension along ¯v is a con- +stant. On the other hand, the scaling of the crescent’s +and the ring’s width is the same, ∝ (κm − 1)3/2. The +different scaling of the transverse width then will be re- +sponsible for the new scaling of the pinning force density, +Fpin ∝ (κm − 1)5/2. +D. +Comparison to isotropic situation +Let us compare the unstable domains U ˜R for the +isotropic and anisotropic defects in Figs. 5(c) and 6, re- +spectively. +In the isotropic example of Sec. II A, the +jump- and landing-circles ˜Rjp( ¯R) and ˜Rlp( ¯R) are con- +nected to different phases, e.g., free (colored in blue at +˜Rjp = ˜Rf−) and pinned (colored in red at ˜Rlp = ˜Rp−) +associated with ¯R−. Furthermore, the topology is differ- +ent, with the unstable ring domain separating the two +distinct phases, free and pinned ones. As a result, a sec- +ond pair of jump- and landing-positions associated with +the asymptotic circle ¯R+ appears along the vortex tra- +jectory of Fig. 5(c); these are the located at the radii +˜Rjp = ˜Rp+ and ˜Rlp = ˜Rf+ and describe the depinning +process from the pinned branch back to the free branch +(while the previous pair at radii ˜Rf− and ˜Rp− describes +the pinning process from the free to the pinned branch). +The pinning (at ¯R−) and depinning (at ¯R+) processes +in the asymptotic coordinates are shown in figure 5(d). +The bistable area B ¯R with coexisting free and pinned +states has a ring-shape as well (colored in magenta, the +superposition of blue and red); the two pairs of jump and +landing points in tip space have collapsed to two pinning +and depinning points in asymptotic space. +In the present situation describing the strong pinning +onset for a generic anisotropic potential, the unstable do- +main U ˜R grows out of an isolated point (in fact, ˜Rm) and +assumes the shape of an ellipse that is simply connected; +as a result, a vortex incident on the defect undergoes +only a single jump, see Fig. 6. The bistable domain B ¯R +is simply connected as well, but now features two cusps +at the end-points of the crescent, see Fig. 7. The bista- +bility again involves two states, but we cannot associate +them with separated pinned and free phases—we thus de- +note them by ‘blue’-type and ‘red’-type. The two states +approach one another further away from the defect and +are distiguishable only in the region close to bistability; +in Fig. 7, this is indicated with appropriate color cod- +ing. +Note that the Landau-type expansion underlying +the coloring in Fig. 7 fails at large distances; going be- +yond a local expansion near ˜Rm, the distortion of the +vortex vanishes at large distances and red/blue colors +faint away to approach ‘white’. +E. +Topology +The different topologies of unstable and bistable re- +gions appearing in the isotropic and anisotropic situa- +tions are owed to the circular symmetry of the isotropic +defect; we will recover the ring-like topology for the +anisotropic situation later when describing a uniaxially +anisotropic defect at larger values of the Labusch param- +eter κm. Indeed, such an increase in pinning strength will +induce a change in topology with two crescents facing one +another joining into a ring-like shape. +Let us discuss the consequences of the different topolo- +gies that we encountered for the isotropic and anisotropic +defects in the discussion above. Specifically, the precise +number and position of the contact points have an elegant +topological explanation. When a vortex tip touches the +edges ˜Rjp of the unstable domain there are two character- +istic directions: one is given by the unstable eigenvector +v−( ˜Rjp) discussed in Sec. III B along which the tip will +jump initially. The second is the tangent vector to the +boundary ∂U ˜R of the unstable domain, i.e., to the unsta- +ble ellipse. While the former is approximately constant +and parallel to the unstable u-direction along ˜Rjp, the +latter winds around the ellipse exactly once after a full +turn around U ˜R. The contact points ˜Rc,± of the unsta- +ble and stable ellipses then coincide with those points on +the ellipse where the tangent vector are parallel and anti- +parallel to v−; at these points, the tip touches the unsta- +ble ellipse but does not undergo a jump any more. Given +the different winding numbers of v− and of the tangent +vector, there are exactly two points along the circum- +ference of U ˜R where the tangent vector is parallel/anti- + +16 +parallel to the u-direction; these are the points found in +(67). This argument remains valid as long as the contour +∂U ˜R is not deformed to cross/encircle the singular point +of the v−( ˜Rjp) field residing at the defect center. +The same arguments allow us to understand the ab- +sence of contact points in the isotropic scenario: For an +isotropic potential, the winding number nU of the tan- +gent vector around U ˜R remains unchanged, i.e., nU = ±1, +while the unstable direction v− is pointing along the ra- +dius and thus acquires a unit winding number as well. +Indeed, the two directions, tangent and jump, then ro- +tate simultaneously and do not wind around each other +after a full rotation, explaining the absence of contact +points in the isotropic situation. +F. +Energy jumps +Within strong pinning theory, the energy jump ∆epin +associated with the vortex tip jump between bistable vor- +tex configurations at the boundaries of B ¯R determines +the pinning force density Fpin and the critical current +jc, see Eqs. (16) and (17). Formally, the energy jump +∆epin is defined as the difference in energy epin( ˜R; ¯R) at +fixed asymptotic position ¯R ∈ ∂B ¯R between vortex con- +figurations with tips in the jump ( ˜Rjp( ¯R)) and landing +( ˜Rlp( ¯R) = ˜Rjp( ¯R) + ∆ ˜R) positions, +∆epin( ¯R ∈ ∂B ¯R) ≡ epin[ ˜Rjp( ¯R); ¯R] +− epin[ ˜Rlp( ¯R); ¯R]. +(75) +In Sec. III B 2 above, we have found that the jump ∆ ˜R +is mainly forward directed along u. Making use of the +expansion (61) of epin at ˜Rjp and the result (62) for the +jump distance ∆˜u, we find the energy jumps ∆epin in tip- +and asymptotic space in the form (cf. with the isotropic +result Eq. (24)), +∆epin( ¯R) ≈ γ +72∆˜u4 ≈ +� 9 +8γ3 +� +[γ ˜ujp(˜v) + β ˜v]4 +(76) +≈ +� 9 +8γ3 +� � +(γδ − β2) +� +˜v2 +c − ˜v2��2 +≈ +� 9 +8γ3 +� � +(γδ − β2) +(1 + λ+/ ¯C)2 +� +¯v2 +c − ¯v2��2 +. +Here, we have used the parametric shape ˜ujp(˜v) in Eq. +(56) for the jumping ellipse as well as (69) to lowest or- +der, ˜v ≈ ¯v/(1 + λ+/ ¯C), to relate the tip and asymptotic +positions in the last equation. +The energy jump (76) +scales as (κm − 1)2 and is shown in Fig. 8. It depends on +the v coordinate of the asymptotic (or tip) position only +and vanishes at the cusps ¯Rc,±, see Eq. (73) (or at the +touching points ˜Rc,±, see Eq. (67)). To order (κm − 1)2, +the energy jumps are identical at the left and right edges +of the bistable domain B ¯R. +Following the two bistable branches and the associated +energy jumps between them to the inside of B ¯R, the latter +−2 +0 +0.0 +0.1 +0.2 +¯v/ξ√κm − 1 +∆epin/ep(κm − 1)2 +FIG. 8. Energy jump ∆epin along the edges of the bistable +domain B ¯ +R as a function of the transverse coordinate ¯v; we +have used the same parameters as in Fig. 6. The energy jump +vanishes at the cusps ±¯vc, as the bistable tip configurations +become identical and their energies turn equal. +vanish along the branch crossing line ¯R0. In the thermo- +dynamic analogue, this line corresponds to the first-order +equilibrium transition line that is framed by the spinodal +lines; for the isotropic defect, this is the circle with ra- +dius ¯R0 = x0 framed by the spinodal circles with radii +¯R±, see Figs. 4 and 5(d). For the anisotropic defect with +β = 0, this line is trivially given by the centered parabola +of B ¯R, see Eq. (70), and hence +¯u0 ≈ a +2 ¯C +1 +(1 + λ+/ ¯C)2 ¯v2 +0. +(77) +The result for a finite skew parameter β ̸= 0 is given by +Eq. (A27) in Appendix A 1. +G. +Pinning force density +The pinning force density Fpin is defined as the aver- +age force density exerted on a vortex line as it moves +across the superconducting sample. +For the isotropic +case described in Sec. II E, the individual pinning force +fpin( ¯R) = −∇ ¯Repin( ¯R), see Eq. (7), is directed radially +and the force density Fpin is given by the (constant) en- +ergy jump ∆epin ∝ (κ − 1)2 on the edge ∂B ¯R of the +bistable domain and the transverse length t⊥ ∼ ξ, hence, +Fpin ∝ t⊥∆epin scales as (κ − 1)2. +For an anisotropic defect, the pinning force depends +on the vortex direction of motion ˆv = (cos θ, sin θ) rela- +tive to the axis of the bistable region: we choose angles +−π/2 ≤ θ ≤ π/2 measured from the unstable direction +¯u, i.e., vortices incident from the left; the case of larger +impact angles |θ| > π/2 corresponds to vortices incident +from the right and can be reduced to the previous case +by inverting the sign of the parameter a in the expan- +sion (58), i.e., the curvature of the parabola (70); to our +leading order analysis, the results remain the same. The + +17 +pinning force is no longer directed radially but depends +on θ; furthermore, the energy jump (76) is non-uniform +along the boundary B ¯R. +In spite of these complications, we can perform some +simple scaling estimates as a first step: let us assume +a uniform distribution of identical anisotropic defects, +all with their unstable direction pointing along x. The +jumps in energy still scale as ∆epin ∝ (κm − 1)2, how- +ever, the trapping distance is no longer finite but grows +from zero as κm − 1 increases. Due to their elongated +shapes, the bistable domains B ¯R exhibit different exten- +sions along the y and x directions, i.e., ∝ ¯vc ∝ √κm − 1 +along y and ∝ ¯uc ∝ (κm −1) along x, respectively. These +simple considerations then suggest that the pinning force +density exhibits a scaling Fpin ∝ (κm − 1)µ with µ > 2, +different from the setup with isotropic defects. +Even +more, vortices moving along the x or y directions, re- +spectively, will experience different forces F ∥ +pin and F ⊥ +pin +scaling as +F ∥ +pin ∝ (κm − 1)5/2, +F ⊥ +pin ∝ (κm − 1)3 +(78) +near the onset of strong pinning. +While such uniform +anisotropic defects could be created artificially, a more +realistic scenario will involve defects that are randomly +oriented and an additional averaging over angles θ has to +be performed; this will be done at the end of this section. +We first determine the magnitude and orientation of +the pinning force density Fpin(θ) as a function of the +vortex impact angle θ for randomly positioned but uni- +formly oriented (along x) defects of density np. The pin- +ning force density is given by the average over relative +positions between vortices and defects (with a minus sign +following convention; V ¯R denotes the vortex lattice unit +cell), +Fpin(θ) = −np +� +V ¯ +R\B ¯ +R +d2 ¯R +a2 +0 +fpin( ¯R) +(79) +−np +� +B ¯ +R +d2 ¯R +a2 +0 +� +pb( ¯R; θ) f b +pin( ¯R) + pr( ¯R; θ) f r +pin( ¯R) +� +. +Outside of the bistable domain, i.e., in V ¯R \ B ¯R, a sin- +gle stable vortex tip configuration exists and the pinning +force fpin( ¯R) is uniquely defined. Inside B ¯R, the branch +occupation functions pb,r( ¯R; θ) are associated with the +tip positions appertaining to the ‘blue’ and the ‘red’ vor- +tex configurations with different tip positions ˜Rb,r( ¯R), +cf. Figs. 6 and 7. The pinning forces f b,r +pin( ¯R) are evalu- +ated for the corresponding vortex tip positions and are +defined as +f b,r +pin( ¯R) = −∇ ¯Repin[ ˜Rb,r( ¯R); ¯R]. +(80) +Let us now study how vortex lines populate the +bistable domain as a function of the impact angle θ. Ex- +amining Fig. 7, we can distinguish between two different +angular regimes: a frontal-impact regime at angles away +from π/2, |θ| ≤ θ∗, where all the vortices that cross the +bistable domain undergo exactly one jump on the far edge +of B ¯R, see the blue dot and blue boundary ∂Bb +¯R in Fig. +7; and a transverse regime for angles θ∗ ≤ |θ| ≤ π/2, +where vortices crossing the bistable domain undergo ei- +ther no jump, one or two. The angle θ∗ is given by the +(outer) tangent of the bistable domain at the cusps ¯Rc,±; +making use of the lowest order approximation (70) of the +crescent’s geometry, we find that +tan(θ∗) = ∂¯v(0) +∂¯u(0) +��� +¯vc += ( ¯C + λ+) +a +� +γδ − β2 +2γ ¯C(κm − 1), +(81) +implying that π/2 − θ∗ ∝ √κm − 1 is small, +θ∗ ≈ π/2 − +a +( ¯C + λ+) +� +2γ ¯C(κm − 1) +γδ − β2 +. +(82) +1. +Impact angles |θ| < θ∗ +For a frontal impact with |θ| < θ∗, vortices occupy the +‘blue’ branch and remain there throughout the bistable +domain B ¯R until its termination on the far edge ∂Bb +¯R, see +Fig. 7, implying that pb( ¯R ∈ B ¯R) = 1 and pr( ¯R ∈ B ¯R) = +0, independent of θ. As a consequence, the pinning force +Fpin does not depend an the impact angle and is given +by the expression +F< +pin = −np +� +V ¯ +R\B ¯ +R +d2 ¯R +a2 +0 +fpin( ¯R) − np +� +B ¯ +R +d2 ¯R +a2 +0 +f b +pin( ¯R). +Next, Gauss’ formula tells us that for a function e(x), we +can transform +� +V +dnx ∇e(x) = +� +∂V +dn−1 S⊥ e(x), +(83) +with the surface element dn−1 S⊥ oriented perpendicular +to the surface and pointing outside of the domain V. In +applying (83) to the first integral of F< +pin, we can drop +the contribution from the outer boundary ∂V ¯R since we +assume a compact defect potential. The remaining con- +tribution from the crescent’s boundary ∂B ¯R joins up with +the second integral but with an opposite sign, as the two +terms involve the same surface but with opposite orien- +tations. Altogether, we then arrive at the expression +F< +pin = np +� +∂Bb +¯ +R +d S⊥ +a2 +0 +� +eb +pin( ¯R) − epin( ¯R) +� ++ np +� +∂Br +¯ +R +d S⊥ +a2 +0 +� +eb +pin( ¯R) − epin( ¯R) +� +, +(84) +where we have separated the left and right borders ∂Br,b +¯R +of the bistable domain. +Due to continuity, the stable +vortex energy epin( ¯R) will be equal to eb +pin( ¯R) on the + +18 +left border ∂Br +¯R and equal to er +pin( ¯R) on the right border +∂Bb +¯R. The expression (84) for F< +pin then reduces to +F< +pin = np +� +∂Bb +¯ +R +d S⊥ +a2 +0 +� +eb +pin( ¯R) − er +pin( ¯R) +� += np +� ¯vc +−¯vc +d¯v +a0 +∆epin(¯v) +a0 +[1, −∂¯u/∂¯v] += np +�2¯vc +a0 +⟨∆epin⟩ +a0 +, 0 +� +≡ [F ∥ +pin, 0] +(85) +with ⟨∆epin⟩ the average energy jump evaluated along the +v-direction. The force F< +pin is aligned with the unstable +directed along u, with the v-component vanishing due to +the antisymmetry in ¯v ↔ −¯v of the derivative ∂¯u/∂¯v, +and is independent on θ for |θ| < θ∗. +2. +Impact angle |θ| = π/2 +Second, let us find the pinning force density Fπ/2 +pin for +vortices moving along the (positive) v-direction, θ = π/2. +As follows from Fig. 7, vortices occupy the blue branch +and jump to the red one upon hitting the lower half +of the boundary ∂Bb +¯R; vortices that enter B ¯R but do +not cross ∂Bb +¯R undergo no jump and hence do not con- +tribute to Fπ/2 +pin . As vortices in the red branch proceed +upwards, they jump back to the blue branch upon cross- +ing the red boundary ∂Br +¯R. While jumps appear on all +of the lower half of ∂Bb +¯R, a piece of the upper bound- +ary ∂Br +¯R that contributes with a second jump is cut +away (as vortices to the left of ¯u(0) + ¯u(1) do not change +branch from blue to red). The length ∆¯v of this inter- +val scales as ∆¯v/¯vc ∝ (κm − 1)1/4; ignoring this small +jump-free region, we determine Fπ/2 +pin assuming that vor- +tices contributing to Fπ/2 +pin undergo a sequence of two +jumps, from blue to red on the lower half ∂Bb< +¯R and back +from red to blue on the upper half ∂Br> +¯R of the bound- +ary ∂B ¯R. +Repeating the above analysis, we find that +the u-components in Fπ/2 +pin arising from the blue and red +boundaries now cancel, while the v-components add up, +Fπ/2 +pin = np +� +∂Bb< +¯ +R +d S⊥ +a2 +0 +� +eb +pin( ¯R) − er +pin( ¯R) +� ++ np +� +∂Br> +¯ +R +d S⊥ +a2 +0 +� +er +pin( ¯R) − eb +pin( ¯R) +� += 2np +� ¯vc +0 +d¯v +a0 +∆epin(¯v) +a0 +[0, ∂¯u/∂¯v] +(86) += np +� +0, 2¯vc +a0 +⟨∆epin∂¯v¯u⟩ +a0 +� +≡ [0, F ⊥ +pin]. +Making use of the result (76) for ∆epin(¯v) in (85), we +find explicit expressions for the pinning force densities +for impacts parallel and perpendicular to the unstable +direction u, +F ∥ +pin ≈ +� 9np +8 a2 +0γ3 +�� ¯vc +−¯vc +d¯v +� +γδ − β2 +(1 + λ+/ ¯C)2 +� +¯v2 +c − ¯v2��2 +(87) += 24 +5 np +� +2 ¯C/γ +a0 +¯C2 +γa0 +γ(1 + λ+/ ¯C) +� +γδ − β2 +(κm − 1)5/2 +and +F ⊥ +pin ≈ 3 +¯C2 +γa0 +γa/a0 +γδ − β2 (κm − 1)3, +(88) +that confirm the scaling estimates of Eq. (78). +Here, +we have made use of the definition (73) of ¯vc and have +brought the final result into a form similar to the isotropic +result (28) (with the length +� ¯C/γ and the force ¯C2/γa0, +equal to ξ/ +√ +3κ and ep/12κ2 for a Lorentzian poten- +tial). The result (87) provides the pinning force density +Fpin = [F ∥ +pin, 0] for all impact angles |θ| ≤ θ∗ (note that +(87) depends on the curvature a of the crescent via δ, +Eq. (49), that involves a2 only, but higher-order correc- +tions will introduce an asymmetry between left- and right +moving vortices). Within the interval θ∗ < θ < π/2, the +longitudinal force Fpin,u along u decays to zero and the +transverse force Fpin,v along v becomes finite, assuming +the value (88) at θ = π/2. The two force components +have been evaluated numerically over the entire angular +regime and the results are shown in Fig. 9: when mov- +ing away from the angle θ = π/2, the transition from +the blue to the red boundary is moving upwards, with +the relevant boundary turning fully blue at θ = θ∗, thus +smoothly transforming (86) into (85) (we have adopted +the approximation of dropping the jump-free interval ∆¯v +that moves up and becomes smaller as θ decreases from +π/2 to θ∗). +3. +Anisotropic critical force density Fc +When the vortex system is subjected to a current den- +sity j, the associated Lorentz force FL(ϕ) = j ∧ B/c di- +rected along ϕ pushes the vortices across the defects. +When FL is directed along u, we have Fpin = [F ∥ +pin, 0] +and the vortex system gets immobilized at force densi- +ties FL < Fc = F ∥ +pin (or associated current densities jc). +When FL is directed away from u, the driving compo- +nent along v has to be compensated by a finite pinning +force Fpin,v that appears only for angles θ∗ < θ < π/2. +Hence, the angles of force and motion, ϕ associated with +the Lorentz force FL(ϕ) and θ providing the direction of +the pinning force Fpin(θ), are different. We find them, +along with the critical force density Fc(ϕ), by solving +the dynamical force equation (17) at vanishing velocity +v = 0, +Fc(ϕ) = Fpin(θ) +(89) + +19 +0 +π/8 +0 +0.5 +1 +1.5 +0 +π/4 +0 +1 +Fpin,u/[np(ep/a0)(ξ/a0)(κm − 1)5/2] +Fpin,v/[np(ep/a0)(ξ/a0)(κm − 1)3] +F ∥ +pin +F ⊥ +pin +Fc/F ∥ +pin +θ +π/2 +θ∗ +ϕ +π/2 +FIG. 9. Top: scaled pinning force densities Fpin,u and Fpin,v +versus impact angle θ; we have used the same parameters as +in Fig. 6. +The longitudinal (along u) force Fpin,u remains +constant and equal to F ∥ +pin for all angles |θ| < θ∗, while the +transverse (along v) component Fpin,v vanishes in this regime. +The longitudinal force drops and vanishes over the narrow +interval θ∗ < |θ| < π/2, while the transverse force Fpin,v +increases up to F ⊥ +pin. Bottom: critical force density Fc (di- +rected along the Lorentz force FL = j ∧ B/c) versus angle ϕ +of the Lorentz force; the dashed line shows the upper bound +Fc < F ⊥ +pin/ sin(ϕ). +resulting in a critical force density +Fc(ϕ) = +� +F 2 +pin,u(θ) + F 2 +pin,v(θ) +(90) +with angles ϕ and θ related via +tan ϕ = Fpin,u(θ) +Fpin,v(θ). +(91) +Since Fpin,u(θ < θ∗) = 0, the entire interval θ < θ∗ +is compressed to ϕ = 0 and it is the narrow regime +θ∗ < θ < π/2 that determines the angular characteristic +of the critical force density Fc(ϕ). The critical force den- +sity Fc(ϕ) is peaked at ϕ = 0 as shown in Fig. 9 (with a +correspondingly sharp peak in jc at right angles). Comb- +ing Eqs. (90) and (91), we can derive a simple expression +bounding the function Fc(ϕ), +Fc(ϕ) = Fpin,v(θ) +� +1 + cot2(ϕ) ≤ +F ⊥ +pin +sin(ϕ), +(92) +that traces Fc(ϕ) over a wide angular region, see the +dashed line in Fig. 9. At small values of ϕ we cannot +ignore the angular dependence in Fpin,v(θ) any more that +finally cuts off the divergence ∝ 1/ sin(ϕ) at the value +Fc(ϕ → 0) → F ∥ +pin. +4. +Isotropized pinning force density Fpin +In a last step, we assume an ensemble of equal +anisotropic defects that are uniformly distributed in +space and randomly oriented. +In this situation, we +have to perform an additional average over the insta- +bility directions ˆui associated with the different defects +i = 1, . . . N. Neglecting the modification of Fpin(θ) away +from [F ∥ +pin, 0] in the small angular regions θ∗ < |θ| < π/2, +we find that the force along any direction ˆR has the mag- +nitude +Fpin ≈ 1 +N +N +� +i=1 +|(F ∥ +pinˆui) · ˆR| +(93) +≈ F ∥ +pin +� π/2 +−π/2 +dθ +π cos θ = 2 +π F ∥ +pin. +As a result of the averaging over the angular directions, +the pinning force density is now effectively isotropic and +directed against the velocity v of the vortex motion. +IV. +UNIAXIAL DEFECT +In Sec. III, we have analyzed the onset of strong pin- +ning for an arbitrary potential and have determined the +shape of the unstable and bistable domains U ˜R and +B ¯R—with their elliptic and crescent forms, they look +quite different from their ring-shaped counterparts for +the isotropic defect in Figs. 5(c) and (d). +In this sec- +tion, we discuss the situation for a weakly anisotropic +defect with a small uniaxial deformation quantified by +the small parameter ϵ in order to understand how our +previous findings, the results for the isotropic defect and +those describing the strong-pinning onset, relate to one +another. +Our weakly deformed defect is described by equipo- +tential lines that are nearly circular but slightly elon- +gated along y, implying that pinning is strongest in the +x-direction. We will find that the unstable (bistable) do- +main U ˜R (B ¯R) for the uniaxially anisotropic defect starts +out with two ellipses (crescents) on the x-axis as κm +crosses unity. With increasing pinning strength, i.e., κm, +these ellipses (crescents) grow and deform to follow the +equipotential lines, with the end-points approaching one +another until they merge on the ±y-axis. These merger +points, we denote them as ˜Rs and ¯Rs, define a second +class of important points (besides the onset points ˜Rm +and ¯Rm) in the buildup of the strong pinning landscape: +while the onset points ˜Rm are defined as minima of the +Hessian determinant D( ˜R), the merger points ˜Rs turn +out to be associated with saddle points of D( ˜R). Push- +ing across the merger of the deformed ellipses (crescents) +by further increasing the Labusch parameter κm, the un- +stable (bistable) domains U ˜R (B ¯R) undergo a change in +topology, from two separated areas to a ring-like geom- +etry as it appears for the isotropic defect, see Figs. 5(c) + +20 +and (d), thus explaining the interrelation of our results +for isotropic and anisotropic defects. +With this analysis, we thus show how the strong pin- +ning landscape for the weakly uniaxial defect will finally +assume the shape and topology of the isotropic defect +as the pinning strength κm overcomes the anisotropy ϵ. +Second, this discussion will introduce the merger points +˜Rs as a second type of characteristic points of strong +pinning landscapes that we will further study in section +V A using a Landau-type expansion as done in section +III A above; we will find that the geometry of the merger +points ˜Rs is associated with hyperbolas, as that of the +onset points was associated with ellipses. +Our uniaxially anisotropic defect is described by the +stretched (along the y-axis) Lorentzian +ep(˜x, ˜y) = −ep +� +1 + ˜x2 +2ξ2 + +˜y2 +2ξ2 (1 + ϵ)2 +�−1 +, +(94) +with equipotential lines described by ellipses +˜x2 +ξ2 + +˜y2 +ξ2 (1 + ϵ)2 = const, +(95) +and the small parameter 0 < ϵ ≪ 1 quantifying the de- +gree of anisotropy. At fixed radius ˜R2 = ˜x2 + ˜y2, the +potential (94) assumes maxima in energy and in negative +curvature on the x−axis, and corresponding minima on +the y−axis. Along both axes, the pinning force is directed +radially towards the origin and the Labusch criterion (34) +for strong pinning is determined solely by the curvature +along the radial direction. At the onset of strong pin- +ning, the unstable and bistable domains then first emerge +along the x−axis at the points ˜Rm = (± +√ +2ξ, 0) and +¯Rm = (±2 +√ +2ξ, 0) when +κm = +ep +4 ¯Cξ2 = 1. +(96) +Upon increasing the pinning strength κm, e.g., via soft- +ening of the vortex lattice as described by a decrease in +¯C, the unstable and bistable domains U ˜R and B ¯R expand +away from these points, and eventually merge along the +y−axis at ˜Rs = (0, ± +√ +2ξ(1+ϵ)), ¯Rs = (0, ±2 +√ +2ξ(1+ϵ)) +when +κs = +ep +4 ¯Cξ2(1 + ϵ)2 = +κm +(1 + ϵ)2 = 1, +(97) +i.e., for κm = (1+ϵ)2. The evolution of the strong pinning +landscape from onset to merging takes place in the inter- +val κm ∈ [1, (1 + ϵ)2]; pushing κm beyond this interval, +we will analyze the change in topology and appearance +of non-simply connected unstable and bistable domains +after the merging. +The quantity determining the shape of the unstable +domain U ˜R is the Hessian determinant D( ˜R) of the total +vortex energy epin( ˜R; ¯R), see Eqs. (36) and (1), respec- +tively. At onset, the minimum of D( ˜R) touches zero for +the first time; with increasing κm, this minimum drops +below zero and the condition D( ˜R) = 0 determines the +unstable ellipse that expands in ˜R-space. Viewing the +function D( ˜R) as a height function of a landscape in the +˜R plane, this corresponds to filling this landscape, e.g., +with water, up to the height level D = 0 with the result- +ing lake representing the unstable domain. In the present +uniaxially symmetric case, a pair of unstable ellipses grow +simultaneously, bend around the equipotential line near +the radius ∼ +√ +2ξ and finally touch upon merging on the +y-axis. In our geometric interpretation, this corresponds +to the merging of the two (water-filled) valleys that hap- +pens in a saddle-point of the function D( ˜R) at the height +D = 0. Hence, the merger point ˜Rs correspond to sad- +dles in D( ˜R) with +D( ˜Rs) = 0, +∇ ˜R D(R) +�� ˜Rs = 0, +(98) +and +det +� +Hess +� +D( ˜R) +���� ˜Rs < 0, +(99) +cf. Eq. (44). +In our calculation of D( ˜R), we exploit that the Hessian +in (36) does not depend on the asymptotic position ¯R and +we can set it to zero, +D( ˜R) = det +� +Hess[ ¯C ˜R2/2 + e +(i) +p ( ˜R) + δep( ˜R)] +� +, +(100) +where we have split off the anisotropic correction +δep( ˜R) = ep( ˜R) − e(i) +p ( ˜R) away from the isotropic po- +tential e(i) +p ( ˜R) with ϵ = 0. In the following, we perform a +perturbative analysis around the isotropic limit valid in +the limit of weak anisotropy ϵ ≪ 1; this motivates our +use of polar (tip) coordinates ˜R and ˜φ. +The isotropic contribution H(i) to the Hessian matrix +H is diagonal with components +H +(i) +˜ +R ˜ +R( ˜R) ≡ ∂2 +˜ +R[ ¯C ˜R2/2 + e +(i) +p ( ˜R)] += ¯C + ∂2 +˜ +Re +(i) +p ( ˜R) +(101) +and +H +(i) +˜φ ˜φ( ˜R) ≡ ( ˜R−2∂2 +˜φ ˜φ + ˜R−1∂ ˜ +R)[ ¯C ˜R2/2 + e +(i) +p ( ˜R)] += ¯C − f +(i) +p ( ˜R)/ ˜R. +(102) +The radial component H +(i) +˜ +R ˜ +R ∝ (κm − 1) vanishes at on- +set, while H +(i) +˜φ ˜φ remains finite, positive, and approximately +constant. +The anisotropic component δep( ˜R) introduces correc- +tions ∝ ϵ; these significantly modify the radial entry of +the full Hessian while leaving its azimutal component H˜φ ˜φ +approximately unchanged; the off-diagonal entries of the +full Hessian scale as ϵ and hence contribute in second or- +der of ϵ to D( ˜R). As a result, the sign change in the +determinant +D( ˜R) ≈ H ˜ +R ˜ +R( ˜R)H ˜φ ˜φ( ˜R) + O +� +ϵ2� +, +(103) + +21 +FIG. 10. Unstable and bistable domains close to the onset +of strong pinning for a uniaxial defect (94) centered at the +origin, with ϵ = 0.1 and κm −1 = 0.01. The pinning potential +is steepest at angles ˜φ = 0, π and least steep at ˜φ = ±π/2, +hence strong pinning is realized first in a small interval around +˜φ = 0, π (solid black dots) where κm(˜φ) ≥ 1. (a) The unsta- +ble domain U ˜ +R in tip space is bounded by red/blue solid lines +(jump lines J ˜ +R, see Eq. (108)); dashed lines mark the asso- +ciated landing lines L ˜ +R, see (114). (b) Focus on the unstable +domain near ˜φ = 0 in polar coordinates ˜R and ˜φ. The jump- +ing (solid) and landing (dashed) lines have the approximate +shape of ellipses, see Eq. (111), in agreement with our anal- +ysis of Sec. III B. (c) The bistable domain B ¯ +R in asymptotic +space involves symmetric crescents centered at ¯φ = 0, π and +a narrow width ∝ (κm(¯φ)−1)3/2, see Eq. (112), in agreement +with the analysis of Sec. III C. (d) Focus on the bistable do- +main at ¯φ = 0 in polar coordinates ¯R and ¯φ. Red/blue colors +indicate different vortex configurations as quantified through +the order parameter ˜R − ˜Rm(˜φ). +is determined by +H ˜ +R ˜ +R( ˜R) = H +(i) +˜ +R ˜ +R( ˜R) + ∂2 +˜ +Rδep( ˜R) +(104) +for radii close to ˜Rm with δ ˜R = ˜R − ˜Rm ≈ O(√κm − 1). +We expand the potential (94) around the isotropic part +e(i) +p ( ˜R), +δep( ˜R) ≈ −ϵ [∂ ˜ +Re +(i) +p ( ˜R)] ˜R sin2 ˜φ, +(105) +and additionally expand both e(i) +p ( ˜R) and δep( ˜R) around +˜Rm, keeping terms ∝ ϵ +� +(κm − 1). The radial entry of +the anisotropic Hessian matrix then assumes the form +H ˜ +R ˜ +R( ˜R) ≈ ¯C [1 − κm(˜φ)] ++ γ [δ ˜R2/2 − ϵ sin2 ˜φ ˜Rmδ ˜R] +(106) +with γ = ∂4 +˜ +Re(i) +p ( ˜R)| ˜ +Rm and the angle-dependent Labusch +parameter +κm(˜φ) ≡ +max ˜ +R[−∂2 +˜ +Rep( ˜R, ˜φ)| ˜φ] +¯C += κm − 2ϵ sin2 ˜φ. (107) +The edges of the unstable region U ˜R then can be obtained +by imposing the condition H ˜ +R ˜ +R( ˜R) = 0 and the solution +to the corresponding quadratic equation define the jump +positions ˜Rjp(˜φ) (or boundaries ∂U ˜R) +˜Rjp(˜φ) ≈ ˜Rm(˜φ) ± δ ˜R(˜φ). +(108) +These are centered around the (‘large’) ellipse defined by +˜Rm(˜φ) = ˜Rm(1 + ϵ sin2 ˜φ) +(109) +and separated by (cf. Eq. (20)) +2 δ ˜R(˜φ) = +� +8 ¯C +γ (κm(˜φ) − 1) +(110) +along the radius. Making use of the form (107) of κm(˜φ) +and assuming a small value of κm > 1 near onset, we +obtain the jump line in the form of a (‘small’) ellipse +centered at [± ˜Rm, 0], +γ δ ˜R2 + ϵ ¯C ˜φ2 = ¯C(κm − 1). +(111) +Hence, we find that the anisotropic results are obtained +from the isotropic ones by replacing the circle ˜Rm by the +ellipse ˜Rm(˜φ) and substituting κ → κm(˜φ) in the width +(20), see Figs. 10(a) and (b) evaluated for small values +κm − 1 = 0.01 and ϵ = 0.1. +Analogously, the boundaries of the bistable domain B ¯R +can be found by applying the same substitutions to the +result (25), see Figs. 10(c) and (d), +¯R(¯φ) ≈ ¯Rm(¯φ) ± δ ¯R(¯φ) +(112) +with ¯Rm(¯φ) = ¯Rm(1 + ϵ sin2 ¯φ) and the width +2 δ ¯R(¯φ) = 2 +3 +� +8 ¯C +γ (κm(˜φ) − 1)3/2. +(113) +The landing line L ˜R is given by (see Eq. (23) and note +that the jump point is shifted by ˜ujp away from ˜xm, see +Eq. (19)) +˜Rlp(˜φ) ≈ ˜Rm(˜φ) ∓ 2 δ ˜R(˜φ). +(114) +An additional complication is the finite angular exten- +sion of the unstable and bistable domains U ˜R and B ¯R; + +22 +FIG. 11. Unstable and bistable domains before merging for +a uniaxial defect (94) centered at the origin, with ϵ = 0.1 +and 1 − κs ≈ 0.01. Strong pinning is realized everywhere but +in a small interval around ˜φ = ±π/2 where κm(˜φ) < 1. (a) +The unstable domain U ˜ +R in the tip plane is bounded by the +solid red/blue jump lines J ˜ +R, see Eq. (108) and involves two +strongly bent ellipses originating from angles ˜φ = 0, π (black +dots) and approaching one another close to ˜φ = ±π/2 (black +crosses); red/blue dashed lines are landing points as given +by Eqs. (114). (b) Focus (in polar coordinates ˜R, ˜φ) on the +tips of the unstable domain near ˜φ = π/2. (c) The bistable +domain B ¯ +R in the asymptotic space consists of thin symmetric +crescents (colored in magenta) originating from ¯φ = 0, π, with +the delimiting black solid lines given by Eq. (112). (d) Focus +on the cusps of the bistable domain close to ¯φ = π/2 in polar +coordinates ¯R, ¯φ. Red/blue colors indicate different vortex +configurations as quantified through the order parameter ˜R− +˜Rm(¯φ). +these are limited by the condition κm(φmax) = 1, provid- +ing us with the constraint +˜φmax = ¯φmax ≈ ± +� +κm − 1 +2ϵ +(115) +near the strong pinning onset with (κm − 1) ≪ ϵ. The +resulting domains U ˜R have characteristic extensions of +scale ∝ √κm − 1, see Fig. 10. +Close to merging (marked by crosses in the figure) at +φ = ±π/2, we define the deviation δφ = π/2 − φ with +δφ ≪ 1, and imposing the condition κm(φmax) = 1, we +FIG. 12. Unstable and bistable domains for a uniaxial defect +(94) after merging, with ϵ = 0.1 and κs − 1 ≈ 0.01. (a) The +unstable domain U ˜ +R in tip plane is enclosed between the jump +lines J ˜ +R (solid red/blue, see Eq. (108)) and takes the shape +of a deformed ring with a wider (narrower) width at strongest +(weakest) pinning near the solid dots (crosses). +Red/blue +dashed lines mark the landing positions L ˜ +R of the vortex tips +and are given by Eq. (114). (b) Focus on the narrowing in +the unstable domain close to the merger points (crosses) at +˜φ = π/2 in the polar coordinates ˜R, ˜φ. (c) The bistable do- +main B ¯ +R in asymptotic space is a narrow ring (colored in +magenta) thicker (thinner) at points of strongest (weakest) +pinning near ¯φ = 0, π (¯φ = ±π/2); black lines correspond +to Eq. (112). (d) Focus on the constriction in the bistable +domain close to ¯φ = π/2 in polar coordinates ¯R, ¯φ. Red/blue +colors indicate different vortex configurations as quantified +through the order parameter ˜R − ˜Rm(¯φ). +find +δ ˜φmax = δ ¯φmax ≈ +� +1 − κm − 1 +2ϵ +≈ +� +1 − κs +2ϵ +. +(116) +The corresponding geometries of U ˜R and B ¯R are shown +in Fig. 11 for 1 − κs ≈ 0.01 and ϵ = 0.1. Finally, δ ˜φmax +vanishes at merging for κs = 1 (or κm − 1 ≈ 2ϵ), in +agreement, to order ϵ, with the exact result (97). +Pushing the Labusch parameter beyond the merger +with κs > 1 or κm > (1 + ϵ)2 ≈ 1 + 2ϵ, the unstable +and bistable regimes U ˜R and B ¯R change their topology: +they develop a (non-simply connected) ring-like geome- +try with separated inner and outer edges that are a finite +distance apart in the radial direction at all angles ˜φ and +¯φ. The situation after the merger is shown in Fig. 12 for + +23 +κs − 1 ≈ 0.01 and ϵ = 0.1, with the merging points ˜Rs +and ¯Rs marked by crosses. +The merging of the unstable domains at the saddle +point ˜Rs is a general feature of irregular pinning poten- +tials. In the next section, we will analyze the behavior +of the unstable domains close to a saddle point ˜Rs of +the Hessian determinant D( ˜R) and obtain a universal +description of their geometry close to this point. We will +see that the geometry associated with this merger is of +a hyperbolic type described by γ˜u2 + δ˜v2 = 2 ¯C(κs − 1), +γ > 0 and δ < 0 (assuming no skew). The change in +topology then is driven by the sign change in κs − 1: +before merging, κs < 1, the hyperbola is open along the +unstable (radial) direction ˜u, thus separating the two un- +stable regions, while after merging, κs > 1, the hyperbola +is open along the transverse direction ˜v, with the ensuing +passage defining the single, non-simply connected, ring- +like unstable region. +V. +MERGER POINTS +The merging of unstable and bistable domains is a gen- +eral feature of irregular pinning potentials that is relevant +beyond the simple example of a weakly anisotropic uni- +axial defect discussed above. +Indeed, while the exact +geometries of U ˜R and B ¯R depend on the precise shape of +the pinning potential, their behavior close to merging is +universal. Below, we will study this universal behavior +by generalizing the expansions of Sec. III to saddle points +˜Rs of the determinant D( ˜R). As with the onset of strong +pinning, the merger of two domains induces a change in +topology in the unstable and bistable domains; we will +discuss these topological aspects of onsets and mergers +in Secs. V D and VI below. +A. +Expansion near merger +Following the strategy of Sec. III, we expand the en- +ergy functional around a saddle point ˜Rs of the determi- +nant D( ˜R) in order to obtain closed expressions for the +unstable and bistable domains at merging. In doing so, +we again define local coordinate systems (˜u, ˜v) and (¯u, ¯v) +in tip- and asymptotic space centered at ˜Rs and ¯Rs, +where the latter is associated with ˜Rs through the force +balance equation (38) in the original laboratory system. +Furthermore, we fix our axes such that D( ˜Rs) is a local +maximum along the (unstable) u- and a local minimum +along the (stable) v-direction of the saddle; the mixed +term ∝ ˜u˜v is absent from the expansion (as the Hessian +matrix is symmetric). Furthermore, the vanishing slopes +at the saddle point, see (98), imply the absence of terms +∝ ˜u3 and ∝ ˜u2˜v in the expansion and dropping higher- +order terms (corresponding to double-primed terms in +(40)), we arrive to the expression +epin( ˜R; ¯R) = +¯C +2 (1 − κs) ˜u2 + +¯C + λ+,s +2 +˜v2 + as +2 ˜u˜v2 ++ αs +4 ˜u2˜v2 + βs +6 ˜u3˜v + γs +24 ˜u4 − ¯C¯u˜u − ¯C¯v˜v, +(117) +with κs ≡ −λ−( ˜Rs)/ ¯C, λ+,s ≡ λ+( ˜Rs) and the remain- +ing coefficients defined in analogy to Eq. (58). +The most important term in the expansion (117) is the +curvature term ¯C(1 − κs) ˜u2/2 along the unstable direc- +tion u. As before in Sec. III B, see Eq. (58), the coefficient +(1−κs) changes sign at some value of the pinning strength +and will serve as the small parameter in our considera- +tions. The higher-order terms in the expansion (117) are +constrained by the saddle condition (99), implying that +(cf. (48) and (50)) +γsδs − β2 +s < 0 +(118) +with +δs ≡ αs − +2a2 +s +¯C + λ+,s +(119) +(for the saddle point there is no condition on the trace of +the Hessian). The mapping of the two-dimensional pin- +ning energy (117) to an effective one-dimensional Landau +theory (A30) of the van der Waals kind is discussed in +Appendix A 2, both before and after merging. +B. +Unstable domain U ˜ +R +1. +Jump line J ˜ +R +The boundary of the unstable domain U ˜R is deter- +mined by the jump condition D( ˜Rs,jp) = 0. Making use +of the expansion (117) and keeping only terms quadratic +in ˜u, ˜v, the edges δ ˜Rs,jp = (˜us,jp, ˜vs,jp) of U ˜R (measured +relative to ˜Rs) are given by the solutions of the quadratic +form (cf. (53)) +[γs ˜u2 + 2βs ˜u˜v + δs ˜v2] ˜Rs,jp = 2 ¯C(κs − 1). +(120) +Equation (120) describes a hyperbola (centered at ˜Rs) +as its associated determinant is negative, see Eq. (118). +Again, (120) can be cast in the form of a matrix equation +δ ˜RT +s,jpMs,jpδ ˜Rs,jp = ¯C(κs − 1), +(121) +with Ms,jp given by +Ms,jp = +� +γs/2 +βs/2 +βs/2 +δs/2 +� +(122) +with det Ms,jp = (γsδs − β2 +s)/4 < 0. As shown in Fig. 13, +the geometry of the unstable domain U ˜R changes drasti- +cally when 1 − κs changes sign. Before merging, i.e., for + +24 +−3 +0 +−5 +0 +−3 +0 +−25 +0 +(a) +(b) +˜u/ξ√1 − κs +˜v/ξ√1 − κs +˜u/ξ√κs − 1 +˜v/ξ√κs − 1 +FIG. 13. Jump lines J ˜ +R (solid red/blue) and landing lines L ˜ +R +(dashed red/blue) in tip space ˜R (in units of ξ), with the hy- +perbola J ˜ +R defining the edge ∂U ˜ +R of the unstable domain U ˜ +R, +before (a) and after (b) merging, for 1 − κs = ±0.01. Param- +eters are λ−,s = −0.25 ep/ξ2, λ+,s = 0, and as ≈ 0.035 ep/ξ3, +αs = −0.025 ep/ξ4, βs = 0, γs ≈ 0.68 ep/ξ4. A finite skew +parameter βs = 0.025ep/ξ4 tilts the hyperbola away from the +axes (dotted curves). Crosses correspond to the vertices (125) +and (129) of the hyperbola before and after merging. Pairs +of solid and open circles connected via long arrows are ex- +amples of pairs of jumping- and landing tip positions. After +merging, see (b), the unstable domain U ˜ +R is connected along +the ˜v-axis, dividing the tip coordinate plane into two sepa- +rate regions. The jumping and landing hyperbolas coincide +at their vertices before merging, see (a), but not thereafter, +see (b), where the jumping and landing hyperbolas are sepa- +rated (vertices on L ˜ +R are marked with open red/blue stars) +and no contact point is present. Note the rotation by 90 de- +grees of the unstable direction with respect to Figs. 11(b) and +12(b). +1−κs > 0, the unstable domain (top and bottom regions +in Fig. 13(a)) is disconnected along the stable v-direction +and the two red/blue branches of the hyperbola (120) de- +scribe the tips of U ˜R. When κs goes to unity, the tips of +the unstable domain merge at the saddle point ˜Rs. After +merging, the unstable domain extends continuously from +the top to the bottom in Fig. 13(b) with a finite width +along the unstable u-direction, similarly to the isotropic +case shown in Fig. 5(c). Correspondingly, the two (red +and blue) branches of the hyperbola (120) now describe +the edges of U ˜R. +Solving the quadratic equation (120) before merging, +i.e., 1 − κs > 0, we find solutions ˜us,jp(˜v) away from a +gap along the stable v-direction, +˜us,jp(|˜v| ≥ ˜vs,c) = − 1 +γs +� +βs˜v +± +� +2γs ¯C(κs − 1) − (γsδs − β2s)˜v2 +� +, +(123) +i.e., Eq. (123) has real solutions in the (unbounded) in- +terval |˜v| ≥ ˜vs,c, with +˜vs,c = +� +2γs ¯C(1 − κs)/|γsδs − β2s|. +(124) +For the uniaxial defect (94) before merging, this gap cor- +responds to a splitting of U ˜R along the stable angular +direction, producing two separated domains as shown in +Fig. 11(a). The coordinates (˜us,jp(±˜vs,c), ±˜vs,c) give the +positions of the vertices δ ˜R< +s,c,± (relative to ˜Rs) of the +hyperbola before merging, +δ ˜R< +s,c,± = ± (−βs/γs, 1) ˜vs,c. +(125) +These are marked as black crosses in Fig. 13(a) (note the +rotation in the geometry as compared with Fig. 11(a)). +We denote the distance between these vertices by δv<, +defining a gap of width ∝ √1 − κs given by +δv< = 2|δ ˜R< +s,c,±| = 2 +�� +γs + β2s +γs +� ¯C(1 − κs) +|γsδs − β2s|. +(126) +After merging, i.e., for κs − 1 > 0, the (local) topology +of U ˜R has changed as the gap along v closes and reopens +along the unstable u-direction; as a result, the two sepa- +rated domains of U ˜R have merged. The two branches of +the hyperbola derived from (120) are now parametrized +as +˜vs,jp(|˜u| ≥ ˜us,e) = − 1 +δs +� +βs˜u +± +� +2δs ¯C(κs − 1) − (γsδs − β2s)˜u2 +� +, +(127) +with +˜us,e = +� +2δs ¯C(κs − 1)/|γsδs − β2s|. +(128) +The corresponding unstable domain is shown in Fig. +13(b). For the uniaxial defect (94) after merging, this +gap now corresponds to the finite width of U ˜R along the +radial direction, as shown in Fig. 12(a). The coordinates +(±˜us,e, ˜vs,jp(±˜us,e)) for the vertices ˜R> +s,e,± read +δ ˜R> +s,e,± = ± +� +1, −βs +δs +� +˜us,e +(129) +and correspond to the points of closest approach in the +branches of the hyperbola (120); these are again marked +as black crosses in Fig. 13(b) but are no longer associated +with critical points (we index these extremal points by +‘e’). Their distance δu> is given by +δu> = 2|δ ˜R> +s,e,±| = 2 +�� +δs + β2s +δs +� ¯C(κs − 1) +|γsδs − β2s|, +(130) +i.e., the smallest width in U ˜R grows as ∝ √κs − 1. +As discussed above and shown in Fig. 13, the solu- +tions of the quadratic form (120) before and after merg- +ing are unbounded for every value of κs − 1. As a conse- +quence, neglecting the higher order terms in the determi- +nant D( ˜R) is valid only in a narrow neighborhood of the + +25 +saddle ˜Rs, where the boundaries of U ˜R have the shape +of a hyperbola. Away from the saddle, these higher or- +der terms are relevant in determining the specific shape +of the unstable and bistable domain, e.g., the ring-like +structures of U ˜R and B ¯R in Figs. 11 and 12. +2. +Landing line L ˜ +R +To find the second bistable vortex tip configuration +˜Rs,lp associated to the edges of B ¯R before and after merg- +ing, we repeat the steps of Sec. III B 2. +For the jump +vector ∆ ˜Rs = ˜Rs,lp − ˜Rs,jp, we find the result +∆˜us(˜v) = −3 (γs ˜us,jp(˜v) + βs ˜v) /γs, +(131) +∆˜vs(˜v) = − +� +as/( ¯C + λs,+) +� +˜v ∆˜us(˜v), +(132) +cf. Eqs. (65) and (66) above. +Here, we make use of +the parametrization for the jump coordinate ˜us,jp(˜v) in +(123) before merging; after merging, the above result +is still valid but should be expressed in terms of the +parametrization ˜vs,jp(˜u) in Eq. (127). +The landing positions ˜Rs,lp = ˜Rs,jp + ∆ ˜Rs arrange +along the branches L ˜R of a hyperbola in ˜R-space that +are described by the matrix equation +δ ˜RT +s,lpMs,lp δ ˜Rs,lp = ¯C(κs − 1), +(133) +with the landing matrix now given by +Ms,lp = 1 +4Ms,jp + +� +� +0 +0 +0 +3 +4 +�δs +2 − β2 +s +2γs +� +� +� +(134) +with det Ms,lp = (γsδs − β2 +s)/16 < 0. Before merging, +the vertices of the landing and jumping hyperbolas coin- +cide and the jump (131)–(132) vanishes at these points. +Moreover, as for the contact points (67) close to onset of +strong pinning, the tangent to the jumping and landing +hyperbolas at the vertices is parallel to the u-direction, +as is visible in Fig. 13(a). +For κs = 1, the tips of U ˜R merge and both the jumping +and landing hyperbolas coincide at ˜Rs. After merging, +i.e., for κs − 1 > 0, the condition ∆˜us = ∆˜vs = 0 cannot +be realized along the hyperbola (120) and the jumping +and landing lines separate completely; as a result, both +the jumping distance ∆ ˜Rs as well as the jump in energy +∆epin are always finite (see also Appendix A 2). Indeed, +after merging the landing hyperbola (133) has vertices +δ ˜Rs,v,± = ± +� +1, − +γsβs +(4γsδs − 3β2s) +� +˜us,v, +(135) +with +˜us,v = +� +2 ¯C(κs − 1)(4γsδs − 3β2s) +γs(γsδs − β2s) +(136) +0 +2 +−5 +0 +0 +4 +−5 +0 +−5 +0 +5 +¯u/ξ(κm − 1) +¯v/ξ√κm − 1 +(a) +˜u/ξ√κm − 1 +¯u/ξ(κm − 1) +¯v/ξ√κm − 1 +(b) +FIG. 14. Bistable domain B ¯ +R in asymptotic space ¯R before +(a) and after (b) merging, for 1 − κs = ±0.01 and parame- +ters as in Fig. 13. (a) Before merging, the bistable domain +B ¯ +R consists of two parts, corresponding to the two unstable +regions U ˜ +R in Fig. 13(a). +These terminate in the cusps at +¯R< +s,c,± that approach one another along the dashed parabola +(139) to merge at κs = 1. Red/blue colors indicate different +vortex configurations as quantified through the order param- +eter ˜u − ˜um(¯v), while magenta is associated to the bistable +region B ¯ +R. Colored dots mark the asymptotic positions asso- +ciated to the pairs of jump positions in Fig. 13(a). (b) After +merging, the bistable domain is continuously connected; the +cusps/critical points have vanished and the dashed parabola +turns into the branch cutting line. +The black crosses now +mark the positions of strongest pinching of B ¯ +R, the colored +dots mark the asymptotic positions associated to the pairs of +tip positions in Fig. 13(b). +different from the jumping hyperbola in (129). At these +points, the stable and unstable hyperbolas are tangent to +the v-direction, as is visible in Fig. 13(b). +In section Sec. V D below, we will take a step back +from the local analysis of the unstable domain U ˜R close +to a saddle point ˜Rs and consider the evolution of its +geometry across the merging transition from a global +perspective using specific examples. Elaborating on the +analysis of Sec. III E, we will provide a simple argument +explaining the absence of contact points between jump +and landing lines after merging. Furthermore, we discuss +the two possible roles of mergers as changing the number +of components of U ˜R or changing the connectivity of U ˜R +between simply and non-simply connected areas. Before +doing so, we discuss the behavior of the bistable region +B ¯R close to merging. +C. +Bistable domain B ¯ +R +The set of asymptotic positions corresponding to +U ˜R before and after merging, i.e., the bistable do- + +26 +main B ¯R, can be found by systematically repeating the +steps in Sec. III C. Applying the force balance equation +∇Repin(R; ¯R) +��� ˜R = 0 to the energy expansion (117), we +find the counterpart of Eqs. (69), +¯C¯u = ¯C(1 − κs)˜u + as +2 ˜v2 + γs +6 ˜u3 + βs +2 ˜u2˜v + αs +2 ˜u˜v2, +¯C¯v = ( ¯C + λs,+)˜v + as ˜u˜v + βs +6 ˜u3 + αs +2 ˜u2˜v, +(137) +relating tip and asymptotic positions close to merging. +As for the unstable domain, the topology of B ¯R depends +on the sign of 1 − κs. The bistable domain B ¯R before +merging is shown in Fig. 14(a) for 1 − κs = 0.01. +It +consists of two parts, corresponding to the two pieces of +U ˜R for 1−κs > 0, that terminate at the cusps ¯R< +s,c,±. The +latter are related to the vertices ˜R< +s,c,± of the jumping +hyperbola through the force balance equation (137), +δ ¯R< +s,c,± ≈ +�� +as/2 ¯C +� +˜v2 +s,c, ± +� +1 + λs,+/ ¯C +� +˜vs,c +� +. +(138) +For finite values of (1 − κs), the cusps are separated by +a distance 2|δ ¯R< +s,c,±| ≈ 2 +� +1 + λs,+/ ¯C +� +˜vs,c ∝ √1 − κs. +They approach one another along the parabola +¯us,0 ≈ a +2 ¯C +1 +(1 + λ+/ ¯C)2 ¯v2 +s,0, +(139) +see the black dashed line in Fig. 14, with higher-order cor- +rections appearing at finite skew β ̸= 0. After merging, +this line lies within B ¯R and defines the branch crossing +line, cf. Eq. (77). +After merging, when κs − 1 > 0, the cusps have van- +ished and the edges have rearranged to define a connected +bistable region, see Fig. 14(b). The extremal points of +the two edges are found by evaluating the force balance +equation (137) at the vertices ˜R> +s,e,±, Eq. (129), to lowest +order, +δ ¯R> +s,e,± ≈ βs +δs +� as +2 ¯C +βs +δs +˜u2 +s,e, ∓ +� +1 + λs,+ +¯C +� +˜us,e +� +. (140) +For finite values of (κs − 1), these points are separated +by a distance 2|δ ¯R> +s,e,±| ≈ 2 +� +1 + λs,+/ ¯C +� +(βs/δs)˜us,e ∝ +√κs − 1. Note that the extremal points ¯R> +s,e,± are no +longer associated to cusps or critical points as these have +disappeared in the merging process. +When the skew +parameter vanishes as in Fig. 14, βs = 0, higher-order +terms in (κs − 1) in the force-balance equation (137) be- +come relevant in determining the positions ¯R> +s,e,±, sep- +arating them along the unstable u-direction. +In this +case, we obtain a different scaling for their distance, i.e., +|δ ¯R> +s,e,±| ∝ (1 − κs)3/2. +D. +Topological aspect of mergers +In order to discuss the topological aspect of a merger, it +is convenient to consider some specific examples. In Sec. +IV, we have analyzed the case of a uniaxial defect with +a quadrupolar anisotropy δep ∝ ϵ sin2 ˜φ in the pinning +potential, see (105), that produced a degenerate onset +at symmetric points [±˜xm, 0]. Here, we choose again a +weakly anisotropic defect centered in the origin but with +a dipolar deformation δep ∝ ϵ cos ˜φ that results in an +angle-dependent Labusch parameter +κm(˜φ) = κm − ϵ cos ˜φ, +(141) +see Eq. (107). The strong pinning onset of such a defect +then appears in an isolated point on the negative x-axis, +with the unstable ellipse U ˜R deforming with increasing +κm into a horseshoe that is open on the positive x-axis— +the closing of the horseshoe to produce a ring, see Fig. 15, +then corresponds to the local merger shown in Fig. 13. +With this example in mind, we can repeat the discussion +in Sec. III E: The unstable eigenvector v−(Rjp) points ra- +dially outwards from the origin over the entire horseshoe, +including the merging region at positive x. On the other +hand, the tangent to the boundary ∂U ˜R rotates forward +and back along the horseshoe as shown in Fig. 15 (we at- +tribute a direction to ∂U ˜R with the convention of follow- +ing the boundary with the unstable region on the left); in +fact, over most of the boundary, the tangent is simply or- +thogonal to v−, with both vectors rotating together when +going along ∂U ˜R. At the ends of the horseshoe, however, +the tangent locally aligns parallel (anti-parallel) to v− +and the two vectors rotate (anti-clockwise) with respect +to one another, with the total winding equal to 2π. Af- +ter the merger, this winding has disappeared, with the +resulting ring exhibiting no winding in the tangent fields +on the inner/outer boundary; as a result, the contact +points between the jump and landing lines have disap- +peared. +Furthermore, the merger changes the topology of U ˜R +from the simply-connected horseshoe to the non-simply +connected ring, while the number of components in U ˜R +has not changed. Note that the change in the relative +winding is not due to crossing the singularity of the vec- +tor field v− as alluded to in Sec. III E—rather, it is the +merger of the horseshoe tips that rearranges the bound- +aries of U ˜R and make them encircle the singularity. +In the above example, we have discussed a merger that +changes the connectedness of U ˜R. On the other hand, as +we are going to show, a merger might leave the connect- +edness of U ˜R unchanged, while modifying the number of +components, i.e., the number of disconnected parts, in +U ˜R. Let us again consider a specific example in the form +of an anisotropic defect with a warped well shape, pro- +ducing several (in general subsequent) onsets and merg- +ers; in Fig. 16, we consider a situation with three on- +set points and subsequent individual mergers. After the +onset, the three ellipses define an unstable region U ˜R +with three disconnected parts that are simply-connected +each. This configuration is characterized by its number +of components measuring C = 3. As two of the three +ellipses merge, the number of components of U ˜R reduces +to C = 2, the next merger generates a horseshoe that is + +27 +−2 +0 +−2 +0 +−2 +0 +−2 +0 +2 +(a) +(b) +˜y/ξ +˜x/ξ +˜x/ξ +FIG. 15. Left: Unstable region U ˜ +R for a defect with dipolar +asymmetry. Upon the onset of strong pinning, an unstable +ellipse appears to the left of the defect center (black solid dot). +With increasing pinning strength (decreasing ¯C) the ellipse +grows and deforms into a horseshoe geometry. The unstable +eigenvector field v− (red arrows) points radially outward away +from the defect center. The tangent field to the boundary ∂U ˜ +R +(black arrows) follows the unstable direction at an angle of +π/2 over most of ∂U ˜ +R, with the exception of the two turning +points where the tangent rotates by π with respect to v−, +producing a relative winding of 2π. Right: After the merger +of the turning points the unstable region U ˜ +R changes topology +and assumes the shape of a ring. The windings of the tangent +field with respect to the eigenvector-field v− vanish separately +for both boundaries of U ˜ +R. +still simply-connected with C = 1. The final merger pro- +duces a ring; while the number of components remains +unchanged, C = 1, the unstable area assumes a non- +simply connected shape with a ‘hole’; we associate the +index H = 1 with the appearance of this hole within U ˜R. +In physics terms, the last merger producing a hole in U ˜R +is associated with the appearance of a pinned state; the +unstable ring separates stable tip positions that are asso- +ciated with pinned and free vortex configurations residing +at small and large radii, respectively. +Defining the (topological) characteristic χ ≡ C−H, we +see that χ changes by unity at every onset and merger, +either through an increase (for an onset) or decrease (for +a merger) in the number of components C → C ± 1, +or through the appearance of a hole (in a merger) H → +H +1. Indeed, the quantity χ is known as the Euler char- +acteristic of a manifold and describes its global topolog- +ical properties; it generalizes the well known Euler char- +acteristic of a polyhedron to surfaces and manifolds29, +see Sec. VI below. Finally, Morse theory30 connects the +Euler characteristic with the local differential properties +(minima, maxima, saddles) of that manifold, hence es- +tablishing a connection between local onsets and mergers +(at minima and saddles of D( ˜R)) and the global proper- +ties of U ˜R such as the appearance of new pinned states. +In Sec. VI below, we consider the general case of a ran- +dom pinning landscape in two dimensions and discuss the +connection between local differential and global topolog- +ical properties of U ˜R in the light of Morse theory—the +topology of bistable domains B ¯R then follows trivially. +−2 +0 +2 +−2 +0 +−2 +0 +2 +−2 +0 +2 +−2 +0 +−2 +0 +−2 +0 +−2 +0 +2 +(a) +(b) +(c) +(d) +˜y/ξ +˜y/ξ +˜x/ξ +˜x/ξ +C = 3 +H = 0 +χ = 3 +C = 2 +H = 0 +χ = 2 +C = 1 +H = 0 +χ = 1 +C = 1 +H = 1 +χ = 0 +FIG. 16. The unstable domain U ˜ +R starting out with C = 3 +components in (a) changes topology in three steps: after the +first (b) and second (c) mergers the number of components C +has changed from three in (a) to two in (b) to one in (c), lead- +ing to a horseshoe shape of U ˜ +R. The third merger closes the +horseshoe to produce the ring geometry in (d) characterized +by the coefficients C = 1 and H = 1 (H denotes the number +of ‘holes’ in U ˜ +R); the Euler characteristic χ = C − H changes +by unity in every merger. +VI. +U ˜ +R OF A TWO-DIMENSIONAL PINSCAPE +We consider a two-dimensional pinning landscape +ep(R), e.g., as produced by a superposition of several +(anisotropic Lorentzian) defects residing in the z = 0 +plane. +In the figures 17 and 18, we analyse two spe- +cific cases with n = 3 and n = 2 defects as given in Eq. +(94) with ϵ = 0.1 and positions listed in Tables I and +II; these produce unstable landscapes U ˜R of considerable +complexity already, see Figs. 17(a) and 18(a). Our de- +fects are compact with ep(R) → 0 vanishing at R → ∞; +as a result, epin becomes flat at infinity. Note that a dense +assembly of uniformly distributed individual defects pro- +duces a random Gaussian pinning landscape, as has been +shown in Ref. 26. +Here, we are interested in the evolution of the unstable +and bistable domains U ˜R and B ¯R associated with the 2D +pinning landscape epin; we focus on the unstable domain +U ˜R, with the properties of the bistable domain B ¯R fol- +lowing straightforwardly from the solution of the force +balance equation (2). Unlike the analysis above that is +centered on special points of U ˜R, ellipses near onset and +hyperbolas near mergers, here, we are interested in the +global properties of the unstable region produced by a +generic (though still two-dimensional) pinscape. + +28 +TABLE I. +Positions and relative weights of 3 uniaxially +anisotropic Lorentzian defects in Fig. 17 as given by Eq. (94). +x/ξ +y/ξ +weight +defect #1 +1.14 +1.07 +0.65 +defect #2 +−0.98 −0.19 +1 +defect #3 +0.20 +−0.67 +1 +TABLE II. +Positions and relative weights of 2 uniaxially +anisotropic Lorentzian defects in Fig. 18 as given by Eq. (94). +x/ξ +y/ξ +weight +defect #1 +−1.32 +0.33 +1 +defect #2 +1.48 +−0.76 +1 +As discussed in Sec. III above, the unstable region U ˜R +associated with strong pinning is determined by the con- +dition D( ˜R) = 0 of vanishing Hessian determinant, more +precisely, by the competition between the lowest eigen- +value λ−( ˜R) of the Hessian matrix Hij of the pinning +potential ep(R) and the effective elasticity ¯C, see Eq. +(37). In order to avoid the interference with the second +eigenvalue λ+( ˜R) of the Hessian matrix, we consider the +shifted (by ¯C) curvature function +Λ ¯ +C( ˜R) ≡ ¯C + λ−( ˜R), +(142) +i.e., the relevant factor of the determinant D( ˜R) = [ ¯C + +λ−( ˜R)][ ¯C + λ−( ˜R)]. The condition +Λ ¯ +C( ˜R) = 0 +(143) +then determines the boundaries of U ˜R. +The above problem can be mapped to the problem +of cutting a surface, where Λ ¯ +C( ˜R) is interpreted as a +height-function over R2 that is cut at zero level; the elas- +ticity ¯C then plays the role of a shift parameter that +moves the function λ−( ˜R) downwards in height with de- +creasing ¯C (that corresponds to increasing the relative +pinning strength of the pinscape in physical terms). As +¯C is decreased to compensate the absolute minimum of +λ−( ˜R) < 0, ¯C + λ−( ˜R) = 0, strong pinning sets in lo- +cally at ˜Rm for the first time in the form of an unstable +ellipse U ˜R, see Fig. 17(b) for our specific example with +three defects; the Labusch parameter κ( ˜R) evaluated at +the point ˜Rm defines κm, the parameter tuned in Fig. +17. Decreasing ¯C further, this ellipse grows and deforms, +while other local minima of λ−( ˜R) produce new discon- +nected parts of U ˜R, a situation illustrated in Fig. 17(c) +where four ‘ellipses’ have appeared around (local) minima +(blue filled dots). A further increase in pinning strength +(decrease in ¯C) continuous to deform these ‘ellipses’ and +adds three new ones. +As the first saddle drops below +the zero level (red cross), two components merge and the +number of components decreases; in Fig. 17(d), we have +three below-zero saddles and only four components re- +main, C = 4. In Fig. 17(e) four further mergers have +reduced C to 1 as the corresponding saddles drop below +zero level. This produces a single non-simply connected +component, i.e., C = 1 and a hole, increasing the num- +ber of holes H from zero to one. The last merger leading +to (f) finally leaves C = 1 but cuts the stable region in- +side the ring into two, increasing the number of holes to +H = 2. +This sequence of onsets and mergers is conveniently +described in the topographic language introduced in sec- +tion IV that interprets stable tip regions as land mass +(green with bright regions indicating higher mountains +in Fig. 17) and unstable regions as lakes (flat blue with +(below-water) height levels indicated by thin black lines), +with the height Λ ¯ +C = 0 defining the water level. The se- +quence (b) to (f) then shows the flooding of the landscape +as pinning increases ( ¯C decreasing), with white dot min- +ima turning blue at strong pinning onsets and white cross +saddles turning red at mergings; maxima in the landscape +are shown as black open circles. Note that we distinguish +critical points (minima, saddles) residing below (blue and +red) and above (white) water level. Similarly, a (local) +maximum above sea level (black open dot) turns into a +blue open dot as it drops belop sea level; such an event is +missing in Fig. 17 but can be produced with other con- +figurations of defects, see Fig. 18 where the curvature +landscape for two defects is shown. +The above discussion relates the local differential prop- +erties of the function Λ ¯ +C( ˜R) < 0, minima and saddles, +to the global topological properties of U ˜R, its number of +components C(U ˜R) and holes H(U ˜R). This connection +between local and global properties is conveniently dis- +cussed within Morse theory30. Before presenting a gen- +eral mathematical formulation, let us discuss a simple +heuristic argument producing the result relevant in the +present context; in doing so, we make use of the above +topographic language. +Starting with the minima of the function Λ ¯ +C( ˜R), a +new disconnected component appears in U ˜R whenever +the minimum drops below sea level as ¯C is decreased, +that produces an increase C → C + 1. With the further +decrease of ¯C, these disconnected regions expand and +merge pairwise whenever a saddle point of Λ ¯ +C( ˜R) goes +below sea level, thereby inducing a change in the topol- +ogy of U ˜R by either reducing the number of components +C → C −1 (keeping H constant) or leaving it unchanged +(changing H → H + 1), see, e.g., the example with the +horseshoe closing up on itself in Sec. V D. The below sea- +level minima and saddles of Λ ¯ +C( ˜R) can naturally be iden- +tified with the vertices and edges of a graph; the edges in +the graph then define the boundaries of the graph’s faces +(the same way as the vertices are the boundaries of the +edges). For a connected graph, Euler’s formula then tells +us that the number V of vertices, E of edges, and F of +faces are constrained via V − E + F = 1 (not counting +the outer face extending to infinity) and a graph with +C components satisfies the relation C = V − E + F as +follows from simple addition. +We have already identified minima and saddles of + +29 +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4 +κm = 1.1 +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +κm = 1.8 +-4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4 +κm = 2.5 +-4 +-2 +0 +2 +-4 +-2 +0 +2 +κm = 4.4 +-4 +-2 +0 +2 +-4 +-2 +0 +2 +4 +κm = 8.7 +(a) +(b) +(c) +(d) +(e) +(f) +˜y/ξ +˜y/ξ +˜y/ξ +˜x/ξ +˜x/ξ +FIG. 17. (a) Grayscale image of the pinning potential land- +scape ep( ˜R), with the three diamonds marking the positions +of the defects. (b)–(f) Shifted curvature function Λ ¯ +C( ˜R) ver- +sus tip position ˜R for increasing values of κm (decreasing ¯C) +as we proceed from (b) to (f). We make use of the topographic +interpretation with positive values of Λ ¯ +C marked as landmass +(greenish colors, with low/high elevation in dark/light green) +and negative values of Λ ¯ +C constituting U ˜ +R in flat light blue +(height levels are shown by thin black lines). The pinscape +in (a) produces a curvature landscape with 7 minima (solid +dots), 4 maxima (open dots), and 10 saddles (crosses). Sev- +eral unstable regions U ˜ +R appear (solid dots turn blue) and +merge (crosses turn red) to change the topology of U ˜ +R. The +Euler characteristic χ(U ˜ +R) = m − s + M = 1 − 0 + 0 = 1 +in (b) changes to χ(U ˜ +R) = 4 in (c) and (d), drops to +χ(U ˜ +R) = 0 in (e) and χ(U ˜ +R) = −1 in (f); indeed, U ˜ +R in (f) +has one component C = 1 and two holes H = 2, reproducing +χ(U ˜ +R) = C − H = −1. +Λ ¯ +C( ˜R) < 0 with vertices and edges of a graph; denot- +ing the number of below sea-level minima and saddles +by m and s, we have V = m and E = s. It remains +to express the number F of faces in terms of critical +points of the surface Λ ¯ +C( ˜R) < 0. Indeed, the faces of +our graph are associated with maxima of the function +Λ ¯ +C( ˜R): following the boundaries of a face, we cross the +corresponding saddles with the function Λ ¯ +C( ˜R) curving +upwards away from the edges, implying that the faces +of our graph include maxima of Λ ¯ +C( ˜R). These maxima +manifest in two possible ways: either the face contains +a single below sea-level maximum or a single above sea- +level landscape. The above sea-level landscape comprises +at least one maximum but possibly also includes other +extremal points that we cannot analyse with our knowl- +edge of the below sea-level function Λ ¯ +C( ˜R) < 0 only; +we therefore call the above sea-level landscape a (single) +hole. +The appearance of a single maximum or hole is +owed to the fact that faces are not split by a below sea- +level saddle as these have already been accounted for in +setting up the graph. +Let us denote the number of (below sea-level) maxima +by M and the number of holes by H, then F = H + M. +Combining this last expression with Euler’s formula and +regrouping topological coefficients C(U ˜R) and H(U ˜R) on +one side and extremal points m[Λ ¯ +C( ˜R)], s[Λ ¯ +C( ˜R)], and +M[Λ ¯ +C( ˜R)] on the other, we arrive at the Euler charac- +teristic χ ≡ C − H and its representation through local +differential properties, +χ(U ˜R) ≡ [C − H]U ˜ +R = [m − s + M]Λ ¯ +C( ˜R)<0. +(144) +The result (144) follows rigorously from the Euler- +Poincar´e theorem29,30 +in combination with Morse’s +theorem30, with the former expressing the Euler char- +acteristic χ(U ˜R) through the so-called Betti numbers +bi(U ˜R), +χ(U ˜R) ≡ +2 +� +i=0 +(−1)ibi(U ˜R), +(145) +where the i-th Betti number bi(U ˜R) = Dim[Hi(U ˜R)] is +given by the dimension or rank of the i-th (singular) ho- +mology group Hi(U ˜R). +In colloquial terms, the Betti +numbers bi count the number of ‘holes’ in the mani- +fold with different dimensions i: the zeroth Betti number +gives the number of components b0 = C of U ˜R, the first +Betti number b1 = H counts the holes, and the second +Betti number refers to cavities, here b2 = 0 for our open +manifold. Hence, we find that the Euler characteristic is +given by the number of components and holes in U ˜R, +χ(U ˜R) = C(U ˜R) − H(U ˜R), +(146) +in agreement with the discussion in Sec. V D and (144). +Morse theory30 then provides a connection between the +topological properties of the manifold U ˜R and the local +differential properties of the surface Λ ¯ +C( ˜R) < 0 defining +it: with Ci the number of critical points with index i of +the surface Λ ¯ +C( ˜R) < 0 (the index i counts the number of +negative eigenvalues of the Hessian matrix evaluated at +the critical point), the Euler characteristic χ(U ˜R) relates +the manifold’s topology to the number and properties of + +30 +−4 +−2 +0 +2 +4 +−4 +−2 +0 +2 +−4 +−2 +0 +2 +4 +−4 +−2 +0 +2 +4 +κm = 1.1 +−4 +−2 +0 +2 +4 +−4 +−2 +0 +2 +κm = 3.3 +−4 +−2 +0 +2 +4 +−4 +−2 +0 +2 +4 +κm = 4.0 +−4 +−2 +0 +2 +−4 +−2 +0 +2 +κm = 7.1 +−4 +−2 +0 +2 +−4 +−2 +0 +2 +4 +κm = 10.0 +(a) +(b) +(c) +(d) +(e) +(f) +˜y/ξ +˜y/ξ +˜y/ξ +˜x/ξ +˜x/ξ +FIG. 18. (a) Grayscale image of the pinning potential land- +scape ep( ˜R), with the two diamonds marking the positions +of the defects. (b)–(f) Shifted curvature function Λ ¯ +C( ˜R) (in +topographic coloring, see caption of Fig. 17) versus tip po- +sition ˜R for increasing values of κm as we proceed from (b) +to (f). The pinscape in (a) produces a curvature landscape +with 6 minima (solid dots), 4 maxima (open dots), and 9 sad- +dles (crosses). Upon increasing κm, several unstable regions +U ˜ +R appear (solid dots turn blue) and merge (crosses turn +red) to change the topology of U ˜ +R. The Euler characteristic +χ(U ˜ +R) = m − s + M = 1 = C in (b), remains χ(U ˜ +R) = 1 in +(c), but with C = 2 and H = 1, changes to χ(U ˜ +R) = −1 in +(d), and χ(U ˜ +R) = −3 with one component C = 1 and four +holes H = 4 in (e). In going from (e) to (f) two of the max- +ima (black open dots turn blue) drop below zero, producing +a characteristic χ(U ˜ +R) = 6 − 9 + 2 = −1; indeed, U ˜ +R in (f) +has one component C = 1 and two holes H = 2, reproducing +χ(U ˜ +R) = C − H = −1. +critical points, +χ(U ˜R) = +2 +� +i=0 +(−1)iCi(Λ ¯ +C < 0). +(147) +For our 2D manifold the coefficients Ci count the minima +C0 = m, the number of saddles C1 = s, and C2 = M +refers to the number of maxima, hence, +χ(U ˜R) = [m − s + M]Λ ¯ +C<0 +(148) +and the combination with (146) produces the result (144) +anticipated above. +Summarizing, knowing the number of critical points m, +M, and s of the seascape, i.e., its local differential prop- +erties, we can determine the global topological aspects +of the pinning landscape via the evaluation of the Euler +characteristic χ(U ˜R) with the help of Eq. (148). The lat- +ter then informs us about the number C of unstable do- +mains in U ˜R where locally pinned states appear and the +number of holes H in U ˜R where globally distinct pinned +states show up. +Furthermore, the outer boundaries of +the lakes, of which we have C components, are to be as- +sociated with instabilities of the free vortex state, while +inner boundaries (or boundaries of holes, which count H +elements) tell about instabilities of pinned states, hence +the Betti numbers C and H count different types of in- +stabilities. It would then have been nice to determine the +separate topological coefficients C and H individually— +unfortunately, χ(U ˜R) as derived from local differential +properties provides us only with the difference C − H +between locally and globally pinned areas and not their +individual values. Nevertheless, using Morse theory, we +could connect our discussion of local differential proper- +ties of the pinning landscape in Secs. III A and V A with +the global pinning properties of the pinning energy land- +scape as expressed through the topology of the unstable +domain U ˜R. +Regarding our previous examples, the isotropic and +uniaxial defects, we remark that for the latter the two +simultaneous mergers on the y-axis produce a reduction +in C = 2 → 1 and an increase of H = 0 → 1 and hence a +jump from χ = 2 to χ = 0 in one step, as expected for two +simultaneous mergers. The symmetry of the isotropic de- +fect produces a (degenerate) critical line at ˜Rm rather +than a critical point; adding a small perturbation ∝ x3 +breaks this symmetry and produces the horseshoe geom- +etry discussed in Sec. V D above that is amenable to the +standard analysis. +A last remark is in place about the topological prop- +erties in dual space, i.e., of bistable regions B ¯R. Here, +the mergers produce another interesting phenomenon as +viewed from the perspective of its thermodynamic ana- +logue. +Indeed, the merger of deformed ellipses in tip- +space corresponds to the merger of cusps in asymptotic +space, what translates to the vanishing of critical points +and a smooth continuation of the first-order critical and +spinodal lines in the thermodynamic analogue, see also +Sec. V C. We are not aware of a physical example in ther- +modynamics that produces such a merger and disappear- +ance of critical points. + +31 +VII. +SUMMARY AND OUTLOOK +Strong pinning theory is a quantitative theory describ- +ing vortex pinning in the dilute defect limit where this +complex many-body system can be reduced to an effec- +tive single-pin–single-vortex problem. The accuracy of- +fered by this theory then allows for a realistic description +of the shape of the pinning potential ep(R) associated +with the defects. +While previous work focused on the +simplest case of isotropic defects, here, we have general- +ized the strong pinning theory to the description of arbi- +trary anisotropic pinning potentials. Surprisingly, going +from an isotropic to an anisotropic defect has quite aston- +ishing consequences for the physics of strong pinning— +this reminds about other physical examples where the +removal of symmetries or degeneracies produces new ef- +fects. +While the strong pinning problem is quite a complex +one requiring the use of numerical tools in general, we +have identified several generic features that provide the +essential physics of the problem and that are amenable +to an analytic treatment. +Specifically, these are the +points of strong pinning onset and the merger points, +around which the local expansions of the pinning poten- +tial epin( ˜R; ¯R) in the tip coordinate ˜R allow us to find +all the characteristics of strong pinning. In particular, we +identify the instability region U ˜R in the vortex tip space +(with coordinates ˜R) and the bistable region B ¯R in the +space of asymptotic vortex positions ¯R as the main ge- +ometric objects that determine the critical pinning force +density Fpin, from which the critical current density jc, +the technologically most relevant quantity of the super- +conductor, follows straightforwardly. While the relevance +of the bistable region B ¯R was recognized in the past8–10, +the important role played by the unstable region U ˜R went +unnoticed so far. +When going from an isotropic defect to an anisotropic +one, the strong pinning onset changes dramatically: +while the unstable region U ˜R grows out of a circle of +radius ∼ ξ and assumes the shape of a ring at κ > 1 +for the isotropic situation, for an anisotropic defect the +onset appears in a point ˜Rm and grows in the shape of +an ellipse with increasing κm > 1; the location where +this onset appears is given by the Hessian of epin, specif- +ically, the point ˜Rm where its determinant touches zero +first, det{Hess[epin( ˜R; ¯R)| ¯R]} ˜Rm = 0. The boundary of +this ellipse defines the jump positions J ˜R associated with +the strong pinning instabilities; when combined with the +landing ellipse L ˜R, these two ellipses determine the jump +distance δ˜u of the vortex tip, from which follows the jump +in the pinning energy ∆epin ∝ δ˜u4, which in turn deter- +mines Fpin and jc. +The bistable region B ¯R in asymptotic vortex space +comes into play when calculating the average critical +force density Fpin opposing the vortex motion: +while +the vortex tip undergoes a complex trajectory includ- +ing jumps, the vortex motion in asymptotic space ¯R is +described by a straight line. As this trivial trajectory in +¯R-space traverses the bistable region B ¯R, the vortex tip +jumps upon exiting B ¯R, that produces the jump ∆epin +and hence Fpin. Again, the shape of B ¯R changes when +going from the isotropic to the anisotropic defect, assum- +ing a ring of finite width around a circle of radius ∼ ξ in +the former case, while growing in the form of a crescent +out of a point for the anisotropic defect. +The new geometries associated with U ˜R and B ¯R then +produce a qualitative change in the scaling behavior of +the pinning force density Fpin ∝ (κm − 1)µ near onset, +with the exponent µ changing from µ = 2 to µ = 5/2 +when going from the isotropic to the anisotropic defect. +This change is due to the change in the scaling of the +geometric size of B ¯R, with the replacement of the fixed +radius ∼ ξ of the ring by the growing size of the crescent +∼ ξ(κm−1)1/2 [the exponent µ assumes a value µ = 3 for +trajectories cutting the crescent along its short dimension +of size ξ(κm − 1)]. Furthermore, for directed defects, the +pinning force density Fpin(θ) depends on the impact angle +θ relative to the unstable direction u and is aligned with +u, except for a small angular regime close to θ = π/2. +This results in a pronounced anisotropy in the critical +current density jc in the vicinity of the strong pinning +onset. +A fundamental difference between the strong pinning +onsets in the isotropic and in the anisotropic case are +the geometries of the unstable U ˜R and bistable B ¯R re- +gions: these are non-simply connected for the isotropic +case (rings) but simply connected for the anisotropic de- +fect (ellipse and crescent). The resolution of this funda- +mental difference is provided by the second type of special +points, the mergers. Indeed, for a general anisotropic de- +fect, the strong pinning onset appears in a multitude of +points, with unstable and bistable regions growing with +κm > 1 and finally merging into larger areas. Two exam- +ples illustrate this behavior particularly well, the uniaxial +defects with a quadrupolar and a dipolar deformation, +see Secs. IV and V D. In the first case, symmetric on- +set points on the x axis produce two ellipses/crescents +that grow, approach one another, and finally merge in a +ring-shaped geometry that is non-simply connected. In +the case of a dipolar deformation, we have seen U ˜R grow +out of a single point with its ellipse expanding and de- +forming around a circle, assuming a horseshoe geometry, +that finally undergoes a merging of the two tips to pro- +duce again a ring; similar happens when multiple U ˜R +domains grow and merge as in Figs. 16 (a warped defect) +and 18(c) (a 2D pinning landscape where four unstable +domains have merged to enclose an ‘island’). +These merger points are once more amenable to an +analytic study using a proper expansion of epin( ˜R; ¯R) +in ˜R around the merger point ˜Rs, the latter again de- +fined by the local differential properties of the determi- +nant det{Hess[epin( ˜R; ¯R)| ¯R]}, this time not a minimum +but a saddle. Rather than elliptic as at onset, at merger +points the geometry is hyperbolic, with the sign change +associated with increasing κs ≡ κ( ˜Rs) across unity pro- +ducing a reconnection of the jump- and landing lines J ˜R + +32 +and L ˜R. +While the expansions of epin( ˜R; ¯R) are describing the +local pinning landscape near onset and merging (and thus +produce generic results), the study of the combined set of +onset- and merger-points describe the global topological +properties of U ˜R as discussed in Sec. VI: every new (non- +degenerate) onset increases the number of components C +in U ˜R, while every merger either decreases C or increases +H, the number of ‘holes’ or ‘islands’ (or nontrivial loops +in a non-simply connected region) in the pinning land- +scape. It is the ‘last’ merging producing a non-simply +connected domain that properly defines a new pinned +state; in our examples these are the closing of the two +deformed ellipses in the uniaxial defect with quadrupo- +lar deformation and the closing of the horseshoe in the +defect with a dipolar deformation. +Formally, the rela- +tion between the local differential properties of the cur- +vature function Λ ¯ +C( ˜R) = ¯C + λ−( ˜R) [with λ−( ˜R) the +lower eigenvalue of the Hessian of ep( ˜R)], its minima, +saddles, and maxima, are related to the global topologi- +cal properties of U ˜R as described by its Euler characteris- +tic χ = C −H through Morse theory, see Eq. (144). Such +topological structures have recently attracted quite some +interest, e.g., in the context of Fermi surface topologies +and topological Lifshitz transitions31,32. +The physics around the onset points as expressed +through an expansion of epin( ˜R; ¯R) resembles a Landau +theory with ˜R playing the role of an order parameter and +¯R the dual variable corresponding to a driving field— +here, ¯R drives the vortex lattice across the defect and +˜R describes the deformation of the pinned vortex. The +endpoints of the crescent B ¯R correspond to critical end +points as they appear in the Landau theory of a first- +order transition line, e.g., the Ising model in an external +field or the van der Waals gas. The boundary lines of B ¯R +correspond to spinodal lines where phases become un- +stable, e.g., the termination of overheated/undercooled +phases in the van der Waals gas. The existence of criti- +cal end points tells that ‘phases’, here in the form of dif- +ferent pinning branches, are smoothly connected when +going around the critical point, similar as in the gas– +liquid transition of the van der Waals gas. As the ‘last’ +critical point vanishes in a merger, a well defined new +phase, here a new pinned branch, appears. +Perspectives for future theoretical work include the +study of correlations between anisotropic defects (see Ref. +17 addressing isotropic defects) or the inclusion of ther- +mal fluctuations, i.e., creep (see Refs. 13 and 21). Fur- +thermore, our discussion of the extended pinscape in Sec. +VI has been limited to a two-dimensional pinning poten- +tial. +In reality, defects are distributed in all three di- +mensions that considerable complicates the correspond- +ing analysis of a full three-dimensional disordered pinning +potential, with the prospect of interesting new results. +On the experimental side, there are several possible +applications for our study of anisotropic defects. For a +generic anisotropic defect, the inversion symmetry may +be broken. In this case, the pinning force along opposite +directions is different in magnitude, as different jumps +are associated to the boundaries of the bistable region +B ¯R away from onset, i.e., at sufficiently large values of +κm. Reversing the current, the different critical forces +then result in a ratchet effect33,34. This leads to a rec- +tification of an ac current and hence a superconducting +diode effect. While for randomly oriented defects the pin- +ning force is averaged and the symmetry is statistically +restored, for specially oriented defects, the diode effect +will survive. Indeed, introducing nanoholes into the ma- +terial, vortex pinning was enhanced23,35 and a diode ef- +fect has been observed recently36. Generalizing strong +pinning theory to this type of defects then may help in +the design of superconducting metamaterials with inter- +esting functionalities. Furthermore, vortex imaging has +always provided fascinating insights into vortex physics. +Recently, the SQUID-on-tip technique has been success- +ful in mapping out a 2D pinning landscape in a film37 +(including the observation of vortex jumps) that has in- +spired a new characterization of the pinscape through its +Hessian analysis26; the adaptation of this current-driven +purely 2D setup to the 3D situation described in the +present paper is an interesting challenge. +Finally, we recap the main benefits of this work in a +nutshell: For one, we have established a detailed con- +nection of the strong pinning transition with a the con- +cept of first-order phase transitions in thermodynamics, +with the main practical result that the scaling of the +pinning force density Fpin ∝ (κm − 1)µ comes with an +exponent µ = 5/2 when working with generic defects +of arbitrary shapes. +Second, we have found a mecha- +nism, the breaking of a defect’s inversion symmetry, that +produces rachets and a diode effect in superconducting +material. Third, we have uncovered the geometric struc- +ture and its topological features that is underlying strong +pinning theory, including a proper understanding of the +appearance of distinguished pinned states. +While un- +derstanding these geometric structures seems to be of +rather fundamental/scholarly interest at present, future +work may establish further practical consequences that +can be used in the development of superconducting ma- +terials with specific functional properties. +ACKNOWLEDGMENTS +We thank Tom´aˇs Bzduˇsek, Gian Michele Graf, and +Roland Willa for discussions and acknowledge financial +support of the Swiss National Science Foundation, Divi- +sion II. +Appendix A: Effective 1D Landau theory +The Landau-type pinning energies (18) and (117) for +the vector order parameter (˜u, ˜v) involves a soft variable +˜u with a vanishing quadratic term ∝ (1 − κm) ˜u2, as well +as a stiff one, ˜v, characterized by a finite elasticity. By + +33 +eliminating the stiff direction ˜v, we can arrive at a 1D +Landau expansion for the order parameter ˜u that pro- +vides us with the desired results for the unstable and +bistable domains U ˜R and B ¯R near onset and merging in +a very efficient manner. +1. +Close to onset +We start with the two-dimensional Landau-type energy +functional (58) +epin( ˜R; ¯R) = +¯C (1 − κm) +2 +˜u2 + +¯C + λ+ +2 +˜v2 + a +2 ˜u˜v2 ++ α +4 ˜u2˜v2 + β +6 ˜u3˜v + γ +24 ˜u4 − ¯C ¯u˜u − ¯C ¯v˜v +(A1) +written in terms of the tip coordinates ˜u, ˜v measured rel- +ative to ˜Rm, the position of the minimal determinant +D( ˜R) at strong pinning onset, and with ˜u and ˜v aligned +with the stable and unstable directions, respectively. The +expansion (A1) is anisotropic: the quadratic (elastic) co- +efficient along the unstable ˜u-direction vanishes at the +onset of strong pinning, while the one along the stable +˜v-direction stays positive and large, allowing us to ‘inte- +grate out’ the latter. The asymptotic coordinates ¯u, ¯v +assume the role of the driving (conjugate) fields for the +tip positions (or order parameters) ˜u, ˜v; the latter then +are determined by the force equations ∂ ˜Repin( ˜R; ¯R) = 0, +¯C¯u = ¯C(1 − κ)˜u + a +2 ˜v2 + γ +6 ˜u3 + β +2 ˜u2˜v + α +2 ˜u˜v2, +(A2) +¯C¯v = ( ¯C + λ+)˜v + a ˜u˜v + β +6 ˜u3 + α +2 ˜u2˜v, +(A3) +see Eq. (69), with δ ¯R = (¯u, ¯v) measured relative to ¯Rm. +Inspection of Eqs. (A2) and (A3) shows that near the +strong pinning onset, the Ansatz ˜u, ˜v, ¯v ∝ √κm − 1 and +¯u ∝ (κm − 1) produces a consistent solution. Solving the +second equation (A3) for the stiff degree of freedom ˜v, +we then find that +˜v ≈ +¯C¯v +¯C + λ++ a˜u ≈ +¯v +1+λ+/ ¯C +� +1 − +a/ ¯C +1+λ+/ ¯C ˜u +� +(A4) +which is precise to order (κm − 1). Inserting ˜v back into +the force-balance equation (A2) for the unstable compo- +nent ˜u, we find a cubic equation for ˜u (precise to order +(κm −1)3/2) that is driven by a combination of ¯u and ¯v2, +¯C¯u − +(a/2) ¯v2 +(1 + λ+/ ¯C)2 ≈ +� +¯C(1 − κm) + +(δ/2) ¯v2 +(1 + λ+/ ¯C)2 +� +˜u ++ +(β/2) ¯v +(1 + λ+/ ¯C) ˜u2 + γ +6 ˜u3. +(A5) +Upon integration, we finally arrive at the effective one- +dimensional Landau expansion for the 1D order parame- +ter ˜u that is precise to order (κm−1)2 (up to an irrelevant +shift ∝ ¯v2), +eeff +pin(˜u; ¯u, ¯v) = r(¯v) +2 +˜u2 + w(¯v) +6 +˜u3 + γ +24 ˜u4 −h(¯u, ¯v)˜u, (A6) +with the coefficients r, w, and h defined as +r(¯v) = +� +¯C(1 − κm) + δ +2 +¯v2 +(1 + λ+/ ¯C)2 +� +, +w(¯v) = β +¯v +(1 + λ+/ ¯C), +h(¯u, ¯v) = ¯C¯u − a +2 +¯v2 +(1 + λ+/ ¯C)2 . +(A7) +The Landau-type energy function (A6) belongs to the van +der Waals (gas-liquid) universality class; its first-order +transition line maps to the branch crossing line in the +strong pinning problem, its spinodals correspond to the +arcs of the crescent defining the bistable region B ¯R, and +its critical points map to the two cusps of B ¯R, i.e., in the +strong pinning problem, the spinodals end in two critical +points. The cubic term w˜u3/6 is determined by the skew +parameter β; in the absence of such a skew, i.e., for a +±˜v-symmetric unstable ellipse U ˜R, we have β = 0 and +our problem assumes an Ising-type Z2 symmetry. +Let us begin with the determination of the critical co- +efficients rc, wc, and hc. These are found by setting the +first three derivatives of eeff +pin(˜u) to zero [two spinodals +(implying ∂˜ueeff +pin = 0 and ∂2 +˜ueeff +pin = 0) coalescing into a +single point (→ ∂3 +˜ueeff +pin = 0)]. Setting the cubic derivative +to zero, we find the order parameter +˜uc = −wc/γ ≈ −(β/γ)˜vc, +(A8) +where we have used Eq. (A7) and the transformation +¯v ↔ ˜v in (A4) to leading order. +The vanishing of the second derivative relates the crit- +ical coefficients rc and wc, +rc = w2 +c/2γ, +(A9) +(where we have made use of ˜uc). Inserting the dependen- +cies r(¯v) and w(¯v), see Eq. (A7), we find that +¯v2 +c +(1 + λ+/ ¯C)2 = γ ¯C(κm − 1) +2 det Mjp +, +(A10) +with det Mjp = (γδ − β2)/4. Using again Eq. (A4) to +leading order, we find that +˜vc ≈ +� +2γ ¯C(κm − 1) +γδ − β2 +, +(A11) +cf. Eq. (57). The critical endpoints of the 1D Landau +theory then correspond to the touching points (67) of +the unstable domain U ˜R +δ ˜Rc,± = ± (−β/γ, 1) ˜vc, +(A12) +found before, see Eq. (67) with (57). +Finally, the vanishing of the first derivative defines the +critical drive +hc = [r˜u + w˜u2/2 + γ˜u3/6]c = − w3 +c +6γ2 . +(A13) + +34 +Making use of the coefficients (A7), this translates to the +critical drive ¯uc +¯uc = (a/2 ¯C)˜v2 +c − +w3 +c +6 ¯Cγ2 +(A14) +and its combination with the result for ¯vc tells us that +the critical drives match up, to leading order, with the +cusps (73) of the bistable domain at ¯Rc,±, +δ ¯Rc,± = (¯uc, ±¯vc) +(A15) +≈ +�� +a/2 ¯C +� +˜v2 +c, ±(1 + λ+/ ¯C)˜vc +� +. +Next, we find the entire boundary of the unstable re- +gion U ˜R that is defined as the points where local minima +and maxima of eeff +pin coalesce, i.e., where ∂2 +ueeff +pin = 0, +r + w˜ujp + γ +2 ˜u2 +jp = 0. +(A16) +Making use of the Landau coefficients (A7) as well as the +relation between ˜v and ¯v in (A4), we recover the equation +(53) for the ellipse (we drop corrections ∝ (κm − 1)3/2) +γ˜u2 +jp + 2β˜ujp˜vjp + δ˜v2 +jp ≈ 2 ¯C(κm − 1). +(A17) +In order to find the shape of the bistable region B ¯R, +we exploit the fact that for fixed drives ¯u and ¯v, the +bistable and the unstable vortex tip configurations are +local extrema of eeff +pin, implying that ∂˜ueeff +pin = 0 and hence +r˜u + w +2 ˜u2 + γ +6 ˜u3 = h, +(A18) +what corresponds to the force-balance equation (A5) ex- +pressed in terms of the coefficients (A7). The cubic equa- +tion (A18) with its left side ∝ (κm − 1)3/2 depends on ¯u +through the drive h. According to (A7), the two terms in +the drive are of order (κm − 1) and hence have to cancel +one another to lowest order. As a result, we find that the +bistable domain is centered around the parabola +¯u = a +2 ¯C +¯v2 +(1 + λ+/ ¯C)2 , +(A19) +that matches up with Eq. (70) found in Sec. III. Finding +the precise form of the bistable region B ¯R, we have to +solve Eq. (A18) to cubic order in √κm − 1 with the help +of an expansion around the center parabola (A19), what +amounts to repeating the analysis leading to the results +(71) and (72) in Sec. III C. +Finally, we find the landing line L ˜R defined as the sec- +ond bistable tip position at fixed ¯u and ¯v. +We make +use of the cubic equation (A18) and represent it in the +factorized form (with the inflection point at ˜ujp having +multiplicity two) +(˜u − ˜ujp)2(˜u − ˜ulp) = 0, +(A20) +and ˜ulp the landing position of the tip introduced in +Sec. III B 2. A somewhat tedious but straightforward cal- +culation shows that the stable solution ˜ulp satisfies the +quadratic equation +r − 3 +8 +w2 +γ + w +4 ˜ulp + γ +8 ˜u2 +lp = 0 +(A21) +and thus arranges along the ellipse +γ +8 ˜u2 +lp + β +4 ˜ulp˜vlp + +�δ +2 − 3 +8 +β2 +γ +� +˜v2 +lp = ¯C(κm − 1) (A22) +when expressed in the original two-dimensional tip space; +this coincides with the original result (63). +In a last step, we may go over to an Ising-type Lan- +dau expansion by measuring the order parameter ¯u with +reference to the skewed line +˜um(¯v) = +� +−β +γ +� +¯v +(1 + λ+/ ¯C), +(A23) +i.e., +˜u′ = ˜u − ˜um(¯v). +(A24) +The 1D effective Landau expansion now reads, with pre- +cision to order (κm − 1)2, +eeff +pin(˜u′; ¯u, ¯v) = r′ +2 ˜u′2 + γ +24 ˜u′4 − h′˜u′, +(A25) +with the new coefficients +r′ = r − w2 +2γ , +h′ = h − w3 +3γ2 + rw +γ . +(A26) +The condition h′ = 0 now defines the equilibrium state +of the thermodynamic problem that translates into the +branch crossing line where the bistable vortex tip posi- +tions have equal energy. Using the definitions (A7) and +(A26) for h and h′, we find that the branch crossing line +¯u0(¯v0) in the original two-dimensional asymptotic space +reads +¯u0 = a +2 ¯C +¯v2 +0 +(1 + λ+/ ¯C)2 − β +γ +� +(κm − 1) +¯v0 +1 + λ+/ ¯C ++ +�δ +2 − β2 +3γ +� 1 +¯C +¯v3 +0 +(1 + λ+/ ¯C)3 +� +, +(A27) +extending the result (77) from Sec. III to finite values of +β with an additional term ∝ (κm − 1)3/2. +2. +Close to merging +Let us study the strong pinning problem close to merg- +ing, as described by the two-dimensional Landau-type +energy functional (117), +epin( ˜R; ¯R) = +¯C(1 − κs) +2 +˜u2 + +¯C + λ+,s +2 +˜v2 + as +2 ˜u˜v2 ++ αs +4 ˜u2˜v2 + βs +6 ˜u3˜v + γs +24 ˜u4 − ¯C¯u˜u − ¯C¯v˜v. +(A28) + +35 +As found before for strong pinning close to onset, the +energy functional (A28) is anisotropic with respect to +vortex displacements in the stable and unstable direction. +Following the strategy of Sec. A 1, we can use the force- +balance equation (137) to relate the tip position along +the v-axis to ¯v and ˜u, +˜v ≈ +¯v +1 + λ+,s/ ¯C +� +1 − +as/ ¯C +1 + λ+,s/ ¯C ˜u +� +. +(A29) +Inserting (A29) into the force-balance equation for the +unstable component ˜u and integrating, we find that the +resulting effective 1D Landau theory is identical in form +to the one close to onset, +eeff +pin(˜u; ¯u, ¯v) = rs +2 ˜u2 + ws +6 ˜u3 + γs +24 ˜u4 − hs˜u, +(A30) +with a proper replacement of all coefficients involving the +parameters appropriate at merging, +rs = +� +¯C(1 − κs) − |δs| +2 +¯v2 +(1 + λ+,s/ ¯C)2 +� +, +ws = βs +¯v +(1 + λ+,s/ ¯C), +hs = ¯C¯u − as +2 +¯v2 +(1 + λ+,s/ ¯C)2 . +(A31) +The difference to (A7) is the sign change in the term +∝ |δs|¯v2. This implies a modification of the main equa- +tion determining the shape of U ˜R (from which B ¯R fol- +lows via the force balance equation (38)), with the elliptic +equation (A17) transforming to the hyperbolic expression +γs˜u2 +jp + 2βs˜ujp˜vjp − |δs|˜v2 +jp ≈ 2 ¯C(κs − 1). +(A32) +The results for the jumping and landing hyperbolas in ˜R- +space and for the edges of the bistable domain in ¯R-space +before and after merging can be derived by following the +strategy of Sec. A 1 above and agree with the correspond- +ing results from Sec. V A. +We close with a final remark on the disappearance +of critical points after merging. The critical points are +found in the standard manner by setting the first three +derivatives of eeff +pin(˜u; ¯u, ¯v) to zero. This works fine before +merging when 1 − κs > 0 and we find that criticality +is realized for tip and asymptotic positions as given by +Eqs. (125) and (138) in Sec. V A. However, after merg- +ing, the cubic derivative ∂3 +˜ueeff +pin never vanishes, signalling +the absence of a critical point, in agreement with the dis- +cussion in Secs. V C and V B 2. The merger thus leads +to the disappearance of the two critical (end-)points in +asymptotic space, with the attached first-order lines (the +branch crossing line) joining up into a single line that is +framed by two separated spinodals. We are not aware of +such a disappearance of critical points in a merging pro- +cess within the standard discussion of thermodynamic +phase transitions. +1 A. Campbell and J. Evetts, Advances in Physics 21, 199 +(1972). +2 M. E. Kassner, Fundamentals of Creep in Metals and Al- +loys (Elsevier Science & Technology Books, Amsterdam, +2015). +3 J. Gorchon, S. Bustingorry, J. Ferr´e, V. Jeudy, A. Kolton, +and T. Giamarchi, Phys. Rev. Lett. 113, 027205 (2014). +4 A. Abrikosov, Sov.Phys.JETP 5 (1957) 5, 1174 (1957). +5 J. Burgers, Proc. Phys. Soc. 52, 23 (1940). +6 F. Bloch, Z. Physik 74, 295 (1932). +7 L. D. Landau and E. Lifshitz, Phys. Z. Sowjet 8, 153 +(1935). +8 R. Labusch, Crystal Lattice Defects 1, 1 (1969). +9 A. I. Larkin and Y. N. Ovchinnikov, Journal of Low Tem- +perature Physics 34, 409 (1979). +10 G. Blatter, V. B. Geshkenbein, and J. A. G. Koopmann, +Phys. Rev. Lett. 92, 067009 (2004). +11 A. U. Thomann, V. B. Geshkenbein, and G. Blatter, Phys- +ical Review Letters 108, 217001 (2012). +12 R. Willa, V. Geshkenbein, R. Prozorov, +and G. Blatter, +Physical Review Letters 115, 207001 (2015). +13 M. Buchacek, R. Willa, V. B. Geshkenbein, and G. Blat- +ter, Physical Review B 100, 014501 (2019). +14 D. Ertas and D. Nelson, Physica C 272, 79 (1996). +15 V. Vinokur, B. Khaykovich, E. Zeldov, M. Konczykowski, +R. Doyle, and P. Kes, Physica C 295, 209 (1998). +16 M. Buchacek, Z. L. Xiao, S. Dutta, E. Y. Andrei, P. Ray- +chaudhuri, V. B. Geshkenbein, and G. Blatter, Phys. Rev. +B 100, 224502 (2019). +17 M. Buchacek, V. B. Geshkenbein, and G. Blatter, Physical +Review Research 2, 043266 (2020). +18 G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. +Larkin, +and V. M. Vinokur, Rev. Mod. Phys. 66, 1125 +(1994). +19 A. U. Thomann, V. B. Geshkenbein, and G. Blatter, Phys- +ical Review B 96, 144516 (2017). +20 R. Willa, V. B. Geshkenbein, and G. Blatter, Phys. Rev. +B 93, 064515 (2016). +21 F. Gaggioli, G. Blatter, +and V. B. Geshkenbein, Phys. +Rev. Research 4, 013143 (2022). +22 M. Buchacek, R. Willa, V. B. Geshkenbein, and G. Blat- +ter, Physical Review B 98, 094510 (2018). +23 W.-K. Kwok, U. Welp, A. Glatz, A. E. Koshelev, K. J. +Kihlstrom, +and G. W. Crabtree, Rep. Prog. Phys. 79, +116501 (2016). +24 R. Willa, A. Koshelev, I. Sadovskyy, and A. Glatz, Super- +cond. Sci. Technol. 31, 014001 (2018). +25 R. Willa, A. E. Koshelev, I. A. Sadovskyy, and A. Glatz, +Physical Review B 98, 054517 (2018). +26 R. Willa, V. B. Geshkenbein, and G. Blatter, Phys. Rev. +B 105, 144504 (2022). +27 M. Buchacek, The Strong Pinning Paradigm Applied to +Vortices in Type-II Superconductors, PhD Thesis (Insti- +tute for Theoretical Physics, ETH Zurich, Switzerland, + +36 +2020). +28 J. Bardeen and M. J. Stephen, Physical Review 140, +A1197 (1965). +29 M. Nakahara, Geometry, Topology and Physics (Graduate +Student Series in Physics) (Taylor and Francis, 2003) pp. +88,118. +30 C. Nash and S. Sen, Topology and Geometry for Physicists +(Dover Publications Inc., 2011) pp. 105, 228. +31 G. Volovik, Low Temp. Phys. 43, 47 (2017). +32 C. Kane, Phys. Rev. Lett. 128, 076801 (2022). +33 J. Villegas, S. Savel’ev, F. Nori, E. Gonzalez, J. Anguita, +R. Garc´ıa, and J. Vicent, Science 302, 1188 (2003). +34 C. de Souza Silva, J. Van de Vondel, M. Morelle, +and +V. Moshchalkov, Nature 440, 651 (2006). +35 Y. Wang, M. Latimer, Z. Xiao, R. Divan, L. Ocola, +G. Crabtree, and W. Kwok, Phys. Rev. B 87, 220501(R) +(2013). +36 Y. Lyu, J. Jiang, Y. Wang, Z. Xiao, S. Dong, Q. Chen, +M. Miloˇsevi´c, H. Wang, R. Divan, J. Pearson, P. Wu, +F. Peeters, +and W.-K. Kwok, Nature Communications +12, 2703 (2021). +37 L. Embon, Y. Anahory, A. Suhov, D. Halbertal, J. Cup- +pens, A. Yakovenko, A. Uri, Y. Myasoedov, M. L. Rappa- +port, M. E. Huber, A. Gurevich, and E. Zeldov, Scientific +Reports 5, 7598 (2015). + diff --git a/S9E0T4oBgHgl3EQfUwBZ/content/tmp_files/load_file.txt b/S9E0T4oBgHgl3EQfUwBZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..32589e09959d75533b5091e374ebffb2c80a4501 --- /dev/null +++ b/S9E0T4oBgHgl3EQfUwBZ/content/tmp_files/load_file.txt @@ -0,0 +1,1755 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf,len=1754 +page_content='Strong pinning transition with arbitrary defect potentials Filippo Gaggioli,1 Gianni Blatter,1 Martin Buchacek,1 and Vadim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein1 1Institut f¨ur Theoretische Physik, ETH Z¨urich, CH-8093 Z¨urich, Switzerland (Dated: January 9, 2023) Dissipation-free current transport in type II superconductors requires vortices, the topological defects of the superfluid, to be pinned by defects in the underlying material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinning capacity of a defect is quantified by the Labusch parameter κ ∼ fp/ξ ¯C, measuring the pinning force fp relative to the elasticity ¯C of the vortex lattice, with ξ denoting the coherence length (or vortex core size) of the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The critical value κ = 1 separates weak from strong pinning, with a strong defect at κ > 1 able to pin a vortex on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' So far, this weak-to-strong pinning transition has been studied for isotropic defect potentials, resulting in a critical exponent µ = 2 for the onset of the strong pinning force density Fpin ∼ npfp(ξ/a0)2(κ − 1)µ, with np denoting the density of defects and a0 the intervortex distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This result is owed to the special rotational symmetry of the defect producing a finite trapping area Strap ∼ ξ2 at the strong-pinning onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The behavior changes dramatically when studying anisotropic defects with no special symmetries: the strong pinning then originates out of isolated points with length scales growing as ξ(κ − 1)1/2, resulting in a different force exponent µ = 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Our analysis of the strong pinning onset for arbitrary defect potentials ep(R), with R a planar coordinate, makes heavy use of the Hessian matrix describing its curvature and leads us to interesting geometrical structures: the strong pinning onset is characterized by the appearance of unstable areas of elliptical shape whose boundaries mark the locations where vortices jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The associated locations of asymptotic vortex positions define areas of bistable vortex states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these bistable regions assume the shape of a crescent with boundaries that correspond to the spinodal lines in a thermodynamic first-order transition and cusps corresponding to critical end- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Both, unstable and bistable areas grow with κ > 1 and join up into larger domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' for a uniaxially anisotropic defect, two face to face crescents merge into the ring-shaped area previously encountered for the isotropic defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Both, onset and merger points are defined by local differential properties of the Hessian’s determinant D(R), specifically, its minima and saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Extending our analysis to the case of a random two-dimensional pinning landscape, we discuss the topological properties of unstable and bistable regions as expressed through the Euler characteristic, with the latter related to the local differential properties of D(R) through Morse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' INTRODUCTION Vortex pinning by material defects1 determines the phenomenological properties of all technically rele- vant (type II) superconducting materials, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', their dissipation-free transport or magnetic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Similar applies to the pinning of dislocations in metals2 or do- main walls in magnets3, with the commonalities found in the topological defects of the ordered phase being pinned by defects in the host material: these topolog- ical defects are the vortices4, dislocations5, or domain walls6,7 appearing within the respective ordered phases— superconducting, crystalline, or magnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The theory describing the pinning of topological defects has been furthest developed in superconductors, with the strong pinning paradigm8,9 having been strongly pushed during the last decade10–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In its simplest form, it boils down to the setup involving a single vortex subject to one defect and the cage potential14,15 of other vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While still exhibiting a remarkable complexity, it produces quanti- tative results which benefit the comparison between the- oretical predictions and experimental findings16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' So far, strong pinning has focused on isotropic defects, with the implicit expectation that more general potential shapes would produce small changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This is not the case, as first demonstrated by Buchacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='17 in their study of correlation effects between defects that can be mapped to the problem of a string pinned to an anisotropic pin- ning potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the present work, we generalize strong pinning theory to defect potentials of arbitrary shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We find that this simple generalization has pronounced (geometric) effects near the onset of strong pinning that even change the growth of the pinning force density Fpin ∝ (κ − 1)µ with increasing pinning strength κ > 1 in a qualitative manner, changing the exponent µ from µ = 2 for isotropic defects8,10 to µ = 5/2 for general anisotropic pinning potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinning of topological defects poses a rather complex problem that has been attacked within two paradigms, weak-collective- and strong pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These have been developed in several stages: originating in the sixties of the last century, weak pinning and creep9 has been further developed with the discovery of high tem- perature superconductors as a subfield of vortex matter physics18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Strong pinning was originally introduced by Labusch8 and by Larkin and Ovchinnikov9 and has been further developed recently with several works studying critical currents10, current–voltage characteristics11,19, magnetic field penetration12,20,21, and creep13,21,22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' re- sults on numerical simulations involving strong pins have been reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 23–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The two theories come to- gether at the onset of strong pinning: an individual defect is qualified as weak if it is unable to pin a vortex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', a vortex traverses the pin smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Crossing a strong pin, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='02254v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='supr-con] 5 Jan 2023 2 however, the vortex undergoes jumps that mathemati- cally originate in bistable distinct vortex configurations, ‘free’ and ‘pinned’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Quantitatively, the onset of strong pinning is given by the Labusch criterion κ = 1, with the Labusch parameter κ ≡ max[−e′′ p]/ ¯C ∼ fp/ξ ¯C, the dimensionless ratio of the negative curvature e′′ p of the isotropic pinning potential and the effective elasticity ¯C of the vortex lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Strong pinning appears for κ > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', when the lattice is soft compared to the curvatures in the pinning landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' So far, the strong pinning transition at κ = 1 has been described for defects with isotropic pinning potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' it can be mapped10 to the magnetic transition in the h-T (field–temperature) space, with the strong-pinning phenomenology at κ > 1 corresponding to the first-order Ising magnetic transition at T < Tc and the critical point at T = Tc corresponding to the strong pinning transition at κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The role of the reduced temperature T/Tc is then assumed by the Labusch parameter κ and the bistabilities associated with the ferromagnetic phases at T/Tc < 1 translate to the bistable pinned and free vor- tex states at κ > 1, with the bistability disappearing on approaching the critical point, T/Tc = 1 and κ = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A first attempt to account for correlations between defects has been done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The latter analysis takes into account the enhanced pinning force excerted by pairs of isotropic defects that can be cast in the form of anisotropic effective pinning centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Besides shifting the onset of strong pinning to κ = 1/2 (with κ defined for one individual defect), the analysis unravelled quite astonishing (geometric) features that appeared as a con- sequence of the symmetry reduction in the pinning po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the present paper, we take a step back and study the transition to strong pinning for anisotropic de- fect potentials ep(R), with R a planar coordinate, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that collective effects of many weak defects can add up to effectively strong pins that smoothen the transition at κ = 1, thereby turning the strong pinning transition into a weak-to-strong pinning crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We find that the onset of strong pinning proceeds quite differently when going from the isotropic defect to the anisotropic potential of a generic defect without spe- cial symmetries and further on to a general random pin- ning landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The simplest comparison is between an isotropic and a uniaxially anisotropic defect, acting on a vortex lattice that is directed along the magnetic field B ∥ ez chosen parallel to the z-axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' for convenience, we place the defect at the origin of our coordinate system r = (R, z) and have it act only in the z = 0-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In this setup, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1, the pinning potential ep(R) acts on the nearest vortex with a force fp(R) = −∇Rep|z=0 at- tracting the vortex to the defect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the presence of the other vortices constituting the lattice renormalizes the vortex elasticity ¯C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With the pinning potential acting in the z = 0 plane, the vortex is deformed with a pronounced cusp at z = 0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we denote the tip position of the vortex where the cusp appears by ˜R, while the asymp- ˜R ¯R defect tip vortex asymptotic x y z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sketch of a vortex interacting with a defect located at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The vortex approaches the asymptotic position ¯R at z → ±∞ and is attracted to the defect residing at the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the cusp at z = 0 defines the tip position ˜R and its angle quantifies the pinning strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' totic position of the vortex at z → ±∞ is fixed at ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With this setup the problem can be reduced to a planar one, with the tip coordinate ˜R and the asymptotic co- ordinate ¯R determining the location and full shape (and hence the pinning force) of the vortex line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the case of an isotropic pin, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', produced by a point-like defect11, strong pinning first appears on a cir- cle of finite radius Rm ∼ ξ, typically of order of the vortex core radius ξ, see left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This is owed to the fact that, given the radial symmetry, the Labusch cri- terion κ = maxR[−e′′ p(R)]/ ¯C = 1 is satisfied on a circle R = Rm where the (negative) curvature −e′′ p > 0 is max- imal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Associated with the radius Rm where the tip is lo- cated at κ = 1, ˜R(κ = 1) ≡ ˜Rm = Rm, there is an asymp- totic vortex position ¯R(κ = 1) = ¯Rm > ˜Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Increasing the Labusch parameter beyond κ = 1, the circle of ra- dius ¯Rm transforms into a ring ¯R− < ¯R < ¯R+ of finite width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices placed inside the ring at small distances ¯R < ¯R− near the defect are qualified as ‘pinned’, while vortices at large distances ¯R > ¯R+ away from the pin are described as ‘free’, see right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' phys- ically, we denote a vortex configuration as ‘free’ when it is smoothly connected to the asymptotic undeformed state, while a ‘pinned’ vortex is localized to a finite region around the defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices placed inside the bistable ring at ¯R− < ¯R < ¯R+ acquire two possible states, pinned and free (colored magenta in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2, the superposition of red (pinned state) and blue (free state) colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The onset of strong pinning for the uniaxially 3 anisotropic defect proceeds in several stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Let us con- sider an illustrative example and assume a defect with an anisotropy aligned with the axes and a steeper po- tential along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In this situation, strong pinning as de- fined by the criterion κm = 1, with a properly gener- alized Labusch parameter κm, appears out of two points (±¯xm, 0) where the Labusch criterion κm = 1 is met first, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2(b) left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Increasing κm > 1 beyond unity, two bistable domains spread around these points and develop two crescent-shaped areas (with their large extent along ¯y) in asymptotic ¯R-space, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2(b) right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices with asymptotic positions within these crescent-shaped regions experience bistability, while outside these regions the vortex state is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Classifying the bistable solu- tions as ‘free’ and ‘pinned’ is not possible, with the sit- uation resembling the one around the gas–liquid critical point with a smooth crossover (from blue to white to red) between phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With κm increasing further, the cusps of the crescents approach one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As the arms of the two crescents touch and merge at a sufficiently large value of κm, the topology of the bistable area changes: the two merged crescents now define a ring-like geometry and separate ¯R-space into an inside region where vortices are pinned, an outside region where vortices are free and the bistable region with pinned and free states inside the ring-like region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As a result, the pinning geometry of the isotropic defect is recovered, though with the perfect ring replaced by a deformed ring with varying width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Us- ing the language describing a thermodynamic first-order transition, the cusps of the crescents correspond to criti- cal points while its boundaries map to spinodal lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the merging of critical points changing the topology of the bistable regions of the pinning landscape goes beyond the standard thermodynamic analogue of phase diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bistable area is defining the trapping area where vortices get pinned to the defect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this trapping area is one of the relevant quantities determining the pinning force density Fpin, the other being the jumps in energy associ- ated with the difference between the bistable states8,10, see the discussion in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II C, II E, and III G below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' It is the change in the bistable- and hence trapping geom- etry that modifies the exponent µ in Fpin ∝ (κ − 1)µ, replacing the exponent µ = 2 for isotropic defects by the new exponent µ = 5/2 for general anisotropic pinning potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While the existence of bistable regions B ¯R in the space of asymptotic vortex positions ¯R is an established ele- ment of strong pinning theory by now, in the present pa- per, we introduce the new concept of unstable domains U ˜R in tip-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The two coordinates ˜R and ¯R represent dual variables in the sense of the thermodynamic analog, with the asymptotic coordinate ¯R corresponding to the driving field h in the Ising model and the tip position ˜R replacing the magnetic response m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' from a thermody- namic perspective it is then quite natural to change view by going back and forth between intensive (h) and exten- sive (m) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In tip space ˜R, the onset of pinning appears at isolated points ˜Rm that grow into ellipses as −3 0 −3 0 −3 0 −3 0 −3 0 −3 0 −3 0 −3 0 −1 0 2 (a) (b) ˜ R/ξ − ˜ Rm( ˜φ)/ξ ¯y/ξ ¯x/ξ ¯y/ξ ¯x/ξ ¯y/ξ ¯x/ξ ¯y/ξ ¯x/ξ ¯Rm ¯R− ¯R+ ¯R0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Illustration of bistable regions in asymptotic ¯R-space for a vortex pinned to a defect located at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) For an isotropic defect (Lorentzian shape with κ = 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5), pinning appears at κ = 1 along a ring with radius ¯Rm, with the red area corresponding to pinned states and free states colored in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With increasing pinning strength κ, see right panel at κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5, a bistable region (in magenta) appears in a ring ge- ometry, with vortices residing inside, ¯R < ¯R−, being pinned and vortices outside, ¯R > ¯R+, remaining free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices with asymptotic positions inside the ring ( ¯R− < ¯R < ¯R+) exhibit bistable states, pinned and free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The dashed circle ¯R0 marks the crossing of pinned and free branches, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b) For a uniaxially anisotropic defect, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (94) with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='3 and largest (negative) curvature along x, pinning appears in two points (±¯xm, 0) along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As the pinning strength increases beyond unity, see right panel, bistable regions (ma- genta) develop in a crescent-shape geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pinned- and free-like states are smoothly connected as indicated by the crossover of colors (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III C for the precise description of coloring in terms of an ‘order parameter’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As κm fur- ther increases, the cusps of the two crescents merge on the y-axis, changing the topology of the ¯R-plane through sepa- ration into inner and outer regions (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A ring-like bistable region appears as in (a), with the inner (outer) region corresponding to unique vortex states that are pinned (free), while vortices residing inside the ring-shaped domain exhibit bistable states, pinned and free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' κ is increased beyond unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These ellipses describe un- stable areas U ˜R in the ˜R-plane across which vortex tips jump when flipping between bistable states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they relate to the bistable crescent-shaped areas B ¯R in asymptotic space through the force balance equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the latter determines the vortex shape with elastic and pinning forces compen- 4 sating one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The unstable regions U ˜R in tip space are actually more directly accessible than the bistable re- gions B ¯R in asymptotic space and play an equally central role in the discussion of the strong pinning landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The simplification introduced by the concept of unsta- ble domains U ˜R in tip space ˜R is particularly evident when going from individual defects as described above to a generic pinning landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, we focus on a model pinning potential landscape (or short pinscape) confined to the two-dimensional (2D) R plane at z = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' such a pin- scape can be produced, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', by defects that reside in the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinned vortex tip ˜R then still resides in the z = 0 plane as well and the strong pinning problem remains two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For a 2D random pinscape, unstable ellipses appear sequentially out of different (iso- lated) points and at different pinning strength κm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' their assembly defines the unstable area U ˜R, with each newly appearing ellipse changing the topology of U ˜R, specif- ically, its number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Increasing κm, the ellipses first grow in size, then deform away from their original elliptical shapes, and finally touch and merge in a hyperbolic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Such mergers change, or more pre- cisely reduce, the number of components in U ˜R and hence correspond again to topological transitions as described by a change in the Euler characteristic χ associated with the shape of U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, these mergers tend to produce U ˜R shapes that are non-simply connected, again implying a topological transition in U ˜R with a change in χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Such non-simply connected parts of U ˜R separate the tip space into ‘inner’ and ‘outer’ regions that allows to define proper ‘pinned’ states (localized near a poten- tial minimum) in the ‘inner’ of U ˜R, while ‘free’ states (smoothly connected to asymptotically undeformed vor- tices) occupy the regions outside of U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The discussion below is dominated by three mathemat- ical tools: for one, it is the Hessian matrix H(R) of the pinning potential17,26 ep(R), its eigenvalues λ±(R) and eigenvectors v±(R), its determinant det[H](R) and trace tr[H](R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The Hessian matrix involves the curvatures Hij = ∂i∂jep(R), i, j ∈ {x, y}, of the pinning potential, that in turn are the quantities determining strong pin- ning, as can be easily conjectured from the form of the Labusch parameter κ ∝ −e′′ p for the isotropic defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The second tool is the Landau-type expansion of the total pin- ning energy near the strong-pinning onset around ˜Rm at κm = 1 (appearance of a critical point) as well as near merging around ˜Rs at κ( ˜Rs) ≡ κs = 1 (disappearance of a pair of critical points);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the standard manipulations as they are known from the description of a thermody- namic first-order phase transition produce most of the new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Third, the topological structure of the un- stable domain U ˜R associated with a generic 2D pinning landscape, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', its components and their connectedness, is conveniently described through its Euler characteristic χ with the help of Morse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The structure of the paper is as follows: In Section II, we briefly introduce the concepts of strong pinning theory with a focus on the isotropic defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The onset of strong pinning by a defect of arbitrary shape is pre- sented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we start with a translation and ex- tension of the strong pinning ideas from the isotropic situation to a general anisotropic one, that leads us to the Hessian analysis of the pinning potential as our ba- sic mathematical tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Close to onset, we find (using a Landau-type expansion, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III A) that the unstable (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III B) and bistable (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III C) domains are asso- ciated with minima of the determinant of the Hessian curvature matrix and assume the shape of an ellipse and a crescent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Due to the anisotropy, the ge- ometry of the trapping region depends non-trivially on the Labusch parameter and the critical exponent for the pinning force is changed from µ = 2 to µ = 5/2, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The analytic solution of the strong pinning onset for a weakly uniaxial defect presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' IV leads us to define new hyperbolic points associated with sad- dle points of the determinant of the Hessian curvature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These hyperbolic points describe the merging of unstable and bistable domains, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V A, and allow us to relate the new results for the anisotropic defect to our established understanding of isotropic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In a final step, we extend the local perspective on the pin- scape, as acquired through the analysis of minima and saddles of the determinant of the Hessian curvature ma- trix, to a global description in terms of the topological characteristics of the unstable domain U ˜R: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' VI, we discuss strong pinning in a two-dimensional pinning potential of arbitrary shape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', as it appears when mul- tiple pinning defects overlap (though all located in one plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We follow the evolution of the unstable domain U ˜R with increasing pinning strength κm and express its topological properties through the Euler characteristic χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the latter is related to the local differential properties of the pinscape’s curvature, its minima, saddles, and max- ima, through Morse theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, in Appendix A, we map the two-dimensional Landau-type theories (involv- ing two order parameters) describing onset and merging, to effective one-dimensional Landau theories and rederive previous results following standard statistical mechanics calculations as they are performed in the analysis of the critical point in the van der Waals gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' STRONG PINNING THEORY We start with a brief introduction to strong pinning theory, keeping a focus on the transition region at mod- erate values of κ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We consider an isotropic defect (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II A) and determine the unstable and bistable ring domains for this situation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We derive the general expression for the pinning force density Fpin in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II C, determine the relevant scales of the strong pin- ning characteristic near the crossover in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II D, and apply the results to derive the scaling Fpin ∝ (κ − 1)2 for the isotropic defect (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II F, we relate the strong pinning theory for the isotropic defect to the Landau mean-field description for the Ising model in a 5 magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Isotropic defect The standard strong-pinning setup involves a vortex lattice directed along z with a lattice constant a0 deter- mined by the induction B = φ0/a2 0 that is interacting with a dilute set of randomly arranged defects of den- sity np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This many-body problem can be reduced10,13,20 to a much simpler effective problem involving an elastic string with effective elasticity ¯C that is pinned by a de- fect potential ep(R) acting in the origin, as described by the energy function epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) = ¯C 2 ( ˜R − ¯R)2 + ep( ˜R) (1) depending on the tip- and asymptotic coordinates ˜R and ¯R of the vortex, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The energy (or Hamilto- nian) epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) of this setup involves an elastic term and the pinning energy ep(R) evaluated at the location ˜R of the vortex tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We denote the depth of the pin- ning potential by ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A specific example is the point- like defect that produces an isotropic pinning potential which is determined by the form of the vortex11 and as- sumes a Lorentzian shape ep(R) = −ep/(1 + R2/2ξ2) with R = |R|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III below, we will consider pin- ning potentials of arbitrary shape ep(R) but assume a small (compared to the coherence length ξ) extension along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ‘Integrating out’ the vortex lattice, the re- maining string or vortex is described by the effective elasticity ¯C ≈ νε(a2 0/λL) � c66c44(0) ∼ εε0/a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, ε0 = (φ0/4πλL)2 is the vortex line energy, λL denotes the London penetration depth, ε < 1 is the anisotropy parameter for a uniaxial material18, and ν is a numerical, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 23 and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The most simple pinning geometry is for a vortex that traverses the defect through its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Given the rota- tional symmetry of the isotropic defect, we choose a vor- tex that impacts the defect in a head-on collision from the left with asymptotic coordinate ¯R = (¯x, 0) and increase ¯x along the x-axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' finite impact parameters ¯y ̸= 0 will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The geometry then simplifies consid- erably and involves the asymptotic vortex position ¯x and the tip position ˜x of the vortex, reducing the problem to a one-dimensional one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the full geometry of the deformed string can be determined straightforwardly20 once the tip position ˜x has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The latter follows from mini- mizing (1) with respect to ˜x at fixed asymptotic position ¯x and leads to the non-linear equation ¯C(˜x − ¯x) = −∂xep|x=˜x = fp(˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (2) This can be solved graphically, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3, and produces either a single solution or multiple solutions—the appear- ance of multiple tip solutions is the signature of strong pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The relevant parameter that distinguishes the ¯C(˜x − ¯x+) ¯C(˜x − ¯x−) ˜xp− ˜xp+ ˜xf− ˜xf+ ˜x fp(˜x) 0 ξ ¯x− ¯x+ 0 ˜xm ¯xm ˜x fp(˜x) κ < 1 ¯C(˜x − ¯xm) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Graphical illustration13 of the self-consistent solu- tion of the microscopic force-balance equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (2) for a Lorentzian potential with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The vortex coordinates ˜x and ¯x are expressed in units of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When moving the asymp- totic vortex position ¯x across the bistable interval [¯x−, ¯x+], we obtain three solutions describing pinned ˜xp ≲ ξ, free ˜xf close to ¯x, and unstable ˜xus states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they define the corre- sponding pinned (red), free (blue), and unstable (black dot- ted) branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The tip-positions at the edges of the bistable interval denoted by ˜xp+ and ˜xf− denote jump points where the vortex tip turns unstable, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they are defined by the condition f ′ p(˜xp+) = f ′ p(˜xf−) = ¯C (black solid dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The associated positions ˜xf+ and ˜xp− denote the tip landing points after the jump (open circles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they are given by the second solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (2) at the same asymptotic position ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The open red/blue circles and the cross mark the positions of metastable minima and the unstable maximum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The lower right inset shows the weak-pinning situation at κ < 1, here implemented with a larger ¯C, where the tip solution ˜x is unique for all ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' two cases is found by taking the derivative of (2) with respect to ¯x that leads to ∂¯x˜x = 1 1 − f ′p(˜x)/ ¯C , (3) where prime denotes the derivative, f ′ p(x) = ∂xfp(x) = −∂2 xep(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Strong pinning involves vortex instabilities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', jumps in the tip coordinate ˜x, that appear when the denominator in (3) vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this leads us to the strong pinning parameter κ first introduced by Labusch8, κ = max ˜x f ′ p(˜x) ¯C = f ′ p(˜xm) ¯C , (4) with ˜xm defined as the position of maximal force deriva- tive f ′ p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', f ′′ p (˜xm) = 0, or maximal negative curva- ture −e′′ p of the defect potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Defining the force scale fp ≡ ep/ξ and estimating the force derivative or curva- ture f ′ p = −e′′ p ∼ fp/ξ produces a Labusch parameter κ ∼ ep/ ¯Cξ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' for the Lorentzian potential, we find that f ′ p(˜xm) = ep/4ξ2 at ˜xm = √ 2 ξ and hence κ = ep/4 ¯Cξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We see that strong pinning is realized for either large pinning energy ep or small effective elasticity ¯C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6 As follows from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3 (inset), for κ < 1 (large ¯C) the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (2) is unique for all values of ¯x and pinning is weak, while for κ > 1 (small ¯C), multiple solutions appear in the vicinity of ˜xm and pinning is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These multiple solutions appear in a finite interval ¯x ∈ [¯x−, ¯x+] and we denote them by ˜x = ˜xf, ˜xp, ˜xus, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they are associated with free (weakly deformed vortex with ˜xf close to ¯x), pinned (strongly deformed vortex with ˜xp < ξ), and unstable vortex states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inserting the solutions ˜x(¯x) = ˜xf(¯x), ˜xp(¯x), ˜xus(¯x) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (2) at a given vortex position ¯x back into the pinning energy epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x), we find the energies of the correspond- ing branches, ei pin(¯x) ≡ epin[˜xi(¯x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x], i = f, p, us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (5) The pair ep(˜x) and ei pin(¯x) of energies in tip- and asymp- totic spaces then has its correspondence in the force: as- sociated with fp(˜x) in tip space are the force branches f i pin(¯x) in asymptotic ¯x-space defined as f i pin(¯x) = fp[˜xi(¯x)], i = f, p, us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (6) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (2), it turns out that the force fpin can be written as the total derivative of epin, fpin(¯x) = −depin[˜x(¯x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x] d¯x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (7) The multiple branches ei pin and f i pin associated with a strong pinning situation at κ > 1 are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4 and 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Unstable and bistable domains U ˜ R and B ¯ R Next, we identify the unstable (in ˜x) and bistable (in ¯x) domains of the pinning landscape that appear as sig- natures of strong pinning when κ increases beyond unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Figure 5(a) shows the force profile fp(˜x) as experienced by the tip coordinate ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A vortex passing the defect on a head-on trajectory from left to right undergoes a forward jump in the tip from −˜xf− to −˜xp−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' subsequently, the tip follows the pinned branch until ˜xp+ and then returns back to the free state with a forward jump from ˜xp+ to ˜xf+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The jump positions (later indexed by a subscript ‘jp’) are determined by the two solutions of the equation f ′ p(x) ��� −˜xf−,˜xp+ = ¯C (8) that involves the curvature of the pinning potential ep(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the landing positions −˜xp− and ˜xf+ (later indexed by a subscript ‘lp’), on the other hand, are given by the second solution of the force-balance equation (2) that involves the driving term ¯C(˜x − ¯x) and hence depends on the asymptotic position ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, the positions in asymp- totic space ¯x where the vortex tip jumps are obtained again from the force balance equation (2), ¯x− = ˜xf− − fp(˜xf−)/ ¯C, (9) ¯x+ = ˜xp+ − fp(˜xp+)/ ¯C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' epin ¯x epin 0 −¯x0 ∆efp pin −¯x− ξ ∆epf pin ¯x+ ˜x ˜xp ˜xus ˜xf FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Multi-valued pinning energy landscape ei pin(¯x) for a defect producing a Lorentzian-shaped potential with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the branches i = p, f, us correspond to the pinned (red), free (blue), and unstable (black dotted) vortex states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bista- bility extends over the intervals |¯x| ∈ [¯x−, ¯x+] where the dif- ferent branches coexist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' pinned and free vortex branches cut at the branch crossing point ¯x = ¯x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A vortex traversing the defect from left to right assumes the free and pinned states marked with thick colored lines and undergoes jumps ∆efp pin and ∆epf pin in energy (vertical black solid lines) at the bound- aries −¯x− and ¯x+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The asymmetric occupation of states pro- duces a finite pinning force density Fpin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inset: Total energy epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) versus vortex tip position ˜x for a fixed vortex po- sition ¯x (vertical dashed line in the main figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The points ˜xf, ˜xp, and ˜xus mark the free, pinned, and unstable solutions of the force-balance equation (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they correspond to local minima and the maximum in epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) and are marked with corresponding symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that the two pairs of tip jump and landing posi- tions, ˜xp+, ˜xf+ and ˜xf−, ˜xp− are associated with only two asymptotic positions ¯x+ and ¯x−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Let us generalize the geometry and consider a vortex moving parallel to ¯x, impacting the defect at a finite dis- tance ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We then have to extend the above discussion to the entire z = 0 plane, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For an isotropic de- fect, the jump- and landing points now define jump cir- cles with radii ˜Rjp given by ˜Rf− = ˜xf− and ˜Rp+ = ˜xp+ (solid circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c)) and landing circles with radii ˜Rlp given by ˜Rf+ = ˜xf+, ˜Rp− = ˜xp− (dashed circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Their combination defines an unstable ring ˜Rp+ < ˜R < ˜Rf− in tip space where tips cannot reside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The existence of unstable domains U ˜R in tip space is a signature of strong pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Figures 5(b) and (d) show the corresponding results in asymptotic coordinates ¯x and ¯R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinning force fpin(¯x) = fp[˜x(¯x)] shown in (b) is simply an ‘outward tilted’ version of fp(˜x), with S-shaped over- hangs that generate bistable intervals [−¯x+, −¯x−] and [¯x−, ¯x+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Extending them to the asymptotic ¯R-plane with radii ¯R− ≡ ¯x− and ¯R+ ≡ ¯x+, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(d), we obtain a ring ¯R− < ¯R < ¯R+ that marks the location of bistability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Again, the appearance of bistable domains 7 (a) (c) (b) (d) ˜x ˜xp+ −˜xp− ξ ˜xf+ −˜xf− ˜x fp ˜y ¯x ¯x+ −¯x− ¯x ξ ξ fpin ¯y ¯R− ¯R+ ¯R0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) and (b): Force profiles fp(˜x) and fpin(¯x) in tip- and asymptotic coordinates for a Lorentzian-shaped poten- tial with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The tip of a vortex moving from left to right along the x-axis approaches the defect on the free branch (thick blue line) undergoes a jump (arrow) from −˜xf− to −˜xp−, follows the pinned branch (red) until ˜xp+ and then jumps back (arrow) to the free (blue) state at ˜xf+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Extend- ing these jump positions to the (˜x, ˜y)-plane, see (c), defines jump (solid) and landing (dashed) circles, with the jump cir- cles enclosing an unstable domain U ˜ R characteristic of strong pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The force profile fpin(¯x) in (b) includes free (blue), pinned (red), and unstable branches (black dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (d) Ex- tending the bistable intervals [−¯x+, −¯x−] and [¯x−, ¯x+] to the [¯x, ¯y]-plane defines a bistable ring B ¯ R (magenta), again a strong pinning characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The dashed circle of radius ¯R0 in (d) marks the branch crossing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices passing the defect with a finite impact parameter ¯y ̸= 0 move on a straight line in asymptotic space, see (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the associated trajectory in tip space is nontrivial, see (c) and undergoes jumps at pinning (circle ˜Rf−) and depinning (circle ˜Rp+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B ¯R in asymptotic space is a signature of strong pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Both, the size of the unstable- and bistable rings depend on the Labusch parameter κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they appear out of circles with radii ˜R = ˜xm and ¯R = ¯xm = ˜xm − fp(˜xm)/ ¯C at κ = 1 and grow in radius and width when κ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The unstable and bistable domains U ˜R and B ¯R (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 27) will exhibit interesting non-trivial behavior as a func- tion of κ when generalizing the analysis to defect poten- tials of arbitrary shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Alternative strong pinning formulation An alternative formulation of strong pinning physics is centered on the local differential properties of the pinning energy epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', its extremal points in ˜x at different values of the asymptotic coordinate ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We start from equation (1) restricted to one dimension and rearrange terms to arrive at the expression epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) = eeff(˜x) − ¯C¯x ˜x + ¯C¯x2/2 (10) with the effective pinning energy eeff(˜x) = ep(˜x) + ¯C˜x2/2 (11) involving both pinning and elastic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Equation (10) describes a particle at position ˜x subject to the potential eeff(˜x) and the force term f ˜x = − ¯C¯x ˜x, see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The potential eeff(˜x) can trap two particle states if there is a protecting maximum with negative curva- ture ∂2 ˜xeeff = ∂2 ˜xepin < 0, preventing its escape from the metastable state at forces f = ± ¯C¯x with ¯x ∈ [¯x+, ¯x−];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the maximum in epin at ˜xus then separates two minima in epin defining distinct branches with different tip coor- dinates ˜xp and ˜xf, see the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As the asymptotic position ¯x approaches the bound- aries ¯x±, one of the minima joins up with the maximum to define an inflection point with [∂2 ˜xeeff]˜xjp = [∂2 ˜xepin]˜xjp = 0, (12) that corresponds to the instability condition (8) where the vortex tip jumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the persistent second minimum in epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) defines the landing position ˜xlp and the condi- tion for a flat inflection point [∂˜xepin]˜xjp = 0 defines the associated asymptotic coordinate ±¯x±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, strong pinning vanishes at the Labusch point κ = 1, with the inflection point in eeff(˜x) coalescing with the second minimum at ˜xm, hence [∂2 ˜xeeff]˜xm = 0 and (13) [∂3 ˜xeeff]˜xm = [∂3 ˜xep]˜xm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note the subtle use of epin versus eeff versus ep in the above discussion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' as we go to higher derivatives, first the asymptotic coordinate ¯x turns irrelevant in the sec- ond derivative ∂2 ˜xepin = ∂2 ˜xeeff and then all of the elas- tic response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', ¯C, drops out in the third derivative [∂3 ˜xepin] = [∂3 ˜xep].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The above alternative formulation of strong pinning turns out helpful in several discussions below, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the derivation of strong pinning characteristics near the tran- sition in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II D and III A and in the generalization of the instability condition to an anisotropic defect in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III and furthermore provides an inspiring link to the Lan- dau theory of phase transitions discussed below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pinning force density Fpin Next, we determine the pinning force density Fpin at strong pinning, assuming a random homogeneous distri- bution of pins with a small density np, npa0ξ2 ≪ 1, see 8 Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The derivation of Fpin is conveniently done in asymptotic ¯R coordinates where vortex trajec- tories follow simple straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices approach the pin by following the free branch until its termination, jump to the pinned branch to again follow this to its termination, and finally jump back to the free branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This produces an asymmetric pinned-branch occupation pc( ¯R) that leads to the pinning force density (we assume vortices approaching the defect along ¯x from the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' fol- lowing convention, we include a minus sign) Fc = −np � d2 ¯R a2 0 � pc( ¯R)f p pin( ¯R) + (1 − pc( ¯R))f f pin( ¯R) � = −np � d2 ¯R a2 0 pc( ¯R)[∂x∆efp pin( ¯R)] e¯x, (14) with the energy difference ∆efp pin( ¯R) = ef pin( ¯R) − ep pin( ¯R) and e¯x the unit vector along ¯x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the ¯y-component of the pinning force density vanishes due to the antisymmetry in fpin,¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For the isotropic defect, the jumps ∆efp pin( ¯R) in energy appearing upon changing branches are inde- pendent of angle and the average in (14) separates in ¯x and ¯y coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' note that the energy jumps are no longer constant for an anisotropic defect and hence such a separation does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, i) all vor- tices approaching the defect within the transverse length |¯y| < ¯R− get pinned, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' while those passing further away follow a smooth (weak pinning) trajectory that does not undergo jumps and hence do not contribute to the pinning force,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' and ii) all vortices that get pinned contribute the same force that is most easily evaluated for a head-on vortex–defect collision on the ¯x-axis with pc(¯x) = Θ(¯x + ¯x−) − Θ(¯x − ¯x+) and ⟨fpin⟩ = − � a0/2 −a0/2 d¯x a0 � pc(¯x)f p pin(¯x) + (1 − pc(¯x))f f pin(¯x) � = ∆efp pin(−¯x−) + ∆epf pin(¯x+) a0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (15) where we have replaced −∆efp pin(¯x+) by ∆epf pin(¯x+) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Hence, the average pinning force ⟨fpin⟩ is given by the jumps in the pinning energy ei pin(¯x) associated with dif- ferent branches i = p, f, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, accounting for trajectories with finite impact parameter |¯y| < ¯R−, we arrive at the result for the pin- ning force density Fpin acting on the vortex system, Fpin = np 2 ¯R− a0 ⟨fpin⟩ = np 2 ¯R− a0 ∆efp pin + ∆epf pin a0 , (16) where the factor 2 ¯R−/a0 accounts for the averaging of the pinning force along the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As strong pins act independently, a consequence of the small defect density np, the pinning force density is linear in the defect den- sity, Fpin ∝ np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' If pinning is weak, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', κ < 1, we have no jumps, ⟨fpin⟩ = 0, and Fpin|strong = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A finite pinning force then only arises from correlations between pinning defects and scales in density as9,10 Fpin|weak ∝ n2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This contribution to the pinning force density Fpin continues beyond κ = 1, hence, while the strong pinning onset at κ = 1 can be formulated in terms of a transition, weak pinning goes to strong pinning in a smooth crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Knowing the pinning force density Fpin, the motion of the vortex lattice follows from the bulk dynamical equa- tion ηv = FL(j) − Fpin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (17) Here, η = BHc2/ρnc2 is the Bardeen-Stephen viscosity28 (per unit volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ρn is the normal state resistivity) and FL = j × B/c is the Lorentz force density driving the vortex system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinning force density Fpin is directed along v, in our case along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Next, we determine the strong pinning characteristics ¯x−, ¯x+, ˜xf±, ˜xp±, ∆efp pin and ∆epf pin as a function of the Labusch parameter κ close to the strong pinning transi- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', κ ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Strong pinning characteristics near the transition Near the strong pinning transition at κ ≳ 1, we can derive quantitative results for the strong pinning char- acteristics by expanding the pinning energy epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) in ˜x at fixed ¯x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this reminds about the Landau expansion of the free energy f(φ, h) in the order parameter φ at a fixed field h in a thermodynamic transition, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II F below for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We expand epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) in ˜x around the point of first instability ˜xm by introducing the relative tip and asymp- totic positions ˜u = ˜x − ˜xm and ¯u = ¯x − ¯xm and make use of our alternative strong pinning formulation sum- marized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At ˜xm and close to κ = 1, we have [∂2 ˜xepin]˜xm = [∂2 ˜xep]˜xm + ¯C = ¯C(1−κ) and [∂3 ˜xepin]˜xm = 0, hence, epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) ≈ ¯C 2 (1 − κ) ˜u2 + γ 24 ˜u4 − ¯C¯u˜u, (18) where we have introduced the shape parameter γ = [∂4 xep]˜xm describing the quartic term in the expansion and we have made use of the force balance equation (2) to rewrite fp(˜xm) = ¯C(˜xm − ¯xm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' furthermore, we have dropped all irrelevant terms that do not depend on ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We find the jump and landing positions ˜xjp and ˜xlp ex- ploiting the differential properties of epin(˜x) at a fixed ¯x: As discussed above, the vortex tip jumps at the bound- aries ¯x± of the bistable regime, where epin develops a flat inflection point at ˜xjp with one minimum joining up with the unstable maximum and the second minimum at the landing position ˜xlp staying isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Within our fourth- order expansion the jump positions at (de)pinning are placed symmetrically with respect to the onset at ˜xm, ˜xp+ = ˜xm + ˜ujp, ˜xf− = ˜xm − ˜ujp (19) 9 and imposing the condition [∂2 ˜uepin]˜xjp = 0 (that is equiv- alent to the jump condition f ′ p[˜xf−] = f ′ p[˜xp+] = ¯C of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (8), see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3), we find that ˜ujp ≈ − � 2 ¯C γ (κ − 1)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (20) In order to find the (symmetric) landing positions, it is convenient to shift the origin of the expansion to the jump position, ˜u → ˜u − ˜ujp ≡ ˜u′, and define the jump distance ∆˜u, ˜xf+ = ˜xp+ + ∆˜u, ˜xp− = ˜xf− − ∆˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (21) At the jump position, the linear and quadratic terms in ˜u′ vanish, resulting in the expansion (up to an irrelevant constant) epin(˜xp+ + ˜u′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x+) ≈ γ 6 ˜ujp˜u′ 3 + γ 24 ˜u′ 4 (22) and similar at ˜xf− and ¯x− for a left moving vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This expression is minimal at the landing position ˜xf+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', at ˜u′ = ∆˜u, [∂˜u′epin]∆˜u = 0, and we find the jump distance ∆˜u = −3˜ujp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (23) Inserting this result back into (22), we obtain the jump in energy ∆epf pin = epin(˜xp+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x+) − epin(˜xf+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x+), ∆epf pin(¯x+) ≈ γ 72(∆˜u)4 ≈ 9 ¯C2 2γ (κ − 1)2, (24) and similar at ¯x−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that all these results have been obtained without explicit knowledge of the asymptotic coordinates ¯x± where these tip jumps are triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The latter follow from the force equation (2) that corresponds to the condition [∂˜xepin]˜xjp = 0 for a flat inflection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Using the expansion (18) of the pinning energy, we find that ¯x± − ¯xm = ∓2 3 ˜ujp(κ − 1) = ±2 3 � 2 ¯C γ (κ − 1)3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (25) The pair ¯xm and ˜xm of asymptotic and tip positions depends on the details of the potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' while ˜xm derives solely from the shape ep(˜x), ¯xm as given by (2) involves ¯C and shifts ∝ (κ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For a Lorentzian potential, we find that ˜xm = √ 2ξ, ¯xm = 2 √ 2ξ + √ 2ξ(κ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (26) The shape coefficient is γ = 3ep/4ξ4 and the Labusch parameter is given by κ = ep/4 ¯Cξ2 (hence ¯C2/γ = ep/12κ2), providing us with the results ˜ujp ≈ −ξ [2(κ−1)/3]1/2 and ∆epf pin ≈ 3 8ep(κ−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (27) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pinning force density for the isotropic defect Using the results of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II D in the expression (16) for the pinning force density, we find, to leading order in κ − 1, Fpin = 9np ¯xm a0 ¯C2 γa0 (κ − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (28) The scaling Fpin ∼ np(ξ/a0)2fp(κ − 1)2 (with ¯Cξ2/ep ∼ 1/κ, up to a numerical) uniquely derives from the scaling ∝ (κ − 1)2 of the energy jumps in (24), as the asymp- totic trapping length ¯x− ∼ ξ remains finite as κ → 1 for the isotropic defect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this will change for the anisotropic defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Relation to Landau’s theory of phase transitions The expansion (18) of the pinning energy epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) around the inflection point ˜xm of the force takes the same form as the Landau free energy of a phase transition10, f(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' h) = r0 2 (T/Tc − 1)φ2 + uφ4 − hφ, (29) with the straightforward transcription ˜u ↔ φ, ¯C(1−κ) ↔ r0(T/Tc − 1), γ/24 ↔ u and the conjugate field ¯C¯u ↔ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The functional (29) describes a one-component oder pa- rameter φ driven by h, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', an Ising model with magne- tization density φ in an external magnetic field h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This model develops a mean-field transition with a first-order line in the h–T phase diagram that terminates in a criti- cal point at T = Tc and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The translation to strong pinning describes a strong pinning region at large κ that terminates (upon decreasing κ) at κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The ferromag- netic phases with φ = ± � r0(1 − T/Tc)/4u correspond to pinned and unpinned states, the paramagnetic phase at T > Tc with φ = 0 translates to the unpinned domain at κ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The spinodals associated with the hysteresis in the first-order magnetic transition correspond to the ter- mination of the free and pinned branches at ¯x±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' indeed, the flat inflection points appearing in epin(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯x) at the boundaries of the bistable region B ¯R as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II B correspond to the disappearance of metastable mag- netic phases in (29) at the spinodals of the first-order transition where ∂φf(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' h) = ∂2 φf(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When in- cluding correlations between defects, the unpinned phase at κ < 1 transforms into a weakly pinned phase that continues beyond κ = 1 into the strongly pinned phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Including such correlations, the strong-pinning transition at the onset of strong pinning at κ = 1 transforms into a weak-to-strong pinning crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ANISOTROPIC DEFECTS Let us generalize the above analysis to make it fit for the ensuing discussion of an arbitrary pinning landscape 10 or short, pinscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Central to the discussion are the unstable and bistable domains U ˜R and B ¯R in tip- and asymptotic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The boundary of the unstable domain U ˜R in tip space is determined by the jump positions of the vortex tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The latter follows from the local differential properties of epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) at fixed asymptotic coordinate ¯R, for the isotropic defect, the appearence of an inflection point [∂2 ˜xepin(˜x, ¯x)] = 0, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In generalizing this condition to the anisotropic situation, we have to study the Hessian matrix of epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (1), � Hess � epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R)| ¯R �� ij = ¯Cδij + Hij( ˜R) (30) with Hij( ˜R) = ∂˜xi∂˜xjep( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) (31) the Hessian matrix associated with the defect potential ep( ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The vortex tip jumps when the pinning landscape epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) at fixed ¯R opens up in an unstable direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', develops an inflection point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this happens when the lower eigenvalue λ−( ˜R) < 0 of the Hessian matrix Hij( ˜R) matches up with ¯C, λ−( ˜R) + ¯C = 0, (32) and strong pinning appears in the location where this happens first, say in the point ˜Rm, implying that the eigenvalue λ−( ˜R) has a minimum at ˜Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, the eigenvector v−( ˜Rm) associated with the eigenvalue λ−( ˜Rm) provides the unstable direction in the pinscape epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) along which the vortex tip escapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Defining the reduced curvature function κ( ˜R) ≡ −λ−( ˜R) ¯C , (33) we find the generalized Labusch parameter κm ≡ κ( ˜Rm), (34) and the Labusch criterion takes the form κm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (35) The latter has to be read as a double condition: i) find the location ˜Rm where the smaller eigenvalue λ−( ˜R) is negative and largest, from which ii), one obtains the crit- ical elasticity ¯C where strong pinning sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A useful variant of the strong pinning condition (32) is provided by the representation of the determinant of the Hessian matrix, D( ˜R) ≡ det � Hess � epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R)| ¯R �� , (36) in terms of its eigenvalues λ±( ˜R), D( ˜R) = [ ¯C + λ−( ˜R)][ ¯C + λ+( ˜R)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' near onset, the second factor ¯C + λ+( ˜R) stays positive and the strong pinning onset ap- pears in the point ˜Rm where D( ˜R) has a minimum which touches zero for the first time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the two conditions ∇D( ˜R)| ˜Rm = 0 and D( ˜Rm) = 0 are satisfied simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The latter conditions make sure that the minima of λ−( ˜R) and D( ˜R) line up at ˜Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that the Hes- sian determinant D( ˜R) does not depend on the asymp- totic coordinate ¯R as it involves only second derivatives of epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The Labusch criterion defines the situation where jumps of vortex tips appear for the first time in the iso- lated point ˜Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Increasing the pinning strength, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', by decreasing the elasticity ¯C for a fixed pinning poten- tial ep(R) (alternatively, the pinning scale ep could be increased at fixed ¯C) the condition (32) is satisfied on the boundary of a finite domain and we can define the unstable domain U ˜R through (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 27) U ˜R = � ˜R | λ−( ˜R) + ¯C ≤ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (37) Once the latter has been determined, the bistable do- main B ¯R follows straightforwardly from the force balance equation ¯C( ˜R − ¯R) = fp( ˜R) = fpin( ¯R), (38) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=',27 B ¯R = � ¯R = ˜R − fp( ˜R)/ ¯C | ˜R ∈ U ˜R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (39) In a last step, one then evaluates the energy jumps ap- pearing at the boundary of B ¯R and proper averaging pro- duces the pinning force density Fpin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Let us apply the above generalized formulation to the isotropic situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Choosing cylindrical coordinates (r, ϕ), the Hessian matrix Hij is already diagonal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' close to the inflection point ˜Rm, where e′′′ p ( ˜Rm) = 0, the eigenval- ues are λ−( ˜R) = e′′ p( ˜R) < 0 and λ+( ˜R) = e′ p( ˜R)/ ˜R > 0, producing results in line with our discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Expansion near strong pinning onset With our focus on the strong pinning transition near κ( ˜Rm) = 1, we can obtain quantitative results using the expansion of the pinning energy epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (1), close to ˜Rm, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Hence, we construct the Landau-type pinning energy corresponding to (29) for the case of an anisotropic pinning potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', we generalize (18) to the two-dimensional situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When generalizing the strong pinning problem to the anisotropic situation, we are free to define local coor- dinate systems (˜u, ˜v) and (¯u, ¯v) in tip- and asymptotic space centered at ˜Rm and ¯Rm, where the latter is asso- ciated with ˜Rm through the force balance equation (38) in the original laboratory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, we fix our axes such that the unstable direction coincides with the u-axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the eigenvector v−( ˜Rm) associated with λ−( ˜Rm) points along u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' as a result, the mixed term ∝ ˜u˜v is absent from the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Keeping all potentially 11 relevant terms up to fourth order in ˜u and ˜v in the ex- pansion, we then have to deal with an expression of the form epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) = ¯C + λ− 2 ˜u2 + ¯C + λ+ 2 ˜v2 − ¯C ¯u˜u − ¯C ¯v˜v + a 2 ˜u˜v2 + a′ 2 ˜u2˜v + b′ 6 ˜u3 + b′′ 6 ˜v3 (40) + α 4 ˜u2˜v2 + β 6 ˜u3˜v + β′′ 6 ˜u˜v3 + γ 24 ˜u4 + γ′′ 24 ˜v4, with λ± = λ±( ˜Rm), ˜R = ˜Rm + δ ˜R, δ ˜R = (˜u, ˜v), (41) ¯R = ¯Rm + δ ¯R, δ ¯R = (¯u, ¯v), and coefficients given by the corresponding derivatives of ep(R), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', a ≡ ∂u∂2 vep(R)| ˜Rm, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' , γ′′ ≡ ∂4 vep(R)| ˜Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As we are going to see, the primed terms in this expan- sion vanish due to the condition of a minimal Hessian determinant at the onset of strong pinning, while double- primed terms will turn out irrelevant to leading order in the small distortions ˜u and ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The first term in (40) drives the strong pinning tran- sition as it changes sign when λ− = − ¯C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Making use of the Labusch parameter κm defined in (34), we can replace (see also (18)) ¯C + λ− → ¯C(1 − κm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (42) In our further considerations below, the quantity κm − 1 ≪ 1 acts as the small parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' it assumes the role of the distance 1 − T/Tc to the critical point in the Landau expansion of a thermodynamic phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The second term in (40) stabilizes the theory along the v direction as ¯C +λ+ > 0 close to the Labusch point, while the sign of the cubic term a ˜u˜v2/2 determines the direction of the instability along x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', to the right (a > 0) or left (a < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The quartic terms ∝ α, γ > 0 bound the pinning energy at large distances, while the term ∝ β determines the skew angle in the shape of the unstable domain U ˜R, see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, we have used the force balance equation (38) in the derivation of the driving terms ¯C ¯u˜u and ¯C ¯v˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The parameters in (40) are constrained by the require- ment of a minimal determinant D( ˜R) at the strong pin- ning onset ˜R = ˜Rm and κm = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', its gradient has to vanish, ∇ ˜R D( ˜R) �� ˜Rm = 0, (43) and its Hessian Hess[D( ˜R)] has to satisfy the relations det � Hess � D( ˜R) ���� ˜Rm > 0, (44) tr � Hess � D( ˜R) ���� ˜Rm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (45) Making use of the expansion (40), the determinant D( ˜R) reads D( ˜R) = � [∂2 ˜uepin][∂2 ˜vepin] − [∂˜u∂˜vepin]2� ˜R (46) with ∂2 ˜uepin = ¯C (1−κm) + a′˜v + b′˜u + α˜v2/2 + β˜u˜v + γ˜u2/2, ∂2 ˜vepin = ¯C + λ+ + a˜u + b′′˜v + α˜u2/2 + β′′˜u˜v + γ′′˜v2/2, ∂˜u∂˜vepin = a˜v + a′˜u + α˜u˜v + β˜u2/2 + β′′˜v2/2, and produces the gradient ∇ ˜R D( ˜R) ��� ˜Rm = ( ¯C + λ+)(b′, a′), (47) hence the primed parameters indeed vanish, a′ = 0 and b′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The Hessian then takes the form Hess � D( ˜R) ���� ˜Rm = ( ¯C + λ+) � γ β β δ � (48) at the Labusch point κm = 1, where we have introduced the parameter δ ≡ α − 2a2 ¯C 1 1 + λ+/ ¯C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (49) The stability conditions (44) and (45) translate, respec- tively, to γδ − β2 > 0 (50) (implying δ > 0) and γ + δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (51) The Landau-type theory (40) involves the two ‘order parameters’ ˜u and ˜v and is driven by the dual coordinates ¯u and ¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This n = 2 theory involves a soft order param- eter ˜u and the stiff ˜v, allowing us to integrate out ˜v and reformulate the problem as an effective one-dimensional Landau theory (A6) of the van der Waals kind—the way of solving the strong pinning problem near onset in this 1D formulation is presented in Appendix A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Unstable domain U ˜ R Next, we determine the unstable domain U ˜R in tip space as defined in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We will find that, up to quadratic order, the boundary of U ˜R has the shape of an ellipse with the semiaxes lengths scaling as √κm − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Jump line J ˜ R We find the unstable domain U ˜R by determining its boundary ∂U ˜R that is given by the set of jump positions ˜Rjp making up the jump line J ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The boundary ∂U ˜R is determined by the condition ¯C +λ− = 0 or, equivalently, the vanishing of the determinant D( ˜Rjp) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (52) 12 The latter condition guarantees the existence of an un- stable direction parallel to the eigenvector v−( ˜Rjp) as- sociated with the eigenvalue λ−( ˜Rjp) where the energy (40) turns flat, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' our discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The edges of the unstable domain U ˜R therefore correspond to a line of inflection points in epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) along which one of the bistable tip configurations of the force balance equation (38) coalesces with the unstable solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Near the onset of strong pinning, the unstable domain U ˜R is closely con- fined around the point ˜Rm where v−( ˜Rm) ∥ ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The un- stable direction v−( ˜Rjp) is therefore approximately ho- mogeneous within the unstable domain U ˜R and is parallel to the u axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This fact will be of importance later, when determining the topological properties of the unstable domain U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inspection of the condition (52) with D( ˜R) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (46) shows that the components of δ ˜Rjp scale as √κm − 1: in the product [∂2 ˜uepin][∂2 ˜vepin], the first fac- tor involves the small constant ¯C(1 − κm) plus quadratic terms (as a′ = 0 and b′ = 0), while the second factor comes with the large constant ¯C + λ+ plus corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The leading term in [∂˜u∂˜vepin] is linear in ˜v with the re- maining terms providing corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' To leading order, the condition of vanishing determinant then produces the quadratic form [γ ˜u2 + 2β ˜u˜v + δ ˜v2] ˜Rjp = 2 ¯C (κm − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (53) With γ and δ positive, this form is associated with an elliptic geometry of extent ∝ √κm − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For later conve- nience, we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (53) in matrix form δ ˜RT jpMjp δ ˜Rjp = ¯C(κm − 1) (54) with Mjp = � γ/2 β/2 β/2 δ/2 � (55) and det Mjp = (γδ − β2)/4 > 0, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The jump line J ˜R can be expressed in the parametric form ˜ujp(|˜v| < ˜vc) = − 1 γ � β˜v ± � 2γ ¯C(κm − 1) − (γδ − β2)˜v2 � , (56) with ˜vc = � 2γ ¯C(κm − 1)/(γδ − β2) (57) and is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6 for the example of an anisotropic potential inspired by the uniaxial defect in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' IV with 10 % anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The associated unstable domain U ˜R as- sumes a compact elliptic shape, with the parameter β de- scribing the ellipse’s skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Comparing with the isotropic defect, this ellipse assumes the role of the ring bounded by solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c), see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III E for a discussion of its different topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' −2 −1 0 1 −2 0 ˜u/ξ√κm − 1 ˜v/ξ√κm − 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Jump line J ˜ R (solid red/blue, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (54)) and landing line (dashed red/blue, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (63)) L ˜ R in tip space ˜R (in units of ξ), with the ellipse J ˜ R representing the edge ∂U ˜ R of the unstable domain U ˜ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We choose parameters κm − 1 = 10−2, with λ− = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='25 ep/ξ2, λ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='05 ep/ξ2, and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='07 ep/ξ3, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 ep/ξ4, β = 0, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='75 ep/ξ4 inspired by the choice of the uniaxial defect with 10 % anisotropy in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' IV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the dotted ellipse shows the effect of a finite skew parameter β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='05 ep/ξ4 on the jump ellipse J ˜ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Along the edges of U ˜ R, one of the stable tip configurations coalesces with the unstable solution of (38) and the total pinning energy epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) develops an inflection line in the tip coordinate ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Crosses correspond to the contact points (67) between the two ellipses J ˜ R and L ˜ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blue and red colors identify dif- ferent types of vortex deformations upon jump and landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pairs of solid and open circles connected via long arrows are, respectively, examples of pairs of jumping- and landing tip po- sitions for vortices approaching the defect from the left (top) and right (bottom), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c) for the isotropic problem’s counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The unstable direction v−( ˜Rjp), shown as short black arrows for different points on the ellipse, always points in the u−direction and are parallel to the tangent vector of the unstable ellipse at the contact points (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' An additional result of the above discussion concerns the terms that we need to keep in the expansion of the pinning energy (40): indeed, dropping corrections amounts to dropping terms with double-primed coeffi- cients and we find that the simplified expansion epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) = ¯C 2 (1 − κm) ˜u2 + ¯C + λ+ 2 ˜v2 + a 2 ˜u˜v2 + α 4 ˜u2˜v2 + β 6 ˜u3˜v + γ 24 ˜u4 − ¯C ¯u˜u − ¯C ¯v˜v (58) produces all of our desired results to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Landing line L ˜ R We find the landing positions ˜Rlp by extending the discussion of the isotropic situation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II D to two dimensions: we shift the origin of the expansion (58) to the jump point ˜Rjp and find the landing point ˜Rlp = 13 ˜Rjp + ∆ ˜R by minimizing the total energy epin(∆ ˜R) at the landing position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Below, we use ∆ ˜R both as a vari- able and as the jump distance to avoid introducing more coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We exploit the differential properties of epin at the jump and landing positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At landing, epin( ˜Rjp + ∆ ˜R) has a minimum, hence, the configuration is force free, in particular along ˜v, ∂˜vepin( ˜Rjp + ∆ ˜R) ≈ [∂˜v∂˜uepin] ˜Rjp∆˜u + [∂2 ˜vepin] ˜Rjp∆˜v = 0, from which we find that ∆˜u and ∆˜v are related via ∆˜v ≈ − [∂˜v∂˜uepin] ˜Rjp [∂2 ˜vepin] ˜Rjp ∆˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (59) Here, we have dropped higher order terms in the expan- sion, assuming that the jump is mainly directed along the unstable u-direction—indeed, using the expansion (58), we find that ∆˜v ≈ − a˜vjp ¯C + λ+ ∆˜u ∝ √ κm − 1 ∆˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (60) Note that we cannot interchange the roles of ˜u and ˜v in this force analysis, as higher order terms in the expression for the force along ˜u cannot be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At the jump position ˜Rjp, the state is force-free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the derivatives [∂˜uepin] ˜Rjp and [∂˜vepin] ˜Rjp vanish, and the Hessian determinant vanishes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' There- fore, the expansion of epin( ˜Rjp + ∆ ˜R) has no linear terms and the second order terms [∂2 ˜uepin] ˜Rjp∆˜u2/2 + [∂˜u∂˜vepin] ˜Rjp∆˜u∆˜v + [∂2 ˜vepin] ˜Rjp∆˜v2/2 combined with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (59) can be expressed through the Hessian determi- nant, {[∂2 ˜uepin][∂2 ˜vepin] − [∂˜u∂˜vepin]2} ˜Rjp∆˜u2/2 = 0, that vanishes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Therefore, the expansion of epin around ˜Rjp starts at third order in ∆ ˜R ≈ (∆˜u, 0) and takes the form (we make use of (60), dropping terms ∝ ∆˜v and a constant) epin( ˜Rjp + ∆ ˜R) ≈ 1 6 � γ˜ujp + β˜vjp � ∆˜u3 + γ 24∆˜u4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (61) Minimizing this expression with respect to ∆˜u (as epin is minimal at ˜Rlp), we obtain the result ∆˜u ≈ −3(γ˜ujp + β˜vjp)/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (62) Making use of the quadratic form (54), we can show that the equation for the landing position ˜Rlp = ˜Rjp + ∆ ˜R can be cast into a similar quadratic form (with δ ˜Rlp measured relative to ˜Rm) δ ˜RT lpMlp δ ˜Rlp = ¯C(κm − 1), (63) but with the landing matrix now given by Mlp = 1 4Mjp + � � 0 0 0 3 4 �δ 2 − β2 2γ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (64) In the following, we will refer to the solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (63) as the ‘landing’ or ‘stable’ ellipse ˜Rlp and write the jump distance in a parametric form involving the shape ˜ujp(˜v) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (56) of the jumping ellipse, ∆˜u(˜v) = −3 [γ ˜ujp(˜v) + β ˜v] /γ, (65) ∆˜v(˜v) = − � a/( ¯C + λ+) � ˜v ∆˜u(˜v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (66) The landing line derived from (63) is displayed as a dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Two tip jumps connected by an arrow are shown for illustration, with solid dots marking the jump position ˜Rjp of the tip and open dots its land- ing position ˜Rlp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they describe tip jumps for a vortex approaching the unstable ellipse once from the left (up- per pair) and another time from the right (lower pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The different topologies associated with jumps and land- ing showing up for the isotropic defect in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c) (two concentric circles) and for the generic onset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6 (two touching ellipses) will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inspecting the matrix equation (63),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we can gain sev- eral insights on the landing ellipse L ˜R: (i) the matrix Mjp/4 on the right-hand side of (64) corresponds to an ellipse with the same geometry as for J ˜R but double in size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (ii) the remaining matrix with vanishing entries in the off-diagonal and the Mxx elements leaves the size dou- bling of the stable ellipse L ˜R at ˜v = 0 unchanged,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' and (iii) the finite Myy component exactly counterbalances the doubling along the v−direction encountered in (i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the definiton (55) of Mjp, up to a term proportional to the skew parameter β accounting for deviations of the semiaxis from the v−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Altogether, the stable ellipse L ˜R extends with a double width along the u−axis and smoothly overlaps with the unstable ellipse at the two contact points ˜vc,±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The latter are found by imposing the condition ∆˜u = ∆˜v = 0 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (65) and (66);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we find them located (relative to ˜Rm) at δ ˜Rc,± = ± (−β/γ, 1) ˜vc, (67) with the endpoint coordinate ˜vc given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (57), and mark them with crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As anticipated, the contact points are off-set with respect to the v−axis for a finite skew parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At these points, the unstable and the stable tip configurations coincide and the vortex tip undergoes no jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, the vector tan- gent to the jump (or landing) ellipse is parallel to the u−direction at the contact points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' To see that, we con- sider (56) and find that ∂˜u ∂˜v ��� ˜v→±˜vc≈ ± � � � ˜v2c − 2γ ¯C(κm − 1) γβ − δ2 � � −1 → ±∞, (68) hence, the corresponding tangents ∂˜u˜v vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The asymptotic positions ¯R where the vortex tips jump and land belong to the boundary of the bistable region B ¯R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' for the isotropic case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(d) these cor- respond to the circles with radii ¯R− (pinning) and ¯R+ 14 (depinning) with jump and landing radii ˜Rf−( ¯R−) and ˜Rp−( ¯R−) and ˜Rp+( ¯R+) and ˜Rf+( ¯R+), respectively, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For the anisotropic defect, we have only a single jump/landing event at one asymptotic position ¯R that we are going to determine in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Bistable domain B ¯ R The set of asymptotic positions ¯R corresponding to the tip positions ˜Rjp along the edges of U ˜R forms the boundary ∂B ¯R of the bistable domain B ¯R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they are re- lated through the force-balance equation (38), with ev- ery vortex tip position ˜Rjp ∈ ∂U ˜R defining an associated asymptotic position ¯R( ˜Rjp) ∈ ∂B ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At the onset of strong pinning, the bistable domain cor- responds to the isolated point ¯Rm, related to ˜Rm through (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Beyond the Labusch point, B ¯R expands out of ¯Rm and its geometry is found by evaluating the force bal- ance equation (38) at a given tip position ˜Rjp ∈ ∂U ˜R, ¯R( ˜Rjp) = ˜Rjp − fp( ˜Rjp)/ ¯C ∈ ∂B ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Using the expansion (58) for epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R), this force equation can be expressed as ∇Repin(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) �� ˜R = 0, or explicitly (we remind that we measure ¯R = ¯Rm + (¯u, ¯v) relative to ¯Rm), ¯C¯u = ¯C(1 − κm)˜u + a 2 ˜v2 + γ 6 ˜u3 + β 2 ˜u2˜v + α 2 ˜u˜v2, ¯C¯v = ( ¯C + λ+)˜v + a ˜u˜v + β 6 ˜u3 + α 2 ˜u2˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (69) Inserting the results for the jump ellipse J ˜R, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (56), into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (69), we find the crescent-shape bistable domain B ¯R shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' let us briefly derive the origin of this shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Solving (69) to leading order, ¯C¯u(0) ≈ (a/2)˜v2 and ¯C¯v(0) ≈ ( ¯C + λ+)˜v, we find the parabolic approximation ¯u (0) ≈ a 2 ¯C 1 (1 + λ+/ ¯C)2 ¯v (0) 2, (70) telling that the extent of B ¯R scales as (κm − 1) along ¯u and ∝ (κm − 1)1/2 along ¯v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', we find a flat parabola opening towards positive ¯u for a > 0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In order to find the width of B ¯R, we have to solve (69) to the next higher order, ¯u = ¯u(0) + ¯u(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' for β = 0, we find the correction ¯u (1) = (1 − κm)˜u + γ 6 ¯C ˜u3 + α 2 ¯C ˜u˜v2 (71) that produces a ¯v ↔ −¯v symmetric crescent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inserting the two branches (56) of the jump ellipse, we arrive at the width of the crescent that scales as (κm − 1)3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The correction to ¯v is ∝ (κm − 1) and we find the closed form ¯v ≈ [1 + (λ+ + a˜u)/ ¯C] ˜v (72) with a small antisymmetric (in ˜u) correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For a finite β ̸= 0, the correction ¯u(1) picks up an additional term 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5 −2 0 −2 0 −2 0 −10 −5 0 5 π/2 − θ∗ θ∗ ¯u/ξ(κm − 1) ¯v/ξ√κm − 1 (a) ¯u/ξ√κm − 1 ¯v/ξ√κm − 1 (b) ˜u/ξ√κm − 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) Bistable domain B ¯ R in asymptotic ¯R-space mea- sured in units of ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the same parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6 have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note the different scaling of the axes in κm − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the right panel (b) shows B ¯ R in isotropic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bistable domain B ¯ R is elongated along the transverse direction ¯v and narrow/bent along the unstable direction ¯u, giving B ¯ R its pe- culiar crescent-like shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The branch crossing line ¯R0, see (77), is shown as a dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Black crosses mark the cusps of B ¯ R and are associated with the contact points of U ˜ R through the force balance equation (38);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they corre- spond to critical end-points in the thermodynamic Ising ana- logue, while the boundaries ∂B ¯ R map to spinodals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blue and red colors identify different characters of vortex tip configu- rations as quantified through the ‘order parameter’ ˜u of the Landau expansion (at β = 0), see text, while magenta is as- sociated to the bistable area B ¯ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the blue and red branches extend to the far side of the crescent and terminate in the blue and red colored boundaries ∂Bb ¯ R and ∂Br ¯ R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Thin horizontal lines show vortex trajectories that proceed smoothly in asymptotic space, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blue and red dots mark the asymptotic positions associated with vor- tex tip jumps that happen at the exit of B ¯ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' they correspond to the pairs of tip positions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b) Bistable domain B ¯ R in isotropic scaled coordinates ¯u and ¯v showing the ‘true’ shape of B ¯ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices impacting on the bistable domain with an angle |θ| ≤ θ∗ undergo a single jump on the far side of B ¯ R, with the pinning force density directed along u and scaling as F ∥ pin ∝ (κ − 1)5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vortices crossing B ¯ R at large angles close to π/2 jump either never, once, or twice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' at θ = π/2 the pinning force density is small, F ⊥ pin ∝ (κ − 1)3, and directed along v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (β/2 ¯C) ˜u2˜v that breaks the ¯v ↔ −¯v symmetry and the crescent is distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Viewing the boundary ∂B ¯R as a parametric curve in the variable ˜v with ˜u = ˜ujp(˜v) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (56), we obtain the boundary ∂B ¯R in the form of two separate arcs that define the crescent-shaped domain B ¯R in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The two arcs merge in two cusps at ¯Rc,± that are associated to the touching points (67) in dual space and derive from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (69);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' measured with respect to ¯Rm, 15 these cusps are located at δ ¯Rc,± = (¯uc, ±¯vc) (73) ≈ �� a/2 ¯C � ˜v2 c, ±(1 + λ+/ ¯C)˜vc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The coloring in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7 indicates the characters ‘red’ and ‘blue’ of the vortex states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these are defined in terms of the ‘order parameter’ ˜u− ˜um(¯v) of the Landau functional (58) that changes sign at the branch crossing line Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (77), with the shift ˜um(¯v) = −β γ ˜v(¯v) ≈ −β γ ¯v 1 + λ+/ ¯C , (74) ˜um(¯v) = 0 for our symmetric case with β = 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Going beyond the cusps (or critical points) at ¯Rc,±, the two states smoothly crossover between ‘red’ and ‘blue’ (indicated by the smooth blue–white–red transition), as known for the van der Waals gas (or Ising magnet) above the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Within the bistable region B ¯R, both ‘red’ and ‘blue’ states coexist and we color this region in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The geometry of the bistable domain B ¯R is very differ- ent from the ring-shaped geometry of the isotropic prob- lem discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II A, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in the discussion of the uniaxial anisotropic defect below, we will learn how these two geometries are interrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Comparing the overall dimensions of the crescent with the ring in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(d), we find the following scaling behavior in κm − 1: while the crescent B ¯R grows along ¯v as (κm − 1)1/2, the isotropic ring involves the characteristic size ξ of the de- fect, ¯R− ∼ ξ and hence its extension along ¯v is a con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' On the other hand, the scaling of the crescent’s and the ring’s width is the same, ∝ (κm − 1)3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The different scaling of the transverse width then will be re- sponsible for the new scaling of the pinning force density, Fpin ∝ (κm − 1)5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Comparison to isotropic situation Let us compare the unstable domains U ˜R for the isotropic and anisotropic defects in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c) and 6, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the isotropic example of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II A, the jump- and landing-circles ˜Rjp( ¯R) and ˜Rlp( ¯R) are con- nected to different phases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', free (colored in blue at ˜Rjp = ˜Rf−) and pinned (colored in red at ˜Rlp = ˜Rp−) associated with ¯R−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, the topology is differ- ent, with the unstable ring domain separating the two distinct phases, free and pinned ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As a result, a sec- ond pair of jump- and landing-positions associated with the asymptotic circle ¯R+ appears along the vortex tra- jectory of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these are the located at the radii ˜Rjp = ˜Rp+ and ˜Rlp = ˜Rf+ and describe the depinning process from the pinned branch back to the free branch (while the previous pair at radii ˜Rf− and ˜Rp− describes the pinning process from the free to the pinned branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinning (at ¯R−) and depinning (at ¯R+) processes in the asymptotic coordinates are shown in figure 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bistable area B ¯R with coexisting free and pinned states has a ring-shape as well (colored in magenta, the superposition of blue and red);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the two pairs of jump and landing points in tip space have collapsed to two pinning and depinning points in asymptotic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the present situation describing the strong pinning onset for a generic anisotropic potential, the unstable do- main U ˜R grows out of an isolated point (in fact, ˜Rm) and assumes the shape of an ellipse that is simply connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' as a result, a vortex incident on the defect undergoes only a single jump, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bistable domain B ¯R is simply connected as well, but now features two cusps at the end-points of the crescent, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bista- bility again involves two states, but we cannot associate them with separated pinned and free phases—we thus de- note them by ‘blue’-type and ‘red’-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The two states approach one another further away from the defect and are distiguishable only in the region close to bistability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7, this is indicated with appropriate color cod- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that the Landau-type expansion underlying the coloring in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7 fails at large distances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' going be- yond a local expansion near ˜Rm, the distortion of the vortex vanishes at large distances and red/blue colors faint away to approach ‘white’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Topology The different topologies of unstable and bistable re- gions appearing in the isotropic and anisotropic situa- tions are owed to the circular symmetry of the isotropic defect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we will recover the ring-like topology for the anisotropic situation later when describing a uniaxially anisotropic defect at larger values of the Labusch param- eter κm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, such an increase in pinning strength will induce a change in topology with two crescents facing one another joining into a ring-like shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Let us discuss the consequences of the different topolo- gies that we encountered for the isotropic and anisotropic defects in the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Specifically, the precise number and position of the contact points have an elegant topological explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When a vortex tip touches the edges ˜Rjp of the unstable domain there are two character- istic directions: one is given by the unstable eigenvector v−( ˜Rjp) discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III B along which the tip will jump initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The second is the tangent vector to the boundary ∂U ˜R of the unstable domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', to the unsta- ble ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While the former is approximately constant and parallel to the unstable u-direction along ˜Rjp, the latter winds around the ellipse exactly once after a full turn around U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The contact points ˜Rc,± of the unsta- ble and stable ellipses then coincide with those points on the ellipse where the tangent vector are parallel and anti- parallel to v−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' at these points, the tip touches the unsta- ble ellipse but does not undergo a jump any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Given the different winding numbers of v− and of the tangent vector, there are exactly two points along the circum- ference of U ˜R where the tangent vector is parallel/anti- 16 parallel to the u-direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these are the points found in (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This argument remains valid as long as the contour ∂U ˜R is not deformed to cross/encircle the singular point of the v−( ˜Rjp) field residing at the defect center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The same arguments allow us to understand the ab- sence of contact points in the isotropic scenario: For an isotropic potential, the winding number nU of the tan- gent vector around U ˜R remains unchanged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', nU = ±1, while the unstable direction v− is pointing along the ra- dius and thus acquires a unit winding number as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, the two directions, tangent and jump, then ro- tate simultaneously and do not wind around each other after a full rotation, explaining the absence of contact points in the isotropic situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Energy jumps Within strong pinning theory, the energy jump ∆epin associated with the vortex tip jump between bistable vor- tex configurations at the boundaries of B ¯R determines the pinning force density Fpin and the critical current jc, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (16) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Formally, the energy jump ∆epin is defined as the difference in energy epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) at fixed asymptotic position ¯R ∈ ∂B ¯R between vortex con- figurations with tips in the jump ( ˜Rjp( ¯R)) and landing ( ˜Rlp( ¯R) = ˜Rjp( ¯R) + ∆ ˜R) positions, ∆epin( ¯R ∈ ∂B ¯R) ≡ epin[ ˜Rjp( ¯R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R] − epin[ ˜Rlp( ¯R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (75) In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III B 2 above, we have found that the jump ∆ ˜R is mainly forward directed along u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Making use of the expansion (61) of epin at ˜Rjp and the result (62) for the jump distance ∆˜u, we find the energy jumps ∆epin in tip- and asymptotic space in the form (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' with the isotropic result Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (24)), ∆epin( ¯R) ≈ γ 72∆˜u4 ≈ � 9 8γ3 � [γ ˜ujp(˜v) + β ˜v]4 (76) ≈ � 9 8γ3 � � (γδ − β2) � ˜v2 c − ˜v2��2 ≈ � 9 8γ3 � � (γδ − β2) (1 + λ+/ ¯C)2 � ¯v2 c − ¯v2��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, we have used the parametric shape ˜ujp(˜v) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (56) for the jumping ellipse as well as (69) to lowest or- der, ˜v ≈ ¯v/(1 + λ+/ ¯C), to relate the tip and asymptotic positions in the last equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The energy jump (76) scales as (κm − 1)2 and is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' It depends on the v coordinate of the asymptotic (or tip) position only and vanishes at the cusps ¯Rc,±, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (73) (or at the touching points ˜Rc,±, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (67)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' To order (κm − 1)2, the energy jumps are identical at the left and right edges of the bistable domain B ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Following the two bistable branches and the associated energy jumps between them to the inside of B ¯R, the latter −2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='2 ¯v/ξ√κm − 1 ∆epin/ep(κm − 1)2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Energy jump ∆epin along the edges of the bistable domain B ¯ R as a function of the transverse coordinate ¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we have used the same parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The energy jump vanishes at the cusps ±¯vc, as the bistable tip configurations become identical and their energies turn equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' vanish along the branch crossing line ¯R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the thermo- dynamic analogue, this line corresponds to the first-order equilibrium transition line that is framed by the spinodal lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' for the isotropic defect, this is the circle with ra- dius ¯R0 = x0 framed by the spinodal circles with radii ¯R±, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4 and 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For the anisotropic defect with β = 0, this line is trivially given by the centered parabola of B ¯R, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (70), and hence ¯u0 ≈ a 2 ¯C 1 (1 + λ+/ ¯C)2 ¯v2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (77) The result for a finite skew parameter β ̸= 0 is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A27) in Appendix A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pinning force density The pinning force density Fpin is defined as the aver- age force density exerted on a vortex line as it moves across the superconducting sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For the isotropic case described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' II E, the individual pinning force fpin( ¯R) = −∇ ¯Repin( ¯R), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (7), is directed radially and the force density Fpin is given by the (constant) en- ergy jump ∆epin ∝ (κ − 1)2 on the edge ∂B ¯R of the bistable domain and the transverse length t⊥ ∼ ξ, hence, Fpin ∝ t⊥∆epin scales as (κ − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For an anisotropic defect, the pinning force depends on the vortex direction of motion ˆv = (cos θ, sin θ) rela- tive to the axis of the bistable region: we choose angles −π/2 ≤ θ ≤ π/2 measured from the unstable direction ¯u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', vortices incident from the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the case of larger impact angles |θ| > π/2 corresponds to vortices incident from the right and can be reduced to the previous case by inverting the sign of the parameter a in the expan- sion (58), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the curvature of the parabola (70);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' to our leading order analysis, the results remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The 17 pinning force is no longer directed radially but depends on θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' furthermore, the energy jump (76) is non-uniform along the boundary B ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In spite of these complications, we can perform some simple scaling estimates as a first step: let us assume a uniform distribution of identical anisotropic defects, all with their unstable direction pointing along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The jumps in energy still scale as ∆epin ∝ (κm − 1)2, how- ever, the trapping distance is no longer finite but grows from zero as κm − 1 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Due to their elongated shapes, the bistable domains B ¯R exhibit different exten- sions along the y and x directions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', ∝ ¯vc ∝ √κm − 1 along y and ∝ ¯uc ∝ (κm −1) along x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These simple considerations then suggest that the pinning force density exhibits a scaling Fpin ∝ (κm − 1)µ with µ > 2, different from the setup with isotropic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Even more, vortices moving along the x or y directions, re- spectively, will experience different forces F ∥ pin and F ⊥ pin scaling as F ∥ pin ∝ (κm − 1)5/2, F ⊥ pin ∝ (κm − 1)3 (78) near the onset of strong pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While such uniform anisotropic defects could be created artificially, a more realistic scenario will involve defects that are randomly oriented and an additional averaging over angles θ has to be performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this will be done at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We first determine the magnitude and orientation of the pinning force density Fpin(θ) as a function of the vortex impact angle θ for randomly positioned but uni- formly oriented (along x) defects of density np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pin- ning force density is given by the average over relative positions between vortices and defects (with a minus sign following convention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V ¯R denotes the vortex lattice unit cell), Fpin(θ) = −np � V ¯ R\\B ¯ R d2 ¯R a2 0 fpin( ¯R) (79) −np � B ¯ R d2 ¯R a2 0 � pb( ¯R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' θ) f b pin( ¯R) + pr( ¯R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' θ) f r pin( ¯R) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Outside of the bistable domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', in V ¯R \\ B ¯R, a sin- gle stable vortex tip configuration exists and the pinning force fpin( ¯R) is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inside B ¯R, the branch occupation functions pb,r( ¯R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' θ) are associated with the tip positions appertaining to the ‘blue’ and the ‘red’ vor- tex configurations with different tip positions ˜Rb,r( ¯R), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinning forces f b,r pin( ¯R) are evalu- ated for the corresponding vortex tip positions and are defined as f b,r pin( ¯R) = −∇ ¯Repin[ ˜Rb,r( ¯R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (80) Let us now study how vortex lines populate the bistable domain as a function of the impact angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Ex- amining Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7, we can distinguish between two different angular regimes: a frontal-impact regime at angles away from π/2, |θ| ≤ θ∗, where all the vortices that cross the bistable domain undergo exactly one jump on the far edge of B ¯R, see the blue dot and blue boundary ∂Bb ¯R in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' and a transverse regime for angles θ∗ ≤ |θ| ≤ π/2, where vortices crossing the bistable domain undergo ei- ther no jump, one or two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The angle θ∗ is given by the (outer) tangent of the bistable domain at the cusps ¯Rc,±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' making use of the lowest order approximation (70) of the crescent’s geometry, we find that tan(θ∗) = ∂¯v(0) ∂¯u(0) ��� ¯vc = ( ¯C + λ+) a � γδ − β2 2γ ¯C(κm − 1), (81) implying that π/2 − θ∗ ∝ √κm − 1 is small, θ∗ ≈ π/2 − a ( ¯C + λ+) � 2γ ¯C(κm − 1) γδ − β2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (82) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Impact angles |θ| < θ∗ For a frontal impact with |θ| < θ∗, vortices occupy the ‘blue’ branch and remain there throughout the bistable domain B ¯R until its termination on the far edge ∂Bb ¯R, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7, implying that pb( ¯R ∈ B ¯R) = 1 and pr( ¯R ∈ B ¯R) = 0, independent of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As a consequence, the pinning force Fpin does not depend an the impact angle and is given by the expression F< pin = −np � V ¯ R\\B ¯ R d2 ¯R a2 0 fpin( ¯R) − np � B ¯ R d2 ¯R a2 0 f b pin( ¯R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Next, Gauss’ formula tells us that for a function e(x), we can transform � V dnx ∇e(x) = � ∂V dn−1 S⊥ e(x), (83) with the surface element dn−1 S⊥ oriented perpendicular to the surface and pointing outside of the domain V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In applying (83) to the first integral of F< pin, we can drop the contribution from the outer boundary ∂V ¯R since we assume a compact defect potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The remaining con- tribution from the crescent’s boundary ∂B ¯R joins up with the second integral but with an opposite sign, as the two terms involve the same surface but with opposite orien- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Altogether, we then arrive at the expression F< pin = np � ∂Bb ¯ R d S⊥ a2 0 � eb pin( ¯R) − epin( ¯R) � + np � ∂Br ¯ R d S⊥ a2 0 � eb pin( ¯R) − epin( ¯R) � , (84) where we have separated the left and right borders ∂Br,b ¯R of the bistable domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Due to continuity, the stable vortex energy epin( ¯R) will be equal to eb pin( ¯R) on the 18 left border ∂Br ¯R and equal to er pin( ¯R) on the right border ∂Bb ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The expression (84) for F< pin then reduces to F< pin = np � ∂Bb ¯ R d S⊥ a2 0 � eb pin( ¯R) − er pin( ¯R) � = np � ¯vc −¯vc d¯v a0 ∆epin(¯v) a0 [1, −∂¯u/∂¯v] = np �2¯vc a0 ⟨∆epin⟩ a0 , 0 � ≡ [F ∥ pin, 0] (85) with ⟨∆epin⟩ the average energy jump evaluated along the v-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The force F< pin is aligned with the unstable directed along u, with the v-component vanishing due to the antisymmetry in ¯v ↔ −¯v of the derivative ∂¯u/∂¯v, and is independent on θ for |θ| < θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Impact angle |θ| = π/2 Second, let us find the pinning force density Fπ/2 pin for vortices moving along the (positive) v-direction, θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As follows from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7, vortices occupy the blue branch and jump to the red one upon hitting the lower half of the boundary ∂Bb ¯R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' vortices that enter B ¯R but do not cross ∂Bb ¯R undergo no jump and hence do not con- tribute to Fπ/2 pin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As vortices in the red branch proceed upwards, they jump back to the blue branch upon cross- ing the red boundary ∂Br ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While jumps appear on all of the lower half of ∂Bb ¯R, a piece of the upper bound- ary ∂Br ¯R that contributes with a second jump is cut away (as vortices to the left of ¯u(0) + ¯u(1) do not change branch from blue to red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The length ∆¯v of this inter- val scales as ∆¯v/¯vc ∝ (κm − 1)1/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ignoring this small jump-free region, we determine Fπ/2 pin assuming that vor- tices contributing to Fπ/2 pin undergo a sequence of two jumps, from blue to red on the lower half ∂Bb< ¯R and back from red to blue on the upper half ∂Br> ¯R of the bound- ary ∂B ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Repeating the above analysis, we find that the u-components in Fπ/2 pin arising from the blue and red boundaries now cancel, while the v-components add up, Fπ/2 pin = np � ∂Bb< ¯ R d S⊥ a2 0 � eb pin( ¯R) − er pin( ¯R) � + np � ∂Br> ¯ R d S⊥ a2 0 � er pin( ¯R) − eb pin( ¯R) � = 2np � ¯vc 0 d¯v a0 ∆epin(¯v) a0 [0, ∂¯u/∂¯v] (86) = np � 0, 2¯vc a0 ⟨∆epin∂¯v¯u⟩ a0 � ≡ [0, F ⊥ pin].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Making use of the result (76) for ∆epin(¯v) in (85), we find explicit expressions for the pinning force densities for impacts parallel and perpendicular to the unstable direction u, F ∥ pin ≈ � 9np 8 a2 0γ3 �� ¯vc −¯vc d¯v � γδ − β2 (1 + λ+/ ¯C)2 � ¯v2 c − ¯v2��2 (87) = 24 5 np � 2 ¯C/γ a0 ¯C2 γa0 γ(1 + λ+/ ¯C) � γδ − β2 (κm − 1)5/2 and F ⊥ pin ≈ 3 ¯C2 γa0 γa/a0 γδ − β2 (κm − 1)3, (88) that confirm the scaling estimates of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, we have made use of the definition (73) of ¯vc and have brought the final result into a form similar to the isotropic result (28) (with the length � ¯C/γ and the force ¯C2/γa0, equal to ξ/ √ 3κ and ep/12κ2 for a Lorentzian poten- tial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The result (87) provides the pinning force density Fpin = [F ∥ pin, 0] for all impact angles |θ| ≤ θ∗ (note that (87) depends on the curvature a of the crescent via δ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (49), that involves a2 only, but higher-order correc- tions will introduce an asymmetry between left- and right moving vortices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Within the interval θ∗ < θ < π/2, the longitudinal force Fpin,u along u decays to zero and the transverse force Fpin,v along v becomes finite, assuming the value (88) at θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The two force components have been evaluated numerically over the entire angular regime and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 9: when mov- ing away from the angle θ = π/2, the transition from the blue to the red boundary is moving upwards, with the relevant boundary turning fully blue at θ = θ∗, thus smoothly transforming (86) into (85) (we have adopted the approximation of dropping the jump-free interval ∆¯v that moves up and becomes smaller as θ decreases from π/2 to θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Anisotropic critical force density Fc When the vortex system is subjected to a current den- sity j, the associated Lorentz force FL(ϕ) = j ∧ B/c di- rected along ϕ pushes the vortices across the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When FL is directed along u, we have Fpin = [F ∥ pin, 0] and the vortex system gets immobilized at force densi- ties FL < Fc = F ∥ pin (or associated current densities jc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When FL is directed away from u, the driving compo- nent along v has to be compensated by a finite pinning force Fpin,v that appears only for angles θ∗ < θ < π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Hence, the angles of force and motion, ϕ associated with the Lorentz force FL(ϕ) and θ providing the direction of the pinning force Fpin(θ), are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We find them, along with the critical force density Fc(ϕ), by solving the dynamical force equation (17) at vanishing velocity v = 0, Fc(ϕ) = Fpin(θ) (89) 19 0 π/8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5 0 π/4 0 1 Fpin,u/[np(ep/a0)(ξ/a0)(κm − 1)5/2] Fpin,v/[np(ep/a0)(ξ/a0)(κm − 1)3] F ∥ pin F ⊥ pin Fc/F ∥ pin θ π/2 θ∗ ϕ π/2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Top: scaled pinning force densities Fpin,u and Fpin,v versus impact angle θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we have used the same parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The longitudinal (along u) force Fpin,u remains constant and equal to F ∥ pin for all angles |θ| < θ∗, while the transverse (along v) component Fpin,v vanishes in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The longitudinal force drops and vanishes over the narrow interval θ∗ < |θ| < π/2, while the transverse force Fpin,v increases up to F ⊥ pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Bottom: critical force density Fc (di- rected along the Lorentz force FL = j ∧ B/c) versus angle ϕ of the Lorentz force;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the dashed line shows the upper bound Fc < F ⊥ pin/ sin(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' resulting in a critical force density Fc(ϕ) = � F 2 pin,u(θ) + F 2 pin,v(θ) (90) with angles ϕ and θ related via tan ϕ = Fpin,u(θ) Fpin,v(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (91) Since Fpin,u(θ < θ∗) = 0, the entire interval θ < θ∗ is compressed to ϕ = 0 and it is the narrow regime θ∗ < θ < π/2 that determines the angular characteristic of the critical force density Fc(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The critical force den- sity Fc(ϕ) is peaked at ϕ = 0 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 9 (with a correspondingly sharp peak in jc at right angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Comb- ing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (90) and (91), we can derive a simple expression bounding the function Fc(ϕ), Fc(ϕ) = Fpin,v(θ) � 1 + cot2(ϕ) ≤ F ⊥ pin sin(ϕ), (92) that traces Fc(ϕ) over a wide angular region, see the dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At small values of ϕ we cannot ignore the angular dependence in Fpin,v(θ) any more that finally cuts off the divergence ∝ 1/ sin(ϕ) at the value Fc(ϕ → 0) → F ∥ pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Isotropized pinning force density Fpin In a last step, we assume an ensemble of equal anisotropic defects that are uniformly distributed in space and randomly oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In this situation, we have to perform an additional average over the insta- bility directions ˆui associated with the different defects i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Neglecting the modification of Fpin(θ) away from [F ∥ pin, 0] in the small angular regions θ∗ < |θ| < π/2, we find that the force along any direction ˆR has the mag- nitude Fpin ≈ 1 N N � i=1 |(F ∥ pinˆui) · ˆR| (93) ≈ F ∥ pin � π/2 −π/2 dθ π cos θ = 2 π F ∥ pin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As a result of the averaging over the angular directions, the pinning force density is now effectively isotropic and directed against the velocity v of the vortex motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' UNIAXIAL DEFECT In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III, we have analyzed the onset of strong pin- ning for an arbitrary potential and have determined the shape of the unstable and bistable domains U ˜R and B ¯R—with their elliptic and crescent forms, they look quite different from their ring-shaped counterparts for the isotropic defect in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In this sec- tion, we discuss the situation for a weakly anisotropic defect with a small uniaxial deformation quantified by the small parameter ϵ in order to understand how our previous findings, the results for the isotropic defect and those describing the strong-pinning onset, relate to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Our weakly deformed defect is described by equipo- tential lines that are nearly circular but slightly elon- gated along y, implying that pinning is strongest in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We will find that the unstable (bistable) do- main U ˜R (B ¯R) for the uniaxially anisotropic defect starts out with two ellipses (crescents) on the x-axis as κm crosses unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With increasing pinning strength, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', κm, these ellipses (crescents) grow and deform to follow the equipotential lines, with the end-points approaching one another until they merge on the ±y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These merger points, we denote them as ˜Rs and ¯Rs, define a second class of important points (besides the onset points ˜Rm and ¯Rm) in the buildup of the strong pinning landscape: while the onset points ˜Rm are defined as minima of the Hessian determinant D( ˜R), the merger points ˜Rs turn out to be associated with saddle points of D( ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Push- ing across the merger of the deformed ellipses (crescents) by further increasing the Labusch parameter κm, the un- stable (bistable) domains U ˜R (B ¯R) undergo a change in topology, from two separated areas to a ring-like geom- etry as it appears for the isotropic defect, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c) 20 and (d), thus explaining the interrelation of our results for isotropic and anisotropic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With this analysis, we thus show how the strong pin- ning landscape for the weakly uniaxial defect will finally assume the shape and topology of the isotropic defect as the pinning strength κm overcomes the anisotropy ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Second, this discussion will introduce the merger points ˜Rs as a second type of characteristic points of strong pinning landscapes that we will further study in section V A using a Landau-type expansion as done in section III A above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we will find that the geometry of the merger points ˜Rs is associated with hyperbolas, as that of the onset points was associated with ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Our uniaxially anisotropic defect is described by the stretched (along the y-axis) Lorentzian ep(˜x, ˜y) = −ep � 1 + ˜x2 2ξ2 + ˜y2 2ξ2 (1 + ϵ)2 �−1 , (94) with equipotential lines described by ellipses ˜x2 ξ2 + ˜y2 ξ2 (1 + ϵ)2 = const, (95) and the small parameter 0 < ϵ ≪ 1 quantifying the de- gree of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At fixed radius ˜R2 = ˜x2 + ˜y2, the potential (94) assumes maxima in energy and in negative curvature on the x−axis, and corresponding minima on the y−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Along both axes, the pinning force is directed radially towards the origin and the Labusch criterion (34) for strong pinning is determined solely by the curvature along the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At the onset of strong pin- ning, the unstable and bistable domains then first emerge along the x−axis at the points ˜Rm = (± √ 2ξ, 0) and ¯Rm = (±2 √ 2ξ, 0) when κm = ep 4 ¯Cξ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (96) Upon increasing the pinning strength κm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', via soft- ening of the vortex lattice as described by a decrease in ¯C, the unstable and bistable domains U ˜R and B ¯R expand away from these points, and eventually merge along the y−axis at ˜Rs = (0, ± √ 2ξ(1+ϵ)), ¯Rs = (0, ±2 √ 2ξ(1+ϵ)) when κs = ep 4 ¯Cξ2(1 + ϵ)2 = κm (1 + ϵ)2 = 1, (97) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', for κm = (1+ϵ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The evolution of the strong pinning landscape from onset to merging takes place in the inter- val κm ∈ [1, (1 + ϵ)2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' pushing κm beyond this interval, we will analyze the change in topology and appearance of non-simply connected unstable and bistable domains after the merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The quantity determining the shape of the unstable domain U ˜R is the Hessian determinant D( ˜R) of the total vortex energy epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R), see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (36) and (1), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At onset, the minimum of D( ˜R) touches zero for the first time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' with increasing κm, this minimum drops below zero and the condition D( ˜R) = 0 determines the unstable ellipse that expands in ˜R-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Viewing the function D( ˜R) as a height function of a landscape in the ˜R plane, this corresponds to filling this landscape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', with water, up to the height level D = 0 with the result- ing lake representing the unstable domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the present uniaxially symmetric case, a pair of unstable ellipses grow simultaneously, bend around the equipotential line near the radius ∼ √ 2ξ and finally touch upon merging on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In our geometric interpretation, this corresponds to the merging of the two (water-filled) valleys that hap- pens in a saddle-point of the function D( ˜R) at the height D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Hence, the merger point ˜Rs correspond to sad- dles in D( ˜R) with D( ˜Rs) = 0, ∇ ˜R D(R) �� ˜Rs = 0, (98) and det � Hess � D( ˜R) ���� ˜Rs < 0, (99) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In our calculation of D( ˜R), we exploit that the Hessian in (36) does not depend on the asymptotic position ¯R and we can set it to zero, D( ˜R) = det � Hess[ ¯C ˜R2/2 + e (i) p ( ˜R) + δep( ˜R)] � , (100) where we have split off the anisotropic correction δep( ˜R) = ep( ˜R) − e(i) p ( ˜R) away from the isotropic po- tential e(i) p ( ˜R) with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the following, we perform a perturbative analysis around the isotropic limit valid in the limit of weak anisotropy ϵ ≪ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this motivates our use of polar (tip) coordinates ˜R and ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The isotropic contribution H(i) to the Hessian matrix H is diagonal with components H (i) ˜ R ˜ R( ˜R) ≡ ∂2 ˜ R[ ¯C ˜R2/2 + e (i) p ( ˜R)] = ¯C + ∂2 ˜ Re (i) p ( ˜R) (101) and H (i) ˜φ ˜φ( ˜R) ≡ ( ˜R−2∂2 ˜φ ˜φ + ˜R−1∂ ˜ R)[ ¯C ˜R2/2 + e (i) p ( ˜R)] = ¯C − f (i) p ( ˜R)/ ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (102) The radial component H (i) ˜ R ˜ R ∝ (κm − 1) vanishes at on- set, while H (i) ˜φ ˜φ remains finite, positive, and approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The anisotropic component δep( ˜R) introduces correc- tions ∝ ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these significantly modify the radial entry of the full Hessian while leaving its azimutal component H˜φ ˜φ approximately unchanged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the off-diagonal entries of the full Hessian scale as ϵ and hence contribute in second or- der of ϵ to D( ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As a result, the sign change in the determinant D( ˜R) ≈ H ˜ R ˜ R( ˜R)H ˜φ ˜φ( ˜R) + O � ϵ2� , (103) 21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Unstable and bistable domains close to the onset of strong pinning for a uniaxial defect (94) centered at the origin, with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 and κm −1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinning potential is steepest at angles ˜φ = 0, π and least steep at ˜φ = ±π/2, hence strong pinning is realized first in a small interval around ˜φ = 0, π (solid black dots) where κm(˜φ) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) The unsta- ble domain U ˜ R in tip space is bounded by red/blue solid lines (jump lines J ˜ R, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (108));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' dashed lines mark the asso- ciated landing lines L ˜ R, see (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b) Focus on the unstable domain near ˜φ = 0 in polar coordinates ˜R and ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The jump- ing (solid) and landing (dashed) lines have the approximate shape of ellipses, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (111), in agreement with our anal- ysis of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (c) The bistable domain B ¯ R in asymptotic space involves symmetric crescents centered at ¯φ = 0, π and a narrow width ∝ (κm(¯φ)−1)3/2, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (112), in agreement with the analysis of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (d) Focus on the bistable do- main at ¯φ = 0 in polar coordinates ¯R and ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Red/blue colors indicate different vortex configurations as quantified through the order parameter ˜R − ˜Rm(˜φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' is determined by H ˜ R ˜ R( ˜R) = H (i) ˜ R ˜ R( ˜R) + ∂2 ˜ Rδep( ˜R) (104) for radii close to ˜Rm with δ ˜R = ˜R − ˜Rm ≈ O(√κm − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We expand the potential (94) around the isotropic part e(i) p ( ˜R), δep( ˜R) ≈ −ϵ [∂ ˜ Re (i) p ( ˜R)] ˜R sin2 ˜φ, (105) and additionally expand both e(i) p ( ˜R) and δep( ˜R) around ˜Rm, keeping terms ∝ ϵ � (κm − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The radial entry of the anisotropic Hessian matrix then assumes the form H ˜ R ˜ R( ˜R) ≈ ¯C [1 − κm(˜φ)] + γ [δ ˜R2/2 − ϵ sin2 ˜φ ˜Rmδ ˜R] (106) with γ = ∂4 ˜ Re(i) p ( ˜R)| ˜ Rm and the angle-dependent Labusch parameter κm(˜φ) ≡ max ˜ R[−∂2 ˜ Rep( ˜R, ˜φ)| ˜φ] ¯C = κm − 2ϵ sin2 ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (107) The edges of the unstable region U ˜R then can be obtained by imposing the condition H ˜ R ˜ R( ˜R) = 0 and the solution to the corresponding quadratic equation define the jump positions ˜Rjp(˜φ) (or boundaries ∂U ˜R) ˜Rjp(˜φ) ≈ ˜Rm(˜φ) ± δ ˜R(˜φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (108) These are centered around the (‘large’) ellipse defined by ˜Rm(˜φ) = ˜Rm(1 + ϵ sin2 ˜φ) (109) and separated by (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (20)) 2 δ ˜R(˜φ) = � 8 ¯C γ (κm(˜φ) − 1) (110) along the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Making use of the form (107) of κm(˜φ) and assuming a small value of κm > 1 near onset, we obtain the jump line in the form of a (‘small’) ellipse centered at [± ˜Rm, 0], γ δ ˜R2 + ϵ ¯C ˜φ2 = ¯C(κm − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (111) Hence, we find that the anisotropic results are obtained from the isotropic ones by replacing the circle ˜Rm by the ellipse ˜Rm(˜φ) and substituting κ → κm(˜φ) in the width (20), see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 10(a) and (b) evaluated for small values κm − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Analogously, the boundaries of the bistable domain B ¯R can be found by applying the same substitutions to the result (25), see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 10(c) and (d), ¯R(¯φ) ≈ ¯Rm(¯φ) ± δ ¯R(¯φ) (112) with ¯Rm(¯φ) = ¯Rm(1 + ϵ sin2 ¯φ) and the width 2 δ ¯R(¯φ) = 2 3 � 8 ¯C γ (κm(˜φ) − 1)3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (113) The landing line L ˜R is given by (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (23) and note that the jump point is shifted by ˜ujp away from ˜xm, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (19)) ˜Rlp(˜φ) ≈ ˜Rm(˜φ) ∓ 2 δ ˜R(˜φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (114) An additional complication is the finite angular exten- sion of the unstable and bistable domains U ˜R and B ¯R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 22 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Unstable and bistable domains before merging for a uniaxial defect (94) centered at the origin, with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 and 1 − κs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Strong pinning is realized everywhere but in a small interval around ˜φ = ±π/2 where κm(˜φ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) The unstable domain U ˜ R in the tip plane is bounded by the solid red/blue jump lines J ˜ R, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (108) and involves two strongly bent ellipses originating from angles ˜φ = 0, π (black dots) and approaching one another close to ˜φ = ±π/2 (black crosses);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' red/blue dashed lines are landing points as given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b) Focus (in polar coordinates ˜R, ˜φ) on the tips of the unstable domain near ˜φ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (c) The bistable domain B ¯ R in the asymptotic space consists of thin symmetric crescents (colored in magenta) originating from ¯φ = 0, π, with the delimiting black solid lines given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (112).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (d) Focus on the cusps of the bistable domain close to ¯φ = π/2 in polar coordinates ¯R, ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Red/blue colors indicate different vortex configurations as quantified through the order parameter ˜R− ˜Rm(¯φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these are limited by the condition κm(φmax) = 1, provid- ing us with the constraint ˜φmax = ¯φmax ≈ ± � κm − 1 2ϵ (115) near the strong pinning onset with (κm − 1) ≪ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The resulting domains U ˜R have characteristic extensions of scale ∝ √κm − 1, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Close to merging (marked by crosses in the figure) at φ = ±π/2, we define the deviation δφ = π/2 − φ with δφ ≪ 1, and imposing the condition κm(φmax) = 1, we FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Unstable and bistable domains for a uniaxial defect (94) after merging, with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 and κs − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) The unstable domain U ˜ R in tip plane is enclosed between the jump lines J ˜ R (solid red/blue, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (108)) and takes the shape of a deformed ring with a wider (narrower) width at strongest (weakest) pinning near the solid dots (crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Red/blue dashed lines mark the landing positions L ˜ R of the vortex tips and are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b) Focus on the narrowing in the unstable domain close to the merger points (crosses) at ˜φ = π/2 in the polar coordinates ˜R, ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (c) The bistable do- main B ¯ R in asymptotic space is a narrow ring (colored in magenta) thicker (thinner) at points of strongest (weakest) pinning near ¯φ = 0, π (¯φ = ±π/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' black lines correspond to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (112).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (d) Focus on the constriction in the bistable domain close to ¯φ = π/2 in polar coordinates ¯R, ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Red/blue colors indicate different vortex configurations as quantified through the order parameter ˜R − ˜Rm(¯φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' find δ ˜φmax = δ ¯φmax ≈ � 1 − κm − 1 2ϵ ≈ � 1 − κs 2ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (116) The corresponding geometries of U ˜R and B ¯R are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 11 for 1 − κs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, δ ˜φmax vanishes at merging for κs = 1 (or κm − 1 ≈ 2ϵ), in agreement, to order ϵ, with the exact result (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pushing the Labusch parameter beyond the merger with κs > 1 or κm > (1 + ϵ)2 ≈ 1 + 2ϵ, the unstable and bistable regimes U ˜R and B ¯R change their topology: they develop a (non-simply connected) ring-like geome- try with separated inner and outer edges that are a finite distance apart in the radial direction at all angles ˜φ and ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The situation after the merger is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 12 for 23 κs − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1, with the merging points ˜Rs and ¯Rs marked by crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The merging of the unstable domains at the saddle point ˜Rs is a general feature of irregular pinning poten- tials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the next section, we will analyze the behavior of the unstable domains close to a saddle point ˜Rs of the Hessian determinant D( ˜R) and obtain a universal description of their geometry close to this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We will see that the geometry associated with this merger is of a hyperbolic type described by γ˜u2 + δ˜v2 = 2 ¯C(κs − 1), γ > 0 and δ < 0 (assuming no skew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The change in topology then is driven by the sign change in κs − 1: before merging, κs < 1, the hyperbola is open along the unstable (radial) direction ˜u, thus separating the two un- stable regions, while after merging, κs > 1, the hyperbola is open along the transverse direction ˜v, with the ensuing passage defining the single, non-simply connected, ring- like unstable region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' MERGER POINTS The merging of unstable and bistable domains is a gen- eral feature of irregular pinning potentials that is relevant beyond the simple example of a weakly anisotropic uni- axial defect discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, while the exact geometries of U ˜R and B ¯R depend on the precise shape of the pinning potential, their behavior close to merging is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Below, we will study this universal behavior by generalizing the expansions of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III to saddle points ˜Rs of the determinant D( ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As with the onset of strong pinning, the merger of two domains induces a change in topology in the unstable and bistable domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we will discuss these topological aspects of onsets and mergers in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V D and VI below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Expansion near merger Following the strategy of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III, we expand the en- ergy functional around a saddle point ˜Rs of the determi- nant D( ˜R) in order to obtain closed expressions for the unstable and bistable domains at merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In doing so, we again define local coordinate systems (˜u, ˜v) and (¯u, ¯v) in tip- and asymptotic space centered at ˜Rs and ¯Rs, where the latter is associated with ˜Rs through the force balance equation (38) in the original laboratory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, we fix our axes such that D( ˜Rs) is a local maximum along the (unstable) u- and a local minimum along the (stable) v-direction of the saddle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the mixed term ∝ ˜u˜v is absent from the expansion (as the Hessian matrix is symmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, the vanishing slopes at the saddle point, see (98), imply the absence of terms ∝ ˜u3 and ∝ ˜u2˜v in the expansion and dropping higher- order terms (corresponding to double-primed terms in (40)), we arrive to the expression epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) = ¯C 2 (1 − κs) ˜u2 + ¯C + λ+,s 2 ˜v2 + as 2 ˜u˜v2 + αs 4 ˜u2˜v2 + βs 6 ˜u3˜v + γs 24 ˜u4 − ¯C¯u˜u − ¯C¯v˜v, (117) with κs ≡ −λ−( ˜Rs)/ ¯C, λ+,s ≡ λ+( ˜Rs) and the remain- ing coefficients defined in analogy to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The most important term in the expansion (117) is the curvature term ¯C(1 − κs) ˜u2/2 along the unstable direc- tion u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As before in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III B, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (58), the coefficient (1−κs) changes sign at some value of the pinning strength and will serve as the small parameter in our considera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The higher-order terms in the expansion (117) are constrained by the saddle condition (99), implying that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (48) and (50)) γsδs − β2 s < 0 (118) with δs ≡ αs − 2a2 s ¯C + λ+,s (119) (for the saddle point there is no condition on the trace of the Hessian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The mapping of the two-dimensional pin- ning energy (117) to an effective one-dimensional Landau theory (A30) of the van der Waals kind is discussed in Appendix A 2, both before and after merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Unstable domain U ˜ R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Jump line J ˜ R The boundary of the unstable domain U ˜R is deter- mined by the jump condition D( ˜Rs,jp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Making use of the expansion (117) and keeping only terms quadratic in ˜u, ˜v, the edges δ ˜Rs,jp = (˜us,jp, ˜vs,jp) of U ˜R (measured relative to ˜Rs) are given by the solutions of the quadratic form (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (53)) [γs ˜u2 + 2βs ˜u˜v + δs ˜v2] ˜Rs,jp = 2 ¯C(κs − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (120) Equation (120) describes a hyperbola (centered at ˜Rs) as its associated determinant is negative, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Again, (120) can be cast in the form of a matrix equation δ ˜RT s,jpMs,jpδ ˜Rs,jp = ¯C(κs − 1), (121) with Ms,jp given by Ms,jp = � γs/2 βs/2 βs/2 δs/2 � (122) with det Ms,jp = (γsδs − β2 s)/4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13, the geometry of the unstable domain U ˜R changes drasti- cally when 1 − κs changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Before merging, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', for 24 −3 0 −5 0 −3 0 −25 0 (a) (b) ˜u/ξ√1 − κs ˜v/ξ√1 − κs ˜u/ξ√κs − 1 ˜v/ξ√κs − 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Jump lines J ˜ R (solid red/blue) and landing lines L ˜ R (dashed red/blue) in tip space ˜R (in units of ξ), with the hy- perbola J ˜ R defining the edge ∂U ˜ R of the unstable domain U ˜ R, before (a) and after (b) merging, for 1 − κs = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Param- eters are λ−,s = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='25 ep/ξ2, λ+,s = 0, and as ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='035 ep/ξ3, αs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='025 ep/ξ4, βs = 0, γs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='68 ep/ξ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A finite skew parameter βs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='025ep/ξ4 tilts the hyperbola away from the axes (dotted curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Crosses correspond to the vertices (125) and (129) of the hyperbola before and after merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pairs of solid and open circles connected via long arrows are ex- amples of pairs of jumping- and landing tip positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' After merging, see (b), the unstable domain U ˜ R is connected along the ˜v-axis, dividing the tip coordinate plane into two sepa- rate regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The jumping and landing hyperbolas coincide at their vertices before merging, see (a), but not thereafter, see (b), where the jumping and landing hyperbolas are sepa- rated (vertices on L ˜ R are marked with open red/blue stars) and no contact point is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note the rotation by 90 de- grees of the unstable direction with respect to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 11(b) and 12(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1−κs > 0, the unstable domain (top and bottom regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(a)) is disconnected along the stable v-direction and the two red/blue branches of the hyperbola (120) de- scribe the tips of U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When κs goes to unity, the tips of the unstable domain merge at the saddle point ˜Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' After merging, the unstable domain extends continuously from the top to the bottom in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(b) with a finite width along the unstable u-direction, similarly to the isotropic case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Correspondingly, the two (red and blue) branches of the hyperbola (120) now describe the edges of U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Solving the quadratic equation (120) before merging, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', 1 − κs > 0, we find solutions ˜us,jp(˜v) away from a gap along the stable v-direction, ˜us,jp(|˜v| ≥ ˜vs,c) = − 1 γs � βs˜v ± � 2γs ¯C(κs − 1) − (γsδs − β2s)˜v2 � , (123) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (123) has real solutions in the (unbounded) in- terval |˜v| ≥ ˜vs,c, with ˜vs,c = � 2γs ¯C(1 − κs)/|γsδs − β2s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (124) For the uniaxial defect (94) before merging, this gap cor- responds to a splitting of U ˜R along the stable angular direction, producing two separated domains as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The coordinates (˜us,jp(±˜vs,c), ±˜vs,c) give the positions of the vertices δ ˜R< s,c,± (relative to ˜Rs) of the hyperbola before merging, δ ˜R< s,c,± = ± (−βs/γs, 1) ˜vs,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (125) These are marked as black crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(a) (note the rotation in the geometry as compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 11(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We denote the distance between these vertices by δv<, defining a gap of width ∝ √1 − κs given by δv< = 2|δ ˜R< s,c,±| = 2 �� γs + β2s γs � ¯C(1 − κs) |γsδs − β2s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (126) After merging, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', for κs − 1 > 0, the (local) topology of U ˜R has changed as the gap along v closes and reopens along the unstable u-direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' as a result, the two sepa- rated domains of U ˜R have merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The two branches of the hyperbola derived from (120) are now parametrized as ˜vs,jp(|˜u| ≥ ˜us,e) = − 1 δs � βs˜u ± � 2δs ¯C(κs − 1) − (γsδs − β2s)˜u2 � , (127) with ˜us,e = � 2δs ¯C(κs − 1)/|γsδs − β2s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (128) The corresponding unstable domain is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For the uniaxial defect (94) after merging, this gap now corresponds to the finite width of U ˜R along the radial direction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 12(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The coordinates (±˜us,e, ˜vs,jp(±˜us,e)) for the vertices ˜R> s,e,± read δ ˜R> s,e,± = ± � 1, −βs δs � ˜us,e (129) and correspond to the points of closest approach in the branches of the hyperbola (120);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these are again marked as black crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(b) but are no longer associated with critical points (we index these extremal points by ‘e’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Their distance δu> is given by δu> = 2|δ ˜R> s,e,±| = 2 �� δs + β2s δs � ¯C(κs − 1) |γsδs − β2s|, (130) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the smallest width in U ˜R grows as ∝ √κs − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As discussed above and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13, the solu- tions of the quadratic form (120) before and after merg- ing are unbounded for every value of κs − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As a conse- quence, neglecting the higher order terms in the determi- nant D( ˜R) is valid only in a narrow neighborhood of the 25 saddle ˜Rs, where the boundaries of U ˜R have the shape of a hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Away from the saddle, these higher or- der terms are relevant in determining the specific shape of the unstable and bistable domain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the ring-like structures of U ˜R and B ¯R in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Landing line L ˜ R To find the second bistable vortex tip configuration ˜Rs,lp associated to the edges of B ¯R before and after merg- ing, we repeat the steps of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For the jump vector ∆ ˜Rs = ˜Rs,lp − ˜Rs,jp, we find the result ∆˜us(˜v) = −3 (γs ˜us,jp(˜v) + βs ˜v) /γs, (131) ∆˜vs(˜v) = − � as/( ¯C + λs,+) � ˜v ∆˜us(˜v), (132) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (65) and (66) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, we make use of the parametrization for the jump coordinate ˜us,jp(˜v) in (123) before merging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' after merging, the above result is still valid but should be expressed in terms of the parametrization ˜vs,jp(˜u) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (127).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The landing positions ˜Rs,lp = ˜Rs,jp + ∆ ˜Rs arrange along the branches L ˜R of a hyperbola in ˜R-space that are described by the matrix equation δ ˜RT s,lpMs,lp δ ˜Rs,lp = ¯C(κs − 1), (133) with the landing matrix now given by Ms,lp = 1 4Ms,jp + � � 0 0 0 3 4 �δs 2 − β2 s 2γs � � � (134) with det Ms,lp = (γsδs − β2 s)/16 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Before merging, the vertices of the landing and jumping hyperbolas coin- cide and the jump (131)–(132) vanishes at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Moreover, as for the contact points (67) close to onset of strong pinning, the tangent to the jumping and landing hyperbolas at the vertices is parallel to the u-direction, as is visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For κs = 1, the tips of U ˜R merge and both the jumping and landing hyperbolas coincide at ˜Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' After merging, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', for κs − 1 > 0, the condition ∆˜us = ∆˜vs = 0 cannot be realized along the hyperbola (120) and the jumping and landing lines separate completely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' as a result, both the jumping distance ∆ ˜Rs as well as the jump in energy ∆epin are always finite (see also Appendix A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, after merging the landing hyperbola (133) has vertices δ ˜Rs,v,± = ± � 1, − γsβs (4γsδs − 3β2s) � ˜us,v, (135) with ˜us,v = � 2 ¯C(κs − 1)(4γsδs − 3β2s) γs(γsδs − β2s) (136) 0 2 −5 0 0 4 −5 0 −5 0 5 ¯u/ξ(κm − 1) ¯v/ξ√κm − 1 (a) ˜u/ξ√κm − 1 ¯u/ξ(κm − 1) ¯v/ξ√κm − 1 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Bistable domain B ¯ R in asymptotic space ¯R before (a) and after (b) merging, for 1 − κs = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01 and parame- ters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) Before merging, the bistable domain B ¯ R consists of two parts, corresponding to the two unstable regions U ˜ R in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These terminate in the cusps at ¯R< s,c,± that approach one another along the dashed parabola (139) to merge at κs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Red/blue colors indicate different vortex configurations as quantified through the order param- eter ˜u − ˜um(¯v), while magenta is associated to the bistable region B ¯ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Colored dots mark the asymptotic positions asso- ciated to the pairs of jump positions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b) After merging, the bistable domain is continuously connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the cusps/critical points have vanished and the dashed parabola turns into the branch cutting line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The black crosses now mark the positions of strongest pinching of B ¯ R, the colored dots mark the asymptotic positions associated to the pairs of tip positions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' different from the jumping hyperbola in (129).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At these points, the stable and unstable hyperbolas are tangent to the v-direction, as is visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In section Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V D below, we will take a step back from the local analysis of the unstable domain U ˜R close to a saddle point ˜Rs and consider the evolution of its geometry across the merging transition from a global perspective using specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Elaborating on the analysis of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III E, we will provide a simple argument explaining the absence of contact points between jump and landing lines after merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, we discuss the two possible roles of mergers as changing the number of components of U ˜R or changing the connectivity of U ˜R between simply and non-simply connected areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Before doing so, we discuss the behavior of the bistable region B ¯R close to merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Bistable domain B ¯ R The set of asymptotic positions corresponding to U ˜R before and after merging, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the bistable do- 26 main B ¯R, can be found by systematically repeating the steps in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Applying the force balance equation ∇Repin(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) ��� ˜R = 0 to the energy expansion (117), we find the counterpart of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (69), ¯C¯u = ¯C(1 − κs)˜u + as 2 ˜v2 + γs 6 ˜u3 + βs 2 ˜u2˜v + αs 2 ˜u˜v2, ¯C¯v = ( ¯C + λs,+)˜v + as ˜u˜v + βs 6 ˜u3 + αs 2 ˜u2˜v, (137) relating tip and asymptotic positions close to merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As for the unstable domain, the topology of B ¯R depends on the sign of 1 − κs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bistable domain B ¯R before merging is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 14(a) for 1 − κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' It consists of two parts, corresponding to the two pieces of U ˜R for 1−κs > 0, that terminate at the cusps ¯R< s,c,±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The latter are related to the vertices ˜R< s,c,± of the jumping hyperbola through the force balance equation (137), δ ¯R< s,c,± ≈ �� as/2 ¯C � ˜v2 s,c, ± � 1 + λs,+/ ¯C � ˜vs,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (138) For finite values of (1 − κs), the cusps are separated by a distance 2|δ ¯R< s,c,±| ≈ 2 � 1 + λs,+/ ¯C � ˜vs,c ∝ √1 − κs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' They approach one another along the parabola ¯us,0 ≈ a 2 ¯C 1 (1 + λ+/ ¯C)2 ¯v2 s,0, (139) see the black dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 14, with higher-order cor- rections appearing at finite skew β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' After merging, this line lies within B ¯R and defines the branch crossing line, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' After merging, when κs − 1 > 0, the cusps have van- ished and the edges have rearranged to define a connected bistable region, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 14(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The extremal points of the two edges are found by evaluating the force balance equation (137) at the vertices ˜R> s,e,±, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (129), to lowest order, δ ¯R> s,e,± ≈ βs δs � as 2 ¯C βs δs ˜u2 s,e, ∓ � 1 + λs,+ ¯C � ˜us,e � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (140) For finite values of (κs − 1), these points are separated by a distance 2|δ ¯R> s,e,±| ≈ 2 � 1 + λs,+/ ¯C � (βs/δs)˜us,e ∝ √κs − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that the extremal points ¯R> s,e,± are no longer associated to cusps or critical points as these have disappeared in the merging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When the skew parameter vanishes as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 14, βs = 0, higher-order terms in (κs − 1) in the force-balance equation (137) be- come relevant in determining the positions ¯R> s,e,±, sep- arating them along the unstable u-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In this case, we obtain a different scaling for their distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', |δ ¯R> s,e,±| ∝ (1 − κs)3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Topological aspect of mergers In order to discuss the topological aspect of a merger, it is convenient to consider some specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' IV, we have analyzed the case of a uniaxial defect with a quadrupolar anisotropy δep ∝ ϵ sin2 ˜φ in the pinning potential, see (105), that produced a degenerate onset at symmetric points [±˜xm, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, we choose again a weakly anisotropic defect centered in the origin but with a dipolar deformation δep ∝ ϵ cos ˜φ that results in an angle-dependent Labusch parameter κm(˜φ) = κm − ϵ cos ˜φ, (141) see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The strong pinning onset of such a defect then appears in an isolated point on the negative x-axis, with the unstable ellipse U ˜R deforming with increasing κm into a horseshoe that is open on the positive x-axis— the closing of the horseshoe to produce a ring, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 15, then corresponds to the local merger shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With this example in mind, we can repeat the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III E: The unstable eigenvector v−(Rjp) points ra- dially outwards from the origin over the entire horseshoe, including the merging region at positive x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' On the other hand, the tangent to the boundary ∂U ˜R rotates forward and back along the horseshoe as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 15 (we at- tribute a direction to ∂U ˜R with the convention of follow- ing the boundary with the unstable region on the left);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in fact, over most of the boundary, the tangent is simply or- thogonal to v−, with both vectors rotating together when going along ∂U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' At the ends of the horseshoe, however, the tangent locally aligns parallel (anti-parallel) to v− and the two vectors rotate (anti-clockwise) with respect to one another, with the total winding equal to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Af- ter the merger, this winding has disappeared, with the resulting ring exhibiting no winding in the tangent fields on the inner/outer boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' as a result, the contact points between the jump and landing lines have disap- peared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, the merger changes the topology of U ˜R from the simply-connected horseshoe to the non-simply connected ring, while the number of components in U ˜R has not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that the change in the relative winding is not due to crossing the singularity of the vec- tor field v− as alluded to in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III E—rather, it is the merger of the horseshoe tips that rearranges the bound- aries of U ˜R and make them encircle the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the above example, we have discussed a merger that changes the connectedness of U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' On the other hand, as we are going to show, a merger might leave the connect- edness of U ˜R unchanged, while modifying the number of components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the number of disconnected parts, in U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Let us again consider a specific example in the form of an anisotropic defect with a warped well shape, pro- ducing several (in general subsequent) onsets and merg- ers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 16, we consider a situation with three on- set points and subsequent individual mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' After the onset, the three ellipses define an unstable region U ˜R with three disconnected parts that are simply-connected each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This configuration is characterized by its number of components measuring C = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As two of the three ellipses merge, the number of components of U ˜R reduces to C = 2, the next merger generates a horseshoe that is 27 −2 0 −2 0 −2 0 −2 0 2 (a) (b) ˜y/ξ ˜x/ξ ˜x/ξ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Left: Unstable region U ˜ R for a defect with dipolar asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Upon the onset of strong pinning, an unstable ellipse appears to the left of the defect center (black solid dot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With increasing pinning strength (decreasing ¯C) the ellipse grows and deforms into a horseshoe geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The unstable eigenvector field v− (red arrows) points radially outward away from the defect center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The tangent field to the boundary ∂U ˜ R (black arrows) follows the unstable direction at an angle of π/2 over most of ∂U ˜ R, with the exception of the two turning points where the tangent rotates by π with respect to v−, producing a relative winding of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Right: After the merger of the turning points the unstable region U ˜ R changes topology and assumes the shape of a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The windings of the tangent field with respect to the eigenvector-field v− vanish separately for both boundaries of U ˜ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' still simply-connected with C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The final merger pro- duces a ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' while the number of components remains unchanged, C = 1, the unstable area assumes a non- simply connected shape with a ‘hole’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we associate the index H = 1 with the appearance of this hole within U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In physics terms, the last merger producing a hole in U ˜R is associated with the appearance of a pinned state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the unstable ring separates stable tip positions that are asso- ciated with pinned and free vortex configurations residing at small and large radii, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Defining the (topological) characteristic χ ≡ C−H, we see that χ changes by unity at every onset and merger, either through an increase (for an onset) or decrease (for a merger) in the number of components C → C ± 1, or through the appearance of a hole (in a merger) H → H +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, the quantity χ is known as the Euler char- acteristic of a manifold and describes its global topolog- ical properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' it generalizes the well known Euler char- acteristic of a polyhedron to surfaces and manifolds29, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' VI below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, Morse theory30 connects the Euler characteristic with the local differential properties (minima, maxima, saddles) of that manifold, hence es- tablishing a connection between local onsets and mergers (at minima and saddles of D( ˜R)) and the global proper- ties of U ˜R such as the appearance of new pinned states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' VI below, we consider the general case of a ran- dom pinning landscape in two dimensions and discuss the connection between local differential and global topolog- ical properties of U ˜R in the light of Morse theory—the topology of bistable domains B ¯R then follows trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' −2 0 2 −2 0 −2 0 2 −2 0 2 −2 0 −2 0 −2 0 −2 0 2 (a) (b) (c) (d) ˜y/ξ ˜y/ξ ˜x/ξ ˜x/ξ C = 3 H = 0 χ = 3 C = 2 H = 0 χ = 2 C = 1 H = 0 χ = 1 C = 1 H = 1 χ = 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The unstable domain U ˜ R starting out with C = 3 components in (a) changes topology in three steps: after the first (b) and second (c) mergers the number of components C has changed from three in (a) to two in (b) to one in (c), lead- ing to a horseshoe shape of U ˜ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The third merger closes the horseshoe to produce the ring geometry in (d) characterized by the coefficients C = 1 and H = 1 (H denotes the number of ‘holes’ in U ˜ R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the Euler characteristic χ = C − H changes by unity in every merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' U ˜ R OF A TWO-DIMENSIONAL PINSCAPE We consider a two-dimensional pinning landscape ep(R), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', as produced by a superposition of several (anisotropic Lorentzian) defects residing in the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the figures 17 and 18, we analyse two spe- cific cases with n = 3 and n = 2 defects as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (94) with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 and positions listed in Tables I and II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' these produce unstable landscapes U ˜R of considerable complexity already, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17(a) and 18(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Our de- fects are compact with ep(R) → 0 vanishing at R → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' as a result, epin becomes flat at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that a dense assembly of uniformly distributed individual defects pro- duces a random Gaussian pinning landscape, as has been shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, we are interested in the evolution of the unstable and bistable domains U ˜R and B ¯R associated with the 2D pinning landscape epin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we focus on the unstable domain U ˜R, with the properties of the bistable domain B ¯R fol- lowing straightforwardly from the solution of the force balance equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Unlike the analysis above that is centered on special points of U ˜R, ellipses near onset and hyperbolas near mergers, here, we are interested in the global properties of the unstable region produced by a generic (though still two-dimensional) pinscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 28 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Positions and relative weights of 3 uniaxially anisotropic Lorentzian defects in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17 as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' x/ξ y/ξ weight defect #1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='65 defect #2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='19 1 defect #3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='67 1 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Positions and relative weights of 2 uniaxially anisotropic Lorentzian defects in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 18 as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' x/ξ y/ξ weight defect #1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='33 1 defect #2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='76 1 As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III above, the unstable region U ˜R associated with strong pinning is determined by the con- dition D( ˜R) = 0 of vanishing Hessian determinant, more precisely, by the competition between the lowest eigen- value λ−( ˜R) of the Hessian matrix Hij of the pinning potential ep(R) and the effective elasticity ¯C, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In order to avoid the interference with the second eigenvalue λ+( ˜R) of the Hessian matrix, we consider the shifted (by ¯C) curvature function Λ ¯ C( ˜R) ≡ ¯C + λ−( ˜R), (142) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the relevant factor of the determinant D( ˜R) = [ ¯C + λ−( ˜R)][ ¯C + λ−( ˜R)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The condition Λ ¯ C( ˜R) = 0 (143) then determines the boundaries of U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The above problem can be mapped to the problem of cutting a surface, where Λ ¯ C( ˜R) is interpreted as a height-function over R2 that is cut at zero level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the elas- ticity ¯C then plays the role of a shift parameter that moves the function λ−( ˜R) downwards in height with de- creasing ¯C (that corresponds to increasing the relative pinning strength of the pinscape in physical terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As ¯C is decreased to compensate the absolute minimum of λ−( ˜R) < 0, ¯C + λ−( ˜R) = 0, strong pinning sets in lo- cally at ˜Rm for the first time in the form of an unstable ellipse U ˜R, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17(b) for our specific example with three defects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the Labusch parameter κ( ˜R) evaluated at the point ˜Rm defines κm, the parameter tuned in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Decreasing ¯C further, this ellipse grows and deforms, while other local minima of λ−( ˜R) produce new discon- nected parts of U ˜R, a situation illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17(c) where four ‘ellipses’ have appeared around (local) minima (blue filled dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A further increase in pinning strength (decrease in ¯C) continuous to deform these ‘ellipses’ and adds three new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As the first saddle drops below the zero level (red cross), two components merge and the number of components decreases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17(d), we have three below-zero saddles and only four components re- main, C = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17(e) four further mergers have reduced C to 1 as the corresponding saddles drop below zero level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This produces a single non-simply connected component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', C = 1 and a hole, increasing the num- ber of holes H from zero to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The last merger leading to (f) finally leaves C = 1 but cuts the stable region in- side the ring into two, increasing the number of holes to H = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This sequence of onsets and mergers is conveniently described in the topographic language introduced in sec- tion IV that interprets stable tip regions as land mass (green with bright regions indicating higher mountains in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17) and unstable regions as lakes (flat blue with (below-water) height levels indicated by thin black lines), with the height Λ ¯ C = 0 defining the water level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The se- quence (b) to (f) then shows the flooding of the landscape as pinning increases ( ¯C decreasing), with white dot min- ima turning blue at strong pinning onsets and white cross saddles turning red at mergings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' maxima in the landscape are shown as black open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Note that we distinguish critical points (minima, saddles) residing below (blue and red) and above (white) water level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Similarly, a (local) maximum above sea level (black open dot) turns into a blue open dot as it drops belop sea level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' such an event is missing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17 but can be produced with other con- figurations of defects, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 18 where the curvature landscape for two defects is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The above discussion relates the local differential prop- erties of the function Λ ¯ C( ˜R) < 0, minima and saddles, to the global topological properties of U ˜R, its number of components C(U ˜R) and holes H(U ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This connection between local and global properties is conveniently dis- cussed within Morse theory30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Before presenting a gen- eral mathematical formulation, let us discuss a simple heuristic argument producing the result relevant in the present context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in doing so, we make use of the above topographic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Starting with the minima of the function Λ ¯ C( ˜R), a new disconnected component appears in U ˜R whenever the minimum drops below sea level as ¯C is decreased, that produces an increase C → C + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' With the further decrease of ¯C, these disconnected regions expand and merge pairwise whenever a saddle point of Λ ¯ C( ˜R) goes below sea level, thereby inducing a change in the topol- ogy of U ˜R by either reducing the number of components C → C −1 (keeping H constant) or leaving it unchanged (changing H → H + 1), see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the example with the horseshoe closing up on itself in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The below sea- level minima and saddles of Λ ¯ C( ˜R) can naturally be iden- tified with the vertices and edges of a graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the edges in the graph then define the boundaries of the graph’s faces (the same way as the vertices are the boundaries of the edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For a connected graph, Euler’s formula then tells us that the number V of vertices, E of edges, and F of faces are constrained via V − E + F = 1 (not counting the outer face extending to infinity) and a graph with C components satisfies the relation C = V − E + F as follows from simple addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We have already identified minima and saddles of 29 4 2 0 2 4 4 2 0 2 4 2 0 2 4 4 2 0 2 4 κm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 4 2 0 2 4 4 2 0 2 κm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='8 4 2 0 2 4 4 2 0 2 4 κm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='5 4 2 0 2 4 2 0 2 κm = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='4 4 2 0 2 4 2 0 2 4 κm = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='7 (a) (b) (c) (d) (e) (f) ˜y/ξ ˜y/ξ ˜y/ξ ˜x/ξ ˜x/ξ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) Grayscale image of the pinning potential land- scape ep( ˜R), with the three diamonds marking the positions of the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b)–(f) Shifted curvature function Λ ¯ C( ˜R) ver- sus tip position ˜R for increasing values of κm (decreasing ¯C) as we proceed from (b) to (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We make use of the topographic interpretation with positive values of Λ ¯ C marked as landmass (greenish colors, with low/high elevation in dark/light green) and negative values of Λ ¯ C constituting U ˜ R in flat light blue (height levels are shown by thin black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinscape in (a) produces a curvature landscape with 7 minima (solid dots), 4 maxima (open dots), and 10 saddles (crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sev- eral unstable regions U ˜ R appear (solid dots turn blue) and merge (crosses turn red) to change the topology of U ˜ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The Euler characteristic χ(U ˜ R) = m − s + M = 1 − 0 + 0 = 1 in (b) changes to χ(U ˜ R) = 4 in (c) and (d), drops to χ(U ˜ R) = 0 in (e) and χ(U ˜ R) = −1 in (f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' indeed, U ˜ R in (f) has one component C = 1 and two holes H = 2, reproducing χ(U ˜ R) = C − H = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Λ ¯ C( ˜R) < 0 with vertices and edges of a graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' denot- ing the number of below sea-level minima and saddles by m and s, we have V = m and E = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' It remains to express the number F of faces in terms of critical points of the surface Λ ¯ C( ˜R) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, the faces of our graph are associated with maxima of the function Λ ¯ C( ˜R): following the boundaries of a face, we cross the corresponding saddles with the function Λ ¯ C( ˜R) curving upwards away from the edges, implying that the faces of our graph include maxima of Λ ¯ C( ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These maxima manifest in two possible ways: either the face contains a single below sea-level maximum or a single above sea- level landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The above sea-level landscape comprises at least one maximum but possibly also includes other extremal points that we cannot analyse with our knowl- edge of the below sea-level function Λ ¯ C( ˜R) < 0 only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' we therefore call the above sea-level landscape a (single) hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The appearance of a single maximum or hole is owed to the fact that faces are not split by a below sea- level saddle as these have already been accounted for in setting up the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Let us denote the number of (below sea-level) maxima by M and the number of holes by H, then F = H + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Combining this last expression with Euler’s formula and regrouping topological coefficients C(U ˜R) and H(U ˜R) on one side and extremal points m[Λ ¯ C( ˜R)], s[Λ ¯ C( ˜R)], and M[Λ ¯ C( ˜R)] on the other, we arrive at the Euler charac- teristic χ ≡ C − H and its representation through local differential properties, χ(U ˜R) ≡ [C − H]U ˜ R = [m − s + M]Λ ¯ C( ˜R)<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (144) The result (144) follows rigorously from the Euler- Poincar´e theorem29,30 in combination with Morse’s theorem30, with the former expressing the Euler char- acteristic χ(U ˜R) through the so-called Betti numbers bi(U ˜R), χ(U ˜R) ≡ 2 � i=0 (−1)ibi(U ˜R), (145) where the i-th Betti number bi(U ˜R) = Dim[Hi(U ˜R)] is given by the dimension or rank of the i-th (singular) ho- mology group Hi(U ˜R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In colloquial terms, the Betti numbers bi count the number of ‘holes’ in the mani- fold with different dimensions i: the zeroth Betti number gives the number of components b0 = C of U ˜R, the first Betti number b1 = H counts the holes, and the second Betti number refers to cavities, here b2 = 0 for our open manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Hence, we find that the Euler characteristic is given by the number of components and holes in U ˜R, χ(U ˜R) = C(U ˜R) − H(U ˜R), (146) in agreement with the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V D and (144).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Morse theory30 then provides a connection between the topological properties of the manifold U ˜R and the local differential properties of the surface Λ ¯ C( ˜R) < 0 defining it: with Ci the number of critical points with index i of the surface Λ ¯ C( ˜R) < 0 (the index i counts the number of negative eigenvalues of the Hessian matrix evaluated at the critical point),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the Euler characteristic χ(U ˜R) relates the manifold’s topology to the number and properties of 30 −4 −2 0 2 4 −4 −2 0 2 −4 −2 0 2 4 −4 −2 0 2 4 κm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 −4 −2 0 2 4 −4 −2 0 2 κm = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='3 −4 −2 0 2 4 −4 −2 0 2 4 κm = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='0 −4 −2 0 2 −4 −2 0 2 κm = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='1 −4 −2 0 2 −4 −2 0 2 4 κm = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='0 (a) (b) (c) (d) (e) (f) ˜y/ξ ˜y/ξ ˜y/ξ ˜x/ξ ˜x/ξ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (a) Grayscale image of the pinning potential land- scape ep( ˜R), with the two diamonds marking the positions of the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (b)–(f) Shifted curvature function Λ ¯ C( ˜R) (in topographic coloring, see caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17) versus tip po- sition ˜R for increasing values of κm as we proceed from (b) to (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The pinscape in (a) produces a curvature landscape with 6 minima (solid dots), 4 maxima (open dots), and 9 sad- dles (crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Upon increasing κm, several unstable regions U ˜ R appear (solid dots turn blue) and merge (crosses turn red) to change the topology of U ˜ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The Euler characteristic χ(U ˜ R) = m − s + M = 1 = C in (b), remains χ(U ˜ R) = 1 in (c), but with C = 2 and H = 1, changes to χ(U ˜ R) = −1 in (d), and χ(U ˜ R) = −3 with one component C = 1 and four holes H = 4 in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In going from (e) to (f) two of the max- ima (black open dots turn blue) drop below zero, producing a characteristic χ(U ˜ R) = 6 − 9 + 2 = −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' indeed, U ˜ R in (f) has one component C = 1 and two holes H = 2, reproducing χ(U ˜ R) = C − H = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' critical points, χ(U ˜R) = 2 � i=0 (−1)iCi(Λ ¯ C < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (147) For our 2D manifold the coefficients Ci count the minima C0 = m, the number of saddles C1 = s, and C2 = M refers to the number of maxima, hence, χ(U ˜R) = [m − s + M]Λ ¯ C<0 (148) and the combination with (146) produces the result (144) anticipated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Summarizing, knowing the number of critical points m, M, and s of the seascape, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', its local differential prop- erties, we can determine the global topological aspects of the pinning landscape via the evaluation of the Euler characteristic χ(U ˜R) with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (148).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The lat- ter then informs us about the number C of unstable do- mains in U ˜R where locally pinned states appear and the number of holes H in U ˜R where globally distinct pinned states show up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, the outer boundaries of the lakes, of which we have C components, are to be as- sociated with instabilities of the free vortex state, while inner boundaries (or boundaries of holes, which count H elements) tell about instabilities of pinned states, hence the Betti numbers C and H count different types of in- stabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' It would then have been nice to determine the separate topological coefficients C and H individually— unfortunately, χ(U ˜R) as derived from local differential properties provides us only with the difference C − H between locally and globally pinned areas and not their individual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Nevertheless, using Morse theory, we could connect our discussion of local differential proper- ties of the pinning landscape in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III A and V A with the global pinning properties of the pinning energy land- scape as expressed through the topology of the unstable domain U ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Regarding our previous examples, the isotropic and uniaxial defects, we remark that for the latter the two simultaneous mergers on the y-axis produce a reduction in C = 2 → 1 and an increase of H = 0 → 1 and hence a jump from χ = 2 to χ = 0 in one step, as expected for two simultaneous mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The symmetry of the isotropic de- fect produces a (degenerate) critical line at ˜Rm rather than a critical point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' adding a small perturbation ∝ x3 breaks this symmetry and produces the horseshoe geom- etry discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V D above that is amenable to the standard analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A last remark is in place about the topological prop- erties in dual space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', of bistable regions B ¯R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Here, the mergers produce another interesting phenomenon as viewed from the perspective of its thermodynamic ana- logue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, the merger of deformed ellipses in tip- space corresponds to the merger of cusps in asymptotic space, what translates to the vanishing of critical points and a smooth continuation of the first-order critical and spinodal lines in the thermodynamic analogue, see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We are not aware of a physical example in ther- modynamics that produces such a merger and disappear- ance of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 31 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' SUMMARY AND OUTLOOK Strong pinning theory is a quantitative theory describ- ing vortex pinning in the dilute defect limit where this complex many-body system can be reduced to an effec- tive single-pin–single-vortex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The accuracy of- fered by this theory then allows for a realistic description of the shape of the pinning potential ep(R) associated with the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While previous work focused on the simplest case of isotropic defects, here, we have general- ized the strong pinning theory to the description of arbi- trary anisotropic pinning potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Surprisingly, going from an isotropic to an anisotropic defect has quite aston- ishing consequences for the physics of strong pinning— this reminds about other physical examples where the removal of symmetries or degeneracies produces new ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While the strong pinning problem is quite a complex one requiring the use of numerical tools in general, we have identified several generic features that provide the essential physics of the problem and that are amenable to an analytic treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Specifically, these are the points of strong pinning onset and the merger points, around which the local expansions of the pinning poten- tial epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) in the tip coordinate ˜R allow us to find all the characteristics of strong pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In particular, we identify the instability region U ˜R in the vortex tip space (with coordinates ˜R) and the bistable region B ¯R in the space of asymptotic vortex positions ¯R as the main ge- ometric objects that determine the critical pinning force density Fpin, from which the critical current density jc, the technologically most relevant quantity of the super- conductor, follows straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While the relevance of the bistable region B ¯R was recognized in the past8–10, the important role played by the unstable region U ˜R went unnoticed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' When going from an isotropic defect to an anisotropic one, the strong pinning onset changes dramatically: while the unstable region U ˜R grows out of a circle of radius ∼ ξ and assumes the shape of a ring at κ > 1 for the isotropic situation, for an anisotropic defect the onset appears in a point ˜Rm and grows in the shape of an ellipse with increasing κm > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the location where this onset appears is given by the Hessian of epin, specif- ically, the point ˜Rm where its determinant touches zero first, det{Hess[epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R)| ¯R]} ˜Rm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The boundary of this ellipse defines the jump positions J ˜R associated with the strong pinning instabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' when combined with the landing ellipse L ˜R, these two ellipses determine the jump distance δ˜u of the vortex tip, from which follows the jump in the pinning energy ∆epin ∝ δ˜u4, which in turn deter- mines Fpin and jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The bistable region B ¯R in asymptotic vortex space comes into play when calculating the average critical force density Fpin opposing the vortex motion: while the vortex tip undergoes a complex trajectory includ- ing jumps, the vortex motion in asymptotic space ¯R is described by a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As this trivial trajectory in ¯R-space traverses the bistable region B ¯R, the vortex tip jumps upon exiting B ¯R, that produces the jump ∆epin and hence Fpin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Again, the shape of B ¯R changes when going from the isotropic to the anisotropic defect, assum- ing a ring of finite width around a circle of radius ∼ ξ in the former case, while growing in the form of a crescent out of a point for the anisotropic defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The new geometries associated with U ˜R and B ¯R then produce a qualitative change in the scaling behavior of the pinning force density Fpin ∝ (κm − 1)µ near onset, with the exponent µ changing from µ = 2 to µ = 5/2 when going from the isotropic to the anisotropic defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This change is due to the change in the scaling of the geometric size of B ¯R, with the replacement of the fixed radius ∼ ξ of the ring by the growing size of the crescent ∼ ξ(κm−1)1/2 [the exponent µ assumes a value µ = 3 for trajectories cutting the crescent along its short dimension of size ξ(κm − 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, for directed defects, the pinning force density Fpin(θ) depends on the impact angle θ relative to the unstable direction u and is aligned with u, except for a small angular regime close to θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This results in a pronounced anisotropy in the critical current density jc in the vicinity of the strong pinning onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A fundamental difference between the strong pinning onsets in the isotropic and in the anisotropic case are the geometries of the unstable U ˜R and bistable B ¯R re- gions: these are non-simply connected for the isotropic case (rings) but simply connected for the anisotropic de- fect (ellipse and crescent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The resolution of this funda- mental difference is provided by the second type of special points, the mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, for a general anisotropic de- fect, the strong pinning onset appears in a multitude of points, with unstable and bistable regions growing with κm > 1 and finally merging into larger areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Two exam- ples illustrate this behavior particularly well, the uniaxial defects with a quadrupolar and a dipolar deformation, see Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' IV and V D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the first case, symmetric on- set points on the x axis produce two ellipses/crescents that grow, approach one another, and finally merge in a ring-shaped geometry that is non-simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In the case of a dipolar deformation, we have seen U ˜R grow out of a single point with its ellipse expanding and de- forming around a circle, assuming a horseshoe geometry, that finally undergoes a merging of the two tips to pro- duce again a ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' similar happens when multiple U ˜R domains grow and merge as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 16 (a warped defect) and 18(c) (a 2D pinning landscape where four unstable domains have merged to enclose an ‘island’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These merger points are once more amenable to an analytic study using a proper expansion of epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) in ˜R around the merger point ˜Rs, the latter again de- fined by the local differential properties of the determi- nant det{Hess[epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R)| ¯R]}, this time not a minimum but a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rather than elliptic as at onset, at merger points the geometry is hyperbolic, with the sign change associated with increasing κs ≡ κ( ˜Rs) across unity pro- ducing a reconnection of the jump- and landing lines J ˜R 32 and L ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While the expansions of epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) are describing the local pinning landscape near onset and merging (and thus produce generic results), the study of the combined set of onset- and merger-points describe the global topological properties of U ˜R as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' VI: every new (non- degenerate) onset increases the number of components C in U ˜R, while every merger either decreases C or increases H, the number of ‘holes’ or ‘islands’ (or nontrivial loops in a non-simply connected region) in the pinning land- scape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' It is the ‘last’ merging producing a non-simply connected domain that properly defines a new pinned state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in our examples these are the closing of the two deformed ellipses in the uniaxial defect with quadrupo- lar deformation and the closing of the horseshoe in the defect with a dipolar deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Formally, the rela- tion between the local differential properties of the cur- vature function Λ ¯ C( ˜R) = ¯C + λ−( ˜R) [with λ−( ˜R) the lower eigenvalue of the Hessian of ep( ˜R)], its minima, saddles, and maxima, are related to the global topologi- cal properties of U ˜R as described by its Euler characteris- tic χ = C −H through Morse theory, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (144).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Such topological structures have recently attracted quite some interest, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', in the context of Fermi surface topologies and topological Lifshitz transitions31,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The physics around the onset points as expressed through an expansion of epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) resembles a Landau theory with ˜R playing the role of an order parameter and ¯R the dual variable corresponding to a driving field— here, ¯R drives the vortex lattice across the defect and ˜R describes the deformation of the pinned vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The endpoints of the crescent B ¯R correspond to critical end points as they appear in the Landau theory of a first- order transition line, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the Ising model in an external field or the van der Waals gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The boundary lines of B ¯R correspond to spinodal lines where phases become un- stable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', the termination of overheated/undercooled phases in the van der Waals gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The existence of criti- cal end points tells that ‘phases’, here in the form of dif- ferent pinning branches, are smoothly connected when going around the critical point, similar as in the gas– liquid transition of the van der Waals gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As the ‘last’ critical point vanishes in a merger, a well defined new phase, here a new pinned branch, appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Perspectives for future theoretical work include the study of correlations between anisotropic defects (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17 addressing isotropic defects) or the inclusion of ther- mal fluctuations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', creep (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13 and 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Fur- thermore, our discussion of the extended pinscape in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' VI has been limited to a two-dimensional pinning poten- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In reality, defects are distributed in all three di- mensions that considerable complicates the correspond- ing analysis of a full three-dimensional disordered pinning potential, with the prospect of interesting new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' On the experimental side, there are several possible applications for our study of anisotropic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' For a generic anisotropic defect, the inversion symmetry may be broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In this case, the pinning force along opposite directions is different in magnitude, as different jumps are associated to the boundaries of the bistable region B ¯R away from onset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', at sufficiently large values of κm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Reversing the current, the different critical forces then result in a ratchet effect33,34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This leads to a rec- tification of an ac current and hence a superconducting diode effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While for randomly oriented defects the pin- ning force is averaged and the symmetry is statistically restored, for specially oriented defects, the diode effect will survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Indeed, introducing nanoholes into the ma- terial, vortex pinning was enhanced23,35 and a diode ef- fect has been observed recently36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Generalizing strong pinning theory to this type of defects then may help in the design of superconducting metamaterials with inter- esting functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Furthermore, vortex imaging has always provided fascinating insights into vortex physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Recently, the SQUID-on-tip technique has been success- ful in mapping out a 2D pinning landscape in a film37 (including the observation of vortex jumps) that has in- spired a new characterization of the pinscape through its Hessian analysis26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the adaptation of this current-driven purely 2D setup to the 3D situation described in the present paper is an interesting challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, we recap the main benefits of this work in a nutshell: For one, we have established a detailed con- nection of the strong pinning transition with a the con- cept of first-order phase transitions in thermodynamics, with the main practical result that the scaling of the pinning force density Fpin ∝ (κm − 1)µ comes with an exponent µ = 5/2 when working with generic defects of arbitrary shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Second, we have found a mecha- nism, the breaking of a defect’s inversion symmetry, that produces rachets and a diode effect in superconducting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Third, we have uncovered the geometric struc- ture and its topological features that is underlying strong pinning theory, including a proper understanding of the appearance of distinguished pinned states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' While un- derstanding these geometric structures seems to be of rather fundamental/scholarly interest at present, future work may establish further practical consequences that can be used in the development of superconducting ma- terials with specific functional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Tom´aˇs Bzduˇsek, Gian Michele Graf, and Roland Willa for discussions and acknowledge financial support of the Swiss National Science Foundation, Divi- sion II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Appendix A: Effective 1D Landau theory The Landau-type pinning energies (18) and (117) for the vector order parameter (˜u, ˜v) involves a soft variable ˜u with a vanishing quadratic term ∝ (1 − κm) ˜u2, as well as a stiff one, ˜v, characterized by a finite elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' By 33 eliminating the stiff direction ˜v, we can arrive at a 1D Landau expansion for the order parameter ˜u that pro- vides us with the desired results for the unstable and bistable domains U ˜R and B ¯R near onset and merging in a very efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Close to onset We start with the two-dimensional Landau-type energy functional (58) epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) = ¯C (1 − κm) 2 ˜u2 + ¯C + λ+ 2 ˜v2 + a 2 ˜u˜v2 + α 4 ˜u2˜v2 + β 6 ˜u3˜v + γ 24 ˜u4 − ¯C ¯u˜u − ¯C ¯v˜v (A1) written in terms of the tip coordinates ˜u, ˜v measured rel- ative to ˜Rm, the position of the minimal determinant D( ˜R) at strong pinning onset, and with ˜u and ˜v aligned with the stable and unstable directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The expansion (A1) is anisotropic: the quadratic (elastic) co- efficient along the unstable ˜u-direction vanishes at the onset of strong pinning, while the one along the stable ˜v-direction stays positive and large, allowing us to ‘inte- grate out’ the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The asymptotic coordinates ¯u, ¯v assume the role of the driving (conjugate) fields for the tip positions (or order parameters) ˜u, ˜v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' the latter then are determined by the force equations ∂ ˜Repin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) = 0, ¯C¯u = ¯C(1 − κ)˜u + a 2 ˜v2 + γ 6 ˜u3 + β 2 ˜u2˜v + α 2 ˜u˜v2, (A2) ¯C¯v = ( ¯C + λ+)˜v + a ˜u˜v + β 6 ˜u3 + α 2 ˜u2˜v, (A3) see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (69), with δ ¯R = (¯u, ¯v) measured relative to ¯Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inspection of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A2) and (A3) shows that near the strong pinning onset, the Ansatz ˜u, ˜v, ¯v ∝ √κm − 1 and ¯u ∝ (κm − 1) produces a consistent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Solving the second equation (A3) for the stiff degree of freedom ˜v, we then find that ˜v ≈ ¯C¯v ¯C + λ++ a˜u ≈ ¯v 1+λ+/ ¯C � 1 − a/ ¯C 1+λ+/ ¯C ˜u � (A4) which is precise to order (κm − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inserting ˜v back into the force-balance equation (A2) for the unstable compo- nent ˜u, we find a cubic equation for ˜u (precise to order (κm −1)3/2) that is driven by a combination of ¯u and ¯v2, ¯C¯u − (a/2) ¯v2 (1 + λ+/ ¯C)2 ≈ � ¯C(1 − κm) + (δ/2) ¯v2 (1 + λ+/ ¯C)2 � ˜u + (β/2) ¯v (1 + λ+/ ¯C) ˜u2 + γ 6 ˜u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A5) Upon integration, we finally arrive at the effective one- dimensional Landau expansion for the 1D order parame- ter ˜u that is precise to order (κm−1)2 (up to an irrelevant shift ∝ ¯v2), eeff pin(˜u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯u, ¯v) = r(¯v) 2 ˜u2 + w(¯v) 6 ˜u3 + γ 24 ˜u4 −h(¯u, ¯v)˜u, (A6) with the coefficients r, w, and h defined as r(¯v) = � ¯C(1 − κm) + δ 2 ¯v2 (1 + λ+/ ¯C)2 � , w(¯v) = β ¯v (1 + λ+/ ¯C), h(¯u, ¯v) = ¯C¯u − a 2 ¯v2 (1 + λ+/ ¯C)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A7) The Landau-type energy function (A6) belongs to the van der Waals (gas-liquid) universality class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' its first-order transition line maps to the branch crossing line in the strong pinning problem, its spinodals correspond to the arcs of the crescent defining the bistable region B ¯R, and its critical points map to the two cusps of B ¯R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', in the strong pinning problem, the spinodals end in two critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The cubic term w˜u3/6 is determined by the skew parameter β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' in the absence of such a skew, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', for a ±˜v-symmetric unstable ellipse U ˜R, we have β = 0 and our problem assumes an Ising-type Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Let us begin with the determination of the critical co- efficients rc, wc, and hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' These are found by setting the first three derivatives of eeff pin(˜u) to zero [two spinodals (implying ∂˜ueeff pin = 0 and ∂2 ˜ueeff pin = 0) coalescing into a single point (→ ∂3 ˜ueeff pin = 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Setting the cubic derivative to zero, we find the order parameter ˜uc = −wc/γ ≈ −(β/γ)˜vc, (A8) where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A7) and the transformation ¯v ↔ ˜v in (A4) to leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The vanishing of the second derivative relates the crit- ical coefficients rc and wc, rc = w2 c/2γ, (A9) (where we have made use of ˜uc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Inserting the dependen- cies r(¯v) and w(¯v), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A7), we find that ¯v2 c (1 + λ+/ ¯C)2 = γ ¯C(κm − 1) 2 det Mjp , (A10) with det Mjp = (γδ − β2)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Using again Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A4) to leading order, we find that ˜vc ≈ � 2γ ¯C(κm − 1) γδ − β2 , (A11) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The critical endpoints of the 1D Landau theory then correspond to the touching points (67) of the unstable domain U ˜R δ ˜Rc,± = ± (−β/γ, 1) ˜vc, (A12) found before, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (67) with (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, the vanishing of the first derivative defines the critical drive hc = [r˜u + w˜u2/2 + γ˜u3/6]c = − w3 c 6γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A13) 34 Making use of the coefficients (A7), this translates to the critical drive ¯uc ¯uc = (a/2 ¯C)˜v2 c − w3 c 6 ¯Cγ2 (A14) and its combination with the result for ¯vc tells us that the critical drives match up, to leading order, with the cusps (73) of the bistable domain at ¯Rc,±, δ ¯Rc,± = (¯uc, ±¯vc) (A15) ≈ �� a/2 ¯C � ˜v2 c, ±(1 + λ+/ ¯C)˜vc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Next, we find the entire boundary of the unstable re- gion U ˜R that is defined as the points where local minima and maxima of eeff pin coalesce, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', where ∂2 ueeff pin = 0, r + w˜ujp + γ 2 ˜u2 jp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A16) Making use of the Landau coefficients (A7) as well as the relation between ˜v and ¯v in (A4), we recover the equation (53) for the ellipse (we drop corrections ∝ (κm − 1)3/2) γ˜u2 jp + 2β˜ujp˜vjp + δ˜v2 jp ≈ 2 ¯C(κm − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A17) In order to find the shape of the bistable region B ¯R, we exploit the fact that for fixed drives ¯u and ¯v, the bistable and the unstable vortex tip configurations are local extrema of eeff pin, implying that ∂˜ueeff pin = 0 and hence r˜u + w 2 ˜u2 + γ 6 ˜u3 = h, (A18) what corresponds to the force-balance equation (A5) ex- pressed in terms of the coefficients (A7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The cubic equa- tion (A18) with its left side ∝ (κm − 1)3/2 depends on ¯u through the drive h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' According to (A7), the two terms in the drive are of order (κm − 1) and hence have to cancel one another to lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' As a result, we find that the bistable domain is centered around the parabola ¯u = a 2 ¯C ¯v2 (1 + λ+/ ¯C)2 , (A19) that matches up with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (70) found in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finding the precise form of the bistable region B ¯R, we have to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A18) to cubic order in √κm − 1 with the help of an expansion around the center parabola (A19), what amounts to repeating the analysis leading to the results (71) and (72) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Finally, we find the landing line L ˜R defined as the sec- ond bistable tip position at fixed ¯u and ¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We make use of the cubic equation (A18) and represent it in the factorized form (with the inflection point at ˜ujp having multiplicity two) (˜u − ˜ujp)2(˜u − ˜ulp) = 0, (A20) and ˜ulp the landing position of the tip introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A somewhat tedious but straightforward cal- culation shows that the stable solution ˜ulp satisfies the quadratic equation r − 3 8 w2 γ + w 4 ˜ulp + γ 8 ˜u2 lp = 0 (A21) and thus arranges along the ellipse γ 8 ˜u2 lp + β 4 ˜ulp˜vlp + �δ 2 − 3 8 β2 γ � ˜v2 lp = ¯C(κm − 1) (A22) when expressed in the original two-dimensional tip space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' this coincides with the original result (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' In a last step, we may go over to an Ising-type Lan- dau expansion by measuring the order parameter ¯u with reference to the skewed line ˜um(¯v) = � −β γ � ¯v (1 + λ+/ ¯C), (A23) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', ˜u′ = ˜u − ˜um(¯v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A24) The 1D effective Landau expansion now reads, with pre- cision to order (κm − 1)2, eeff pin(˜u′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯u, ¯v) = r′ 2 ˜u′2 + γ 24 ˜u′4 − h′˜u′, (A25) with the new coefficients r′ = r − w2 2γ , h′ = h − w3 3γ2 + rw γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A26) The condition h′ = 0 now defines the equilibrium state of the thermodynamic problem that translates into the branch crossing line where the bistable vortex tip posi- tions have equal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Using the definitions (A7) and (A26) for h and h′, we find that the branch crossing line ¯u0(¯v0) in the original two-dimensional asymptotic space reads ¯u0 = a 2 ¯C ¯v2 0 (1 + λ+/ ¯C)2 − β γ � (κm − 1) ¯v0 1 + λ+/ ¯C + �δ 2 − β2 3γ � 1 ¯C ¯v3 0 (1 + λ+/ ¯C)3 � , (A27) extending the result (77) from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' III to finite values of β with an additional term ∝ (κm − 1)3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Close to merging Let us study the strong pinning problem close to merg- ing, as described by the two-dimensional Landau-type energy functional (117), epin( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯R) = ¯C(1 − κs) 2 ˜u2 + ¯C + λ+,s 2 ˜v2 + as 2 ˜u˜v2 + αs 4 ˜u2˜v2 + βs 6 ˜u3˜v + γs 24 ˜u4 − ¯C¯u˜u − ¯C¯v˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A28) 35 As found before for strong pinning close to onset, the energy functional (A28) is anisotropic with respect to vortex displacements in the stable and unstable direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Following the strategy of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A 1, we can use the force- balance equation (137) to relate the tip position along the v-axis to ¯v and ˜u, ˜v ≈ ¯v 1 + λ+,s/ ¯C � 1 − as/ ¯C 1 + λ+,s/ ¯C ˜u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A29) Inserting (A29) into the force-balance equation for the unstable component ˜u and integrating, we find that the resulting effective 1D Landau theory is identical in form to the one close to onset, eeff pin(˜u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯u, ¯v) = rs 2 ˜u2 + ws 6 ˜u3 + γs 24 ˜u4 − hs˜u, (A30) with a proper replacement of all coefficients involving the parameters appropriate at merging, rs = � ¯C(1 − κs) − |δs| 2 ¯v2 (1 + λ+,s/ ¯C)2 � , ws = βs ¯v (1 + λ+,s/ ¯C), hs = ¯C¯u − as 2 ¯v2 (1 + λ+,s/ ¯C)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A31) The difference to (A7) is the sign change in the term ∝ |δs|¯v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This implies a modification of the main equa- tion determining the shape of U ˜R (from which B ¯R fol- lows via the force balance equation (38)), with the elliptic equation (A17) transforming to the hyperbolic expression γs˜u2 jp + 2βs˜ujp˜vjp − |δs|˜v2 jp ≈ 2 ¯C(κs − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (A32) The results for the jumping and landing hyperbolas in ˜R- space and for the edges of the bistable domain in ¯R-space before and after merging can be derived by following the strategy of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A 1 above and agree with the correspond- ing results from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We close with a final remark on the disappearance of critical points after merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The critical points are found in the standard manner by setting the first three derivatives of eeff pin(˜u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' ¯u, ¯v) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' This works fine before merging when 1 − κs > 0 and we find that criticality is realized for tip and asymptotic positions as given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' (125) and (138) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' However, after merg- ing, the cubic derivative ∂3 ˜ueeff pin never vanishes, signalling the absence of a critical point, in agreement with the dis- cussion in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V C and V B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' The merger thus leads to the disappearance of the two critical (end-)points in asymptotic space, with the attached first-order lines (the branch crossing line) joining up into a single line that is framed by two separated spinodals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' We are not aware of such a disappearance of critical points in a merging pro- cess within the standard discussion of thermodynamic phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Campbell and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Evetts, Advances in Physics 21, 199 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kassner, Fundamentals of Creep in Metals and Al- loys (Elsevier Science & Technology Books, Amsterdam, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Gorchon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Bustingorry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Ferr´e, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Jeudy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kolton, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Giamarchi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 113, 027205 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Abrikosov, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='JETP 5 (1957) 5, 1174 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 5 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Burgers, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 52, 23 (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Bloch, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Physik 74, 295 (1932).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 7 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Lifshitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sowjet 8, 153 (1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Labusch, Crystal Lattice Defects 1, 1 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Larkin and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Ovchinnikov, Journal of Low Tem- perature Physics 34, 409 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Koopmann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 92, 067009 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Thomann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, Phys- ical Review Letters 108, 217001 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 12 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Willa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Prozorov, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, Physical Review Letters 115, 207001 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 13 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Buchacek, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Willa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blat- ter, Physical Review B 100, 014501 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 14 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Ertas and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Nelson, Physica C 272, 79 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 15 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vinokur, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Khaykovich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Zeldov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Konczykowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Doyle, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kes, Physica C 295, 209 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Buchacek, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Xiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Dutta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Andrei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Ray- chaudhuri, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B 100, 224502 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 17 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Buchacek, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, Physical Review Research 2, 043266 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 18 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Feigel’man, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Larkin, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vinokur, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 66, 1125 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 19 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Thomann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, Phys- ical Review B 96, 144516 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 20 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Willa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B 93, 064515 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 21 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Gaggioli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Research 4, 013143 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 22 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Buchacek, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Willa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blat- ter, Physical Review B 98, 094510 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 23 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kwok, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Welp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Glatz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Koshelev, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kihlstrom, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Crabtree, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 79, 116501 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 24 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Willa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Koshelev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sadovskyy, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Glatz, Super- cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 31, 014001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 25 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Willa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Koshelev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sadovskyy, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Glatz, Physical Review B 98, 054517 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 26 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Willa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Geshkenbein, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Blatter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B 105, 144504 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 27 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Buchacek, The Strong Pinning Paradigm Applied to Vortices in Type-II Superconductors, PhD Thesis (Insti- tute for Theoretical Physics, ETH Zurich, Switzerland, 36 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Bardeen and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Stephen, Physical Review 140, A1197 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 29 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Nakahara, Geometry, Topology and Physics (Graduate Student Series in Physics) (Taylor and Francis, 2003) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 88,118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Nash and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Sen, Topology and Geometry for Physicists (Dover Publications Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=', 2011) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 105, 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 31 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Volovik, Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 43, 47 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 32 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kane, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 128, 076801 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 33 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Villegas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Savel’ev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Nori, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Gonzalez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Anguita, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Garc´ıa, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Vicent, Science 302, 1188 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 34 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' de Souza Silva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Van de Vondel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Morelle, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Moshchalkov, Nature 440, 651 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 35 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Latimer, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Xiao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Divan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Ocola, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Crabtree, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kwok, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' B 87, 220501(R) (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 36 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Lyu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Xiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Dong, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Miloˇsevi´c, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Divan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Pearson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Wu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Peeters, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Kwok, Nature Communications 12, 2703 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' 37 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Embon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Anahory, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Suhov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Halbertal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Cup- pens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Yakovenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Uri, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Myasoedov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Rappa- port, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Huber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Gurevich, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} +page_content=' Zeldov, Scientific Reports 5, 7598 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E0T4oBgHgl3EQfUwBZ/content/2301.02254v1.pdf'} diff --git a/StAzT4oBgHgl3EQf0f7W/content/2301.01786v1.pdf b/StAzT4oBgHgl3EQf0f7W/content/2301.01786v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f8f26eadc7631b17648e0ade53a5c8b4c1a5d5be --- /dev/null +++ b/StAzT4oBgHgl3EQf0f7W/content/2301.01786v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:67f69c1536d58915f16db58e25252f5ae0df2df4caab463f39ddc78532241b75 +size 164086 diff --git a/StAzT4oBgHgl3EQf0f7W/vector_store/index.pkl b/StAzT4oBgHgl3EQf0f7W/vector_store/index.pkl new file mode 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contributions of its individual parts. We introduce the task of proactively forecasting popularities of sentences within online news +documents solely utilizing their natural language content. We model sentence-specific popularity forecasting as a sequence regression +task. For training our models, we curate InfoPop, the first dataset containing popularity labels for over 1.7 million sentences from +over 50,000 online news documents. To the best of our knowledge, this is the first dataset automatically created using streams of +incoming search engine queries to generate sentence-level popularity annotations. We propose a novel transfer learning approach +involving sentence salience prediction as an auxiliary task. Our proposed technique coupled with a BERT-based neural model exceeds +nDCG values of 0.8 for proactive sentence-specific popularity forecasting. Notably, our study presents a non-trivial takeaway: though +popularity and salience are different concepts, transfer learning from salience prediction enhances popularity forecasting. We release +InfoPop and make our code publicly available1. +CCS Concepts: • Information systems → Content ranking; • Computing methodologies → Neural networks; +Additional Key Words and Phrases: Sentence Popularity Forecasting, Sentence Salience Prediction, Supervised Transfer Learning +ACM Reference Format: +Sayar Ghosh Roy, Anshul Padhi, Risubh Jain, Manish Gupta, and Vasudeva Varma. 2022. Towards Proactively Forecasting Sentence- +Specific Information Popularity within Online News Documents. In Proceedings of the 33rd ACM Conference on Hypertext and Social +Media (HT ’22), June 28-July 1, 2022, Barcelona, Spain. ACM, New York, NY, USA, 16 pages. https://doi.org/10.1145/3511095.3531268 +1 +INTRODUCTION +In a typical document popularity prediction task, the objective is to estimate the would-be popularity of some content, +say, a news article that is published on an online platform. A popularity prediction system might evaluate the number +of pageviews that a particular online document would receive, or a social-media popularity prediction model might +estimate the number of signatures, comments, shares, or likes that a specific social media post would accrue over a +certain period of time. +Though popularity prediction is a well-studied machine learning task, with approaches ranging from traditional +feature-based methods to recent Deep Neural Network architectures, all of the existing works on popularity prediction +1https://github.com/sayarghoshroy/InfoPopularity +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2022 Association for Computing Machinery. +Manuscript submitted to ACM +1 +arXiv:2301.00152v1 [cs.CL] 31 Dec 2022 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +Local carriers and drivers will +be able to book and move +their first loads with Uber +Freight in the next few weeks, +CEO Lior Ron wrote in a blog +posted +Wednesday. +Uber +Freight plans to expand to +more European countries this +year. The EU and U.S. freight +markets +have +problematic +similarities. +They’re +both +huge — the EU truckload +market +is +a +$400 +billion +marketplace and third after +China and the U.S. — and +inefficient. “The European ... +Local carriers and drivers will be able to book ... +0.1099 +Uber Freight plans to expand to more Euro... +0.2046 +The EU and U.S. freight markets have problem... +0.0092 +They’re both huge — the EU truckload marke... +0.0067 +“The European trucking market is experienc... +0.0110 +Inefficiency of this scale results in shippers ... +0.0132 +Uber Freight has been scaling up its business ... +0.1356 +The company has offices in San Francisco and ... +0.0158 +In August, Uber announced that it would ... +0.2703 +It’s also made some key hires, one of which ... +0.0076 +Uber Freight had about 30,000 active users ... +0.1923 +... +Fig. 1. Task outline. Looking solely at document text as input, forecast prospective sentence-specific popularity scores +have focused exclusively on the document-level prediction of prospective popularity labels [18, 31]. When a system +predicts that a specific article will be popular over time, a natural follow-up is to ask why. Which information pieces +in the document would receive the most notice? Could we identify the informative sentences that would dominate +the popularity level of the article? Being able to generate such insights proactively would have wide-ranging business +applications in article promotion, popularity-guided summarization and pull quote selection, etc. +But sentence-specific forecasting of popularity has not been attempted, not even in a simplified binary classification +of sentences into popular or not setting. The difficulty in obtaining reliable sentence-level labeled data for information +popularity has been a crucial driving factor for the same. +An online article’s popularity can be defined in terms of its page views [34, 40], which is a by-product of the +regular Internet browsing activities of the global population. Similarly, within a particular online document, information +popularity of a unit sentence is proportional to the number of received requests actively seeking its contained information. +To capture this, we leverage incoming search queries encountered by a popular search engine over a sufficient period +to ascertain sentence-specific popularity labels based on how queried-after each sentence is within a document. We +present InfoPop, a dataset of over 50,000 online news articles with over 1.7M sentences, mapped to popularity scores. +Taking inspiration from existing research on document-level popularity prediction, we frame the problem of sentence- +specific popularity forecasting as a regression task similar to previous studies [17, 35] (who model document-level +popularity) as opposed to a simple binary classification of a document’s sentences into popular or not. Unlike some +document-popularity forecasting approaches that rely on post-publication signals like pageview hits in the first half-hour +after publication [2, 17], we forecast sentence popularity scores proactively (before a document’s publication). Note +that we utilize incoming search queries only to define sentence-specific popularity and construct the InfoPop dataset. +However, our problem formulation is one of query-insensitive relative normalized scoring of sentences. Formally, given +just a document as input, our task (Fig. 1) is of assigning normalized scores in the range [0, 1], to every sentence +indicating their intra-document relative information popularity, without utilizing any external signals. +The key novelty of our approach lies in the design of auxiliary transfer learning subtasks adapting the STILTs +(Supplementary Training on Intermediate Labeled-data Tasks) [28] technique. In brief, we create auxiliary subtasks of +sentence salience prediction in the domain of online news, framed under an identical problem formulation as that of our +primary task of popularity forecasting (Sec. 5). We experiment with a baseline neural model using RNNs (Recurrent +Neural Networks) and a robust sequence regression architecture based on BERT (Bidirectional Encoder Representations +from Transformers) [6], showcasing our proposed approach’s efficacy. Also, we handle arbitrarily long documents +2 + +Sentence Specific Popularity Forecasting +HT ’22, June 28-July 1, 2022, Barcelona, Spain +with our neural models employing a sliding window mechanism over a document’s sentences. In our analysis, we +illustrate the differences in the concepts of popularity and salience, both qualitatively and quantitatively (Sec. 8). The +empirically demonstrated efficacy of our approach brings out an interesting yet non-trivial takeaway – though salience +and popularity are varying concepts, transfer learning from salience prediction improves popularity forecasting. +Overall, in this paper, we make the following contributions. +• We introduce the task of proactively forecasting relative information popularities of sentences within online +news documents solely based on their natural language content, without the use of any external features. +• We present InfoPop, the first labeled dataset containing over 50,000 online news documents from 26 news +websites with over 1.7M sentences, each mapped to supervised popularity scores. +• Through a novel STILTs-based transfer learning approach, we build high-performance neural models reaching +nDCG scores over 0.8 for sentence-specific popularity forecasting. +The remainder of the paper is organized as follows. In Section 2, we discuss related work on document-level popularity +prediction, automatic text summarization and snippet generation. We formally present the concept of sentence-specific +information popularity in Section 3. We describe details of curation of our dataset, InfoPop in Section 4. Next, we +outline our proposed methods in Section 5. Further, we discuss evaluation metrics in Section 6, experimental details in +Section 7 and present detailed results with analysis in Section 8. Finally, we conclude with a brief summary in Section 9. +2 +RELATED WORK +In this section, we discuss related work on document-level popularity prediction, automatic text summarization and +snippet generation. +2.1 +Document-level Popularity Prediction +Existing works in popularity prediction have considered a piece of content as an atomic unit whose prospective +popularity is inferred. Based on the choice of popularity surrogate, we distinguish between two kinds of document +popularity prediction problems. +• Popularity based on Internet browsing: Studies on forecasting popularities of online news documents typi- +cally consider the number of received page load requests over sufficient time (accumulated pageviews or hit +count) as a surrogate for article-level popularity [17, 18, 40]. +• Social Media Popularity: These approaches aim to estimate the prospective engagement of a piece of content +put up on a particular social media platform. Previous works have dealt with multi-modal posts [41], images [42], +movies [1], petitions [35], etc. and have utilized user-behavior based markers such as the number of comments [32, +36], shares [25, 38], etc. as surrogates of social-media popularity. There has also been limited work on time- +aware [14] and time-series prediction of social-media popularity [20, 29]. +Unlike all previous approaches that determine popularity at the document-level, we focus on sentence-specific +popularity forecasting (as shown in Fig. 1) with our domain of information being online news, not social-media posts. +For informative documents (including online news), a preferred surrogate of popularity has been the number of +pageview hits [3, 31, 34]. Intuitively, pageviews captures the generic browsing trends of the population not limited +3 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +to social media actions. Our task, however, requires knowledge of more fine-grained Internet browsing activity for +annotating specific sentences with popularity labels. +2.2 +Automatic Text Summarization +Text summarization aims to recognize information that is central to the core idea of one or more document(s). Approaches +are roughly divided into two categories, namely, abstractive: generate a concise, coherent, and cogent summary that +captures the central idea expressed by the text piece [19, 33], and extractive: select and arrange salient plus diverse +ranges of text to form a summary [43, 44]. Extractive summarization is often framed as a sequence classification +problem – binary labels are assigned to sentences indicating their presence in the summary [11, 21]. In contrast, our task +necessitates sequential regression (numerical sentence-specific scores are forecasted). Also, text summarization has the +additional objective of generating text without repeated information. In that, an otherwise summary inclusion-worthy +sentence may have a binary label of 0 if a similar sentence is already present in the oracle summary [15] (in an extractive +setting). However, we expect almost identical sentences in an article to have very similar information popularity values. +2.3 +Snippet Generation +A page snippet can be a document excerpt that lets a user understand whether a document is pertinent to their query +without accessing it in its entirety. Snippet-generation [4] has a ranking-based problem formulation somewhat related +to sentence popularity forecasting. However, our task is one of query-insensitive scoring of sentences. In contrast, tasks +such as snippet generation (and document retrieval [12]) are query-sensitive [10] – typical methods require a particular +query tailored to which an appropriate snippet is generated (except for naïve approaches that output initial tokens). +3 +SENTENCE-SPECIFIC INFORMATION POPULARITY +Document popularity magnitudes and popular-or-not labels based on pageviews capture the amount of notice that +the document receives on the Internet over a period of time [18, 40] and is a consequence of the everyday Internet +browsing actions of the worldwide population. +With increased Internet penetration, the average number of queries encountered daily by commercial search engines +has exceeded the billion mark2. Google Trends3 showcases popular topics of interest segregated by region and timespan +based on the pool of encountered search queries, and the collection of worldwide queries serves as a guide to mark +information of universal interest. Correspondingly, within the local context of a single news document (say 𝐷), if an +information piece, say 𝐼1, is more queried-after by the global population than, say, piece 𝐼2 in 𝐷, then: 𝑝𝑜𝑝𝑢𝑙𝑎𝑟𝑖𝑡𝑦(𝐼1) +> 𝑝𝑜𝑝𝑢𝑙𝑎𝑟𝑖𝑡𝑦(𝐼2). Popularity levels of sentences within 𝐷 can be derived based on this base principle, wherein the +popularity of a specific sentence (say 𝑠) in 𝐷 is incremented based on the lexical similarity between 𝑠 and an encountered +query. +For a news document 𝐷 with sentences [𝑠1, 𝑠2, ..., 𝑠𝑁 ], comparing each sentence {𝑠𝑖}𝑁 +𝑖=1 to every search query +encountered by a commercial search engine would be computationally infeasible. Moreover, only a negligibly small +percentage of all encountered queries would positively contribute to sentences’ popularities within 𝐷. Such queries +would precisely be the ones for which 𝐷 could be deemed relevant. Thus, to derive scores of 𝑠𝑖’s within 𝐷, we filter +incoming queries and consider the sublist, say 𝑄 = [𝑞1, 𝑞2, ..., 𝑞|𝑄 |], for which the document 𝐷 was significantly relevant. +2https://searchengineland.com/google-now-handles-2-999-trillion-searches-per-year-250247 +3https://trends.google.com +4 + +Sentence Specific Popularity Forecasting +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Table 1. Selected sentences from a document in InfoPop with their true and forecasted popularities and predicted salience. Popularity +forecasts are from our best performing model on nDCG (BERTReg with TL = SL). Salience predictions are based on BERTReg trained +on S1. +[TPL: True Popularity Label, FPL: Forecasted Popularity Label, PSL: Predicted Salience Label, TPR: True Popularity Rank, FPR: +Forecasted Popularity Rank, PSR: Predicted Salience Rank] +Sentence +TPL +FPL +PSL +TPR +FPR +PSR +Local carriers and drivers will be able to book and move their first loads with Uber +Freight in the next few weeks, CEO Lior Ron wrote in a blog posted Wednesday. +0.1099 +0.1522 +0.1237 +5 +4 +1 +Uber Freight plans to expand to more European countries this year. +0.2046 +0.1914 +0.0924 +2 +2 +4 +The EU and U.S. freight markets have problematic similarities. +0.0092 +0.0274 +0.0736 +10 +6 +7 +They’re both huge – the EU truckload market is a $400 billion marketplace and +third after China and the U.S. – and inefficient. +0.0067 +0.0048 +0.0818 +12 +7 +6 +“The European trucking market is experiencing a severe shortage of drivers, and +of the time drivers are on the road, 21 percent of total kilometers travelled are +empty”, Ron wrote. +0.0110 +0.0041 +0.0892 +9 +8 +5 +Inefficiency of this scale results in shippers struggling to find available drivers to +move their goods. +0.0132 +0.0015 +0.0530 +7 +12 +10 +Uber Freight has been scaling up its business since launching in May 2017, growing +from limited regional operations in Texas to the rest of the continental U.S. +0.1356 +0.1301 +0.1018 +4 +5 +2 +The company has offices in San Francisco and Chicago. +0.0158 +0.0024 +0.0590 +6 +10 +9 +In August, Uber announced that it would make Uber Freight a separate unit and +more than double its investment into the business. +0.2703 +0.1968 +0.0988 +1 +1 +3 +It’s also made some key hires, one of which intimated the company’s global +ambitions. +0.0076 +0.0015 +0.0382 +11 +11 +12 +The company has made headway breaking into the U.S. market. +0.0116 +0.0025 +0.0675 +8 +9 +8 +Uber Freight had about 30,000 active users last quarter. +0.1923 +0.1906 +0.0427 +3 +3 +11 +We regard 𝐷 as a significantly relevant document for query 𝑞 if 𝐷 was shown within the top 10 search results (which +roughly translates to the first page of news results) when the search engine encountered 𝑞. We assign a base score of +�|𝑄 | +𝑗=1 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑞𝑗,𝑠𝑖) to each sentence {𝑠𝑖}𝑁 +𝑖=1. Sentence popularity scores are then normalized to a [0, 1] range such +that the popularity labels of sentences within 𝐷 sum up to 1. +4 +INFOPOP DATASET +In this section, we outline the creation of the InfoPop dataset and present its basic statistics. +4.1 +Data Acquisition +We scraped 82,540 news documents from 26 reputed online news websites (websites listed in Fig. 2). We accessed +incoming queries of Microsoft Bing Search and mapped each document to the global assemblage of queries that deemed +the document as significantly relevant (Sec. 3) at the time when it was encountered. We extracted each article’s text +content and split that into an ordered list of sentences. With news articles as unit data points, individual sentences +served as our atomic information pieces. We discounted articles with over 100 sentences as we found many of these to +be long non-news documents. +5 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +897 +75 +40 +146 +371 +3252 +1806 +2522 +1638 +411 +1190 +2616 +1038 +44 +1490 +1917 +1064 +1453 +2509 +189 +10677 +2388 +6570 +142 +4979 +2346 +0 +2500 +5000 +7500 +10000 +denverpost +huffingtonpost +washington.cbslocal +sanfrancisco.cbslocal +hindustantimes +npr +newsweek +cbslocal +cnn +livemint +phys +reuters +chicagotribune +crictracker +abcnews.go +latimes +techcrunch +nydailynews +nypost +timesnownews +nytimes +foxnews +usatoday +theguardian +cbsnews +nbcnews +Fig. 2. InfoPop: news sources (on x-axis) with their corresponding #documents (on y-axis) +4.2 +Preprocessing +We encountered a considerable amount of noise in the extracted webpage texts. These primarily manifested as strings +of seemingly random word tokens. We removed such heavily non-grammatical sentences using two dependency +parsing-based heuristics. +• Remove a sentence if its dependency graph is not a tree. The dependency graph of a grammatical sentence is +supposed to be a tree and not a forest. We evaluated whether the produced dependency graph of a sentence was +both connected and acyclic or not. +• Remove a sentence if its dependency parse contains some xcomp branch leading to a single participle. The X +complement or xcomp is a dependency label used to mark dependent clauses that do not contain a subject. +Consider a noisy unit such as ‘I would like to, I would like to, I would like to.’ The parse tree of such a text string +contains an xcomp branch leading to a participle at the leaf node. We observed a positive correlation between +such phenomena and text repetition. +Though the above cleaning heuristics are computationally expensive, we employ these owing to the one-time +nature of the operation. After these cleanup steps, we further removed articles containing less than three grammatical +sentences. +4.3 +Popularity Labels +For a particular online document 𝐷 (say), we considered the collection of all queries, say 𝑄 = [𝑞1, 𝑞2, ..., 𝑞|𝑄 |], for which +𝐷 was significantly relevant. We generated supervised popularity labels for every sentence in 𝐷 using the method +6 + +Sentence Specific Popularity Forecasting +HT ’22, June 28-July 1, 2022, Barcelona, Spain +# Sentences per document +# Documents +0 +2500 +5000 +7500 +10000 +12500 +0 - 10 +11 - 20 +21 - 30 +31 - 40 +41 - 50 +51 - 60 +61 - 70 +71 - 80 +81 - 90 +91 - 100 +Fig. 3. InfoPop: Distribution of #sentences per document +outlined in Sec. 3. More formally, 𝑃𝑖 ← +�|𝑄| +𝑗=1 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑞𝑗,𝑠𝑖) +�𝑁 +𝑖=1 +�|𝑄| +𝑗=1 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑞𝑗,𝑠𝑖) , ∀𝑖 ∈ {1, 2, ..., 𝑁 } where 𝑃𝑖 denotes the normalized +popularity label assigned to the 𝑖𝑡ℎ sentence in 𝐷, assuming 𝐷 contains 𝑁 sentences: 𝑠1, 𝑠2, ..., 𝑠𝑁 . +We used cosine-similarity between corresponding TF-IDF vectors as the measure of similarity. Though our similarity +function might feel primitive, we uphold that utilizing lexical similarity is optimum in this case since search engine +ranking algorithms still utilize word match as the basic principle and most relevant pages typically contain the query +terms (that usually do not have well defined semantic embeddings) themselves. That said, we plan to explore semantic +embedding based similarity as part of future work. +4.4 +InfoPop Dataset Statistics +InfoPop contains 51,770 news documents containing 1,711,890 sentences annotated with normalized popularity scores. +Each document contains a minimum of 3 and a maximum of 100 sentences, with the average number of sentences +per article being 33.07. Fig. 3 shows the distribution of the number of sentences per document. On average, each +sentence contains 18.23 word tokens. The minimum and maximum document lengths stand at 15 and 2,516 tokens, +respectively, with 602.92 tokens being the average document length. The news articles are sourced from 26 reputed +news websites, with the average number of articles per website being 1991.15. Fig. 2 illustrates the specific number +of documents sourced from each news website. We observe that popularity scores and sentence lengths have a weak +positive correlation of 0.168. We divide our dataset into train, validation, and test splits in an 8:1:1 ratio. +To ensure reproducibility, we make InfoPop (and our defined train, validation, and test splits) publicly available here4. +Table 1 shows an excerpt from a document within InfoPop with sentence-specific popularity labels (TPL). +5 +METHODS +We describe various methods for proactive sentence-specific popularity forecasting, including unsupervised sentence +ranking baselines, neural sentence sequence regression architectures, and our proposed supervised Transfer Learning +approach from an auxiliary task of salience prediction. +4https://github.com/sayarghoshroy/InfoPopularity +7 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +5.1 +Unsupervised sentence ranking baselines +News articles typically utilize the pyramid structure of reporting with primary information contained within the initial +sentences. We experiment with a position baseline where we score sentences in descending order of their position from +the beginning of the article. Specifically, we assign the score of 1 − 𝑖 +𝑛 to the 𝑖𝑡ℎ sentence of an article with 𝑛 sentences. +We also evaluate the performance of TextRank [24] and LexRank [8] algorithms that exploit graph-based similarities +between all sentence pairs through PageRank. +5.2 +Our Proposed Approach +The primary novelty of our proposed approach lies in the design of a STILTs-based [28] Transfer Learning (TL) setup +using a constructed task of sentence salience prediction. Recent advances in NLP have utilized transfer learning in two +broad ways, (a) unsupervised pre-training of text encoders using specific Language Modeling objectives [9, 23] and (b) +supervised pre-training of architectures on tasks with similar problem formulations [28, 30]. In STILTs (Supplementary +Training on Intermediate Labeled-data Tasks), a model pre-trained on unsupervised corpora (like BERT) is further +pre-trained on an intermediate, supervised task for which ample labeled data is available before fine-tuning on its +primary task. +We observe how sentence salience is well studied in context of text summarization, an area that enjoys availability +of sizeable amounts of news domain data and hypothesize that STILTs-based transfer learning from an auxiliary task of +salience prediction would boost our models’ popularity forecasting ability. +5.2.1 +Auxiliary Transfer Learning (TL) Subtasks. For document summarization, a sentence is considered salient if it +contains information related to the primary semantics of the document and can be a worthy inclusion for the document’s +summary [26]. Given an article and its gold standard summary, the salience of an individual sentence is computed as +the ROUGE overlap between sentence and summary. For oracle creation in a typical extractive summarization setting, +binary summary-inclusion labels are computed for sentences greedily to maximize the ROUGE overlap between the +complete oracle and the true summary [26]. Such a greedy labeling scheme implicitly takes minimization of information +redundancy into account. In that, a salient sentence might receive a summary-inclusion label of 0 if a lexically similar +sentence was previously included in the oracle summary [15]. However, we only capture the salience of sentences and +desire very similar sentences to have similar labels. Similar to sentence-specific popularity forecasting, we package our +auxiliary task as a sentence sequence regression problem. This lets us (a) train our architectures adapting the STILTs +approach and (b) perform an empirical cross-task evaluation (Sec. 8.2.2). +Since our domain of interest is online news, we utilize the publicly available CNN-DailyMail news summarization +dataset (with the same splits as in [11]). We compute three weakly supervised salience scores for each sentence based +on its ROUGE 1, ROUGE 2, and ROUGE L overlap with the corresponding article’s summary. Like InfoPop labels, we +normalize salience labels across a document by dividing by the sum of individual sentence scores. Thus, we create three +auxiliary subtasks due to the three labeling schemes, which we tag as S1, S2, and SL, respectively. +5.2.2 +Supervised Neural Architectures. We frame information popularity forecasting as a sentence sequence regression +task where a sentence’s score is relative to the entire article. Thus, effective neural models require the global context. We +pre-train a neural architecture on a supervised TL subtask (S1, S2, or SL), and then fine-tune the model for popularity +forecasting. High-level overviews of BaseReg, our rudimentary neural baseline, and BERTReg, our BERT-based sentence- +sequence regression model are illustrated in Fig. 4 and Fig. 5, respectively. We use MSE (Mean Squared Error) loss between +8 + +Sentence Specific Popularity Forecasting +HT ’22, June 28-July 1, 2022, Barcelona, Spain +S1 +S2 +S3 +Sn +... +GRU +GRU +GRU +GRU +... +... +pool +FC +Doc +score1 +score2 +score3 +scoren +... +Sentences +CNN +Layers +bi-RNN +Layer +Fig. 4. BaseReg: RNN for Sentence Sequence Regression +true and inferred sentence scores for training our neural models. Note that we handle arbitrarily large documents us- +ing both BERTReg and BaseReg employing a sliding window mechanism over a document’s sentences with a preset stride. +BaseReg: As a simple baseline neural model for sentence scoring, we build upon the rudimentary SummaRunner [16, 26] +architecture with Convolutional Neural Network (CNN) based sentence encoders. We input the concatenated padded +sequence of GloVe [27] embeddings for tokens in a sentence to a two-layered CNN sentence vectorizer. Internally, a +CNN layer applies a sequence of convolution and batch normalization followed by a Leaky ReLU activation. The output +from the two stacked CNNs is then max pooled. Inside, three CNN stacks with kernel sizes of 3, 4, and 5 are used whose +outputs are concatenated to form the sentence embedding. +Individual CNN-based sentence vectors are passed through a layer of bi-GRUs (bidirectional Gated Recurrent +Units) [5] that produce a sequence of contextual sentence embeddings. All such contextual embeddings are max pooled +and then passed through a fully connected (FC) layer to derive a document embedding vector that captures the global +context of the complete document. Each sentence score is then computed based on aspects such as their content, novelty, +etc., [26] which are in turn expressed as functions of their contextual embedding and the document embedding vector. +BERTReg: The use of Transformer-based [39] models such as BERT [6] have yielded state-of-the-art results on various +NLP tasks. We adapt BertSumExt’s [22] architecture of sentence classification to our sequential regression setting. To +generate contextual sentence embeddings, we add [SEP] tokens to mark sentence ends and distinctive [REG] tokens at +sentence beginnings. Downstream, the contextual embeddings of the [REG] tokens are used for sequence regression. +For each input token, its token, position, and segment embeddings are generated [6]. Here, the segment embeddings +allow differentiation between odd and even positioned sentences within BERT (since traditional BERT has been trained +with only two segments). The token, position, and segment embeddings of the defined input tokens are passed to +BERT, which produces a sequence of refined token embeddings. Now, the sequence of BERT-based sentence embeddings +(embeddings corresponding to the [REG] tokens) is summed with their sinusoidal sentence positional embeddings +and passed through a two-layer Transformer model to yield contextual sentence embeddings aware of multi-sentence +discourse. [21] showed the usefulness of adding a two-layer Transformer that captures inter-sentence relationships as +opposed to directly using BERT-generated embeddings for sentence-sequence labeling. Finally, the generated contextual +9 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +[REG] 1st + sentence +[SEP] [REG] 2nd sentence [SEP] +[REG] nth sentence +[SEP] +T[REG] T1st +Tsent T[SEP] T[REG] T2nd Tsent T[SEP] +T[REG] Tnth Tsent T[SEP] +S0 +S0 +S0 +S0 +S1 +S1 +S1 +S0 +S0 +S0 +S0 +E1 +E5 +E4n-3 +SP1 +E2 +E3 +E4 +E6 +E7 +E8 +SPn +E4n-3 E4n-3 +E4n +... +Linear +Activation +score1 +Input +Token +Embeddings +Segment +Embeddings +BERT +Embeddings +Sentence +Positional +Embeddings +... +... +... +... +C1 +Contextual +Embeddings +... +P0 +P1 +P2 +P3 +P4 +P5 +P6 +P7 +P4n-3 P4n-2 P4n-1 +P4n +Position +Embeddings +... +Linear +Activation +score2 +C2 +Linear +Activation +scoren +Cn +SP2 +S1 +Transformer Layers +BERT +Fig. 5. BERTReg: BERT for Sentence Sequence Regression +embeddings of the individual sentences are passed through a linear layer with a sigmoid activation to generate regression +scores ∈ [0, 1]. Fig. 5 shows an overview of BERTReg. +6 +EVALUATION METRICS +We utilize the following automatic evaluation metrics to measure the performance of our models. +• Top 𝑘 overlap Let 𝐴𝑘 and 𝑃𝑘 be the sets of actual and predicted top-𝑘 highest scored sentences, respectively. +We define top 𝑘 overlap (top𝑘) as |𝐴𝑘 ∩ 𝑃𝑘 |/𝑘 and report its percentage. Thus, top1 marks whether a method +correctly identifies the highest scored sentence. +• Regression errors We report Mean Squared Error (MSE) and Mean Absolute Error (MAE) between arrays of +actual and predicted sentence labels. +• Rank Correlation Metrics Top 𝑘 overlap focuses only on the identification of the highest scored sentences. +To get a picture of how well various algorithms rank and order the complete set of sentences in a document, we +use Spearman’s rank correlation (𝜌) and Kendall’s Tau (𝜏). 𝜌,𝜏 ∈ [−1, 1]. +• nDCG Top 𝑘 overlap, 𝜌, and 𝜏 only consider sentence ranking, while regression errors only account for sentence +scores. Normalized Discounted Cumulative Gain (nDCG) [13] captures the normalized gain or usefulness of a +sentence based on both its position in the inferred rank list and its actual score. nDCG ∈ [0, 1]. +Intuitively, the penalty for downscoring and inverting ranks of two truly high scored sentences should be lower +than that of inverting ranks of two negligibly scored ones. Rank Correlation metrics do not take scores into +10 + +Sentence Specific Popularity Forecasting +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Table 2. Sentence Popularity Forecasting Results +Method +TL +Top1 +Top2 +Top3 +MSE +MAE +𝜏 +𝜌 +nDCG +Position +− +6.92 +10.81 +16.46 +0.0079 +0.0530 +0.0334 +0.0424 +0.5804 +TextRank +− +6.08 +12.58 +19.08 +0.0316 +0.1486 +0.0345 +0.0474 +0.6313 +LexRank +− +18.76 +29.95 +37.84 +0.0072 +0.0503 +0.0545 +0.0705 +0.7324 +BaseReg +× +9.12 +15.49 +20.93 +0.0083 +0.0452 +0.0534 +0.0746 +0.6228 +S1 +9.35 +14.91 +20.49 +0.0075 +0.0478 +0.0428 +0.0592 +0.6323 +S2 +10.08 +16.63 +22.83 +0.0073 +0.0468 +0.0545 +0.0751 +0.6465 +SL +9.16 +14.85 +20.76 +0.0075 +0.0475 +0.0465 +0.0638 +0.6307 +BERTReg +× +27.53 +38.89 +45.89 +0.0055 +0.0335 +0.0704 +0.0955 +0.7921 +S1 +27.54 +38.73 +45.86 +0.0052 +0.0342 +0.0734 +0.0988 +0.8009 +S2 +28.34 +39.08 +46.17 +0.0053 +0.0332 +0.0646 +0.0876 +0.8007 +SL +28.11 +39.53 +46.96 +0.0053 +0.0331 +0.0510 +0.0674 +0.8025 +account and fail to capture such insights. Whereas regression errors only focus on true and forecasted scores +ignoring the score-induced rank ordering. Thus, broadly, nDCG is the most holistic metric for our task. +7 +EXPERIMENTAL DETAILS +We used publicly available implementations of TextRank5 and LexRank6. We normalized inferred LexRank scores to a +[0, 1] range (by dividing by the sum of all sentence scores) for computing regression errors. To create the auxiliary +transfer learning subtasks, we utilized the non-anonymized version of CNN-DailyMail dataset [11]. We used spaCy’s7 +dependency parser and NLTK’s8 sent_tokenize function during preprocessing. +We used a maximum sequence limit of 100 sentences in the layer of bi-GRUs within our BaseReg model. The CNN- +sentence encoder had an upper input limit of 100 tokens per sentence. For training our BaseReg model, we used Adam +optimizer with a batch size of 256 documents, an initial learning rate of 10−5, with default hyperparameters. We used +the 6-layer bert-base-uncased model in our BERTReg experiments and considered a maximum sequence length of 1536 +tokens. Adam optimizer was used with a learning rate of 2.10−3, with other hyperparameters set to their default values. +We split documents exceeding the maximum sequence length into overlapping sliding windows with a predefined +maximum stride of 10 sentences. During inference, we adopted the same splitting technique. For a sentence 𝑠 within +the stride of two consecutive windows, we scored 𝑠 as the mean of the computed scores from the two windows. +We trained all of our models on NVIDIA RTX 2080Ti(s). BaseReg models were trained on a single GPU for a maximum +of 4 epochs with early stopping. A single epoch on our transfer learning task took over 6 hours, while one epoch for +popularity forecasting training took close to 45 minutes. We trained BERTReg based models on 4 GPUs for 50, 000 +optimizer steps with early stopping turned off for both the popularity forecasting and the transfer learning tasks. The +training time ranged between 10 to 12 hours. +8 +RESULTS +In this section, we analyze the performance of baselines and our proposed approach for proactive sentence-specific +popularity forecasting. We then share some insights on the interrelations between our primary task of sentence +popularity forecasting and our auxiliary Transfer Learning (TL) task of salience prediction. +5https://github.com/summanlp/textrank +6https://github.com/crabcamp/lexrank +7https://spacy.io/ +8https://www.nltk.org/ +11 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +Table 3. Performance of various approaches on auxiliary transfer learning subtasks (S1, S2, SL) +Task +Approach +Top1 +Top2 +Top3 +MSE +MAE +𝜏 +𝜌 +nDCG +S1 +Position +12.86 +26.30 +34.04 +0.0010 +0.0215 +0.2030 +0.2855 +0.8682 +TextRank +15.55 +24.28 +30.89 +0.0243 +0.1487 +0.0771 +0.1082 +0.8999 +LexRank +13.14 +21.91 +28.45 +0.0007 +0.0178 +0.0561 +0.0798 +0.8740 +BaseReg +17.01 +26.60 +34.00 +0.0006 +0.0167 +0.1387 +0.1967 +0.8933 +BERTReg +26.41 +36.64 +43.20 +0.0004 +0.0130 +0.1372 +0.1891 +0.9274 +S2 +Position +11.24 +25.03 +32.77 +0.0044 +0.0418 +0.1496 +0.2101 +0.7113 +TextRank +9.29 +17.59 +23.43 +0.0280 +0.1541 +0.0407 +0.0578 +0.6665 +LexRank +11.74 +20.68 +26.88 +0.0045 +0.0422 +0.0473 +0.0669 +0.6847 +BaseReg +17.19 +27.56 +35.83 +0.0041 +0.0373 +0.1391 +0.1989 +0.7382 +BERTReg +23.32 +36.26 +43.68 +0.0034 +0.0332 +0.1108 +0.1559 +0.7946 +SL +Position +13.60 +27.57 +35.18 +0.0009 +0.0211 +0.2050 +0.2881 +0.8760 +TextRank +11.72 +20.05 +26.77 +0.0245 +0.1488 +0.0719 +0.1016 +0.8776 +LexRank +12.40 +21.71 +27.95 +0.0007 +0.0182 +0.0546 +0.0778 +0.8657 +BaseReg +15.13 +24.75 +32.36 +0.0007 +0.0175 +0.1385 +0.1966 +0.8780 +BERTReg +24.24 +34.96 +41.76 +0.0005 +0.0141 +0.1329 +0.1847 +0.9152 +8.1 +Sentence-Specific Popularity Forecasting +8.1.1 +Unsupervised sentence ranking. Table 2 shows performance of unsupervised sentence ranking baselines on +sentence-specific popularity forecasting. LexRank turns out the best unsupervised method beating Position and +TextRank on all metrics. +8.1.2 +Proposed Methods. We present results of our supervised models in Table 2. As expected, BERTReg is the best +architecture for the task. We see performance upgrades across the board upon employment of the Transfer Learning +setup. Though there is no consistent winner among the three Transfer Learning subtasks9, the best result for each +metric is achieved using some form of transfer learning. BERTReg with TL = SL boosted average nDCG over 1% from +plain BERTReg and over 7% from the best unsupervised baseline. We performed 𝑡-tests which showed that BERTReg +with TL = SL significantly outperformed vanilla BERTReg on nDCG at significance level 𝑝 < 0.01. Though BaseReg is +ineffective for popularity forecasting, we observe the same trend of transfer learning as a positive addition. These results +demonstrate that transfer learning from salience prediction significantly improves sentence popularity forecasting. +We attribute the performance enhancement to two factors. Firstly, datasets used for both tasks are sourced from +online news documents, and transfer learning allows the model to witness more domain-specific data [9]. Secondly, +though popularity forecasting differs from salience prediction (Sec. 8.2), they have certain similarities, such as penalizing +not lexically dense sentences or understanding that specific sentences do not carry any notable information. Our +proposed approach grounds the network to capture such relationships. +8.2 +Popularity and Salience +Salient sentences effectively capture summary-inclusion worthy ideas that are central to the core semantics of an +article [44]. Unlike salient information bits, an information piece might deviate significantly from an article’s primary +topic yet be popular. Consider the following sentence from a particular article10 (with ID 34499) within InfoPop: +‘Weinsheimer has spent 27 years at DOJ, where he tried homicide and public corruption cases.’ The sentence is not +9We prefer TL = SL as it outperforms others on Top𝑘 (𝑘 = 2, 3), MAE, and most importantly, nDCG. A downstream application focused purely on accurate +ranking could use TL = S1, while another requiring identification of the most popular sentence could use TL = S2. +10https://www.npr.org/2018/07/03/625581627/another-top-justice-department-lawyer-steps-down-following-earlier-departures +12 + +Sentence Specific Popularity Forecasting +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Table 4. Cross-task evaluation − performance of BERTReg trained for popularity forecasting (PF) evaluated on salience prediction and +vice-versa +Train +Eval +Top 1 +Top 2 +Top 3 +MSE +MAE +𝜏 +𝜌 +nDCG +S1 +PF +10.97 +18.31 +25.69 +0.0068 +0.0475 +0.0430 +0.0598 +0.6864 +S2 +10.74 +19.83 +27.89 +0.0077 +0.0455 +0.0380 +0.0531 +0.6942 +SL +11.44 +19.09 +27.02 +0.0068 +0.0476 +0.0428 +0.0595 +0.6937 +PF +S1 +8.36 +16.63 +22.64 +0.0020 +0.0301 +0.0600 +0.0847 +0.8603 +S2 +8.51 +16.34 +22.53 +0.0053 +0.0430 +0.0373 +0.0525 +0.6579 +SL +9.07 +16.62 +22.86 +0.0020 +0.0304 +0.0563 +0.0798 +0.8524 +salient enough to be included in a summary as it is barely related to the article’s core topic (Scott Schools’ resignation). +But, it contains one of the most popular information pieces in the document. +8.2.1 +Quantitative analysis. We evaluate our unsupervised baselines and supervised neural models on the transfer +learning subtasks (Table 3). The position baseline achieves the best rank correlation scores, explainable based on the +pyramid structure of news reports. BERTReg beats other approaches on nDCG and in identifying the most salient +sentences (Top 𝑘). +Comparing with Table 2, we see how sentence ranking baselines are more capable of capturing information salience +than forecasting information popularity. Position baseline’s scores for salience prediction are significantly greater than +for popularity forecasting, indicating that the salience of the initial sentences in news articles is typically higher than +their popularity. +Comparison with Table 2 shows that supervised salience prediction models achieve much better results on 𝜌, 𝜏, +and nDCG compared to popularity forecasting models utilizing the same underlying architecture. Also, note how +BaseReg, which performed poorly on popularity forecasting, achieves respectable scores for salience prediction. These +experiments identify salience prediction as the less complicated problem, i.e., given a document, it is easier to compute +how central and summary-worthy a contained information piece is than forecasting its popularity. Table 1 shows an +excerpt from an InfoPop article with actual and forecasted popularity plus predicted salience labels for sentences +showcasing the variation between information popularity and salience within the same document. +8.2.2 +Empirical cross-task evaluation. We performed an empirical cross-task analysis [7, 37] to capture the degree of +relatedness between primary and auxiliary tasks. We used the BERTReg popularity forecasting model to infer sentences’ +labels for documents in the salience-prediction dataset. We evaluated these scores against actual sentence salience labels +of the three types – S1, S2, and SL. Similarly, we evaluated models trained on S1, S2, and SL upon our primary popularity +forecasting task. We show results in Table 4. Predictably, we observe a massive drop in performance when we switch to +the cross-task setting. Values across all metrics fall below those achieved by some unsupervised ranking baseline. This +further experimentally showcases the distinction between information popularity and information salience. +9 +CONCLUSION +In this work, we introduced the task of proactively forecasting sentence-specific information popularity. We contribute +InfoPop, a dataset containing 51,770 news articles from 26 news websites with over 1.7 million sentences labeled with +normalized popularity scores. We experimented with both unsupervised and supervised baselines and demonstrated +the efficacy of our novel STILTs-based Transfer Learning approach involving an auxiliary supervised task of salience +prediction. Our best models achieved nDCG values over 0.8 for sentence-specific popularity forecasting. Our analysis +13 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +presents an interesting takeaway: though popularity forecasting and salience prediction are quite different problems, +transferring the learning from a salience prediction task enhances a model’s popularity forecasting proficiency. In +future, we aim to explore potential business applications of sentence popularity forecasting in problems such as pull +quote extraction, popularity-guided text summarization, etc. We also plan on exploring a multi-task learning approach +considering the popularity forecasting and salience prediction problems. +REFERENCES +[1] Raza Abidi, Yonglin Xu, Jianyue Ni, Wang Xiangmeng, and wu Zhang. 2020. Popularity prediction of movies: from statistical modeling to machine +learning techniques. Multimedia Tools and Applications 79 (12 2020), 1–35. https://doi.org/10.1007/s11042-019-08546-5 +[2] Mohamed Ahmed, Stella Spagna, Felipe Huici, and Saverio Niccolini. 2013. A Peek into the Future: Predicting the Evolution of Popularity in User +Generated Content. In Proceedings of the Sixth ACM International Conference on Web Search and Data Mining (Rome, Italy) (WSDM ’13). Association +for Computing Machinery, New York, NY, USA, 607–616. https://doi.org/10.1145/2433396.2433473 +[3] Anthony Chen, Pallavi Gudipati, Shayne Longpre, Xiao Ling, and Sameer Singh. 2021. Evaluating Entity Disambiguation and the Role of Popularity +in Retrieval-Based NLP. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International +Joint Conference on Natural Language Processing (Volume 1: Long Papers). Association for Computational Linguistics, Online, 4472–4485. https: +//doi.org/10.18653/v1/2021.acl-long.345 +[4] Wei-Fan Chen, Shahbaz Syed, Benno Stein, Matthias Hagen, and Martin Potthast. 2020. Abstractive Snippet Generation. Proceedings of The Web +Conference 2020 (2020). +[5] Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio. 2014. Empirical Evaluation of Gated Recurrent Neural Networks on +Sequence Modeling. arXiv:1412.3555 [cs.NE] +[6] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. Bert: Pre-training of deep bidirectional transformers for language +understanding. arXiv preprint arXiv:1810.04805 (2018). +[7] Elozino Egonmwan, Vittorio Castelli, and Md Arafat Sultan. 2019. Cross-Task Knowledge Transfer for Query-Based Text Summarization. In +Proceedings of the 2nd Workshop on Machine Reading for Question Answering. Association for Computational Linguistics, Hong Kong, China, 72–77. +https://doi.org/10.18653/v1/D19-5810 +[8] Günes Erkan and Dragomir R Radev. 2004. Lexrank: Graph-based lexical centrality as salience in text summarization. Journal of artificial intelligence +research 22 (2004), 457–479. +[9] Suchin Gururangan, Ana Marasović, Swabha Swayamdipta, Kyle Lo, Iz Beltagy, Doug Downey, and Noah A. Smith. 2020. Don’t Stop Pretraining: +Adapt Language Models to Domains and Tasks. arXiv:2004.10964 [cs.CL] +[10] Jing He, Pablo Duboue, and Jian-Yun Nie. 2012. Bridging the gap between intrinsic and perceived relevance in snippet generation. In Proceedings of +COLING 2012. 1129–1146. +[11] Karl Moritz Hermann, Tomás Kociský, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom. 2015. Teaching +Machines to Read and Comprehend. In NIPS. 1693–1701. http://papers.nips.cc/paper/5945-teaching-machines-to-read-and-comprehend +[12] Sebastian Hofstätter, Bhaskar Mitra, Hamed Zamani, Nick Craswell, and Allan Hanbury. 2021. Intra-Document Cascading: Learning to Select +Passages for Neural Document Ranking. arXiv preprint arXiv:2105.09816 (2021). +[13] Kalervo Järvelin and Jaana Kekäläinen. 2002. Cumulated Gain-Based Evaluation of IR Techniques. ACM Trans. Inf. Syst. 20, 4 (Oct. 2002), 422–446. +https://doi.org/10.1145/582415.582418 +[14] Yitong Ji, Aixin Sun, Jie Zhang, and Chenliang Li. 2020. A Re-Visit of the Popularity Baseline in Recommender Systems. In Proceedings of the +43rd International ACM SIGIR Conference on Research and Development in Information Retrieval (Virtual Event, China) (SIGIR ’20). Association for +Computing Machinery, New York, NY, USA, 1749–1752. https://doi.org/10.1145/3397271.3401233 +[15] Ruipeng Jia, Yanan Cao, Haichao Shi, Fang Fang, Yanbing Liu, and Jianlong Tan. 2020. DistilSum: Distilling the Knowledge for Extractive +Summarization. In Proceedings of the 29th ACM International Conference on Information and Knowledge Management (Virtual Event, Ireland) (CIKM +’20). Association for Computing Machinery, New York, NY, USA, 2069–2072. https://doi.org/10.1145/3340531.3412078 +[16] Chris Kedzie, Kathleen McKeown, and Hal Daumé III. 2018. Content Selection in Deep Learning Models of Summarization. In Proceedings of the +2018 Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, Brussels, Belgium, 1818–1828. +https://doi.org/10.18653/v1/D18-1208 +[17] Yaser Keneshloo, Shuguang Wang, E. Han, and Naren Ramakrishnan. 2016. Predicting the Popularity of News Articles. In SDM. +[18] Sotiris Lamprinidis, Daniel Hardt, and Dirk Hovy. 2018. Predicting News Headline Popularity with Syntactic and Semantic Knowledge Using +Multi-Task Learning. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing. Association for Computational +Linguistics, Brussels, Belgium, 659–664. https://doi.org/10.18653/v1/D18-1068 +[19] Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer. +2020. BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension. In Proceedings of +the 58th Annual Meeting of the Association for Computational Linguistics. Association for Computational Linguistics, Online, 7871–7880. https: +14 + +Sentence Specific Popularity Forecasting +HT ’22, June 28-July 1, 2022, Barcelona, Spain +//doi.org/10.18653/v1/2020.acl-main.703 +[20] Dongliang Liao, Jin Xu, Gongfu Li, Weijie Huang, Weiqing Liu, and Jing Li. 2019. Popularity Prediction on Online Articles with Deep Fusion of +Temporal Process and Content Features. In AAAI. +[21] Yang Liu. 2019. Fine-tune BERT for Extractive Summarization. arXiv:1903.10318 [cs.CL] +[22] Yang Liu and Mirella Lapata. 2019. Text Summarization with Pretrained Encoders. In Proceedings of the 2019 Conference on Empirical Methods in Natural +Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). Association for Computational +Linguistics, Hong Kong, China, 3730–3740. https://doi.org/10.18653/v1/D19-1387 +[23] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. +RoBERTa: A Robustly Optimized BERT Pretraining Approach. arXiv:1907.11692 [cs.CL] +[24] Rada Mihalcea and Paul Tarau. 2004. Textrank: Bringing order into text. In Proceedings of the 2004 conference on empirical methods in natural +language processing. 404–411. +[25] Nuno Moniz and Luís Torgo. 2018. Multi-Source Social Feedback of Online News Feeds. CoRR https://archive.ics.uci.edu/ml/datasets/abs/1801.07055 +(2018). +[26] Ramesh Nallapati, Feifei Zhai, and Bowen Zhou. 2016. Summarunner: A recurrent neural network based sequence model for extractive summarization +of documents. arXiv preprint arXiv:1611.04230 (2016). +[27] Jeffrey Pennington, Richard Socher, and Christopher D Manning. 2014. Glove: Global vectors for word representation. In Proceedings of the 2014 +conference on empirical methods in natural language processing (EMNLP). 1532–1543. +[28] Jason Phang, Thibault Févry, and Samuel R. Bowman. 2018. Sentence Encoders on STILTs: Supplementary Training on Intermediate Labeled-data +Tasks. ArXiv abs/1811.01088 (2018). +[29] Julia Proskurnia, P. A. Grabowicz, Ryota Kobayashi, C. Castillo, P. Cudré-Mauroux, and K. Aberer. 2017. Predicting the Success of Online Petitions +Leveraging Multidimensional Time-Series. Proceedings of the 26th International Conference on World Wide Web (2017). +[30] Yada Pruksachatkun, Jason Phang, Haokun Liu, Phu Mon Htut, Xiaoyi Zhang, Richard Yuanzhe Pang, Clara Vania, Katharina Kann, and Samuel R. +Bowman. 2020. Intermediate-Task Transfer Learning with Pretrained Language Models: When and Why Does It Work?. In ACL. +[31] Alexander Pugachev, Anton Voronov, and Ilya Makarov. 2020. Prediction of News Popularity via Keywords Extraction and Trends Tracking. Recent +Trends in Analysis of Images, Social Networks and Texts 1357 (2020), 37 – 51. +[32] Georgios Rizos, Symeon Papadopoulos, and Yiannis Kompatsiaris. 2016. Predicting News Popularity by Mining Online Discussions. In Proceedings +of the 25th International Conference Companion on World Wide Web (Montréal, Québec, Canada) (WWW ’16 Companion). International World Wide +Web Conferences Steering Committee, Republic and Canton of Geneva, CHE, 737–742. https://doi.org/10.1145/2872518.2890096 +[33] Alexander M. Rush, Sumit Chopra, and Jason Weston. 2015. +A Neural Attention Model for Abstractive Sentence Summarization. +arXiv:1509.00685 [cs.CL] +[34] Yun-Zhu Song, Hong-Han Shuai, Sung-Lin Yeh, Yi-Lun Wu, Lun-Wei Ku, and Wen-Chih Peng. 2020. Attractive or Faithful? Popularity-Reinforced +Learning for Inspired Headline Generation. Proceedings of the AAAI Conference on Artificial Intelligence 34, 05 (Apr. 2020), 8910–8917. +https: +//doi.org/10.1609/aaai.v34i05.6421 +[35] Shivashankar Subramanian, Timothy Baldwin, and Trevor Cohn. 2018. Content-based Popularity Prediction of Online Petitions Using a Deep +Regression Model. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers). Association for +Computational Linguistics, Melbourne, Australia, 182–188. https://doi.org/10.18653/v1/P18-2030 +[36] Alexandru Tatar, Panayotis Antoniadis, Marcelo Dias de Amorim, and Serge Fdida. 2012. Ranking News Articles Based on Popularity Prediction. In +2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining. 106–110. https://doi.org/10.1109/ASONAM.2012.28 +[37] Shubham Toshniwal, Haoyue Shi, Bowen Shi, Lingyu Gao, Karen Livescu, and Kevin Gimpel. 2020. A Cross-Task Analysis of Text Span Representations. +In Proceedings of the 5th Workshop on Representation Learning for NLP. Association for Computational Linguistics, Online, 166–176. +https: +//doi.org/10.18653/v1/2020.repl4nlp-1.20 +[38] Md. Taufeeq Uddin, Muhammed Jamshed Alam Patwary, Tanveer Ahsan, and Mohammed Shamsul Alam. 2016. Predicting the popularity +of online news from content metadata. In 2016 International Conference on Innovations in Science, Engineering and Technology (ICISET). 1–5. +https://doi.org/10.1109/ICISET.2016.7856498 +[39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is +all you need. In Advances in neural information processing systems. 5998–6008. +[40] Anton Voronov, Yao Shen, and Pritom Kumar Mondal. 2019. Forecasting Popularity of News Article by Title Analyzing with BN-LSTM Network. In +Proceedings of the 2019 International Conference on Data Mining and Machine Learning (Hong Kong, Hong Kong) (ICDMML 2019). Association for +Computing Machinery, New York, NY, USA, 19–27. https://doi.org/10.1145/3335656.3335679 +[41] Kai Wang, Penghui Wang, Xin Chen, Qiushi Huang, Zhendong Mao, and Yongdong Zhang. 2020. A Feature Generalization Framework for Social +Media Popularity Prediction. In Proceedings of the 28th ACM International Conference on Multimedia (Seattle, WA, USA) (MM ’20). Association for +Computing Machinery, New York, NY, USA, 4570–4574. https://doi.org/10.1145/3394171.3416294 +[42] Bo Wu, Tao Mei, Wen-Huang Cheng, and Yongdong Zhang. 2016. Unfolding Temporal Dynamics: Predicting Social Media Popularity Using +Multi-Scale Temporal Decomposition. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (Phoenix, Arizona) (AAAI’16). AAAI +Press, 272–278. +15 + +HT ’22, June 28-July 1, 2022, Barcelona, Spain +Ghosh Roy et al., 2022 +[43] Xingxing Zhang, Furu Wei, and Ming Zhou. 2019. HIBERT: Document Level Pre-training of Hierarchical Bidirectional Transformers for Document +Summarization. arXiv:1905.06566 [cs.CL] +[44] Ming Zhong, Pengfei Liu, Yiran Chen, Danqing Wang, Xipeng Qiu, and Xuanjing Huang. 2020. Extractive Summarization as Text Matching. In +Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. Association for Computational Linguistics, Online, 6197–6208. +https://doi.org/10.18653/v1/2020.acl-main.552 +16 + diff --git a/U9AyT4oBgHgl3EQfV_cW/content/tmp_files/load_file.txt b/U9AyT4oBgHgl3EQfV_cW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b663d36ecee501e33a9ed2a86e51f976cd43a31 --- /dev/null +++ b/U9AyT4oBgHgl3EQfV_cW/content/tmp_files/load_file.txt @@ -0,0 +1,1047 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf,len=1046 +page_content='Towards Proactively Forecasting Sentence-Specific Information Popularity within Online News Documents SAYAR GHOSH ROY, IIIT Hyderabad, India ANSHUL PADHI, IIIT Hyderabad, India RISUBH JAIN, IIIT Hyderabad, India MANISH GUPTA, IIIT Hyderabad, India and Microsoft, India VASUDEVA VARMA, IIIT Hyderabad, India Multiple studies have focused on predicting the prospective popularity of an online document as a whole, without paying attention to the contributions of its individual parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We introduce the task of proactively forecasting popularities of sentences within online news documents solely utilizing their natural language content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We model sentence-specific popularity forecasting as a sequence regression task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For training our models, we curate InfoPop, the first dataset containing popularity labels for over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='7 million sentences from over 50,000 online news documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' To the best of our knowledge, this is the first dataset automatically created using streams of incoming search engine queries to generate sentence-level popularity annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We propose a novel transfer learning approach involving sentence salience prediction as an auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Our proposed technique coupled with a BERT-based neural model exceeds nDCG values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='8 for proactive sentence-specific popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Notably, our study presents a non-trivial takeaway: though popularity and salience are different concepts, transfer learning from salience prediction enhances popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We release InfoPop and make our code publicly available1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' CCS Concepts: • Information systems → Content ranking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' • Computing methodologies → Neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Additional Key Words and Phrases: Sentence Popularity Forecasting, Sentence Salience Prediction, Supervised Transfer Learning ACM Reference Format: Sayar Ghosh Roy, Anshul Padhi, Risubh Jain, Manish Gupta, and Vasudeva Varma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Towards Proactively Forecasting Sentence- Specific Information Popularity within Online News Documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 33rd ACM Conference on Hypertext and Social Media (HT ’22), June 28-July 1, 2022, Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' ACM, New York, NY, USA, 16 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/3511095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3531268 1 INTRODUCTION In a typical document popularity prediction task, the objective is to estimate the would-be popularity of some content, say, a news article that is published on an online platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A popularity prediction system might evaluate the number of pageviews that a particular online document would receive, or a social-media popularity prediction model might estimate the number of signatures, comments, shares, or likes that a specific social media post would accrue over a certain period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Though popularity prediction is a well-studied machine learning task, with approaches ranging from traditional feature-based methods to recent Deep Neural Network architectures, all of the existing works on popularity prediction 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='com/sayarghoshroy/InfoPopularity Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' © 2022 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Manuscript submitted to ACM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='00152v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='CL] 31 Dec 2022 HT ’22, June 28-July 1, 2022, Barcelona, Spain Ghosh Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 2022 Local carriers and drivers will be able to book and move their first loads with Uber Freight in the next few weeks, CEO Lior Ron wrote in a blog posted Wednesday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Uber Freight plans to expand to more European countries this year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The EU and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' freight markets have problematic similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' They’re both huge — the EU truckload market is a $400 billion marketplace and third after China and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' — and inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' “The European .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Local carriers and drivers will be able to book .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1099 Uber Freight plans to expand to more Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2046 The EU and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' freight markets have problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0092 They’re both huge — the EU truckload marke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0067 “The European trucking market is experienc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0110 Inefficiency of this scale results in shippers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0132 Uber Freight has been scaling up its business .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1356 The company has offices in San Francisco and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0158 In August, Uber announced that it would .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2703 It’s also made some key hires, one of which .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0076 Uber Freight had about 30,000 active users .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1923 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Task outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Looking solely at document text as input, forecast prospective sentence-specific popularity scores have focused exclusively on the document-level prediction of prospective popularity labels [18, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' When a system predicts that a specific article will be popular over time, a natural follow-up is to ask why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Which information pieces in the document would receive the most notice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Could we identify the informative sentences that would dominate the popularity level of the article?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Being able to generate such insights proactively would have wide-ranging business applications in article promotion, popularity-guided summarization and pull quote selection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' But sentence-specific forecasting of popularity has not been attempted, not even in a simplified binary classification of sentences into popular or not setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The difficulty in obtaining reliable sentence-level labeled data for information popularity has been a crucial driving factor for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' An online article’s popularity can be defined in terms of its page views [34, 40], which is a by-product of the regular Internet browsing activities of the global population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Similarly, within a particular online document, information popularity of a unit sentence is proportional to the number of received requests actively seeking its contained information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' To capture this, we leverage incoming search queries encountered by a popular search engine over a sufficient period to ascertain sentence-specific popularity labels based on how queried-after each sentence is within a document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We present InfoPop, a dataset of over 50,000 online news articles with over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='7M sentences, mapped to popularity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Taking inspiration from existing research on document-level popularity prediction, we frame the problem of sentence- specific popularity forecasting as a regression task similar to previous studies [17, 35] (who model document-level popularity) as opposed to a simple binary classification of a document’s sentences into popular or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Unlike some document-popularity forecasting approaches that rely on post-publication signals like pageview hits in the first half-hour after publication [2, 17], we forecast sentence popularity scores proactively (before a document’s publication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Note that we utilize incoming search queries only to define sentence-specific popularity and construct the InfoPop dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' However, our problem formulation is one of query-insensitive relative normalized scoring of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Formally, given just a document as input, our task (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 1) is of assigning normalized scores in the range [0, 1], to every sentence indicating their intra-document relative information popularity, without utilizing any external signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The key novelty of our approach lies in the design of auxiliary transfer learning subtasks adapting the STILTs (Supplementary Training on Intermediate Labeled-data Tasks) [28] technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In brief, we create auxiliary subtasks of sentence salience prediction in the domain of online news, framed under an identical problem formulation as that of our primary task of popularity forecasting (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We experiment with a baseline neural model using RNNs (Recurrent Neural Networks) and a robust sequence regression architecture based on BERT (Bidirectional Encoder Representations from Transformers) [6], showcasing our proposed approach’s efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Also, we handle arbitrarily long documents 2 Sentence Specific Popularity Forecasting HT ’22, June 28-July 1, 2022, Barcelona, Spain with our neural models employing a sliding window mechanism over a document’s sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In our analysis, we illustrate the differences in the concepts of popularity and salience, both qualitatively and quantitatively (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The empirically demonstrated efficacy of our approach brings out an interesting yet non-trivial takeaway – though salience and popularity are varying concepts, transfer learning from salience prediction improves popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Overall, in this paper, we make the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We introduce the task of proactively forecasting relative information popularities of sentences within online news documents solely based on their natural language content, without the use of any external features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We present InfoPop, the first labeled dataset containing over 50,000 online news documents from 26 news websites with over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='7M sentences, each mapped to supervised popularity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Through a novel STILTs-based transfer learning approach, we build high-performance neural models reaching nDCG scores over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='8 for sentence-specific popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Section 2, we discuss related work on document-level popularity prediction, automatic text summarization and snippet generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We formally present the concept of sentence-specific information popularity in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We describe details of curation of our dataset, InfoPop in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Next, we outline our proposed methods in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Further, we discuss evaluation metrics in Section 6, experimental details in Section 7 and present detailed results with analysis in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Finally, we conclude with a brief summary in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2 RELATED WORK In this section, we discuss related work on document-level popularity prediction, automatic text summarization and snippet generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1 Document-level Popularity Prediction Existing works in popularity prediction have considered a piece of content as an atomic unit whose prospective popularity is inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Based on the choice of popularity surrogate, we distinguish between two kinds of document popularity prediction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Popularity based on Internet browsing: Studies on forecasting popularities of online news documents typi- cally consider the number of received page load requests over sufficient time (accumulated pageviews or hit count) as a surrogate for article-level popularity [17, 18, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Social Media Popularity: These approaches aim to estimate the prospective engagement of a piece of content put up on a particular social media platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Previous works have dealt with multi-modal posts [41], images [42], movies [1], petitions [35], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' and have utilized user-behavior based markers such as the number of comments [32, 36], shares [25, 38], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' as surrogates of social-media popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' There has also been limited work on time- aware [14] and time-series prediction of social-media popularity [20, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Unlike all previous approaches that determine popularity at the document-level, we focus on sentence-specific popularity forecasting (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 1) with our domain of information being online news, not social-media posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For informative documents (including online news), a preferred surrogate of popularity has been the number of pageview hits [3, 31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Intuitively, pageviews captures the generic browsing trends of the population not limited 3 HT ’22, June 28-July 1, 2022, Barcelona, Spain Ghosh Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 2022 to social media actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Our task, however, requires knowledge of more fine-grained Internet browsing activity for annotating specific sentences with popularity labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2 Automatic Text Summarization Text summarization aims to recognize information that is central to the core idea of one or more document(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Approaches are roughly divided into two categories, namely, abstractive: generate a concise, coherent, and cogent summary that captures the central idea expressed by the text piece [19, 33], and extractive: select and arrange salient plus diverse ranges of text to form a summary [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Extractive summarization is often framed as a sequence classification problem – binary labels are assigned to sentences indicating their presence in the summary [11, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In contrast, our task necessitates sequential regression (numerical sentence-specific scores are forecasted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Also, text summarization has the additional objective of generating text without repeated information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In that, an otherwise summary inclusion-worthy sentence may have a binary label of 0 if a similar sentence is already present in the oracle summary [15] (in an extractive setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' However, we expect almost identical sentences in an article to have very similar information popularity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3 Snippet Generation A page snippet can be a document excerpt that lets a user understand whether a document is pertinent to their query without accessing it in its entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Snippet-generation [4] has a ranking-based problem formulation somewhat related to sentence popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' However, our task is one of query-insensitive scoring of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In contrast, tasks such as snippet generation (and document retrieval [12]) are query-sensitive [10] – typical methods require a particular query tailored to which an appropriate snippet is generated (except for naïve approaches that output initial tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 3 SENTENCE-SPECIFIC INFORMATION POPULARITY Document popularity magnitudes and popular-or-not labels based on pageviews capture the amount of notice that the document receives on the Internet over a period of time [18, 40] and is a consequence of the everyday Internet browsing actions of the worldwide population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' With increased Internet penetration, the average number of queries encountered daily by commercial search engines has exceeded the billion mark2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Google Trends3 showcases popular topics of interest segregated by region and timespan based on the pool of encountered search queries, and the collection of worldwide queries serves as a guide to mark information of universal interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Correspondingly, within the local context of a single news document (say 𝐷), if an information piece, say 𝐼1, is more queried-after by the global population than, say, piece 𝐼2 in 𝐷, then: 𝑝𝑜𝑝𝑢𝑙𝑎𝑟𝑖𝑡𝑦(𝐼1) > 𝑝𝑜𝑝𝑢𝑙𝑎𝑟𝑖𝑡𝑦(𝐼2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Popularity levels of sentences within 𝐷 can be derived based on this base principle, wherein the popularity of a specific sentence (say 𝑠) in 𝐷 is incremented based on the lexical similarity between 𝑠 and an encountered query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For a news document 𝐷 with sentences [𝑠1, 𝑠2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 𝑠𝑁 ], comparing each sentence {𝑠𝑖}𝑁 𝑖=1 to every search query encountered by a commercial search engine would be computationally infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Moreover, only a negligibly small percentage of all encountered queries would positively contribute to sentences’ popularities within 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Such queries would precisely be the ones for which 𝐷 could be deemed relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Thus, to derive scores of 𝑠𝑖’s within 𝐷, we filter incoming queries and consider the sublist, say 𝑄 = [𝑞1, 𝑞2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 𝑞|𝑄 |], for which the document 𝐷 was significantly relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2https://searchengineland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='com/google-now-handles-2-999-trillion-searches-per-year-250247 3https://trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='com 4 Sentence Specific Popularity Forecasting HT ’22, June 28-July 1, 2022, Barcelona, Spain Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Selected sentences from a document in InfoPop with their true and forecasted popularities and predicted salience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Popularity forecasts are from our best performing model on nDCG (BERTReg with TL = SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Salience predictions are based on BERTReg trained on S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [TPL: True Popularity Label, FPL: Forecasted Popularity Label, PSL: Predicted Salience Label, TPR: True Popularity Rank, FPR: Forecasted Popularity Rank, PSR: Predicted Salience Rank] Sentence TPL FPL PSL TPR FPR PSR Local carriers and drivers will be able to book and move their first loads with Uber Freight in the next few weeks, CEO Lior Ron wrote in a blog posted Wednesday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0382 11 11 12 The company has made headway breaking into the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0675 8 9 8 Uber Freight had about 30,000 active users last quarter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0427 3 3 11 We regard 𝐷 as a significantly relevant document for query 𝑞 if 𝐷 was shown within the top 10 search results (which roughly translates to the first page of news results) when the search engine encountered 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We assign a base score of �|𝑄 | 𝑗=1 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑞𝑗,𝑠𝑖) to each sentence {𝑠𝑖}𝑁 𝑖=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Sentence popularity scores are then normalized to a [0, 1] range such that the popularity labels of sentences within 𝐷 sum up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 4 INFOPOP DATASET In this section, we outline the creation of the InfoPop dataset and present its basic statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1 Data Acquisition We scraped 82,540 news documents from 26 reputed online news websites (websites listed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We accessed incoming queries of Microsoft Bing Search and mapped each document to the global assemblage of queries that deemed the document as significantly relevant (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 3) at the time when it was encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We extracted each article’s text content and split that into an ordered list of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' With news articles as unit data points, individual sentences served as our atomic information pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We discounted articles with over 100 sentences as we found many of these to be long non-news documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5 HT ’22, June 28-July 1, 2022, Barcelona, Spain Ghosh Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 2022 897 75 40 146 371 3252 1806 2522 1638 411 1190 2616 1038 44 1490 1917 1064 1453 2509 189 10677 2388 6570 142 4979 2346 0 2500 5000 7500 10000 denverpost huffingtonpost washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='cbslocal sanfrancisco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='cbslocal hindustantimes npr newsweek cbslocal cnn livemint phys reuters chicagotribune crictracker abcnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='go latimes techcrunch nydailynews nypost timesnownews nytimes foxnews usatoday theguardian cbsnews nbcnews Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' InfoPop: news sources (on x-axis) with their corresponding #documents (on y-axis) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2 Preprocessing We encountered a considerable amount of noise in the extracted webpage texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' These primarily manifested as strings of seemingly random word tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We removed such heavily non-grammatical sentences using two dependency parsing-based heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Remove a sentence if its dependency graph is not a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The dependency graph of a grammatical sentence is supposed to be a tree and not a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We evaluated whether the produced dependency graph of a sentence was both connected and acyclic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Remove a sentence if its dependency parse contains some xcomp branch leading to a single participle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The X complement or xcomp is a dependency label used to mark dependent clauses that do not contain a subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Consider a noisy unit such as ‘I would like to, I would like to, I would like to.’ The parse tree of such a text string contains an xcomp branch leading to a participle at the leaf node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We observed a positive correlation between such phenomena and text repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Though the above cleaning heuristics are computationally expensive, we employ these owing to the one-time nature of the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' After these cleanup steps, we further removed articles containing less than three grammatical sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3 Popularity Labels For a particular online document 𝐷 (say), we considered the collection of all queries, say 𝑄 = [𝑞1, 𝑞2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 𝑞|𝑄 |], for which 𝐷 was significantly relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We generated supervised popularity labels for every sentence in 𝐷 using the method 6 Sentence Specific Popularity Forecasting HT ’22, June 28-July 1, 2022, Barcelona, Spain # Sentences per document # Documents 0 2500 5000 7500 10000 12500 0 - 10 11 - 20 21 - 30 31 - 40 41 - 50 51 - 60 61 - 70 71 - 80 81 - 90 91 - 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' InfoPop: Distribution of #sentences per document outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' More formally, 𝑃𝑖 ← �|𝑄| 𝑗=1 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑞𝑗,𝑠𝑖) �𝑁 𝑖=1 �|𝑄| 𝑗=1 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑞𝑗,𝑠𝑖) , ∀𝑖 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 𝑁 } where 𝑃𝑖 denotes the normalized popularity label assigned to the 𝑖𝑡ℎ sentence in 𝐷, assuming 𝐷 contains 𝑁 sentences: 𝑠1, 𝑠2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 𝑠𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We used cosine-similarity between corresponding TF-IDF vectors as the measure of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Though our similarity function might feel primitive, we uphold that utilizing lexical similarity is optimum in this case since search engine ranking algorithms still utilize word match as the basic principle and most relevant pages typically contain the query terms (that usually do not have well defined semantic embeddings) themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' That said, we plan to explore semantic embedding based similarity as part of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='4 InfoPop Dataset Statistics InfoPop contains 51,770 news documents containing 1,711,890 sentences annotated with normalized popularity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Each document contains a minimum of 3 and a maximum of 100 sentences, with the average number of sentences per article being 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 3 shows the distribution of the number of sentences per document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' On average, each sentence contains 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='23 word tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The minimum and maximum document lengths stand at 15 and 2,516 tokens, respectively, with 602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='92 tokens being the average document length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The news articles are sourced from 26 reputed news websites, with the average number of articles per website being 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2 illustrates the specific number of documents sourced from each news website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We observe that popularity scores and sentence lengths have a weak positive correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We divide our dataset into train, validation, and test splits in an 8:1:1 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' To ensure reproducibility, we make InfoPop (and our defined train, validation, and test splits) publicly available here4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Table 1 shows an excerpt from a document within InfoPop with sentence-specific popularity labels (TPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5 METHODS We describe various methods for proactive sentence-specific popularity forecasting, including unsupervised sentence ranking baselines, neural sentence sequence regression architectures, and our proposed supervised Transfer Learning approach from an auxiliary task of salience prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='com/sayarghoshroy/InfoPopularity 7 HT ’22, June 28-July 1, 2022, Barcelona, Spain Ghosh Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 2022 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1 Unsupervised sentence ranking baselines News articles typically utilize the pyramid structure of reporting with primary information contained within the initial sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We experiment with a position baseline where we score sentences in descending order of their position from the beginning of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Specifically, we assign the score of 1 − 𝑖 𝑛 to the 𝑖𝑡ℎ sentence of an article with 𝑛 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We also evaluate the performance of TextRank [24] and LexRank [8] algorithms that exploit graph-based similarities between all sentence pairs through PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2 Our Proposed Approach The primary novelty of our proposed approach lies in the design of a STILTs-based [28] Transfer Learning (TL) setup using a constructed task of sentence salience prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Recent advances in NLP have utilized transfer learning in two broad ways, (a) unsupervised pre-training of text encoders using specific Language Modeling objectives [9, 23] and (b) supervised pre-training of architectures on tasks with similar problem formulations [28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In STILTs (Supplementary Training on Intermediate Labeled-data Tasks), a model pre-trained on unsupervised corpora (like BERT) is further pre-trained on an intermediate, supervised task for which ample labeled data is available before fine-tuning on its primary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We observe how sentence salience is well studied in context of text summarization, an area that enjoys availability of sizeable amounts of news domain data and hypothesize that STILTs-based transfer learning from an auxiliary task of salience prediction would boost our models’ popularity forecasting ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1 Auxiliary Transfer Learning (TL) Subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For document summarization, a sentence is considered salient if it contains information related to the primary semantics of the document and can be a worthy inclusion for the document’s summary [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Given an article and its gold standard summary, the salience of an individual sentence is computed as the ROUGE overlap between sentence and summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For oracle creation in a typical extractive summarization setting, binary summary-inclusion labels are computed for sentences greedily to maximize the ROUGE overlap between the complete oracle and the true summary [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Such a greedy labeling scheme implicitly takes minimization of information redundancy into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In that, a salient sentence might receive a summary-inclusion label of 0 if a lexically similar sentence was previously included in the oracle summary [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' However, we only capture the salience of sentences and desire very similar sentences to have similar labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Similar to sentence-specific popularity forecasting, we package our auxiliary task as a sentence sequence regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' This lets us (a) train our architectures adapting the STILTs approach and (b) perform an empirical cross-task evaluation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Since our domain of interest is online news, we utilize the publicly available CNN-DailyMail news summarization dataset (with the same splits as in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We compute three weakly supervised salience scores for each sentence based on its ROUGE 1, ROUGE 2, and ROUGE L overlap with the corresponding article’s summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Like InfoPop labels, we normalize salience labels across a document by dividing by the sum of individual sentence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Thus, we create three auxiliary subtasks due to the three labeling schemes, which we tag as S1, S2, and SL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2 Supervised Neural Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We frame information popularity forecasting as a sentence sequence regression task where a sentence’s score is relative to the entire article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Thus, effective neural models require the global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We pre-train a neural architecture on a supervised TL subtask (S1, S2, or SL), and then fine-tune the model for popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' High-level overviews of BaseReg, our rudimentary neural baseline, and BERTReg, our BERT-based sentence- sequence regression model are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We use MSE (Mean Squared Error) loss between 8 Sentence Specific Popularity Forecasting HT ’22, June 28-July 1, 2022, Barcelona, Spain S1 S2 S3 Sn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' GRU GRU GRU GRU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' pool FC Doc score1 score2 score3 scoren .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Sentences CNN Layers bi-RNN Layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BaseReg: RNN for Sentence Sequence Regression true and inferred sentence scores for training our neural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Note that we handle arbitrarily large documents us- ing both BERTReg and BaseReg employing a sliding window mechanism over a document’s sentences with a preset stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BaseReg: As a simple baseline neural model for sentence scoring, we build upon the rudimentary SummaRunner [16, 26] architecture with Convolutional Neural Network (CNN) based sentence encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We input the concatenated padded sequence of GloVe [27] embeddings for tokens in a sentence to a two-layered CNN sentence vectorizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Internally, a CNN layer applies a sequence of convolution and batch normalization followed by a Leaky ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The output from the two stacked CNNs is then max pooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Inside, three CNN stacks with kernel sizes of 3, 4, and 5 are used whose outputs are concatenated to form the sentence embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Individual CNN-based sentence vectors are passed through a layer of bi-GRUs (bidirectional Gated Recurrent Units) [5] that produce a sequence of contextual sentence embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' All such contextual embeddings are max pooled and then passed through a fully connected (FC) layer to derive a document embedding vector that captures the global context of the complete document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Each sentence score is then computed based on aspects such as their content, novelty, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', [26] which are in turn expressed as functions of their contextual embedding and the document embedding vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BERTReg: The use of Transformer-based [39] models such as BERT [6] have yielded state-of-the-art results on various NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We adapt BertSumExt’s [22] architecture of sentence classification to our sequential regression setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' To generate contextual sentence embeddings, we add [SEP] tokens to mark sentence ends and distinctive [REG] tokens at sentence beginnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Downstream, the contextual embeddings of the [REG] tokens are used for sequence regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For each input token, its token, position, and segment embeddings are generated [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Here, the segment embeddings allow differentiation between odd and even positioned sentences within BERT (since traditional BERT has been trained with only two segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The token, position, and segment embeddings of the defined input tokens are passed to BERT, which produces a sequence of refined token embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Now, the sequence of BERT-based sentence embeddings (embeddings corresponding to the [REG] tokens) is summed with their sinusoidal sentence positional embeddings and passed through a two-layer Transformer model to yield contextual sentence embeddings aware of multi-sentence discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [21] showed the usefulness of adding a two-layer Transformer that captures inter-sentence relationships as opposed to directly using BERT-generated embeddings for sentence-sequence labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Finally, the generated contextual 9 HT ’22, June 28-July 1, 2022, Barcelona, Spain Ghosh Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 2022 [REG] 1st sentence [SEP] [REG] 2nd sentence [SEP] [REG] nth sentence [SEP] T[REG] T1st Tsent T[SEP] T[REG] T2nd Tsent T[SEP] T[REG] Tnth Tsent T[SEP] S0 S0 S0 S0 S1 S1 S1 S0 S0 S0 S0 E1 E5 E4n-3 SP1 E2 E3 E4 E6 E7 E8 SPn E4n-3 E4n-3 E4n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Linear Activation score1 Input Token Embeddings Segment Embeddings BERT Embeddings Sentence Positional Embeddings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' C1 Contextual Embeddings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' P0 P1 P2 P3 P4 P5 P6 P7 P4n-3 P4n-2 P4n-1 P4n Position Embeddings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Linear Activation score2 C2 Linear Activation scoren Cn SP2 S1 Transformer Layers BERT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BERTReg: BERT for Sentence Sequence Regression embeddings of the individual sentences are passed through a linear layer with a sigmoid activation to generate regression scores ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5 shows an overview of BERTReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 6 EVALUATION METRICS We utilize the following automatic evaluation metrics to measure the performance of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Top 𝑘 overlap Let 𝐴𝑘 and 𝑃𝑘 be the sets of actual and predicted top-𝑘 highest scored sentences, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We define top 𝑘 overlap (top𝑘) as |𝐴𝑘 ∩ 𝑃𝑘 |/𝑘 and report its percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Thus, top1 marks whether a method correctly identifies the highest scored sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Regression errors We report Mean Squared Error (MSE) and Mean Absolute Error (MAE) between arrays of actual and predicted sentence labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Rank Correlation Metrics Top 𝑘 overlap focuses only on the identification of the highest scored sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' To get a picture of how well various algorithms rank and order the complete set of sentences in a document, we use Spearman’s rank correlation (𝜌) and Kendall’s Tau (𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 𝜌,𝜏 ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' nDCG Top 𝑘 overlap, 𝜌, and 𝜏 only consider sentence ranking, while regression errors only account for sentence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Normalized Discounted Cumulative Gain (nDCG) [13] captures the normalized gain or usefulness of a sentence based on both its position in the inferred rank list and its actual score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' nDCG ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Intuitively, the penalty for downscoring and inverting ranks of two truly high scored sentences should be lower than that of inverting ranks of two negligibly scored ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Rank Correlation metrics do not take scores into 10 Sentence Specific Popularity Forecasting HT ’22, June 28-July 1, 2022, Barcelona, Spain Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Sentence Popularity Forecasting Results Method TL Top1 Top2 Top3 MSE MAE 𝜏 𝜌 nDCG Position − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='92 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='81 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='5804 TextRank − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='08 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='0674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='8025 account and fail to capture such insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Whereas regression errors only focus on true and forecasted scores ignoring the score-induced rank ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Thus, broadly, nDCG is the most holistic metric for our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 7 EXPERIMENTAL DETAILS We used publicly available implementations of TextRank5 and LexRank6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We normalized inferred LexRank scores to a [0, 1] range (by dividing by the sum of all sentence scores) for computing regression errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' To create the auxiliary transfer learning subtasks, we utilized the non-anonymized version of CNN-DailyMail dataset [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We used spaCy’s7 dependency parser and NLTK’s8 sent_tokenize function during preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We used a maximum sequence limit of 100 sentences in the layer of bi-GRUs within our BaseReg model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The CNN- sentence encoder had an upper input limit of 100 tokens per sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For training our BaseReg model, we used Adam optimizer with a batch size of 256 documents, an initial learning rate of 10−5, with default hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We used the 6-layer bert-base-uncased model in our BERTReg experiments and considered a maximum sequence length of 1536 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Adam optimizer was used with a learning rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='10−3, with other hyperparameters set to their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We split documents exceeding the maximum sequence length into overlapping sliding windows with a predefined maximum stride of 10 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' During inference, we adopted the same splitting technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' For a sentence 𝑠 within the stride of two consecutive windows, we scored 𝑠 as the mean of the computed scores from the two windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We trained all of our models on NVIDIA RTX 2080Ti(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BaseReg models were trained on a single GPU for a maximum of 4 epochs with early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A single epoch on our transfer learning task took over 6 hours, while one epoch for popularity forecasting training took close to 45 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We trained BERTReg based models on 4 GPUs for 50, 000 optimizer steps with early stopping turned off for both the popularity forecasting and the transfer learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The training time ranged between 10 to 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8 RESULTS In this section, we analyze the performance of baselines and our proposed approach for proactive sentence-specific popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We then share some insights on the interrelations between our primary task of sentence popularity forecasting and our auxiliary Transfer Learning (TL) task of salience prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5https://github.' metadata={'source': 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+page_content=' Table 2 shows performance of unsupervised sentence ranking baselines on sentence-specific popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' LexRank turns out the best unsupervised method beating Position and TextRank on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2 Proposed Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We present results of our supervised models in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' As expected, BERTReg is the best architecture for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We see performance upgrades across the board upon employment of the Transfer Learning setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Though there is no consistent winner among the three Transfer Learning subtasks9, the best result for each metric is achieved using some form of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BERTReg with TL = SL boosted average nDCG over 1% from plain BERTReg and over 7% from the best unsupervised baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We performed 𝑡-tests which showed that BERTReg with TL = SL significantly outperformed vanilla BERTReg on nDCG at significance level 𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Though BaseReg is ineffective for popularity forecasting, we observe the same trend of transfer learning as a positive addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' These results demonstrate that transfer learning from salience prediction significantly improves sentence popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We attribute the performance enhancement to two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Firstly, datasets used for both tasks are sourced from online news documents, and transfer learning allows the model to witness more domain-specific data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Secondly, though popularity forecasting differs from salience prediction (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2), they have certain similarities, such as penalizing not lexically dense sentences or understanding that specific sentences do not carry any notable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Our proposed approach grounds the network to capture such relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2 Popularity and Salience Salient sentences effectively capture summary-inclusion worthy ideas that are central to the core semantics of an article [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Unlike salient information bits, an information piece might deviate significantly from an article’s primary topic yet be popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Consider the following sentence from a particular article10 (with ID 34499) within InfoPop: ‘Weinsheimer has spent 27 years at DOJ, where he tried homicide and public corruption cases.’ The sentence is not 9We prefer TL = SL as it outperforms others on Top𝑘 (𝑘 = 2, 3), MAE, and most importantly, nDCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A downstream application focused purely on accurate ranking could use TL = S1, while another requiring identification of the most popular sentence could use TL = S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 10https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='npr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/2018/07/03/625581627/another-top-justice-department-lawyer-steps-down-following-earlier-departures 12 Sentence Specific Popularity Forecasting HT ’22, June 28-July 1, 2022, Barcelona, Spain Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Cross-task evaluation − performance of BERTReg trained for popularity forecasting (PF) evaluated on salience prediction and vice-versa Train Eval Top 1 Top 2 Top 3 MSE MAE 𝜏 𝜌 nDCG S1 PF 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='97 18.' metadata={'source': 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in the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1 Quantitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We evaluate our unsupervised baselines and supervised neural models on the transfer learning subtasks (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' The position baseline achieves the best rank correlation scores, explainable based on the pyramid structure of news reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BERTReg beats other approaches on nDCG and in identifying the most salient sentences (Top 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Comparing with Table 2, we see how sentence ranking baselines are more capable of capturing information salience than forecasting information popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Position baseline’s scores for salience prediction are significantly greater than for popularity forecasting, indicating that the salience of the initial sentences in news articles is typically higher than their popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Comparison with Table 2 shows that supervised salience prediction models achieve much better results on 𝜌, 𝜏, and nDCG compared to popularity forecasting models utilizing the same underlying architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Also, note how BaseReg, which performed poorly on popularity forecasting, achieves respectable scores for salience prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' These experiments identify salience prediction as the less complicated problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', given a document, it is easier to compute how central and summary-worthy a contained information piece is than forecasting its popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Table 1 shows an excerpt from an InfoPop article with actual and forecasted popularity plus predicted salience labels for sentences showcasing the variation between information popularity and salience within the same document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2 Empirical cross-task evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We performed an empirical cross-task analysis [7, 37] to capture the degree of relatedness between primary and auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We used the BERTReg popularity forecasting model to infer sentences’ labels for documents in the salience-prediction dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We evaluated these scores against actual sentence salience labels of the three types – S1, S2, and SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Similarly, we evaluated models trained on S1, S2, and SL upon our primary popularity forecasting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We show results in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Predictably, we observe a massive drop in performance when we switch to the cross-task setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Values across all metrics fall below those achieved by some unsupervised ranking baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' This further experimentally showcases the distinction between information popularity and information salience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 9 CONCLUSION In this work, we introduced the task of proactively forecasting sentence-specific information popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We contribute InfoPop, a dataset containing 51,770 news articles from 26 news websites with over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='7 million sentences labeled with normalized popularity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We experimented with both unsupervised and supervised baselines and demonstrated the efficacy of our novel STILTs-based Transfer Learning approach involving an auxiliary supervised task of salience prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Our best models achieved nDCG values over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='8 for sentence-specific popularity forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Our analysis 13 HT ’22, June 28-July 1, 2022, Barcelona, Spain Ghosh Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 2022 presents an interesting takeaway: though popularity forecasting and salience prediction are quite different problems, transferring the learning from a salience prediction task enhances a model’s popularity forecasting proficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In future, we aim to explore potential business applications of sentence popularity forecasting in problems such as pull quote extraction, popularity-guided text summarization, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' We also plan on exploring a multi-task learning approach considering the popularity forecasting and salience prediction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' REFERENCES [1] Raza Abidi, Yonglin Xu, Jianyue Ni, Wang Xiangmeng, and wu Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Popularity prediction of movies: from statistical modeling to machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Multimedia Tools and Applications 79 (12 2020), 1–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1007/s11042-019-08546-5 [2] Mohamed Ahmed, Stella Spagna, Felipe Huici, and Saverio Niccolini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A Peek into the Future: Predicting the Evolution of Popularity in User Generated Content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the Sixth ACM International Conference on Web Search and Data Mining (Rome, Italy) (WSDM ’13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 607–616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/2433396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2433473 [3] Anthony Chen, Pallavi Gudipati, Shayne Longpre, Xiao Ling, and Sameer Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Evaluating Entity Disambiguation and the Role of Popularity in Retrieval-Based NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Online, 4472–4485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='acl-long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='345 [4] Wei-Fan Chen, Shahbaz Syed, Benno Stein, Matthias Hagen, and Martin Potthast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Abstractive Snippet Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Proceedings of The Web Conference 2020 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [5] Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3555 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='NE] [6] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Bert: Pre-training of deep bidirectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='04805 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [7] Elozino Egonmwan, Vittorio Castelli, and Md Arafat Sultan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Cross-Task Knowledge Transfer for Query-Based Text Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 2nd Workshop on Machine Reading for Question Answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Hong Kong, China, 72–77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/D19-5810 [8] Günes Erkan and Dragomir R Radev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Lexrank: Graph-based lexical centrality as salience in text summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Journal of artificial intelligence research 22 (2004), 457–479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [9] Suchin Gururangan, Ana Marasović, Swabha Swayamdipta, Kyle Lo, Iz Beltagy, Doug Downey, and Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Don’t Stop Pretraining: Adapt Language Models to Domains and Tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='10964 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='CL] [10] Jing He, Pablo Duboue, and Jian-Yun Nie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Bridging the gap between intrinsic and perceived relevance in snippet generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of COLING 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 1129–1146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [11] Karl Moritz Hermann, Tomás Kociský, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Teaching Machines to Read and Comprehend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In NIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 1693–1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' http://papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='nips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='cc/paper/5945-teaching-machines-to-read-and-comprehend [12] Sebastian Hofstätter, Bhaskar Mitra, Hamed Zamani, Nick Craswell, and Allan Hanbury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Intra-Document Cascading: Learning to Select Passages for Neural Document Ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='09816 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [13] Kalervo Järvelin and Jaana Kekäläinen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Cumulated Gain-Based Evaluation of IR Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 20, 4 (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2002), 422–446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/582415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='582418 [14] Yitong Ji, Aixin Sun, Jie Zhang, and Chenliang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A Re-Visit of the Popularity Baseline in Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval (Virtual Event, China) (SIGIR ’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 1749–1752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/3397271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3401233 [15] Ruipeng Jia, Yanan Cao, Haichao Shi, Fang Fang, Yanbing Liu, and Jianlong Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' DistilSum: Distilling the Knowledge for Extractive Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 29th ACM International Conference on Information and Knowledge Management (Virtual Event, Ireland) (CIKM ’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 2069–2072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/3340531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3412078 [16] Chris Kedzie, Kathleen McKeown, and Hal Daumé III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Content Selection in Deep Learning Models of Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Brussels, Belgium, 1818–1828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/D18-1208 [17] Yaser Keneshloo, Shuguang Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Han, and Naren Ramakrishnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Predicting the Popularity of News Articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In SDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [18] Sotiris Lamprinidis, Daniel Hardt, and Dirk Hovy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Predicting News Headline Popularity with Syntactic and Semantic Knowledge Using Multi-Task Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Brussels, Belgium, 659–664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/D18-1068 [19] Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Online, 7871–7880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https: 14 Sentence Specific Popularity Forecasting HT ’22, June 28-July 1, 2022, Barcelona, Spain //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='acl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='703 [20] Dongliang Liao, Jin Xu, Gongfu Li, Weijie Huang, Weiqing Liu, and Jing Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Popularity Prediction on Online Articles with Deep Fusion of Temporal Process and Content Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [21] Yang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Fine-tune BERT for Extractive Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='10318 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='CL] [22] Yang Liu and Mirella Lapata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Text Summarization with Pretrained Encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Hong Kong, China, 3730–3740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/D19-1387 [23] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' RoBERTa: A Robustly Optimized BERT Pretraining Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='11692 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='CL] [24] Rada Mihalcea and Paul Tarau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Textrank: Bringing order into text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 2004 conference on empirical methods in natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 404–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [25] Nuno Moniz and Luís Torgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Multi-Source Social Feedback of Online News Feeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' CoRR https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='edu/ml/datasets/abs/1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='07055 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [26] Ramesh Nallapati, Feifei Zhai, and Bowen Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Summarunner: A recurrent neural network based sequence model for extractive summarization of documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='04230 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [27] Jeffrey Pennington, Richard Socher, and Christopher D Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Glove: Global vectors for word representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 1532–1543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [28] Jason Phang, Thibault Févry, and Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Sentence Encoders on STILTs: Supplementary Training on Intermediate Labeled-data Tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' ArXiv abs/1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='01088 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [29] Julia Proskurnia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Grabowicz, Ryota Kobayashi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Castillo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Cudré-Mauroux, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Aberer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Predicting the Success of Online Petitions Leveraging Multidimensional Time-Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Proceedings of the 26th International Conference on World Wide Web (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [30] Yada Pruksachatkun, Jason Phang, Haokun Liu, Phu Mon Htut, Xiaoyi Zhang, Richard Yuanzhe Pang, Clara Vania, Katharina Kann, and Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Intermediate-Task Transfer Learning with Pretrained Language Models: When and Why Does It Work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='. In ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [31] Alexander Pugachev, Anton Voronov, and Ilya Makarov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Prediction of News Popularity via Keywords Extraction and Trends Tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Recent Trends in Analysis of Images, Social Networks and Texts 1357 (2020), 37 – 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [32] Georgios Rizos, Symeon Papadopoulos, and Yiannis Kompatsiaris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Predicting News Popularity by Mining Online Discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 25th International Conference Companion on World Wide Web (Montréal, Québec, Canada) (WWW ’16 Companion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, CHE, 737–742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/2872518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2890096 [33] Alexander M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Rush, Sumit Chopra, and Jason Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A Neural Attention Model for Abstractive Sentence Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='00685 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='CL] [34] Yun-Zhu Song, Hong-Han Shuai, Sung-Lin Yeh, Yi-Lun Wu, Lun-Wei Ku, and Wen-Chih Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Attractive or Faithful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Popularity-Reinforced Learning for Inspired Headline Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Artificial Intelligence 34, 05 (Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020), 8910–8917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1609/aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='v34i05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='6421 [35] Shivashankar Subramanian, Timothy Baldwin, and Trevor Cohn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Content-based Popularity Prediction of Online Petitions Using a Deep Regression Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Melbourne, Australia, 182–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/P18-2030 [36] Alexandru Tatar, Panayotis Antoniadis, Marcelo Dias de Amorim, and Serge Fdida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Ranking News Articles Based on Popularity Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In 2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 106–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1109/ASONAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='28 [37] Shubham Toshniwal, Haoyue Shi, Bowen Shi, Lingyu Gao, Karen Livescu, and Kevin Gimpel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A Cross-Task Analysis of Text Span Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 5th Workshop on Representation Learning for NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Online, 166–176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='repl4nlp-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='20 [38] Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Taufeeq Uddin, Muhammed Jamshed Alam Patwary, Tanveer Ahsan, and Mohammed Shamsul Alam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Predicting the popularity of online news from content metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In 2016 International Conference on Innovations in Science, Engineering and Technology (ICISET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1109/ICISET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='7856498 [39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Advances in neural information processing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' [40] Anton Voronov, Yao Shen, and Pritom Kumar Mondal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Forecasting Popularity of News Article by Title Analyzing with BN-LSTM Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 2019 International Conference on Data Mining and Machine Learning (Hong Kong, Hong Kong) (ICDMML 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 19–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/3335656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3335679 [41] Kai Wang, Penghui Wang, Xin Chen, Qiushi Huang, Zhendong Mao, and Yongdong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' A Feature Generalization Framework for Social Media Popularity Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 28th ACM International Conference on Multimedia (Seattle, WA, USA) (MM ’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 4570–4574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='1145/3394171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='3416294 [42] Bo Wu, Tao Mei, Wen-Huang Cheng, and Yongdong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Unfolding Temporal Dynamics: Predicting Social Media Popularity Using Multi-Scale Temporal Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (Phoenix, Arizona) (AAAI’16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' AAAI Press, 272–278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 15 HT ’22, June 28-July 1, 2022, Barcelona, Spain Ghosh Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=', 2022 [43] Xingxing Zhang, Furu Wei, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' HIBERT: Document Level Pre-training of Hierarchical Bidirectional Transformers for Document Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='06566 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='CL] [44] Ming Zhong, Pengfei Liu, Yiran Chen, Danqing Wang, Xipeng Qiu, and Xuanjing Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Extractive Summarization as Text Matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' Association for Computational Linguistics, Online, 6197–6208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='18653/v1/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='acl-main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf'} +page_content='552 16' metadata={'source': 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Wilde,1 Oskar Elek,2 Joseph N. Burchett,2, 3 Daisuke Nagai,4 J. Xavier Prochaska,2, 5 +Jessica Werk,1 Sarah Tuttle,1 and Angus G. Forbes2, 6 +1University of Washington, Department of Astronomy, Seattle, WA 98195, USA +2University of California in Santa Cruz, 1156 High St, Santa Cruz, CA 95064, USA +3Department of Astronomy, New Mexico State University, PO Box 30001, MSC 4500, Las Cruces, NM 88001 +4Department of Physics, Yale University, New Haven, CT 06520, USA +5Kavli Institute for the Physics and Mathematics of the Universe, 5-1-5 Kashiwanoha, Kashiwa 277-8583, Japan +6Purdue University, 610 Purdue Mall, West Lafayette, IN 47907, USA +ABSTRACT +The “cosmic web”, the filamentary large-scale structure in a cold dark matter Universe, is readily +apparent via galaxy tracers in spectroscopic surveys. However, the underlying dark matter structure +is as of yet unobservable and mapping the diffuse gas permeating it lies beyond practical observational +capabilities. +A recently developed technique, inspired by the growth and movement of Physarum +polycephalum ‘slime mold’, has been used to map the cosmic web of a low redshift sub-sample of the +SDSS spectroscopic galaxy catalog. This model, the Monte Carlo Physarum Machine (MCPM) was +shown to promisingly reconstruct the cosmic web. Here, we improve the formalism used in calibrating +the MCPM to better recreate the Bolshoi-Planck cosmological simulation’s density distributions and +apply them to a significantly larger cosmological volume than previous works using the Sloan Digital +Sky Survey (SDSS, z < 0.1) and the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) +Luminous Red Galaxy (LRG, z ≲ 0.5) spectroscopic catalogs. We present the ‘Cosmic Slime Value +Added Catalog’ which provides estimates for the cosmic overdensity for the sample of galaxies probed +spectroscopically by the above SDSS surveys. +In addition, we provide the fully reconstructed 3D +density cubes of these volumes. These data products were released as part of Sloan Digital Sky Survey +Data Release 17 and are publicly available. We present the input catalogs and the methodology for +constructing these data products. We also highlight exciting potential applications to galaxy evolution, +cosmology, the intergalactic and circumgalactic medium, and transient phenomenon localization. +1. INTRODUCTION +The cosmic web is an emergent prediction of ΛCDM +cosmology and is ubiquitously reproduced and readily +identifiable in cosmological simulations, where the un- +derlying density distribution is known (e.g., Springel +et al. 2005; Vogelsberger et al. 2014). However, unveiling +the large-scale structure in the observational realm using +galaxies and absorption lines as tracers of the intergalac- +tic medium (IGM) is much less straightforward. +The +underlying dark matter distribution remains unobserv- +able. The two most accessible tracers, such as galaxies +and quasar absorption lines, are limited by the practi- +cal observational constraints of galaxy redshift surveys +and the scarcity of quasars in the universe, respectively. +Corresponding author: Matthew C. Wilde +mwilde@uw.edu +Even when observational tracers are available at rela- +tively high sampling densities, the problem of recon- +structing the cosmic web is highly complex. +We highlight two of the myriad scientific motivations +for cosmic web reconstruction. First, of paramount con- +cern in galaxy astrophysics is the impact of a galaxy’s +environment on its evolution. Correlations between en- +vironmental metrics and galaxy properties, such as mor- +phology (e.g., Dressler 1980), color (e.g., Abell 1965), +and star formation (e.g., Balogh et al. 1999; Peng et al. +2010), have been known about for many decades, but +the physical mechanisms and their relative importance +remain heavily pursued problems. Galaxy-environment +analyses typically fall along one of two paths: +local +environment-centric or large-scale environment-centric. +In the former, one employs an environmental density +metric, such as a nearest-neighbor distance or density +within some aperture (Kauffmann et al. 2004; Peng et al. +arXiv:2301.02719v1 [astro-ph.CO] 6 Jan 2023 + +2 +2010), or galaxies are associated with a local group or +cluster environment (Yang et al. 2007; Berlind et al. +2006) and galaxy properties are studied with respect +to the properties of the group or cluster (Carollo et al. +2013; Catinella et al. 2013; Poggianti et al. 2009). +The latter path is less straightforward, as one must +infer the large-scale structure from tracers, typically the +galaxies themselves, and correlate galaxies back to that +structure in some way. Various methods have been de- +vised to reconstruct the cosmic web from discrete trac- +ers. Libeskind et al. (2018) reviewed a number of these, +and we refer the reader to this valuable resource for an +overview of the techniques employed and comparisons +between them. Once the underlying density field is in- +ferred, one can correlate galaxy properties with this den- +sity field (an approach one can directly employ with the +catalog described here) or attempt to geometrically re- +late a galaxy’s position to the structure identified, e.g., +the distance to a filament. One should appreciate that +filament identification (e.g., DisPerSE; Luber et al. 2019; +Tempel et al. 2014), whether from a density field or some +other methodology, is a separate problem from the in- +ference of the field itself. +Studies of galaxy properties and their dependence on +the cosmic environment report mixed results. Kuutma +et al. (2017) find a higher elliptical-to-spiral ratio and +decreasing star formation rate (SFR) towards filament +spines. +Similarly, Crone Odekon et al. (2018) report +that, at fixed stellar mass, galaxies closer to filaments or +in higher density environments are more deficient in HI. +These large-scale environmental correlations with galax- +ies have also been investigated using modern hydrody- +namical cosmological simulations. Codis et al. (2018) +measure the spin-filament alignment in IllustrisTNG +(Vogelsberger et al. 2014) and find a strong dependence +on spin alignment with galaxy mass. Pasha et al. (2022) +find that the collapse of large-scale structure into sheets +at higher redshifts (z ∼ 3) can create shocks that ex- +plain quenching in dwarf galaxies similar to the effects +seen in the presence of clusters and groups. +Second, in addition to the galaxies themselves, the +IGM studied in context with the cosmic web environ- +ment can yield important insight. Wakker et al. (2015) +measured the Lyα absorption in quasar spectra prob- +ing a foreground visually identified filament, finding in- +creasing absorber equivalent width and linewidth with +decreasing projected distance to the center of the fila- +ment. With a larger archival sample of QSOs and fila- +ments, Bouma et al. (2021) find similar results, with Lyα +absorbers showing both greater incidence and column +density at a small projected distance and velocity offsets +from filaments first identified by Courtois et al. (2013). +In the first application of the reconstruction framework +we use here, Burchett et al. (2020) analyzed the Lyα op- +tical depth as a function of cosmic web density probed +by QSO sightlines. They found three distinct regimes: +(1) a void regime at low matter overdensity with no +detected absorption, (2) an onset of absorption in the +outer skins of filaments with monotonically increasing +optical depth, and (3) the highest-density regime where +the absorption no longer increases with local density but +rather turns over and declines at the highest densities. +Associating the IGM to the cosmic web provides impor- +tant constraints on hydrodynamical processes modeled +in cosmological simulations that may be used to inter- +pret the environmental quenching conundrums. +In this manuscript, we employ the novel method first +introduced in Burchett et al. (2020) described in detail +by Elek et al. (2022), which is based on the morphol- +ogy of the Physarum polycephalum slime mold organism +to map the cosmic density field. +This model implic- +itly traces the cosmic web structure by efficiently finding +optimal pathways between the galaxies that trace fila- +ments. We apply our model to two large galaxy cata- +logs, the NASA Sloan Atlas (NSA) (Blanton et al. 2011) +and the catalogs of Luminous Red Galaxies (LRGs) +from the SDSS-IV Extended Baryon Oscillation Spec- +troscopic Survey (Bautista et al. 2018). +Our method +faithfully reconstructs the cosmic matter density of the +cosmic web throughout the observed volume, allowing +the study of the dark matter distribution with respect +to any objects of interest in the survey footprints, not +just at the input galaxy locations. We have released this +data as part of the SDSS Data Release 17 (DR17) as a +Value Added Catalog (VAC) publicly available for the +community’s use. +Unless stated otherwise, +we adopt the Planck15 +(Planck Collaboration et al. 2016) cosmology as en- +coded in the ASTROPY package (Astropy Collaboration +et al. 2013; Price-Whelan et al. 2018). +2. DATA +We first describe the required inputs for reconstruct- +ing the map of cosmic densities produced by MCPM. +MCPM takes as input a 3D catalog of galaxy posi- +tions with known masses and reproduces a data cube +reconstructing the filamentary structure connecting the +galaxy halos. To optimize the parameters in MCPM, we +also require a known density field from a cosmological +simulation to compare our reconstruction. We then ap- +ply the tuned model to observational catalogs of galaxies +with known masses to reconstruct the physical cosmic +web. + +3 +We +employ +the +dark +matter-only +Bolshoi-Plank +ΛCDM (BP) simulation (described below) as our train- +ing density field. +We then apply our model to spec- +troscopic surveys that provide large samples of precise +redshifts combined with value-added catalogs that esti- +mate the galaxy masses. We use two primary catalogs +for our galaxy positions, the NASA-Sloan Atlas (NSA, +or NSA/SDSS) for galaxies with z < 0.1 and the Large +Scale Structure catalogs from Sloan Digital Sky Survey +(SDSS) for galaxies at higher redshifts (z ≲ 0.5). These +two catalogs each offer advantages and disadvantages +and are described below. Note that no new DR17 data +were used in this VAC. We now describe the galaxy +catalogs and the simulations used as inputs to MCPM. +2.1. NASA Sloan Atlas +The NASA Sloan atlas (NSA) is a value-added cata- +log constructed from reprocessed SDSS ugriz photome- +try combined with Galaxy Evolution Explorer (GALEX) +photometry in the ultraviolet. It was designed to im- +prove the standard SDSS sky subtraction pipeline (Blan- +ton et al. 2011). We use the most recent version of this +catalog, nsa v1 0 1.fits, which contains galaxies out to +z = 0.15. In order to prioritize completeness in this data +set, we we imposed an upper redshift cut to those galax- +ies with z = 0.1 resulting in a catalog of 325321 galaxies. +We will often refer to this catalog in this paper as simply +“NSA/SDSS” to distinguish it from the other catalogs +from BOSS. +2.2. LRG catalogs +For the higher redshift portion of our catalog, we use +a sample of Luminous Red Galaxies (LRGs) from the +Baryon Oscillation Spectroscopic Survey (BOSS). BOSS +was part of the SDSS III project, which at the time of its +release, provided the largest survey of galaxy redshifts +available in terms of the number of redshifts measured +by a single survey and the effective cosmological volume +covered. We chose to use the LRG catalogs as tracers of +the dark matter (DM) density as these catalogs are more +complete at these redshifts with respect the selection +function than using the more general SDSS galaxy sam- +ple. The BOSS LRG sample derives from the large scale +structure catalogs provided by the team and is broken +into Northern and Southern Galactic Cap regions (LRG- +NGC and LRG-SGC, respectively) (Ross et al. 2011; Ho +et al. 2012; Ross et al. 2012). We use the LOWZ cata- +logs, which provide a sample of LRGs to z ≲ 0.5 and are +found in the files1 galaxy DR12v5 LOWZ North.fits.gz +and galaxy DR12v5 LOWZ South.fits.gz. The proce- +1 https://www.sdss.org/dr14/spectro/lss/ +dure to create this catalog is mostly based on Reid et al. +(2016) with modifications to the redshift failure and sys- +tematic corrections described in Bautista et al. (2018). +2.3. Mass Determination +We used the LRG galaxy stellar masses from the +Firefly VAC. The Firefly VAC2 (Comparat et al. 2017) +provides galaxy properties of all SDSS, BOSS, and +eBOSS spectra using the FIREFLY fitting routine +(Wilkinson et al. 2017) (v1 0 4 for DR14 and v1 1 1 +for DR16), which incorporates the stellar population +models of Maraston & Str¨omb¨ack (2011). +The Fire- +fly catalog includes light- and mass-weighted stellar +population properties (age and metallicity), E(B-V) +values, and most crucially to this work, stellar mass +for all galaxies in the catalog. We used the DR14 cat- +alog to determine masses for the galaxies in the file +sdss eboss firefly-dr14.fits. +The lower redshift NSA/SDSS catalog contains many +galaxies that are spatially resolved and require more +careful photometric analysis (e.g., Blanton et al. 2011). +The most recent version of this catalog provides elliptical +Petrosian aperture photometry, which is more accurate +than the standard SDSS pipeline. We adopt the Pet- +rosian aperture-derived mass to estimate the galaxy’s +stellar mass for this sample. +2.4. Bolshoi-Planck Simulations +To calibrate our MCPM density estimates to the +cosmic matter density, we use the dark matter only +Bolshoi-Plank ΛCDM (BP) simulation (Klypin et al. +2016; Rodr´ıguez-Puebla et al. 2016). The BP simula- +tion uses 20483 particles in a volume of 250h−1 Mpc3 +and is based on the 2013 Planck (Planck Collaboration +et al. 2014) cosmological parameters and compatible +with the Planck 2015 parameters (Planck Collaboration +et al. 2016). We utilize density field from the simulation +smoothed by Gaussian kernel over scales of 0.25 Mpc +h−1 (Lee et al. 2017; Goh et al. 2019). +We also em- +ploy the BP halo catalog produced using the Rockstar +algorithm (Behroozi et al. 2012). +3. METHODOLOGY +3.1. The MCPM algorithm +We produced the VAC data with the Monte Carlo +Physarum Machine (MCPM) algorithm implemented +in the Polyphorm software3. +MCPM was first used +in Burchett et al. (2020) to reconstruct a 3D density +2 +https://www.sdss.org/dr16/spectro/eboss-firefly-value- +added-catalog/ +3 https://github.com/CreativeCodingLab/Polyphorm + +4 +0 +500 +1000 +1500 +2000 +2500 +3000 +Luminosity Distance +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +z +0 +5000 +10000 +15000 +20000 +25000 +Ngalaxies +RUN0 +RUN1 +RUN3 +RUN5 +RUN7 +RUN2 +RUN4 +RUN6 +RUN8 +Figure 1. Distribution of the galaxy redshifts for the NSA/SDSS (blue, solid) and LRG-NGC (multi-colored, solid) and LRG- +SGC (multi-colored, dashed) data sets that were used to reconstruct the cosmic density map. The NSA/SDSS catalog includes +all galaxies out to z = 0.1 and is denoted as RUN0 in the MCMP VAC. The LRG catalogs extend to higher redshifts but only +include the rarer LRGs, hence the lower galaxy count. This figure also shows the slicing scheme used to self-consistently fit the +MCPM model in subsets of redshift as the density of galaxies decreases with luminosity distance in comoving Mpc. +field estimate of the large-scale structure spanning 37.6k +SDSS galaxies within the 0.018 < z < 0.038 range. The +detailed description of methodology and analyses are de- +scribed in Elek et al. (2022). We provide a brief sum- +mary of the model here. +MCPM is a massively parallel agent-based model in- +spired by the growth patterns of Physarum polycephalum +slime mold. Its main modalities are visualized in Fig- +ure 2. Using a swarm of millions of particle-like agents, +MCPM iteratively traces the network structures implicit +in the input data: dark matter halos or galaxies repre- +sented as a weighted 3D point cloud. In linear propor- +tion to their halo mass, the data points emit a virtual +marker which the agents navigate toward at every iter- +ation. +The key innovation of this model is the probabilistic +navigation of the agents: the sampling of their trajec- +tories according to PDFs derived from the data-emitted +marker field. For reference, the deterministic baseline +model, where the agents always follow the maximum +marker concentration, leads to the collapse of some fila- +mentary configurations and the omission of a significant +portion of data points, approximately a third as mea- +sured in Burchett et al. (2020). In contrast, MCPM fits +over 99% of all input data points and can reconstruct +configurations where multiple filaments branch out from +a single origin, e.g., in massive galaxy clusters. +MCPM produces two main quantities: the trace field +and the orientation field. The trace field fT : R3 → R+ +accumulates the superimposed trajectories of all active +agents and represents the reconstructed LSS density +field (after statistical standardization, Section 3.5). The +orientation field fO : R3 → R3 ++ records the averaged un- +signed directions of the agents and serves as a clustering +criterion in our FoG compensation step (Section 3.6). +Both are robust (i.e., stable in time) Monte-Carlo esti- +mates of the equilibrium agent distributions. +Compared to our earlier applications of the MCPM +model (Burchett et al. 2020; Simha et al. 2020), we intro- +duce a few methodological and implementation changes +aimed at improving the quality of the fits (more on this +in Section 3.2): +1. Linear accumulation of fT and fO values instead +of the original exponential floating window aver- + +5 +Figure 2. +Overview of MCPM’s operating modalities, demonstrated on the 0.018 < z < 0.038 sample of SDSS galaxies. +Clockwise from top left: input data points and the marker concentration emitted by the data (yellow), reconstructed trace field +fT (purple), corresponding orientation field fO (XYZ directions mapped to RGB colors). +aging. The latter is used for the supervised part of +the fitting when exploring different MCPM config- +urations. After finding the optimal data-specific +set of model parameters, we switch to linear av- +eraging, which dramatically reduces the solution +variance. +2. To avoid numerical errors, we increase the numer- +ical precision from fp16 to fp32 for both fT and +fO. +This slows the implementation by 10-20%, +which is acceptable for maintaining interactivity +during fitting. +3. We redesigned the agent rerouting step. Rerout- +ing is invoked when an agent encounters no data +for too many subsequent steps, indicating either a +boundary of the dataset or a large void. Our orig- +inal rerouting assigned such an agent to a random +location in space; currently, we repositioned it to +the location of a random data point. This change +leads to a significant decrease of background noise +and effectively increases the dynamic range of the +obtained solutions for both fT and fO. +3.2. MCPM fit to Bolshoi-Planck +This section describes how we calibrate the MCPM +algorithm using the Bolshoi-Planck data. We refer read- +ers to Elek et al. (2022) for more details of the fitting +procedure and the impact of the model hyperparame- +ters on the resulting reconstruction geometry. Readers +interested in the catalog data can skip to Section 3.3. +Fitting MCPM to input data (either galaxies or ha- +los) is a semi-supervised procedure, where the opera- +tor focuses on maximizing the fitness function E while +maintaining the connectedness and continuity of the re- +constructed geometry. We define the fitness E of a given +reconstructed trace field fT over a dataset D as +E(fT, D) = +1 +|D| +� +d∈D +fT(dposition) +dmass +. +This results in a maximum likelihood estimator normal- +ized by each data point’s mass to avoid overfitting to +the most massive objects (given the large dispersion of +typical galaxy and halo masses). Since we do not yet +have a precise mathematical description of the fit’s con- +nectedness, we rely on the interactive visualization in +Polyphorm to ensure that the fit does not collapse into +a disconnected set of ‘islands’. Defining this property + +Data +Marker +Trace +Orientation6 +Figure 3. Our reconstructed cosmic web data products and their spatial relation to another. The green bands highlight regions +of overlapping LRG slices. The SDSS portion of the data is magnified to visualize the higher amount of recovered structure +owing to the denser observations. +rigorously and developing a fully automated fitting pro- +cedure remains a future work for us. +To calibrate MCPM’s hyperparameters, we fit the +model to two snapshots of the Bolshoi-Planck simula- +tion dataset (at z = 0 and z = 0.5, both containing +roughly 16M halos extracted with the Rockstar algo- +rithm). We adopted some of the parameter estimates +from our previous work (Burchett et al. 2020), includ- +ing the sensing angle at 20 deg, moving angle at 10 +deg, moving distance at 0.1 Mpc and persistence +of 0.9 (now adjusted to 0.92 due to the finer granu- +larity of halos used here). We focused on constraining +the remaining critical parameters: sampling exponent +(which controls the acuity of obtained structures, espe- +cially filaments) and sensing distance (which deter- +mines the scale of the structures, such as mean segment +length and by transition the diameter of loops, voids, +etc). In addition, we maximize the monotonicity of the +obtained overdensity mapping as shown in Figure 5 as an +additional constraint when determining the sampling +exponent. +Using this fitting procedure, we matched the MCPM +fits to the ground truth densities in Bolshoi-Planck. We +determined the optimal sampling exponent to be 2.5 +at z = 0 and 2.2 at z = 0.5, which is consistent with the +observation that the LSS at higher redshifts is less con- +densed. For the sensing distance, the optimal value +was 2.37 Mpc. It is worth noting that these sampling +exponent and sensing distance values pose lower lim- +its for the values used to fit the observational data, be- +cause of the significantly lower spatial density of data +points in the galaxy catalogs relative to BP simulations, +which was compensated for by proportionally increasing +the two parameters. +In Figure 4, we demonstrate that MCPM reconstructs +not just the halos that we feed into it but the cosmic +structure, including filaments and voids. More quanti- +tative assessments are available in Elek et al. (2022). +3.3. Fit to NASA-Sloan Atlas +The first component of the VAC is based on the +MCPM fit to the NASA-Sloan Atlas catalog for 0 < +z < 0.1, which contains roughly 325k galaxies in lumi- +nosity distances between 44 and 476 Mpc. Similar to + +LRG (NGC) +SDSS +LRG (SGC) +? +250 Mpc +500 Mpc +1000 Mpc +2000 Mpo +3000 Mpc7 +Bolshoi-Planck z=0 +Bolshoi-Planck z=0.5 +MCPM z=0 +MCPM z=0.5 +Figure 4. Comparison of the Bolshoi-Planck simulations (top row; where the density field is known) at redshifts of z = 0.0 +(left) and z = 0.5 (right) to the MCPM trace of the simulations (bottom row; density recovered from halos alone). MCPM +faithfully reconstructs the cosmic structure from the galaxy halo population. +the BP dark matter halos, we treat the galaxies as 3D +point attractors, in this case, weighted by their stellar +masses. +The fits are based on the hyperparameters calibrated +on the BP simulations. Furthermore, to reflect the lower +spatial density of the galaxies in comparison to the ha- +los, we adjust the two critical parameters of MCPM: +sampling exponent to 3.5 and sensing distance to +5.2. To make these adjustments, we again used the semi- +supervised fitting procedure described in Section 3.2. +To verify the consistency of the fit across different z +values, we have split the SDSS catalog into 3 overlap- +ping slices (44-270 Mpc / 250-370 Mpc / 350-476 Mpc, +each containing about 120k galaxies) and fitted them +separately by only adjusting the sensing distance pa- +rameter. +The resulting optimal values (Figure 6) fol- +low a linear trend, implying that the spatial density of +galaxies decreases in corresponding proportion. +How- +ever, the obtained variation of sensing distance (3.8– +5.6) is well within the ability of the model to perform a +consistent fit using a single parameter value. Therefore, +we opt for a single fit to the entire catalog using the +aforementioned sensing distance value of 5.2. +3.4. Fit to LRG Catalogs +The procedure of fitting to the LRG NGC and SGC +catalogs is identical with the SDSS data: +using the +sampling exponent of 3.5 and the BP-calibrated values +for the other hyperparameters, we continued increasing +sensing distance until reaching an optimal fit. +Due to the much lower spatial density of LRG ob- +servations compared to SDSS, the optimal values of +sensing distance end up being considerably higher +(Figure 3). Also, unlike SDSS, the LRG galaxies span +a significantly more extended range of redshifts. The +consequence is nearly a two-fold growth of the optimal + +8 +4 +2 +0 +2 +exp = 2.5 +2 +0 +2 +4 +log m/ +exp = 3.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +log MCPM density +2 +0 +2 +4 +exp = 3.5 +Figure 5. Comparison of different sampling exponents in in- +creasing order from top to bottom. We find that a sampling +exponent of 2.5 produces the most linear mapping between +the MCPM densities and the cosmic matter densities from +the BP simulations, especially at lower densities where pre- +vious versions of the MCPM have generally failed to recover +the lowest density structures. +(see Figure 10 in Burchett +et al. 2020). +sensing distance value across the catalog’s redshift +range (Figure 6). Therefore to construct the VAC, we +split the LRG galaxies into 4 overlapping ‘slices’ of ap- +proximately equal numbers of galaxies (about 70k per +slice for NGC, about 25k per slice for SGC) and fit each +separately. The resulting distance intervals are 0-1000 +Mpc (z ≈ 0 − 0.2), 900-1600 Mpc (z ≈ 0.18 − 0.3), +1500-2100 Mpc (z ≈ 0.28 − 0.38), and 2000-3000 Mpc +(z ≈ 0.36 − 0.51). +Figure 3 shows the visualization of all obtained density +slices and their spatial relations. An added benefit of +this approach is the higher resolution of each slice we +can afford. This is desirable again due to the massive +redshift range of the LRG data. +3.5. Statistical Standardization & Mapping +The MCPM densities that fit each survey slice, al- +though related to the true physical density, are rather +the density of agents in the fit. To translate the MCPM +density to cosmic overdensity, we standardize each dis- +tribution to the MCPM fit of the simulation so that a +mapping between MCPM and cosmic overdensity can +be applied. The MCPM fits to the galaxy surveys differ +Figure 6. Plot of MCPM agents’ sensing distance (the +main feature scaling parameter) as read out from the best +fits for the LRG data, radially sliced into 4 runs at overlap- +ping luminous distance intervals. For comparison, we also +show the best-fit sensing distances for 3 SDSS slices, which +manifest a similar linear growth as we observe in the LRG +data. +from the fits to the BP simulations because they suffer +from luminosity selection functions and are thus much +sparser. +This particularly affects the lowest density +regime of the density distribution. To account for this +effect, we used the Wasserstein distance4 or the “Earth +Movers Distance” to calculate the stretch and shift val- +ues such that the distribution of MCPM densities of the +surveys could be linearly transformed to best fit the BP- +MCPM fit. That is TargetDist = stretch×SurveyDist ++ shift. Where the TargetDist is the BP-MCPM den- +sity distribution, and SurveyDist is the density distri- +bution of each survey slice. The benefit of this method is +that we can impose a lower limit on the density distribu- +tions to only take into account the higher density wing +of the distribution corresponding to densities that con- +tain structure and avoid the empty space in the survey +fits. +In order to retrieve the cosmic matter density, +ρm/⟨ρm⟩, we must map the MCPM trace density to +that of the BP simulations at each redshift. We fit the +BP simulations using the MCPM algorithm and then +apply a mapping from MCPM density to cosmic matter +density. +This mapping was achieved by sampling the +MCPM fits in bins of equal density and then determin- +ing the density from the BP simulations at the same +location. This is shown by the multi-colored stripes in +Figure 7. We then determine the median (and 1σ lim- +its) of each MCPM density bin. The median densities +in each bin were then used to create a mapping func- +tion. We based our mapping function on the rectified +linear activation function (ReLU), where the maximum +change of the median of the bins determines the inflec- +4 https://docs.scipy.org/doc/scipy/reference/generated/ +scipy.stats.wasserstein distance.html + +9 +Figure 7. Mapping of the MCPM derived density to the +cosmic matter density from the BP simulation. The MCPM +densities were binned evenly in MCPM space in bins of 0.1 +dex as demarcated by the colored stripes. The custom ReLU +mapping function fit to the medians of the bins (thick black +line) and 1σ limits (thinner black lines) are plotted on top +of the data. This mapping function provides a translation +from the MCPM density to the cosmic overdensity. +tion point. On the right-hand side of the flat part of the +function, we fit a cubic polynomial to the data, creating +a piece-wise continuous mapping function. This method +was chosen over other methods, such as interpolating +the bins or using a spline function because fitting to +densities above or below those found in the MCPM fits +is not well defined. Our method is illustrated in Figure 7 +where the thicker black line shows the mapping function +applied to the z = 0 simulation. The thinner black lines +show the 1σ limits of our mapping, which correspond to +±0.5 dex in log cosmic matter overdensity, ρm/⟨ρm⟩. +3.6. Correction for Redshift Space Distortions +As MCPM operates in 3D space, applying the algo- +rithm necessitates attaching physical distances to the in- +put dataset. Although distance measurements via more +direct methods (e.g., tip of the red giant branch or Type +Ia supernovae) (Tully et al. 2016) may be available for a +small subset of the galaxies (and therefore tracers of the +underlying density field), we must primarily assume dis- +tances concordant with the Hubble flow. Thus, we ini- +tially attach to each galaxy the luminosity distance given +the adopted cosmology and galaxy redshift. Denser en- +vironments such as galaxy groups and clusters will in- +clude galaxies with large peculiar velocities. These pe- +culiar velocities will result in redshift space distortions +(RSDs), or ‘fingers of god’ (FoG), if adopted directly. +For example, a typical velocity dispersion for a > 1014 +M⊙ galaxy cluster (∼ 1000 km/s) would propagate to +a systematic error in the distance by assuming pure +Hubble flow of > 10 Mpc. This issue plagues our low- +redshift SDSS sample significantly more than the LRG +samples for two reasons: 1) Low-mass galaxies are much +more abundant and likely to be observed at low z in +the magnitude-limited SDSS, which results in many ob- +jects composing apparent false structures along the di- +rection pointing away from (and towards) the observer; +(2) High-mass galaxies, which will dominate the samples +at progressively higher redshifts, preferentially reside as +central galaxies in their local environments (Lan et al. +2016). Therefore, these galaxy samples will be less sub- +ject to systematic error in cosmological distance than +our lowest redshift sample. Thus, we employ an RSD +correction for the z < 0.1 SDSS galaxy sample that we +detail here. +A key feature of MCPM is that the cosmic web re- +construction converges to an equilibrium state but is a +dynamical system nonetheless. The adopted ‘densities’ +are aggregated trajectories of the millions of agents seek- +ing efficient pathways between galaxy tracers. MCPM +also outputs the components of an aggregated three- +dimensional agent velocity vector for each cell in the +volume. +We use these velocities to identify RSDs, as +the agent velocities producing them will be preferen- +tially oriented perpendicular to the plane of the sky +along the line of sight and will be clustered in their +celestial coordinates. +We select points in the MCPM +cube by orientation as follows: we (1) convert each in- +put galaxy’s location in the MCPM-output cube to its +equivalent celestial coordinates, (2) find the three com- +ponents of a unit radial vector parallel to the line of +sight in Cartesian space to match the coordinate sys- +tem of the MCPM velocity vectors, and (3) calculate +the dot product between the aggregated velocity vector +at each galaxy’s position in the cube with the unit ra- +dial vector and assign the result to that galaxy. Galaxies +within an RSD structure (FoG), having either parallel +or antiparallel velocity vectors to the unit radial vec- +tor, should not have dot product absolute values close + +z= 0.5 +3 +2 +log Pm/(p) +1 +0 +-1-Z=0.0 +3 +2 +log pm/p) +0 +-1 +-2 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +log MCPM density10 +to zero. Therefore, we filter out galaxies with dot prod- +uct absolute values less than 10, chosen upon inspecting +the distribution of galaxy dot product values as a con- +servative cut. To identify galaxy positions with similar +velocity orientation and projected location on the sky, +we then employ the Density-Based Spatial Clustering of +Applications with Noise (DBSCAN) algorithm as imple- +mented in the scikit-learn5 python package, feeding +it the sky coordinates and redshift. For this step, we +further filter the galaxy catalog by mass to those with +M∗ > 1010 M⊙, as the completeness of SDSS declines +for less massive galaxies at the upper end of our red- +shift range (z ∼ 0.1). DBSCAN operates by locating +high-density cores in the data, which are the beginnings +of the clusters. The algorithm searches out from these +cores, adding points until no more points are found in +within some distance tolerance (in whatever space the +data occupy). This algorithm contains several advan- +tages over other possible choices, including scalability, +compatibility with non-flat geometries, and the feature +that certain points may not be included in any clus- +ter (they are deemed ‘noise’). Two critical parameters +for DBSCAN are the distance tolerance (eps) and the +minimum number of points to be considered a core in +the data (min samples). We chose min samples=3 as +a minimum number of galaxies (e.g., such as in a group +or cluster) that might form a false RSD structure (FoG) +in the MCPM model. We chose a value of eps=2 upon +experimenting with several values through visual inspec- +tion to balance the inclusion of FoGs (which are read- +ily identified by the eye), containing a relatively small +number of galaxies while minimizing false identification +of filaments not oriented antiparallel to the plane of the +sky as RSD structures. +Figure 8 shows the resulting +clusters identified by DBSCAN in a slice in declination +of our galaxy catalog, with galaxies belonging to the +same cluster having the same color. +From the output clusters identified by DBSCAN, we +find the velocity range spanned by galaxy redshifts +within each cluster using the full-width-half-maximum +(FWHM) of the velocity distribution (vFWHM). +For +clusters with vFWHM > 300 km/s, we adopt new red- +shifts for the associated galaxies to be commensurate +with more realistic physical distance separations inferred +from a simple luminosity distance based on the redshift; +this procedure is as follows. Assuming the cluster mem- +bers are bound to the same virialized structure, we con- +vert the velocity FWHM to a velocity dispersion by the +5 https://scikit-learn.org/stable/ +relation: +σv = vFWHM +2 +√ +ln 2 +. +(1) +We then use this velocity dispersion to infer a virial ra- +dius, R200, of the cluster: +Rinfer +200 = +σv +4/3πG∆200ρcrit +, +(2) +where ∆200 and ρcrit are the overdensity and critical den- +sity, respectively. We then adopt new redshifts (solely +for the purpose of feeding MCPM) about the median +redshift of the cluster members by sampling from a nor- +mal distribution with standard deviation corresponding +to the change in redshift that would result in a lumi- +nosity distance difference equal to the inferred R200. Fi- +nally, we convert these galaxy coordinates and adopted +redshifts to 3D Cartesian space via luminosity distances +based on the new redshifts; these then serve as inputs +to MCPM. +4. DATA PRODUCTS +4.1. The Catalog +The final value-added catalog contains the positions +and redshifts and the stellar mass of the galaxies in the +NASA-Sloan Atlas and the eBOSS Firefly Value-Added +Catalog. We include a column, MASS SOURCE, to indi- +cate which catalog was used to estimate the mass. The +MCPM algorithm uses the galaxy mass to build the mat- +ter density field. The primary field of interest here is +MATTERDENS, the matter density field at the location of +a given galaxy, which was derived from fits of MCPM +models in 3D volumes and mapped to the cosmological +matter density (relative to the mean matter density) us- +ing MCPM fits to the Bolshoi-Planck simulations. The +catalogID is a combination of plate-mjd-fiberid. A +unique identifier is the combination of catalogID and +mcpmRun. Objects with the same value of mcpmRun were +fitted with the MCPM model simultaneously. The data +were sliced in redshift to yield samples producing self- +consistent large-scale structures over the volume in each +slice. mcpmRun = 0 correspond to 0.01 < z < 0.1 SDSS +galaxies with masses from the NASA/Sloan Atlas. Sam- +ples of LOWZ LRGs are marked 1-2 (z < 0.20), 3-4 +(0.18 < z < 0.30), 5-6 (0.28 < z < 0.38), and 7-8 +(0.36 < z < 0.51); each pair (e.g., 3-4), corresponds +to the NGC/SGC samples in some redshift slice, with +odd and even numbers for NGC and SGC, respectively. +The data model for the catalog is described in Table 1. +4.2. 3D Density Cube +In addition to the VAC, which contains the density at +the location of each galaxy, we offer the full 3D density + +11 +Table 1. Data Model +Name +Type +Unit +Description +(1) +(2) +(3) +(4) +catalogID +char[13] +Combination of PLATE-MJD-FIBERID +plate +int32 +Plate number +mjd +int32 +MJD of observation +fiberid +int32 +Fiber identification number +ra +float64 +deg +Right ascension of fiber, J2000 +dec +float64 +deg +Declination of fiber, J2000 +z +float32 +Best redshift +massSource +char[7] +Source of the mass determination (nsa or firefly) +mcpmRun +int8 +Index of galaxy sample fitted simultaneously with MCPM +mstars +float64 +M⊙ +Stellar mass +matterDens +float32 +log10 of the ratio of the matter density relative to the mean matter density +Note—Schema +for +the +MCPM +Value-Added +Catalog, +v1.0.0 +as +found +in +slimeMold galaxy catalog v1 0 0.fits. +field of the relevant volumes, available at https://data. +sdss.org/sas/dr17/eboss/lss/mcpm/v1 0 0/datacube/. +These may be queried using our custom package, +pyslime6. +The data will unzip to a directory which +may be opened by pyslime. This will enable the user +to query the overdensity at arbitrary points in the cube, +allowing the study of voids and filamentary structures +outside the local environment of the input galaxy field. +5. DISCUSSION +5.1. Comparison to Peng et al. (2010) +We can additionally validate our model by comparing +our findings to that of other surveys. Although Burchett +et al. (2020) has demonstrated the efficacy of our model, +we present comparisons to other studies, leveraging our +deeper and larger surveys. +Peng et al. (2010) used a method based on the 5th +nearest galaxy neighbors to estimate the environmental +density and studied the SFR and the quenched frac- +tion of galaxies as a functions of this density metric and +galaxy mass. Burchett et al. (2020) illustrate that the +MCPM method of computing cosmic density qualita- +tively matches the results (see Figure 5 & 6 in Peng +et al. 2010). In Figure 9, we demonstrate the improve- +ment in signal gained with the NSA/SDSS sample as +the increase in the number of galaxies is significant and +the reproduction of their density-stellar mass-sSFR re- +lations. +6 https://github.com/jnburchett/pyslime +5.2. Potential Applications +Our primary aim in this manuscript is to showcase +the dataset and describe its construction. +There are, +however, many exciting applications for this dataset that +are well beyond the scope of this publication. Here, we +list four general areas of application: +• Galaxy evolution in the cosmic web: A vast +amount of galaxy properties measured and in- +ferred from both multiwavelength photometry and +spectroscopy have been cataloged for SDSS galax- +ies (many also released as VACs; e.g., Salim et al. +2016) via straightforward crossmatching with our +catalog, myriad galaxy-environment analyses may +be readily conducted. Figure 9 highlights one di- +rect application of the galaxy-density catalog to +study the possible impacts of a galaxy’s location +within the cosmic web on its evolution. In partic- +ular, Figure 9 shows the dependence of star for- +mation activity as a function of large-scale struc- +ture density. +Our dataset is ideal for compar- +ing effects induced by the more local environment +(groups/clusters) to those induced by the cosmic +web. +• Void finding: +In the linear regime, the sizes +of voids and their correlation statistics are sensi- +tive to cosmology, particularly dark energy (Pisani +et al. 2015). +Although most of the analyses we +have alluded to thus far focus on the denser re- +gions of the cosmic web, namely filaments and +nodes, our density cubes naturally include the un- + +12 +8h +10h +12h +14h +16h +RA +0.02 +0.04 +0.06 +0.08 +0.1 +z +8h +10h +12h +14h +16h +RA +0.02 +0.04 +0.06 +0.08 +0.1 +z +Figure 8. A slice in declination of our input galaxy catalog (grey points, top). RSD structures identified by DBSCAN are +shown in various colors overlayed on the original points (bottom). + +13 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +logM /M +0.0 +0.5 +1.0 +1.5 +2.0 +log m/ +m +10.8 +10.6 +10.4 +10.2 +10.0 +9.8 +9.6 +9.4 +log sSFR [yr 1] +Figure 9. +The dependence of star formation activity on +galaxy environment and stellar mass for the galaxies within +the NSA/SDSS volume (z < 0.1). +The color coding de- +notes sSFR in the population within each mass/environment +bin, where the environmental density is determined from our +MCPM cosmic web reconstruction algorithm. A comparison +with Figure 6 of Peng et al. (2010) shows a similarly increas- +ing red fraction as a function of both mass at fixed density +and density at fixed mass. +derdense regions. Simple centroiding and cluster- +ing algorithms may be readily applied to these +density fields to directly identify and character- +ize the voids, which in turn may be used as inputs +for cosmological parameter estimations using, e.g., +the Alcock-Paczynski effect (Alcock & Paczynski +1979). +• The intergalactic medium: +Hydrodynamical +cosmological simulations predict a rich multi- +phase structure in the intergalactic gas perme- +ating throughout the cosmic web (e.g., Cen & +Ostriker 1999; Dav´e & Tripp 2001; Tepper-Garc´ıa +et al. 2012). In addition to the physical states of +gas resulting from large-scale structure formation +(Bertschinger 1985; Molnar et al. 2009), energetic +feedback from the galaxies themselves might ex- +tend well beyond the virial radius, which is often +adopted as a fiducial extent of a galaxy’s halo (Fin- +lator & Dav´e 2008; Schaye et al. 2015; Nelson et al. +2019). Burchett et al. (2020) used HST-observed +background quasar sightlines through the MCPM +reconstructed volume to find a relationship be- +tween cosmic web density and Lyα optical depth. +A similar analysis could and should be done lever- +aging our higher redshift LRG reconstruction with +other absorption tracers, such as Mg II. +• Multimessenger transient followup: +Tran- +sient phenomena, such as gravitational waves and +fast radio bursts, are typically detected with im- +precise localization, with scales of minutes or de- +grees on the sky (Chen & Holz 2016; CHIME/FRB +Collaboration et al. 2019). +Space-based and +ground-based facilities around the world then +follow up these detections to identify and char- +acterize the sources (e.g., Coulter et al. 2017). As +extragalactic sources are statistically more likely +to be found within the large-scale structure, tran- +sient observers could employ our reconstructed +density field of the cosmic web in follow-up imag- +ing campaigns to prioritize pointings toward re- +gions of the sky most likely to contain the source +counterparts. +5.3. Known Limitations +The VAC volumes have the usual luminosity function +systematics that are present in the underlying SDSS +and LRG catalogs. +Specifically, the sampling density +of galaxies is more significant at lower redshifts. This +is reflected in the trace and can be seen in the SDSS +data as well as each slice of the LRG catalogs, as shown +in Figure 3. This presents itself as an increased density +at the lower redshift end of the volume. However, the +mean matter density at the low and high redshift ends +of each volume is consistent. +Some sub-optimality of the model fit arises from the +fact that the optimal sensing distance grows linearly +according to the data in Figure 6, whereas the catalog +is a piece-wise constant approximation of this. +Due to the differing sensing distance in each slice, +there is a slight discontinuity of the MCPM densities ex- +tracted from the overlaps between the LRG slices. Thus, +we recommend comparing densities on a slice-by-slice +basis and avoiding comparing quantities based on the +density at different redshift slices. +6. CONCLUSION +Herein we leverage the Monte Carlo Physarum Ma- +chine (MCPM) methodology, inspired by the growth +and movement of Physarum polycephalum slime mold, +to map the cosmic web within several sub-samples of +the SDSS spectroscopic galaxy catalogs. +The MCPM +model inputs a galaxy field with known masses and out- +puts the large-scale structure density field. +We train +our model using the Bolshoi-Planck cosmological sur- +vey, producing a reconstruction of the simulated cos- +mic web where the underlying density is known. Using + +14 +the simulation as ground truth, we describe the super- +vised tuning of MCPM parameters to produce an opti- +mal fit. We apply this tuned model to the NASA-Sloan +Atlas and the eBOSS LRG Firefly Value-Added Cata- +logs to create both a 3D density cube and a catalog of +cosmic densities at the location of the galaxies. +The +SDSS NASA-Sloan Atlas catalogs include a more com- +plete galaxy sample at z < 0.1. We describe and employ +a novel method on this dataset to reduce the effect of +peculiar motions on the spectroscopic distances. +The +MCPM fits to the eBOSS LRG North and South Galac- +tic Cap catalogs capture the larger-scale cosmic web out +to z ≲ 0.5. This paper describes the release the Cosmic +Slime Value Added Catalog, part of SDSS DR17, which +is the combination the two galaxy catalogs with den- +sity estimates as well as the resultant 3D density cubes +of the two galaxy samples. Finally, we highlight some +exciting potential applications of this data set, which in- +clude galaxy evolution in the context of the cosmic web, +void finding, studies of the intergalactic medium, and +multimessenger transient followup. +7. ACKNOWLEDGEMENTS +The authors would like to especially acknowledge Joel +Primack and Doug Hellinger for sharing the outputs of +the Boloshoi-Planck simulations. We also gratefully ac- +knowledge the hospitality and support of the 2019 Kavli +Summer Program in Astrophysics at UC Santa Cruz. +JB would like to acknowledge funding support from +the National Science Foundation LEAPS-MPS award +#2137452. +MCW and JKW acknowledge support for this work +from NSF-AST 1812521, NSF-CAREER 2044303, the +Research Corporation for Science Advancement, grant +ID number 26842. +OE is supported by an incubator fellowship of the +Open Source Program Office at UC Santa Cruz funded +by the Alfred P. Sloan Foundation (G-2021-16957). +Funding for the Sloan Digital Sky Survey IV has been +provided by the Alfred P. Sloan Foundation, the U.S. +Department of Energy Office of Science, and the Partici- +pating Institutions. SDSS-IV acknowledges support and +resources from the Center for High-Performance Com- +puting at the University of Utah. The SDSS website is +www.sdss.org. +SDSS-IV is managed by the Astrophysical Research +Consortium for the Participating Institutions of the +SDSS Collaboration including the Brazilian Partici- +pation Group, the Carnegie Institution for Science, +Carnegie Mellon University, +the Chilean Participa- +tion Group, the French Participation Group, Harvard- +Smithsonian Center for Astrophysics, Instituto de As- +trof´ısica de Canarias, The Johns Hopkins University, +Kavli Institute for the Physics and Mathematics of +the Universe (IPMU) / University of Tokyo, the Ko- +rean Participation Group, Lawrence Berkeley National +Laboratory, Leibniz Institut f¨ur Astrophysik Potsdam +(AIP), Max-Planck-Institut f¨ur Astronomie (MPIA Hei- +delberg), Max-Planck-Institut f¨ur Astrophysik (MPA +Garching), Max-Planck-Institut f¨ur Extraterrestrische +Physik (MPE), National Astronomical Observatories of +China, New Mexico State University, New York Univer- +sity, University of Notre Dame, Observat´ario Nacional / +MCTI, The Ohio State University, Pennsylvania State +University, Shanghai Astronomical Observatory, United +Kingdom Participation Group, Universidad Nacional +Aut´onoma de M´exico, University of Arizona, University +of Colorado Boulder, University of Oxford, University of +Portsmouth, University of Utah, University of Virginia, +University of Washington, University of Wisconsin, Van- +derbilt University, and Yale University. +REFERENCES +Abell, G. O. 1965, ARAA, 3, 1 +Alcock, C., & Paczynski, B. 1979, Nature, 281, 358 +Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., +et al. 2013, A&A, 558, A33 +Balogh, M. L., Morris, S. L., Yee, H. K. C., Carlberg, +R. G., & Ellingson, E. 1999, ApJ, 527, 54 +Bautista, J. E., Vargas-Maga˜na, M., Dawson, K. S., et al. +2018, ApJ, 863, 110, doi: 10.3847/1538-4357/aacea5 +Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2012, The +Astrophysical Journal, 762, 109, +doi: 10.1088/0004-637x/762/2/109 +Berlind, A. A., Frieman, J., Weinberg, D. H., et al. 2006, +ApJS, 167, 1, doi: 10.1086/508170 +Bertschinger, E. 1985, ApJS, 58, 39 +Blanton, M. R., Kazin, E., Muna, D., Weaver, B. A., & +Price-Whelan, A. 2011, AJ, 142, 31, +doi: 10.1088/0004-6256/142/1/31 +Bouma, S. J. D., Richter, P., & Wendt, M. 2021, A&A, 647, +A166 +Burchett, J. N., Elek, O., Tejos, N., et al. 2020, ApJL, 891, +L35, doi: 10.3847/2041-8213/ab700c +Carollo, C. M., Cibinel, A., Lilly, S. J., et al. 2013, ApJ, +776, 71 +Catinella, B., Schiminovich, D., Cortese, L., et al. 2013, +MNRAS, 436, 34 +Cen, R., & Ostriker, J. P. 1999, ApJ, 514, 1 + +15 +Chen, H.-Y., & Holz, D. E. 2016, arXiv e-prints, +arXiv:1612.01471 +CHIME/FRB Collaboration, Andersen, B. C., Bandura, K., +et al. 2019, ApJL, 885, L24 +Codis, S., Jindal, A., Chisari, N. E., et al. 2018, MNRAS, +481, 4753 +Comparat, J., Maraston, C., Goddard, D., et al. 2017, +arXiv e-prints, arXiv:1711.06575. +https://arxiv.org/abs/1711.06575 +Coulter, D. A., Foley, R. J., Kilpatrick, C. D., et al. 2017, +Science, 358, 1556 +Courtois, H. M., Pomar`ede, D., Tully, R. B., Hoffman, Y., +& Courtois, D. 2013, AJ, 146, 69 +Crone Odekon, M., Hallenbeck, G., Haynes, M. P., et al. +2018, ApJ, 852, 142 +Dav´e, R., & Tripp, T. M. 2001, ApJ, 553, 528 +Dressler, A. 1980, ApJ, 236, 351 +Elek, O., Burchett, J. N., Prochaska, J. X., & Forbes, A. G. +2022, Artificial Life, 28, 22, doi: 10.1162/artl a 00351 +Finlator, K., & Dav´e, R. 2008, MNRAS, 385, 2181 +Goh, T., Primack, J., Lee, C. T., et al. 2019, MNRAS, 483, +2101, doi: 10.1093/mnras/sty3153 +Ho, S., Cuesta, A., Seo, H.-J., et al. 2012, ApJ, 761, 14, +doi: 10.1088/0004-637X/761/1/14 +Kauffmann, G., White, S. D. M., Heckman, T. M., et al. +2004, MNRAS, 353, 713 +Klypin, A., Yepes, G., Gottl¨ober, S., Prada, F., & Heß, S. +2016, MNRAS, 457, 4340, doi: 10.1093/mnras/stw248 +Kuutma, T., Tamm, A., & Tempel, E. 2017, A&A, 600, L6 +Lan, T.-W., M´enard, B., & Mo, H. 2016, MNRAS, 459, +3998 +Lee, C. T., Primack, J. R., Behroozi, P., et al. 2017, +MNRAS, 466, 3834, doi: 10.1093/mnras/stw3348 +Libeskind, N. I., van de Weygaert, R., Cautun, M., et al. +2018, MNRAS, 473, 1195, doi: 10.1093/mnras/stx1976 +Luber, N., van Gorkom, J. H., Hess, K. M., et al. 2019, AJ, +157, 254 +Maraston, C., & Str¨omb¨ack, G. 2011, MNRAS, 418, 2785, +doi: 10.1111/j.1365-2966.2011.19738.x +Molnar, S. M., Hearn, N., Haiman, Z., et al. 2009, ApJ, +696, 1640 +Nelson, D., Pillepich, A., Springel, V., et al. 2019, MNRAS, +490, 3234 +Pasha, I., Mandelker, N., van den Bosch, F. C., Springel, +V., & van de Voort, F. 2022, arXiv e-prints, +arXiv:2204.04097 +Peng, Y.-j., Lilly, S. J., Kovaˇc, K., et al. 2010, ApJ, 721, 193 +Pisani, A., Sutter, P. M., Hamaus, N., et al. 2015, PhRvD, +92, 083531 +Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. +2014, A&A, 571, A16, doi: 10.1051/0004-6361/201321591 +—. 2016, A&A, 594, A13, +doi: 10.1051/0004-6361/201525830 +Poggianti, B. M., Fasano, G., Bettoni, D., et al. 2009, +ApJL, 697, L137 +Price-Whelan, A. M., Sip˝ocz, B. M., G¨unther, H. M., et al. +2018, AJ, 156, 123 +Reid, B., Ho, S., Padmanabhan, N., et al. 2016, MNRAS, +455, 1553, doi: 10.1093/mnras/stv2382 +Rodr´ıguez-Puebla, A., Behroozi, P., Primack, J., et al. +2016, MNRAS, 462, 893, doi: 10.1093/mnras/stw1705 +Ross, A. J., Ho, S., Cuesta, A. J., et al. 2011, MNRAS, 417, +1350, doi: 10.1111/j.1365-2966.2011.19351.x +Ross, A. J., Percival, W. J., S´anchez, A. G., et al. 2012, +MNRAS, 424, 564, doi: 10.1111/j.1365-2966.2012.21235.x +Salim, S., Lee, J. C., Janowiecki, S., et al. 2016, ApJS, 227, +2 +Schaye, J., Crain, R. A., Bower, R. G., et al. 2015, +MNRAS, 446, 521 +Simha, S., Burchett, J. N., Prochaska, J. X., et al. 2020, +ApJ, 901, 134, doi: 10.3847/1538-4357/abafc3 +Springel, V., White, S. D. M., Jenkins, A., et al. 2005, +Nature, 435, 629 +Tempel, E., Stoica, R. S., Mart´ınez, V. J., et al. 2014, +MNRAS, 438, 3465 +Tepper-Garc´ıa, T., Richter, P., Schaye, J., et al. 2012, +MNRAS, 425, 1640 +Tully, R. B., Courtois, H. M., & Sorce, J. G. 2016, AJ, 152, +50 +Vogelsberger, M., Genel, S., Springel, V., et al. 2014, +MNRAS, 444, 1518 +Vogelsberger, M., Genel, S., Springel, V., et al. 2014, +MNRAS, 444, 1518 +Wakker, B. P., Hernandez, A. K., French, D. M., et al. +2015, ApJ, 814, 40 +Wilkinson, D. M., Maraston, C., Goddard, D., Thomas, D., +& Parikh, T. 2017, MNRAS, 472, 4297, +doi: 10.1093/mnras/stx2215 +Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2007, ApJ, +671, 153 + diff --git a/V9E0T4oBgHgl3EQf3AJU/content/tmp_files/load_file.txt b/V9E0T4oBgHgl3EQf3AJU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eaf53a79842a9cb16c5005aeebd55e359cb55603 --- /dev/null +++ b/V9E0T4oBgHgl3EQf3AJU/content/tmp_files/load_file.txt @@ -0,0 +1,1014 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf,len=1013 +page_content='Draft version January 10, 2023 Typeset using LATEX twocolumn style in AASTeX62 SDSS DR17: The Cosmic Slime Value Added Catalog Matthew C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Wilde,1 Oskar Elek,2 Joseph N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Burchett,2, 3 Daisuke Nagai,4 J.' metadata={'source': 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+page_content=' CT 06520,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' USA 5Kavli Institute for the Physics and Mathematics of the Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 5-1-5 Kashiwanoha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Kashiwa 277-8583,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Japan 6Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 610 Purdue Mall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' IN 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' USA ABSTRACT The “cosmic web”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' the filamentary large-scale structure in a cold dark matter Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' is readily apparent via galaxy tracers in spectroscopic surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' However, the underlying dark matter structure is as of yet unobservable and mapping the diffuse gas permeating it lies beyond practical observational capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A recently developed technique, inspired by the growth and movement of Physarum polycephalum ‘slime mold’, has been used to map the cosmic web of a low redshift sub-sample of the SDSS spectroscopic galaxy catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This model, the Monte Carlo Physarum Machine (MCPM) was shown to promisingly reconstruct the cosmic web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Here, we improve the formalism used in calibrating the MCPM to better recreate the Bolshoi-Planck cosmological simulation’s density distributions and apply them to a significantly larger cosmological volume than previous works using the Sloan Digital Sky Survey (SDSS, z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1) and the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) Luminous Red Galaxy (LRG, z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5) spectroscopic catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We present the ‘Cosmic Slime Value Added Catalog’ which provides estimates for the cosmic overdensity for the sample of galaxies probed spectroscopically by the above SDSS surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In addition, we provide the fully reconstructed 3D density cubes of these volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' These data products were released as part of Sloan Digital Sky Survey Data Release 17 and are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We present the input catalogs and the methodology for constructing these data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We also highlight exciting potential applications to galaxy evolution, cosmology, the intergalactic and circumgalactic medium, and transient phenomenon localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' INTRODUCTION The cosmic web is an emergent prediction of ΛCDM cosmology and is ubiquitously reproduced and readily identifiable in cosmological simulations, where the un- derlying density distribution is known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' However, unveiling the large-scale structure in the observational realm using galaxies and absorption lines as tracers of the intergalac- tic medium (IGM) is much less straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The underlying dark matter distribution remains unobserv- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The two most accessible tracers, such as galaxies and quasar absorption lines, are limited by the practi- cal observational constraints of galaxy redshift surveys and the scarcity of quasars in the universe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Corresponding author: Matthew C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Wilde mwilde@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='edu Even when observational tracers are available at rela- tively high sampling densities, the problem of recon- structing the cosmic web is highly complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We highlight two of the myriad scientific motivations for cosmic web reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' First, of paramount con- cern in galaxy astrophysics is the impact of a galaxy’s environment on its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Correlations between en- vironmental metrics and galaxy properties, such as mor- phology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Dressler 1980), color (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Abell 1965), and star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Balogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2010), have been known about for many decades, but the physical mechanisms and their relative importance remain heavily pursued problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Galaxy-environment analyses typically fall along one of two paths: local environment-centric or large-scale environment-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In the former, one employs an environmental density metric, such as a nearest-neighbor distance or density within some aperture (Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='02719v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='CO] 6 Jan 2023 2 2010), or galaxies are associated with a local group or cluster environment (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Berlind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2006) and galaxy properties are studied with respect to the properties of the group or cluster (Carollo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Catinella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Poggianti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The latter path is less straightforward, as one must infer the large-scale structure from tracers, typically the galaxies themselves, and correlate galaxies back to that structure in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Various methods have been de- vised to reconstruct the cosmic web from discrete trac- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Libeskind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2018) reviewed a number of these, and we refer the reader to this valuable resource for an overview of the techniques employed and comparisons between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Once the underlying density field is in- ferred, one can correlate galaxy properties with this den- sity field (an approach one can directly employ with the catalog described here) or attempt to geometrically re- late a galaxy’s position to the structure identified, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', the distance to a filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' One should appreciate that filament identification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', DisPerSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Luber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Tempel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014), whether from a density field or some other methodology, is a separate problem from the in- ference of the field itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Studies of galaxy properties and their dependence on the cosmic environment report mixed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Kuutma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2017) find a higher elliptical-to-spiral ratio and decreasing star formation rate (SFR) towards filament spines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Similarly, Crone Odekon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2018) report that, at fixed stellar mass, galaxies closer to filaments or in higher density environments are more deficient in HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' These large-scale environmental correlations with galax- ies have also been investigated using modern hydrody- namical cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Codis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2018) measure the spin-filament alignment in IllustrisTNG (Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014) and find a strong dependence on spin alignment with galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Pasha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2022) find that the collapse of large-scale structure into sheets at higher redshifts (z ∼ 3) can create shocks that ex- plain quenching in dwarf galaxies similar to the effects seen in the presence of clusters and groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Second, in addition to the galaxies themselves, the IGM studied in context with the cosmic web environ- ment can yield important insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Wakker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2015) measured the Lyα absorption in quasar spectra prob- ing a foreground visually identified filament, finding in- creasing absorber equivalent width and linewidth with decreasing projected distance to the center of the fila- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' With a larger archival sample of QSOs and fila- ments, Bouma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2021) find similar results, with Lyα absorbers showing both greater incidence and column density at a small projected distance and velocity offsets from filaments first identified by Courtois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In the first application of the reconstruction framework we use here, Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2020) analyzed the Lyα op- tical depth as a function of cosmic web density probed by QSO sightlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' They found three distinct regimes: (1) a void regime at low matter overdensity with no detected absorption, (2) an onset of absorption in the outer skins of filaments with monotonically increasing optical depth, and (3) the highest-density regime where the absorption no longer increases with local density but rather turns over and declines at the highest densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Associating the IGM to the cosmic web provides impor- tant constraints on hydrodynamical processes modeled in cosmological simulations that may be used to inter- pret the environmental quenching conundrums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In this manuscript, we employ the novel method first introduced in Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2020) described in detail by Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2022), which is based on the morphol- ogy of the Physarum polycephalum slime mold organism to map the cosmic density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This model implic- itly traces the cosmic web structure by efficiently finding optimal pathways between the galaxies that trace fila- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We apply our model to two large galaxy cata- logs, the NASA Sloan Atlas (NSA) (Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2011) and the catalogs of Luminous Red Galaxies (LRGs) from the SDSS-IV Extended Baryon Oscillation Spec- troscopic Survey (Bautista et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Our method faithfully reconstructs the cosmic matter density of the cosmic web throughout the observed volume, allowing the study of the dark matter distribution with respect to any objects of interest in the survey footprints, not just at the input galaxy locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We have released this data as part of the SDSS Data Release 17 (DR17) as a Value Added Catalog (VAC) publicly available for the community’s use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Unless stated otherwise, we adopt the Planck15 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016) cosmology as en- coded in the ASTROPY package (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Price-Whelan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' DATA We first describe the required inputs for reconstruct- ing the map of cosmic densities produced by MCPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCPM takes as input a 3D catalog of galaxy posi- tions with known masses and reproduces a data cube reconstructing the filamentary structure connecting the galaxy halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To optimize the parameters in MCPM, we also require a known density field from a cosmological simulation to compare our reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We then ap- ply the tuned model to observational catalogs of galaxies with known masses to reconstruct the physical cosmic web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3 We employ the dark matter-only Bolshoi-Plank ΛCDM (BP) simulation (described below) as our train- ing density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We then apply our model to spec- troscopic surveys that provide large samples of precise redshifts combined with value-added catalogs that esti- mate the galaxy masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We use two primary catalogs for our galaxy positions, the NASA-Sloan Atlas (NSA, or NSA/SDSS) for galaxies with z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 and the Large Scale Structure catalogs from Sloan Digital Sky Survey (SDSS) for galaxies at higher redshifts (z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' These two catalogs each offer advantages and disadvantages and are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Note that no new DR17 data were used in this VAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We now describe the galaxy catalogs and the simulations used as inputs to MCPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' NASA Sloan Atlas The NASA Sloan atlas (NSA) is a value-added cata- log constructed from reprocessed SDSS ugriz photome- try combined with Galaxy Evolution Explorer (GALEX) photometry in the ultraviolet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' It was designed to im- prove the standard SDSS sky subtraction pipeline (Blan- ton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We use the most recent version of this catalog, nsa v1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='fits, which contains galaxies out to z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In order to prioritize completeness in this data set, we we imposed an upper redshift cut to those galax- ies with z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 resulting in a catalog of 325321 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We will often refer to this catalog in this paper as simply “NSA/SDSS” to distinguish it from the other catalogs from BOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' LRG catalogs For the higher redshift portion of our catalog, we use a sample of Luminous Red Galaxies (LRGs) from the Baryon Oscillation Spectroscopic Survey (BOSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' BOSS was part of the SDSS III project, which at the time of its release, provided the largest survey of galaxy redshifts available in terms of the number of redshifts measured by a single survey and the effective cosmological volume covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We chose to use the LRG catalogs as tracers of the dark matter (DM) density as these catalogs are more complete at these redshifts with respect the selection function than using the more general SDSS galaxy sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The BOSS LRG sample derives from the large scale structure catalogs provided by the team and is broken into Northern and Southern Galactic Cap regions (LRG- NGC and LRG-SGC, respectively) (Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We use the LOWZ cata- logs, which provide a sample of LRGs to z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 and are found in the files1 galaxy DR12v5 LOWZ North.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='gz and galaxy DR12v5 LOWZ South.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='gz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The proce- 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='org/dr14/spectro/lss/ dure to create this catalog is mostly based on Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2016) with modifications to the redshift failure and sys- tematic corrections described in Bautista et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Mass Determination We used the LRG galaxy stellar masses from the Firefly VAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The Firefly VAC2 (Comparat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017) provides galaxy properties of all SDSS, BOSS, and eBOSS spectra using the FIREFLY fitting routine (Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017) (v1 0 4 for DR14 and v1 1 1 for DR16), which incorporates the stellar population models of Maraston & Str¨omb¨ack (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The Fire- fly catalog includes light- and mass-weighted stellar population properties (age and metallicity), E(B-V) values, and most crucially to this work, stellar mass for all galaxies in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We used the DR14 cat- alog to determine masses for the galaxies in the file sdss eboss firefly-dr14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The lower redshift NSA/SDSS catalog contains many galaxies that are spatially resolved and require more careful photometric analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The most recent version of this catalog provides elliptical Petrosian aperture photometry, which is more accurate than the standard SDSS pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We adopt the Pet- rosian aperture-derived mass to estimate the galaxy’s stellar mass for this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Bolshoi-Planck Simulations To calibrate our MCPM density estimates to the cosmic matter density, we use the dark matter only Bolshoi-Plank ΛCDM (BP) simulation (Klypin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Rodr´ıguez-Puebla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The BP simula- tion uses 20483 particles in a volume of 250h−1 Mpc3 and is based on the 2013 Planck (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014) cosmological parameters and compatible with the Planck 2015 parameters (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We utilize density field from the simulation smoothed by Gaussian kernel over scales of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='25 Mpc h−1 (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Goh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We also em- ploy the BP halo catalog produced using the Rockstar algorithm (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' METHODOLOGY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The MCPM algorithm We produced the VAC data with the Monte Carlo Physarum Machine (MCPM) algorithm implemented in the Polyphorm software3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCPM was first used in Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2020) to reconstruct a 3D density 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='org/dr16/spectro/eboss-firefly-value- added-catalog/ 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='com/CreativeCodingLab/Polyphorm 4 0 500 1000 1500 2000 2500 3000 Luminosity Distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 z 0 5000 10000 15000 20000 25000 Ngalaxies RUN0 RUN1 RUN3 RUN5 RUN7 RUN2 RUN4 RUN6 RUN8 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Distribution of the galaxy redshifts for the NSA/SDSS (blue, solid) and LRG-NGC (multi-colored, solid) and LRG- SGC (multi-colored, dashed) data sets that were used to reconstruct the cosmic density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The NSA/SDSS catalog includes all galaxies out to z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 and is denoted as RUN0 in the MCMP VAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The LRG catalogs extend to higher redshifts but only include the rarer LRGs, hence the lower galaxy count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This figure also shows the slicing scheme used to self-consistently fit the MCPM model in subsets of redshift as the density of galaxies decreases with luminosity distance in comoving Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' field estimate of the large-scale structure spanning 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='6k SDSS galaxies within the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='018 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='038 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The detailed description of methodology and analyses are de- scribed in Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We provide a brief sum- mary of the model here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCPM is a massively parallel agent-based model in- spired by the growth patterns of Physarum polycephalum slime mold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Its main modalities are visualized in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Using a swarm of millions of particle-like agents, MCPM iteratively traces the network structures implicit in the input data: dark matter halos or galaxies repre- sented as a weighted 3D point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In linear propor- tion to their halo mass, the data points emit a virtual marker which the agents navigate toward at every iter- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The key innovation of this model is the probabilistic navigation of the agents: the sampling of their trajec- tories according to PDFs derived from the data-emitted marker field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' For reference, the deterministic baseline model, where the agents always follow the maximum marker concentration, leads to the collapse of some fila- mentary configurations and the omission of a significant portion of data points, approximately a third as mea- sured in Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In contrast, MCPM fits over 99% of all input data points and can reconstruct configurations where multiple filaments branch out from a single origin, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', in massive galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCPM produces two main quantities: the trace field and the orientation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The trace field fT : R3 → R+ accumulates the superimposed trajectories of all active agents and represents the reconstructed LSS density field (after statistical standardization, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The orientation field fO : R3 → R3 + records the averaged un- signed directions of the agents and serves as a clustering criterion in our FoG compensation step (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Both are robust (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', stable in time) Monte-Carlo esti- mates of the equilibrium agent distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Compared to our earlier applications of the MCPM model (Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Simha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2020), we intro- duce a few methodological and implementation changes aimed at improving the quality of the fits (more on this in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Linear accumulation of fT and fO values instead of the original exponential floating window aver- 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Overview of MCPM’s operating modalities, demonstrated on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='018 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='038 sample of SDSS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Clockwise from top left: input data points and the marker concentration emitted by the data (yellow), reconstructed trace field fT (purple), corresponding orientation field fO (XYZ directions mapped to RGB colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' aging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The latter is used for the supervised part of the fitting when exploring different MCPM config- urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' After finding the optimal data-specific set of model parameters, we switch to linear av- eraging, which dramatically reduces the solution variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To avoid numerical errors, we increase the numer- ical precision from fp16 to fp32 for both fT and fO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This slows the implementation by 10-20%, which is acceptable for maintaining interactivity during fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We redesigned the agent rerouting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Rerout- ing is invoked when an agent encounters no data for too many subsequent steps, indicating either a boundary of the dataset or a large void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Our orig- inal rerouting assigned such an agent to a random location in space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' currently, we repositioned it to the location of a random data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This change leads to a significant decrease of background noise and effectively increases the dynamic range of the obtained solutions for both fT and fO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCPM fit to Bolshoi-Planck This section describes how we calibrate the MCPM algorithm using the Bolshoi-Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We refer read- ers to Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2022) for more details of the fitting procedure and the impact of the model hyperparame- ters on the resulting reconstruction geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Readers interested in the catalog data can skip to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Fitting MCPM to input data (either galaxies or ha- los) is a semi-supervised procedure, where the opera- tor focuses on maximizing the fitness function E while maintaining the connectedness and continuity of the re- constructed geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We define the fitness E of a given reconstructed trace field fT over a dataset D as E(fT, D) = 1 |D| � d∈D fT(dposition) dmass .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This results in a maximum likelihood estimator normal- ized by each data point’s mass to avoid overfitting to the most massive objects (given the large dispersion of typical galaxy and halo masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Since we do not yet have a precise mathematical description of the fit’s con- nectedness, we rely on the interactive visualization in Polyphorm to ensure that the fit does not collapse into a disconnected set of ‘islands’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Defining this property Data Marker Trace Orientation6 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Our reconstructed cosmic web data products and their spatial relation to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The green bands highlight regions of overlapping LRG slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The SDSS portion of the data is magnified to visualize the higher amount of recovered structure owing to the denser observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' rigorously and developing a fully automated fitting pro- cedure remains a future work for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To calibrate MCPM’s hyperparameters, we fit the model to two snapshots of the Bolshoi-Planck simula- tion dataset (at z = 0 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5, both containing roughly 16M halos extracted with the Rockstar algo- rithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We adopted some of the parameter estimates from our previous work (Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2020), includ- ing the sensing angle at 20 deg, moving angle at 10 deg, moving distance at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 Mpc and persistence of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='9 (now adjusted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='92 due to the finer granu- larity of halos used here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We focused on constraining the remaining critical parameters: sampling exponent (which controls the acuity of obtained structures, espe- cially filaments) and sensing distance (which deter- mines the scale of the structures, such as mean segment length and by transition the diameter of loops, voids, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In addition, we maximize the monotonicity of the obtained overdensity mapping as shown in Figure 5 as an additional constraint when determining the sampling exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Using this fitting procedure, we matched the MCPM fits to the ground truth densities in Bolshoi-Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We determined the optimal sampling exponent to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 at z = 0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2 at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5, which is consistent with the observation that the LSS at higher redshifts is less con- densed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' For the sensing distance, the optimal value was 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='37 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' It is worth noting that these sampling exponent and sensing distance values pose lower lim- its for the values used to fit the observational data, be- cause of the significantly lower spatial density of data points in the galaxy catalogs relative to BP simulations, which was compensated for by proportionally increasing the two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In Figure 4, we demonstrate that MCPM reconstructs not just the halos that we feed into it but the cosmic structure, including filaments and voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' More quanti- tative assessments are available in Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Fit to NASA-Sloan Atlas The first component of the VAC is based on the MCPM fit to the NASA-Sloan Atlas catalog for 0 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1, which contains roughly 325k galaxies in lumi- nosity distances between 44 and 476 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Similar to LRG (NGC) SDSS LRG (SGC) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 250 Mpc 500 Mpc 1000 Mpc 2000 Mpo 3000 Mpc7 Bolshoi-Planck z=0 Bolshoi-Planck z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 MCPM z=0 MCPM z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Comparison of the Bolshoi-Planck simulations (top row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' where the density field is known) at redshifts of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 (left) and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 (right) to the MCPM trace of the simulations (bottom row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' density recovered from halos alone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCPM faithfully reconstructs the cosmic structure from the galaxy halo population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' the BP dark matter halos, we treat the galaxies as 3D point attractors, in this case, weighted by their stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The fits are based on the hyperparameters calibrated on the BP simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Furthermore, to reflect the lower spatial density of the galaxies in comparison to the ha- los, we adjust the two critical parameters of MCPM: sampling exponent to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 and sensing distance to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To make these adjustments, we again used the semi- supervised fitting procedure described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To verify the consistency of the fit across different z values, we have split the SDSS catalog into 3 overlap- ping slices (44-270 Mpc / 250-370 Mpc / 350-476 Mpc, each containing about 120k galaxies) and fitted them separately by only adjusting the sensing distance pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The resulting optimal values (Figure 6) fol- low a linear trend, implying that the spatial density of galaxies decreases in corresponding proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' How- ever, the obtained variation of sensing distance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='8– 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='6) is well within the ability of the model to perform a consistent fit using a single parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Therefore, we opt for a single fit to the entire catalog using the aforementioned sensing distance value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Fit to LRG Catalogs The procedure of fitting to the LRG NGC and SGC catalogs is identical with the SDSS data: using the sampling exponent of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 and the BP-calibrated values for the other hyperparameters, we continued increasing sensing distance until reaching an optimal fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Due to the much lower spatial density of LRG ob- servations compared to SDSS, the optimal values of sensing distance end up being considerably higher (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Also, unlike SDSS, the LRG galaxies span a significantly more extended range of redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The consequence is nearly a two-fold growth of the optimal 8 4 2 0 2 exp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 2 0 2 4 log m/ exp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 log MCPM density 2 0 2 4 exp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Comparison of different sampling exponents in in- creasing order from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We find that a sampling exponent of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 produces the most linear mapping between the MCPM densities and the cosmic matter densities from the BP simulations, especially at lower densities where pre- vious versions of the MCPM have generally failed to recover the lowest density structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (see Figure 10 in Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' sensing distance value across the catalog’s redshift range (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Therefore to construct the VAC, we split the LRG galaxies into 4 overlapping ‘slices’ of ap- proximately equal numbers of galaxies (about 70k per slice for NGC, about 25k per slice for SGC) and fit each separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The resulting distance intervals are 0-1000 Mpc (z ≈ 0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2), 900-1600 Mpc (z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='18 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3), 1500-2100 Mpc (z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='28 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='38), and 2000-3000 Mpc (z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='36 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Figure 3 shows the visualization of all obtained density slices and their spatial relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' An added benefit of this approach is the higher resolution of each slice we can afford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This is desirable again due to the massive redshift range of the LRG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Statistical Standardization & Mapping The MCPM densities that fit each survey slice, al- though related to the true physical density, are rather the density of agents in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To translate the MCPM density to cosmic overdensity, we standardize each dis- tribution to the MCPM fit of the simulation so that a mapping between MCPM and cosmic overdensity can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The MCPM fits to the galaxy surveys differ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Plot of MCPM agents’ sensing distance (the main feature scaling parameter) as read out from the best fits for the LRG data, radially sliced into 4 runs at overlap- ping luminous distance intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' For comparison, we also show the best-fit sensing distances for 3 SDSS slices, which manifest a similar linear growth as we observe in the LRG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' from the fits to the BP simulations because they suffer from luminosity selection functions and are thus much sparser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This particularly affects the lowest density regime of the density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To account for this effect, we used the Wasserstein distance4 or the “Earth Movers Distance” to calculate the stretch and shift val- ues such that the distribution of MCPM densities of the surveys could be linearly transformed to best fit the BP- MCPM fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' That is TargetDist = stretch×SurveyDist + shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Where the TargetDist is the BP-MCPM den- sity distribution, and SurveyDist is the density distri- bution of each survey slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The benefit of this method is that we can impose a lower limit on the density distribu- tions to only take into account the higher density wing of the distribution corresponding to densities that con- tain structure and avoid the empty space in the survey fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In order to retrieve the cosmic matter density, ρm/⟨ρm⟩, we must map the MCPM trace density to that of the BP simulations at each redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We fit the BP simulations using the MCPM algorithm and then apply a mapping from MCPM density to cosmic matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This mapping was achieved by sampling the MCPM fits in bins of equal density and then determin- ing the density from the BP simulations at the same location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This is shown by the multi-colored stripes in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We then determine the median (and 1σ lim- its) of each MCPM density bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The median densities in each bin were then used to create a mapping func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We based our mapping function on the rectified linear activation function (ReLU), where the maximum change of the median of the bins determines the inflec- 4 https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='org/doc/scipy/reference/generated/ scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='html 9 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Mapping of the MCPM derived density to the cosmic matter density from the BP simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The MCPM densities were binned evenly in MCPM space in bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 dex as demarcated by the colored stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The custom ReLU mapping function fit to the medians of the bins (thick black line) and 1σ limits (thinner black lines) are plotted on top of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This mapping function provides a translation from the MCPM density to the cosmic overdensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' tion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' On the right-hand side of the flat part of the function, we fit a cubic polynomial to the data, creating a piece-wise continuous mapping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This method was chosen over other methods, such as interpolating the bins or using a spline function because fitting to densities above or below those found in the MCPM fits is not well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Our method is illustrated in Figure 7 where the thicker black line shows the mapping function applied to the z = 0 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The thinner black lines show the 1σ limits of our mapping, which correspond to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 dex in log cosmic matter overdensity, ρm/⟨ρm⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Correction for Redshift Space Distortions As MCPM operates in 3D space, applying the algo- rithm necessitates attaching physical distances to the in- put dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Although distance measurements via more direct methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', tip of the red giant branch or Type Ia supernovae) (Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016) may be available for a small subset of the galaxies (and therefore tracers of the underlying density field), we must primarily assume dis- tances concordant with the Hubble flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Thus, we ini- tially attach to each galaxy the luminosity distance given the adopted cosmology and galaxy redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Denser en- vironments such as galaxy groups and clusters will in- clude galaxies with large peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' These pe- culiar velocities will result in redshift space distortions (RSDs), or ‘fingers of god’ (FoG), if adopted directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' For example, a typical velocity dispersion for a > 1014 M⊙ galaxy cluster (∼ 1000 km/s) would propagate to a systematic error in the distance by assuming pure Hubble flow of > 10 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This issue plagues our low- redshift SDSS sample significantly more than the LRG samples for two reasons: 1) Low-mass galaxies are much more abundant and likely to be observed at low z in the magnitude-limited SDSS, which results in many ob- jects composing apparent false structures along the di- rection pointing away from (and towards) the observer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2) High-mass galaxies, which will dominate the samples at progressively higher redshifts, preferentially reside as central galaxies in their local environments (Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Therefore, these galaxy samples will be less sub- ject to systematic error in cosmological distance than our lowest redshift sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Thus, we employ an RSD correction for the z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 SDSS galaxy sample that we detail here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A key feature of MCPM is that the cosmic web re- construction converges to an equilibrium state but is a dynamical system nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The adopted ‘densities’ are aggregated trajectories of the millions of agents seek- ing efficient pathways between galaxy tracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCPM also outputs the components of an aggregated three- dimensional agent velocity vector for each cell in the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We use these velocities to identify RSDs, as the agent velocities producing them will be preferen- tially oriented perpendicular to the plane of the sky along the line of sight and will be clustered in their celestial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We select points in the MCPM cube by orientation as follows: we (1) convert each in- put galaxy’s location in the MCPM-output cube to its equivalent celestial coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2) find the three com- ponents of a unit radial vector parallel to the line of sight in Cartesian space to match the coordinate sys- tem of the MCPM velocity vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' and (3) calculate the dot product between the aggregated velocity vector at each galaxy’s position in the cube with the unit ra- dial vector and assign the result to that galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Galaxies within an RSD structure (FoG), having either parallel or antiparallel velocity vectors to the unit radial vec- tor, should not have dot product absolute values close z= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 3 2 log Pm/(p) 1 0 1-Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 3 2 log pm/p) 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 log MCPM density10 to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Therefore, we filter out galaxies with dot prod- uct absolute values less than 10, chosen upon inspecting the distribution of galaxy dot product values as a con- servative cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' To identify galaxy positions with similar velocity orientation and projected location on the sky, we then employ the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm as imple- mented in the scikit-learn5 python package, feeding it the sky coordinates and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' For this step, we further filter the galaxy catalog by mass to those with M∗ > 1010 M⊙, as the completeness of SDSS declines for less massive galaxies at the upper end of our red- shift range (z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' DBSCAN operates by locating high-density cores in the data, which are the beginnings of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The algorithm searches out from these cores, adding points until no more points are found in within some distance tolerance (in whatever space the data occupy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This algorithm contains several advan- tages over other possible choices, including scalability, compatibility with non-flat geometries, and the feature that certain points may not be included in any clus- ter (they are deemed ‘noise’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Two critical parameters for DBSCAN are the distance tolerance (eps) and the minimum number of points to be considered a core in the data (min samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We chose min samples=3 as a minimum number of galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', such as in a group or cluster) that might form a false RSD structure (FoG) in the MCPM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We chose a value of eps=2 upon experimenting with several values through visual inspec- tion to balance the inclusion of FoGs (which are read- ily identified by the eye), containing a relatively small number of galaxies while minimizing false identification of filaments not oriented antiparallel to the plane of the sky as RSD structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Figure 8 shows the resulting clusters identified by DBSCAN in a slice in declination of our galaxy catalog, with galaxies belonging to the same cluster having the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' From the output clusters identified by DBSCAN, we find the velocity range spanned by galaxy redshifts within each cluster using the full-width-half-maximum (FWHM) of the velocity distribution (vFWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' For clusters with vFWHM > 300 km/s, we adopt new red- shifts for the associated galaxies to be commensurate with more realistic physical distance separations inferred from a simple luminosity distance based on the redshift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' this procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Assuming the cluster mem- bers are bound to the same virialized structure, we con- vert the velocity FWHM to a velocity dispersion by the 5 https://scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='org/stable/ relation: σv = vFWHM 2 √ ln 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (1) We then use this velocity dispersion to infer a virial ra- dius, R200, of the cluster: Rinfer 200 = σv 4/3πG∆200ρcrit , (2) where ∆200 and ρcrit are the overdensity and critical den- sity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We then adopt new redshifts (solely for the purpose of feeding MCPM) about the median redshift of the cluster members by sampling from a nor- mal distribution with standard deviation corresponding to the change in redshift that would result in a lumi- nosity distance difference equal to the inferred R200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Fi- nally, we convert these galaxy coordinates and adopted redshifts to 3D Cartesian space via luminosity distances based on the new redshifts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' these then serve as inputs to MCPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' DATA PRODUCTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The Catalog The final value-added catalog contains the positions and redshifts and the stellar mass of the galaxies in the NASA-Sloan Atlas and the eBOSS Firefly Value-Added Catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We include a column, MASS SOURCE, to indi- cate which catalog was used to estimate the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The MCPM algorithm uses the galaxy mass to build the mat- ter density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The primary field of interest here is MATTERDENS, the matter density field at the location of a given galaxy, which was derived from fits of MCPM models in 3D volumes and mapped to the cosmological matter density (relative to the mean matter density) us- ing MCPM fits to the Bolshoi-Planck simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The catalogID is a combination of plate-mjd-fiberid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A unique identifier is the combination of catalogID and mcpmRun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Objects with the same value of mcpmRun were fitted with the MCPM model simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The data were sliced in redshift to yield samples producing self- consistent large-scale structures over the volume in each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' mcpmRun = 0 correspond to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='01 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 SDSS galaxies with masses from the NASA/Sloan Atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Sam- ples of LOWZ LRGs are marked 1-2 (z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='20), 3-4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='18 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='30), 5-6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='28 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='38), and 7-8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='36 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='51);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' each pair (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', 3-4), corresponds to the NGC/SGC samples in some redshift slice, with odd and even numbers for NGC and SGC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The data model for the catalog is described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 3D Density Cube In addition to the VAC, which contains the density at the location of each galaxy, we offer the full 3D density 11 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Data Model Name Type Unit Description (1) (2) (3) (4) catalogID char[13] Combination of PLATE-MJD-FIBERID plate int32 Plate number mjd int32 MJD of observation fiberid int32 Fiber identification number ra float64 deg Right ascension of fiber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J2000 dec float64 deg Declination of fiber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J2000 z float32 Best redshift massSource char[7] Source of the mass determination (nsa or firefly) mcpmRun int8 Index of galaxy sample fitted simultaneously with MCPM mstars float64 M⊙ Stellar mass matterDens float32 log10 of the ratio of the matter density relative to the mean matter density Note—Schema for the MCPM Value-Added Catalog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 as found in slimeMold galaxy catalog v1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' field of the relevant volumes, available at https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='org/sas/dr17/eboss/lss/mcpm/v1 0 0/datacube/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' These may be queried using our custom package, pyslime6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The data will unzip to a directory which may be opened by pyslime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This will enable the user to query the overdensity at arbitrary points in the cube, allowing the study of voids and filamentary structures outside the local environment of the input galaxy field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Comparison to Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2010) We can additionally validate our model by comparing our findings to that of other surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Although Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2020) has demonstrated the efficacy of our model, we present comparisons to other studies, leveraging our deeper and larger surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2010) used a method based on the 5th nearest galaxy neighbors to estimate the environmental density and studied the SFR and the quenched frac- tion of galaxies as a functions of this density metric and galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2020) illustrate that the MCPM method of computing cosmic density qualita- tively matches the results (see Figure 5 & 6 in Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In Figure 9, we demonstrate the improve- ment in signal gained with the NSA/SDSS sample as the increase in the number of galaxies is significant and the reproduction of their density-stellar mass-sSFR re- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='com/jnburchett/pyslime 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Potential Applications Our primary aim in this manuscript is to showcase the dataset and describe its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' There are, however, many exciting applications for this dataset that are well beyond the scope of this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Here, we list four general areas of application: Galaxy evolution in the cosmic web: A vast amount of galaxy properties measured and in- ferred from both multiwavelength photometry and spectroscopy have been cataloged for SDSS galax- ies (many also released as VACs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Salim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016) via straightforward crossmatching with our catalog, myriad galaxy-environment analyses may be readily conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Figure 9 highlights one di- rect application of the galaxy-density catalog to study the possible impacts of a galaxy’s location within the cosmic web on its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In partic- ular, Figure 9 shows the dependence of star for- mation activity as a function of large-scale struc- ture density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Our dataset is ideal for compar- ing effects induced by the more local environment (groups/clusters) to those induced by the cosmic web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Void finding: In the linear regime, the sizes of voids and their correlation statistics are sensi- tive to cosmology, particularly dark energy (Pisani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Although most of the analyses we have alluded to thus far focus on the denser re- gions of the cosmic web, namely filaments and nodes, our density cubes naturally include the un- 12 8h 10h 12h 14h 16h RA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 z 8h 10h 12h 14h 16h RA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1 z Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A slice in declination of our input galaxy catalog (grey points, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' RSD structures identified by DBSCAN are shown in various colors overlayed on the original points (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 logM /M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 log m/ m 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='4 log sSFR [yr 1] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The dependence of star formation activity on galaxy environment and stellar mass for the galaxies within the NSA/SDSS volume (z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The color coding de- notes sSFR in the population within each mass/environment bin, where the environmental density is determined from our MCPM cosmic web reconstruction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A comparison with Figure 6 of Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2010) shows a similarly increas- ing red fraction as a function of both mass at fixed density and density at fixed mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' derdense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Simple centroiding and cluster- ing algorithms may be readily applied to these density fields to directly identify and character- ize the voids, which in turn may be used as inputs for cosmological parameter estimations using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', the Alcock-Paczynski effect (Alcock & Paczynski 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The intergalactic medium: Hydrodynamical cosmological simulations predict a rich multi- phase structure in the intergalactic gas perme- ating throughout the cosmic web (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Cen & Ostriker 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Dav´e & Tripp 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Tepper-Garc´ıa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' In addition to the physical states of gas resulting from large-scale structure formation (Bertschinger 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Molnar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2009), energetic feedback from the galaxies themselves might ex- tend well beyond the virial radius, which is often adopted as a fiducial extent of a galaxy’s halo (Fin- lator & Dav´e 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Burchett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' (2020) used HST-observed background quasar sightlines through the MCPM reconstructed volume to find a relationship be- tween cosmic web density and Lyα optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A similar analysis could and should be done lever- aging our higher redshift LRG reconstruction with other absorption tracers, such as Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Multimessenger transient followup: Tran- sient phenomena, such as gravitational waves and fast radio bursts, are typically detected with im- precise localization, with scales of minutes or de- grees on the sky (Chen & Holz 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' CHIME/FRB Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Space-based and ground-based facilities around the world then follow up these detections to identify and char- acterize the sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Coulter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' As extragalactic sources are statistically more likely to be found within the large-scale structure, tran- sient observers could employ our reconstructed density field of the cosmic web in follow-up imag- ing campaigns to prioritize pointings toward re- gions of the sky most likely to contain the source counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Known Limitations The VAC volumes have the usual luminosity function systematics that are present in the underlying SDSS and LRG catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Specifically, the sampling density of galaxies is more significant at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This is reflected in the trace and can be seen in the SDSS data as well as each slice of the LRG catalogs, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This presents itself as an increased density at the lower redshift end of the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' However, the mean matter density at the low and high redshift ends of each volume is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Some sub-optimality of the model fit arises from the fact that the optimal sensing distance grows linearly according to the data in Figure 6, whereas the catalog is a piece-wise constant approximation of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Due to the differing sensing distance in each slice, there is a slight discontinuity of the MCPM densities ex- tracted from the overlaps between the LRG slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Thus, we recommend comparing densities on a slice-by-slice basis and avoiding comparing quantities based on the density at different redshift slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' CONCLUSION Herein we leverage the Monte Carlo Physarum Ma- chine (MCPM) methodology, inspired by the growth and movement of Physarum polycephalum slime mold, to map the cosmic web within several sub-samples of the SDSS spectroscopic galaxy catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The MCPM model inputs a galaxy field with known masses and out- puts the large-scale structure density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We train our model using the Bolshoi-Planck cosmological sur- vey, producing a reconstruction of the simulated cos- mic web where the underlying density is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Using 14 the simulation as ground truth, we describe the super- vised tuning of MCPM parameters to produce an opti- mal fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We apply this tuned model to the NASA-Sloan Atlas and the eBOSS LRG Firefly Value-Added Cata- logs to create both a 3D density cube and a catalog of cosmic densities at the location of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The SDSS NASA-Sloan Atlas catalogs include a more com- plete galaxy sample at z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We describe and employ a novel method on this dataset to reduce the effect of peculiar motions on the spectroscopic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The MCPM fits to the eBOSS LRG North and South Galac- tic Cap catalogs capture the larger-scale cosmic web out to z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' This paper describes the release the Cosmic Slime Value Added Catalog, part of SDSS DR17, which is the combination the two galaxy catalogs with den- sity estimates as well as the resultant 3D density cubes of the two galaxy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Finally, we highlight some exciting potential applications of this data set, which in- clude galaxy evolution in the context of the cosmic web, void finding, studies of the intergalactic medium, and multimessenger transient followup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors would like to especially acknowledge Joel Primack and Doug Hellinger for sharing the outputs of the Boloshoi-Planck simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' We also gratefully ac- knowledge the hospitality and support of the 2019 Kavli Summer Program in Astrophysics at UC Santa Cruz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' JB would like to acknowledge funding support from the National Science Foundation LEAPS-MPS award #2137452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' MCW and JKW acknowledge support for this work from NSF-AST 1812521, NSF-CAREER 2044303, the Research Corporation for Science Advancement, grant ID number 26842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' OE is supported by an incubator fellowship of the Open Source Program Office at UC Santa Cruz funded by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Sloan Foundation (G-2021-16957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Department of Energy Office of Science, and the Partici- pating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' SDSS-IV acknowledges support and resources from the Center for High-Performance Com- puting at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The SDSS website is www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' the Chilean Participa- tion Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' the French Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Harvard- Smithsonian Center for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Instituto de As- trof´ısica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' the Ko- rean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Leibniz Institut f¨ur Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Max-Planck-Institut f¨ur Astronomie (MPIA Hei- delberg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Max-Planck-Institut f¨ur Astrophysik (MPA Garching),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Max-Planck-Institut f¨ur Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' National Astronomical Observatories of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' New York Univer- sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Observat´ario Nacional / MCTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' United Kingdom Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Universidad Nacional Aut´onoma de M´exico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' Van- derbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' REFERENCES Abell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1965, ARAA, 3, 1 Alcock, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Paczynski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1979, Nature, 281, 358 Astropy Collaboration, Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2013, A&A, 558, A33 Balogh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Morris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Yee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Carlberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Ellingson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1999, ApJ, 527, 54 Bautista, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Vargas-Maga˜na, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Dawson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2018, ApJ, 863, 110, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3847/1538-4357/aacea5 Behroozi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Wechsler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 762, 109, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1088/0004-637x/762/2/109 Berlind, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Frieman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Weinberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2006, ApJS, 167, 1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1086/508170 Bertschinger, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1985, ApJS, 58, 39 Blanton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Kazin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Muna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Weaver, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Price-Whelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2011, AJ, 142, 31, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1088/0004-6256/142/1/31 Bouma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Richter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Wendt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2021, A&A, 647, A166 Burchett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Elek, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Tejos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2020, ApJL, 891, L35, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3847/2041-8213/ab700c Carollo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Cibinel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Lilly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2013, ApJ, 776, 71 Catinella, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Schiminovich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Cortese, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2013, MNRAS, 436, 34 Cen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Ostriker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1999, ApJ, 514, 1 15 Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Holz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, arXiv e-prints, arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='01471 CHIME/FRB Collaboration, Andersen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Bandura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019, ApJL, 885, L24 Codis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Jindal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Chisari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2018, MNRAS, 481, 4753 Comparat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Maraston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Goddard, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017, arXiv e-prints, arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='06575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='org/abs/1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='06575 Coulter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Foley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Kilpatrick, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017, Science, 358, 1556 Courtois, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Pomar`ede, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Tully, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Hoffman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Courtois, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2013, AJ, 146, 69 Crone Odekon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Hallenbeck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Haynes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2018, ApJ, 852, 142 Dav´e, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Tripp, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2001, ApJ, 553, 528 Dressler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 1980, ApJ, 236, 351 Elek, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Burchett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Prochaska, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Forbes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2022, Artificial Life, 28, 22, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1162/artl a 00351 Finlator, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Dav´e, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2008, MNRAS, 385, 2181 Goh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Primack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019, MNRAS, 483, 2101, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1093/mnras/sty3153 Ho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Cuesta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Seo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012, ApJ, 761, 14, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1088/0004-637X/761/1/14 Kauffmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2004, MNRAS, 353, 713 Klypin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Yepes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Gottl¨ober, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Prada, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Heß, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, MNRAS, 457, 4340, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1093/mnras/stw248 Kuutma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Tamm, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Tempel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017, A&A, 600, L6 Lan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', M´enard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Mo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, MNRAS, 459, 3998 Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Primack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Behroozi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017, MNRAS, 466, 3834, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1093/mnras/stw3348 Libeskind, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', van de Weygaert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Cautun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2018, MNRAS, 473, 1195, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1093/mnras/stx1976 Luber, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', van Gorkom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Hess, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019, AJ, 157, 254 Maraston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Str¨omb¨ack, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2011, MNRAS, 418, 2785, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='19738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='x Molnar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Hearn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Haiman, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2009, ApJ, 696, 1640 Nelson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Pillepich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2019, MNRAS, 490, 3234 Pasha, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Mandelker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', van den Bosch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & van de Voort, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='04097 Peng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Lilly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Kovaˇc, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2010, ApJ, 721, 193 Pisani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Sutter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Hamaus, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2015, PhRvD, 92, 083531 Planck Collaboration, Ade, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Aghanim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014, A&A, 571, A16, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1051/0004-6361/201321591 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, A&A, 594, A13, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1051/0004-6361/201525830 Poggianti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Fasano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Bettoni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2009, ApJL, 697, L137 Price-Whelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Sip˝ocz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', G¨unther, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2018, AJ, 156, 123 Reid, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Ho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Padmanabhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, MNRAS, 455, 1553, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1093/mnras/stv2382 Rodr´ıguez-Puebla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Behroozi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Primack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, MNRAS, 462, 893, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1093/mnras/stw1705 Ross, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Ho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Cuesta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2011, MNRAS, 417, 1350, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='19351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='x Ross, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Percival, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', S´anchez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012, MNRAS, 424, 564, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='21235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='x Salim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Janowiecki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, ApJS, 227, 2 Schaye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Crain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Bower, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2015, MNRAS, 446, 521 Simha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Burchett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Prochaska, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2020, ApJ, 901, 134, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='3847/1538-4357/abafc3 Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Jenkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2005, Nature, 435, 629 Tempel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Stoica, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Mart´ınez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014, MNRAS, 438, 3465 Tepper-Garc´ıa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Richter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Schaye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2012, MNRAS, 425, 1640 Tully, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Courtois, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Sorce, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2016, AJ, 152, 50 Vogelsberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Genel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014, MNRAS, 444, 1518 Vogelsberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Genel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2014, MNRAS, 444, 1518 Wakker, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Hernandez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', French, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2015, ApJ, 814, 40 Wilkinson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Maraston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Goddard, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Thomas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', & Parikh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2017, MNRAS, 472, 4297, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content='1093/mnras/stx2215 Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', Mo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', van den Bosch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} +page_content=' 2007, ApJ, 671, 153' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQf3AJU/content/2301.02719v1.pdf'} diff --git a/V9E0T4oBgHgl3EQfVgDO/content/tmp_files/2301.02266v1.pdf.txt b/V9E0T4oBgHgl3EQfVgDO/content/tmp_files/2301.02266v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4aec1139bd302bbee148bc5ddb81b44267870277 --- /dev/null +++ b/V9E0T4oBgHgl3EQfVgDO/content/tmp_files/2301.02266v1.pdf.txt @@ -0,0 +1,604 @@ +arXiv:2301.02266v1 [math.LO] 5 Jan 2023 +Implication Algebras and Implication +Semigroups of Binary Relations⋆ +Andrew Lewis-Smith1[0000−0001−9020−4055] and Jaˇs ˇSemrl2[0000−0001−7440−8867] +1 King’s College London andrew.lewis-smith@kcl.ac.uk +http://eecs.qmul.ac.uk/profiles/lewis-smithandrewstephen.html +2 UCL (University College London) j.semrl@cs.ucl.ac.uk +http://www0.cs.ucl.ac.uk/staff/jsemrl/ +Abstract. Representable implication algebras are known to be axioma- +tised by a finite number of equations (making the representation and +finite representation problems decidable here). We show that this also +holds in the context of unary (and binary) relations and present a Stone- +style representation theorem. We then show that the (finite) representa- +tion decision problem is undecidable for implication semigroups, in stark +contrast with implication algebras. +Keywords: Implication Algebras · Implication Semigroups · Representabil- +ity as Binary Relations +1 +Introduction +The variety of implication algebras, so-named by Abbott [1] and studied by Ra- +siowa [12], Diego [2], and their students, forms the algebraic semantics of the +implicational fragment of classical propositional logic. These are Boolean alge- +bras restricted to one operation (→) and a constant (⊤ or 1). The variety of +Relation algebras, alias residuated Boolean algebras with an additional involu- +tion operator (x⌣ or ‘converse of x’, with x understood as a relation), forms the +algebraic semantics of the calculus of relations. By a classical result of Korselt at- +tributed in [10], the variety of relation algebras corresponds to the three-variable +fragment of classical first-order logic, permitting a study of mathematical logic, +particularly set theory [14], via a quantifier-free equational theory. Proofs in this +theory consist of simple manipulations of identities, similar to proofs in abstract +algebra. This situation contrasts with proofs of standard mathematical logic (or +set theory) which can involve complex alternations of quantifiers. Meanwhile, +implication algebras (having one operation, classical implication) yield an alge- +braic analysis of the entailment relation between propositions in classical logic. +Although informed by different motivations, a certain elegance recommends the +study of relation and implication algebras. +⋆ This work was supported by the Engineering and Physical Sciences Research Council +EP/S021566/1 + +The present paper considers a fragment of the signature of relation algebras +we call implication semigroups based on adjoining a semigroup operation (;), +i.e. relational composition, to the implication algebras of Abbott. When the car- +rier set of this structure is a set of binary relations, we obtain the fragment of +relation algebras consisting of (S, →, ; ), i.e. relation algebras with signature re- +stricted to implication and composition. There are good reasons to examine this +signature. For one, it has not been well-explored: practically speaking, most al- +gebraic structures considered in algebraic logic are residuated lattices, groups, or +at least monoids – this can be noticed already in a standard definition of relation +algebras, as residuated lattices [9] – where the implication and monoidal oper- +ations interact via residuation. Algebras featuring implication and semigroup +operations fall out of the mainstream substructural logic literature as the alge- +bra at hand lacks the interdefinability present even in the case of a residuated +monoid. +For algebraic logic (particularly relation algebras) the question of whether +an algebra has a finite representation looms large. One typically asks whether a +given logical system of interest is not just consistent but has finite models, i.e. +models we can inspect within finite time or employing finitely many resources. +The present paper demonstrates the (finite) representation problem for implica- +tion semigroups is undecidable. Our results are curious for two reasons. First, +implication semigroups represent, in a sense, a limiting case of substructural +logics of implication for which the question of decidability of finite representa- +tions, to our knowledge, has not been raised, and certainly not approached from +the angle considered here. This suggests a track of further research in what one +might call substructural relation algebras, exploring the effects of weakening the +Boolean base in relation algebra into other algebras of residuation. This is al- +ready a current area of research by Peter Jipsen and Nikolaos Galatos [5] [4], +and has been broached from another angle in [13], where the signature consid- +ered there bears two residuals and a semigroup operation and is in fact a model +of the famed Lambek Calculus (thus connecting that algebra to the base sys- +tem for infinitely many substructural logics). Second, our results contribute to +a research programme seeking a better grasp of the consequences for relation +algebras when operating in a restricted signature. We are particularly motivated +to understand the effect on representability when moving to subsignatures of the +standard presentation of a relation algebra [6]. +2 +Preliminaries +In this section we present the definitions of the algebraic structures and operators +for binary relations. We begin by defining Abbott’s implication algebras. +Definition 1. An implication algebra3 A is a pair (A, →), with A a set and → +a binary operation on A satisfying the following properties: +3 Also known as Tarski algebras. +2 + +(i) (a → b) → a = a (Contraction) +(ii) (a → b) → b = (b → a) → a (Quasi-commutativity) +(iii) a → (b → c) = b → (a → c) (Exchange) +Trivially, because the class of implication algebras is equationally definable, it +forms a variety. We shall refer to this class as IA. Abbott shows a neat property +about these in [1]. +Proposition 2 (Abbott). Let A = (A, →) be an implication algebra. We can +implicitly define a constant 1 as a → a such that b → 1 = 1 and 1 → b = b, for +all b ∈ A. +This also gives us +Proposition 3. For an implication algebra (A, →), we can define a partial order +as +a ≤ b ⇔ (a → b) = 1 +Proof. Let a + b = (a → b) → b. It is commutative by quasi-commutativity, +idempotent by contraction, has 1 as the top by Proposition 2, and can be shown +associative (see [1][Theorem 12]). So a + b = b forms a partial order. Observe +that if a + b = b then a → b = a → ((a → b) → b) = (a → b) → (a → b) = 1. If +a → b = 1 then (a → b) → b = 1 → b = b. +⊓⊔ +Definition 4. Let ⊤ ⊆ X × X be a binary relation. Define A(⊤) = (℘(⊤), →) +where → is interpreted as proper Boolean implication defined below +a → b = (⊤ \ a) ∪ b +One can check that A(⊤) ∈ IA. Although ⊤ is conventionally an arbitrary +maximal relation, this is not the only possible interpretation of the → operation +for binary relations. We say that the implication operator is absolute if we require +⊤ = X × X, else we say that it is relative. +We say that A ∈ IA is representable if and only if it embeds into A(⊤) for +some ⊤ ⊆ X ×X. The embedding (usually denoted h) is called a representation. +If A embeds into A(⊤) and ⊤ is over a finite base X, then we say A is also +finitely representable. +Another standard presentation of implication algebras is A = (A, →, 1). How- +ever, the constant 1 can be defined as a → a, for any a. Furthermore, the quasi- +commutativity axiom is a consequence of the fact that (a → b) → b is equivalent +to the Boolean join of a + b. +Proposition 5. Let A = (A, →) ∈ IA be representable via h. Then h((a → +b) → b) = h(a) ∪ h(b) and h(1) = h(a → a) = ⊤, for any a, b ∈ A. +Proof. Since 1 = a → a and h is a representation we get h(1) = h(a → a) = +h(a) → h(a) = (⊤ \ h(a)) ∪ h(a) = ⊤. +By h being a representation, DeMorgan’s law, and a ∩ ⊤ = a we also have +h((a → b) → b) = (h(a) → h(b)) → h(b) = ⊤\((⊤\h(a))∪b)∪h(b) = (h(a)∩(⊤\ +h(b)))∪h(b) = (h(a)∪h(b))∩((⊤\h(b))∪h(b)) = (h(a)∪h(b))∩⊤ = h(a)∪h(b). +⊓⊔ +3 + +We now direct our attention to what happens when we add a semigroup +operation (;) to the signature. +Definition 6. An implication semigroup S is a tuple (S, →, ; ), with a carrier +set S and →, ; binary operations on S where +(i) (S, →) is an implication algebra +(ii) (S, ; ) is a semigroup +(iii) ((a → b) → b); c = (a; c → b; c) → b; c (Left quasi-additivity) +(iv) c; ((a → b) → b) = (c; a → c; b) → c; b (Right quasi-additivity) +The class of implication semigroups will be called ISG. Similarly to IA we +also examine structures where the carrier set is a set of binary relations. +Definition 7. Let ⊤ ⊆ X × X be a transitive binary relation. Define S(⊤) = +(℘(⊤), →, ; ) where → is interpreted as proper Boolean implication and ; as proper +relational composition defined as +a; b = {(x, z) | ∃y ∈ X : (x, y) ∈ a, (y, z) ∈ b} +Again checking S(⊤) ∈ ISG is relatively straightforward, note that they are +closed under composition due to the transitivity of ⊤. Similarly to IA, S ∈ ISG +is (finitely) representable if it embeds into S(⊤) for some transitive ⊤ (over a +finite base). +3 +Basic Theory, Stone Representation, and Decidability +for Implication Algebras +We now present the basic theory of implication algebras, the implicational frag- +ment of the implication semigroups discussed in the previous section. We first +consider the more general positive implication algebras, subsuming the impli- +cation algebras. This culminates in a representation theorem for implication +algebras, informing our construction in Section 4. 4 +The axiomatics here are largely in [1] and [12] with some corrections and +modifications. Their presentations of the implication algebras are quite different, +Abbott preferring an equational presentation where Rasiowa utilises a quasiequa- +tional definition. +Definition 8 (Rasiowa 2). A positive implication algebra5 (Postive IA) is a +pair (A, →, 1)6, a set A and → satisfying: +4 The representation result for implication algebras appears to have been known to +Diego [2], perhaps Abbott [1], but the proof is given in full by Rasiowa in [12]. It was +probably known to several others throughout different traditions of algebraic logic. +5 Also known as a Hilbert algebra. +6 With this axiomatisation we cannot omit 1 from the signature. Alternatively, 1 could +be replaced with a → a and an extra axiom added as a → a = b → b. +4 + +(P1) a → (b → a) = 1 +(P2) (a → (b → c)) → ((a → b) → (a → c)) = 1 +(P3) if a → b = 1 and b → a = 1 then a = b +(P4) a → 1 = 1 +Without proof, we state the following lemmas. For proofs, refer to [12]. +Proposition 9 (Rasiowa 2(1)). In any positive implication algebra, the fol- +lowing condition is fulfilled: if a → b = 1 and a = 1, then b = 1. Also, if a = 1, +then b → a = 1 for any b ∈ A. +Proposition 10 (Rasiowa 2.2). For any positive IA A, for all a, b ∈ A, we +can define a partial order ≤ on A as +a ≤ b ⇐⇒ a → b = 1 +and 1 = c → c for all maximal c in the poset (A, ≤). +Proposition 11 (Rasiowa 2.3). The following hold in any positive implication +algebra: +(1) If a ≤ b → c then b ≤ a → c +(2) a ≤ (a → b) → b +(3) 1 → a = a +(4) If b ≤ c, then a → b ≤ a → c +(5) If a ≤ b then b → c ≤ a → c +(6) a → (b → c) = b → (a → c) +Proposition 12 (Distributivity). In any (positive) implication algebra A = +(A, →, 1), we have a → (b → c) = (a → b) → (a → c) +Proof. We have b ≤ a → b by (P1) and Proposition 10. Applying Proposi- +tion 11(5)(6), we get (a → b) → (a → c) ≤ b → (a → c) = a → (b → c). So, +a → (b → c) = (a → b) → (a → c) follows from (P2) and Proposition 10. +The proof that distributivity holds in implication algebras is found in [1, +Theorem 5]. +⊓⊔ +We now show that the class of implication algebras lies below the class of pos- +itive implication algebras. Although the following proposition is not in Abbott +or Rasiowa, it is latent in the published results concerning implicative, positive +implication, and implication algebras. +Proposition 13. Any implication algebra (A, →) is a positive implication alge- +bra. +Proof. (P1) follows from the exchange axiom and Proposition 2, more specifically +a → (b → a) = b → (a → a) = b → 1 = 1. For (P2) follows from Proposition 12 +and Proposition 2. For (P3) see that by Proposition 3 we have the anti-symmetry +for the partial order in implication algebras. Finally (P4) follows directly from +Proposition 2. +⊓⊔ +5 + +Proposition 14. Any positive implication algebra (A, →) satisfying +(a → b) → a = a +for all a, b ∈ A is an implication algebra.7 +Proof. To show the other direction, let (A, →, 1) be a positive implication algebra +satisfying (a → b) → a = a. The first axiom of implication algebras (a → b) → +a = a we have already assumed adjoined to the algebra, and the third axiom, +a → (b → c) = b → (a → c), is found in Proposition 11(6). To show the second +axiom: (a → b) → b = (b → a) → a, we note a → b ≤ 1 = (b → b) = (b → +a) → (b → b) = b → ((b → a) → b) by Proposition 10 and Proposition 11(6). +By Proposition 11(1) we have b ≤ (a → b) → ((b → a) → b) and thus by +Proposition 11(1) and (3) we get (a → b) → b ≤ ((a → b) → (a → b)) → ((b → +a) → b) = 1 → ((b → a) → b) = ((b → a) → b). By a completely analogous +argument, (a → b) → b ≤ (b → a) → a. Hence (a → b) → b = (b → a) → a as +desired. +⊓⊔ +In anticipation of the Stone-style representation theorem, we define some +required notions like that of an implicative filter. +Definition 15 (Abbott). An implicative filter of a (positive) implication alge- +bra A = (A, →) is a subset F ⊆ A such that: +(i) 1 ∈ F +(ii) if a ∈ F and a → b ∈ F then b ∈ F +Definition 16. We say that an implicative filter F is proper if F ̸= A. We say +that a proper implicative filter is irreducible if it is not the intersection of two +proper implicative filters distinct from it, or formally: F is irreducible if for any +two proper implicative filters F1, F2 such that F = F1 ∩ F2, either F = F1 or +F = F2. Finally, a proper implicative filter F is said to be prime if a + b ∈ F +(or equivalently (a → b) → b ∈ F) implies that either a ∈ F or b ∈ F, for all +a, b ∈ A. +The proof of the Stone-like Representation theorem follows the following +steps. For proofs, refer to [12]. +Proposition 17 (Rasiowa 1.8). 8 If in any (positive) implication algebra A = +(A, →) one of the following conditions is satisfied for all a, b, c ∈ A: +(F1) (a → (b → c)) → a → b) → (a → c)) = 1 +7 Note that the contraction identity is not provable from the axioms (P1)–(P4), a +counterexample can be found using Mace4. +8 Rasiowa states this result for implicative algebras, the weakest algebra she considers +in her text. Since all implication algebras are positive implication algebras, and all +positive implication algebras are implicative algebras, we can specialise her result +for the present case. +6 + +(F2) (a → b) → (a → (b → c)) → (a → c)) = 1 +then for every implicative filter F in A and for every a ∈ A, the set Fa∗ = +{x ∈ A : a → x ∈ F} is an implicative filter. If, moreover, for all a, b ∈ A : a → +(b → a) = 1, then Fa∗ is the least implicative filter containing F and a. +Proposition 18 (Rasiowa 3.4). If (A, →) is a (positive) implication algebra, +then for every implicative filter F and for every element a ∈ A the set Fa∗ = +{x ∈ A : a → x ∈ F} is the least implicative filter containing F and a. +Proposition 19 (Rasiowa 6.1). An implicative filter in an implication algebra +is prime if and only if it is irreducible. +Lemma 20 (Rasiowa 1.4). If F0 is an implicative filter in an implicative al- +gebra A such that a0 /∈ F0 for some a0 ∈ A then there exists an irreducible +implicative filter G such that F0 ⊂ G and a0 /∈ G. +Immediately, by Lemma 20 and Proposition 19 we have: +Corollary 21. If F is an implicative filter in an implicative algebra A such that +a /∈ F for some a ∈ A then there exists a prime implicative filter G such that +F ⊂ G and a /∈ G. +This next corollary we prove, as it is not found in any of the literature cited +above and is required for the representation theorem. +Corollary 22. Let F be an implicative filter of an implication algebra A = +(A, →) such that a → b /∈ F for some a, b ∈ A. Then there exists a prime +implicative filter G : F ⊆ G such that a ∈ G and b /∈ G. +Proof. Let Fa∗ be the implicative filter generated by the filter F and a. Suppose +that a → b /∈ F. If b ∈ Fa∗, then we have a → b ∈ F by the definition of Fa∗. +This contradicts our assumption that a → b /∈ F; hence b /∈ Fa∗, and applying +Corollary 21 for Fa∗ and b we have a prime filter G such that Fa∗ ⊆ G and +b /∈ G. Clearly, a ∈ G and F ⊆ G. +⊓⊔ +We have then, as an immediate corollary from Corollary 22 and Proposi- +tion 19, the following: +Corollary 23. Let F be an implicative filter of an implication algebra A = +(A, →) such that a → b /∈ F for some a, b ∈ A. Then there exists an irreducible +implicative filter G : F ⊆ G such that a ∈ G and b /∈ G. +Finally, the culminating representation theorem. Rasiowa presents this for +irreducible implicative filters [12], which given her equivalence result, one can +also state using prime implicative filters, or maximal implicative filters. +Theorem 24 (Rasiowa 7.1). For any implication algebra A = (A, →), there is +a monomorphism h from A to (℘(X), →) of an arbitrary space X with |X| ≥ A. +7 + +From this it follows that every implication algebra is isomorphic to an impli- +cation algebra of sets. Since the focus of the present paper is on representations, +we note a corollary from this last result [1,2,12]: +Corollary 25. For any implication algebra A, if A is finite, then A has a finite +representation. +Proof. Let A be a finite implication algebra. Then by Theorem 24 A is monomor- +phic to the algebra A′ under h, where A′ = (℘(X), →) , an implication algebra +of sets. Now if |X| = A then |℘(X)| = 2|A|, and thus finite. That means h(A), +the subalgebra of A′ under h, is finite. So we have h(A) is a finite implication +algebra (induced by h and A), and hence h is a finite representation of A. +⊓⊔ +Now, the focus of the rest of the paper revolves around the (finite) repre- +sentation decision problem for implication semigroups. In the case of IA, this is +defined as follows: +Definition 26. The (finite) representation decision problem for implication al- +gebras is a decision problem that takes an implication algebra with a (finite) +carrier set as input. The algebra is a yes instance if and only if it is (finitely) +representable. +Closing this section, we note: +Corollary 27. IA is finitely axiomatisable. +Corollary 28. The (finite) representation problem for IA is decidable. +4 +Undecidability Results for Implication Semigroups +In this section we build on results from [8,11,7] to show undecidability of some +decision problems for . We begin by defining the representation and the finite +representation decision problems. +Definition 29. The (finite) representation decision problem for implication semi- +groups is a decision problem that takes an implication semigroup with a finite +carrier set as input. The semigroup is a yes instance if and only if it is (finitely) +representable. +As we mention in Section 2, whether a structure is representable, also depends +on our interpretation of the constant 1. Here we show that the (finite) decision +problem is undecidable in both cases. +4.1 +Representation Problem with Absolute Implication +We begin by examining the case with absolute implication, i.e. we require ⊤ = +X × X for some (finite) base X. +8 + +Definition 30. An implication monoid M = (M, 1′, →, ; ) is an algebra where +(M, →, ; ) is an implication semigroup and 1′ is the monoidal identity for ;. For +some transitive and reflexive ⊤ ⊆ X × X, we define M(⊤) = (℘(⊤), 1′, →, ; ) +where →, ; are proper relational implication and composition respectively and 1′ +is the proper relational identity for X defined as 1′ = {(x, x) | x ∈ X}. +In [8, Section 4] a construction of a Boolean monoid from a square cancellative +partial group G is given. Its implication monoid reduct is denoted M(G) = +(M, 1′, →, ; ). By [8, Proposition 5.1, Example 6.2] M(G) is representable (over +a finite base) if and only if G embeds into a (finite) group. +From the fact that both the group and the finite group embedding problems +are undecidable [3] for finite structures it follows that the (finite) representation +decision problem is undecidable. Thus if we prove that the ISG reduct of M(G) +is (finitely) representable if and only if M(G) is representable, we have shown +that the (finite) representability is undecidable. The right to left implication is +trivial. But we must examine the case where we relax the requirement where we +represent 1′ as the true relational identity, and show that this is still sufficient +for the structure to remain (finitely) representable with 1′ taken as the true +relational identity. +Suppose we have an embedding h from M(G) to S(⊤), i.e. an injective +mapping that preserves →, ;, but not necessarily 1′. +Lemma 31. If (x, y) ∈ h(1′) then (y, x) ∈ h(1′). +Proof. Suppose (y, x) ̸∈ h(1′). That means that (y, x) ∈ h(1′ → 0) = h(1′). By +composition of (x, y) ∈ h(1′) and (y, x) ∈ h(1′) we get that (x, x) ∈ h(1′) and by +composing that with (x, y) ∈ h(1′) we have that (x, y) ∈ h(1′). As (x, y) ∈ h(1′) +and (x, y) ∈ h(1′) = h(1′ → 0), we also have (x, y) ∈ h(0). By a series of +compositions we also get that (y, x) ∈ h(0) and because 0 ≤ 1′ we also get +(y, x) ∈ h(1′) and we’ve reached a contradiction. +⊓⊔ +Lemma 32. h(1′) is an equivalence relation. +Proof. By Lemma 31 we have that h(1′) is symmetric. Furthermore, since all +(x, x) ∈ h(⊤) there must exist a z witnessing 1′; ⊤ = ⊤. Thus (x, z) ∈ h(1′) and +(z, x) ∈ h(1′) and we compose that to get (x, x) ∈ h(1′), so h(1′) is reflexive. +Finally, as 1′ = 1′; 1′ we also have that h(1′) is transitive. +⊓⊔ +Lemma 33. For all x, x′, y, y′ ∈ X where (x′, x), (y, y′) ∈ h(1′) we have for all +a ∈ M(G) that (x, y) ∈ h(a) ⇔ (x′, y′) ∈ h(a). +Proof. If (x, y) ∈ h(a) we have (x, y′) ∈ h(a) by (x′, x), (y, y′) ∈ h(1′) and the +composition of 1′; a; 1′ = a. By Lemma 31, we also have (x, x′), (y′, y) ∈ h(1′) so +similarly if (x, y′) ∈ h(a) then (x, y) ∈ h(a). +⊓⊔ +Theorem 34. The (finite) representation decision problem for ISG is undecid- +able when → is interpreted as absolute implication. +9 + +Proof. As h(1′) is an equivalence relation by Lemma 32, so we can define h′ : +M(G) → X/h(1′) where +h′(a) = {([x]h(1′), [y]h(1′)) | (x, y) ∈ h(a)} +and show that h′ is indeed an embedding of M(G) into M(X × X). +By Lemma 33, we know that if (x, y) ∈ h(a) then for any x′ ∈ [x]h(1′), y′ ∈ +[y]h(1′) we have (x′, y′) ∈ h(a). +Take any a ≤ b. Then there exists (x, y) ∈ h(a) \ h(b). From this follows +([x]h(1′), [y]h(1′)) ∈ h′(a) and if it were the case that ([x]h(1′), [y]h(1′)) ∈ h′(b) +that would mean that there exist some (x′, y′) ∈ h(b) with (x, x′) ∈ h(1′) and +(y′, y) ∈ h(1′) and that would also means that (x, y) ∈ h(b). Thus h′ is injective. +Every composition is witnessed by the equivalence class of the witness for +the composition in h and if (x, y) ∈ h(a) and (y′, z) ∈ h(b) with y′ ∈ [y]h(1′) we +also have (y, y′) ∈ h(1′) and thus we have the composition (x, z) ∈ h(a; 1′; b) = +h(a; b). Thus h′ represents ; correctly. Finally 1′ is represented correctly as a pair +of equivalence classes is in h′(1′) if and only if they are the same equivalence +class. +Thus we have shown that if we have an embedding of M(G) into S(X × X) +then we also have an embedding of M(G) into M(X′ ×X′) where X′ = X/h(1′). +Furthermore if X is finite, so is X′. Trivially if M(G) embeds into M(X × X) +it also embeds into S(X × X) via the same embedding. This, together with the +results presented in [8] shows that the (finite) representation decision problem +for ISG is undecidable. +⊓⊔ +4.2 +Representation Problem with Relative Implication +Now we show the same result for relative implication. +Definition 35. A Boolean semigroup is a tuple B = (B, 0, 1, −, +, ; ) is an +algebraic structure where S is a carrier set +(i) (B, 0, 1, −, +) is a Boolean algebra +(ii) (B, ; ) is a semigroup +(iii) ; is additive over + +(iv) 0; a = a; 0 = 0 +Similarly to , we denote the class of Boolean semigroups BSG and we say that +a Boolean semigroup is representable if and only for some transitive ⊤ ⊆ X × X +it embeds into B(⊤) = (℘(X × X), ∅, ⊤, −, +, ; ) where −a is interpreted as +proper Boolean negation ⊤ \ a, + is interpreted as proper Boolean join ∪ and ; +is interpreted as proper relational composition. +The (finite) representation problem for Boolean semigroups is defined analo- +gous to that for implication semigroups. [7, Theorem 11.2] shows that the repre- +sentation problem for Boolean semigroups is undecidable and [11, Theorem 2.5] +shows that the finite representation problem for Boolean semigroups is undecid- +able. From this we show that the (finite) representation problem for implication +semigroups is also undecidable. +10 + +Note that the above results require an operation · to be defined in the signa- +ture, but much like ⊤ in IA, · is term definable for BSG as a · b = −(−a + −b). +Lemma 36. Let S = (S, →, ; ) be an implication semigroup that contains some +element 0 such that for all a ∈ S we have 0 ≤ a and 0; a = a; 0 = 0. If S is +representable via some representation h then there exists a representation h′ of +S where h′(0) = ∅. If h is defined over a finite base, so is h′. +Proof. Let h(S) be the proper structure defined by h for some ⊤ ⊆ X × X. As +h is a representation, there exists for every pair a ̸≤ b ∈ S a discriminator pair +(ι, o) ∈ ⊤ such that (ι, o) ∈ h(a) \ h(b). +Define Xι,o as +Xι,o = +� +x ∈ X | +� +x = ι ∨ (ι, x) ∈ ⊤ +� +∧ +� +y = o ∨ (y, o) ∈ ⊤ +�� +⊤ι,o as ⊤ ∩ (Xι,o × Xι,o), and a mapping hι,o : S → S(⊤ι,o) where hι,o(a) = +h(a) ∩ ⊤ι,o. +First observe that hι,o(0) = ∅. Suppose that there was a pair (x, y) ∈ hι,o(0). +If x = ι, we have (ι, y) ∈ hι,o(0), else (ι, x) ∈ h(1) = ⊤ and thus (ι, y) ∈ h(0) +since 1; 0 = 0 and h preserves composition. Similarly if y = o we get (ι, o) ∈ h(0), +else by composing (ι, y) ∈ h(0) with (y, z) ∈ h(1) we get (ι, o) ∈ h(0). Since +b ≥ 0 that would mean (ι, o) ∈ h(b) that contradicts the fact that (ι, o) is a +discriminator pair for a ̸≤ b. +Now let us check that hι,o preserves composition. Suppose (x, y) ∈ hι,o(a; b). +This means that there exists z ∈ X such that (x, z) ∈ h(a) and (y, z) ∈ h(b). +If x = ι, we trivially have (x, ι) ∈ h(1) = ⊤. Else, by composing (ι, x) ∈ h(1) +and (x, y) ∈ h(a) we get (ι, y) ∈ h(1) = ⊤ as 1; a ≤ 1. Similarly (y, o) ∈ ⊤ and +thus y ∈ Xι,o. Thus we have (x, z) ∈ hι,o(a), (y, z) ∈ hι,o(b) and we have shown +hι,o(a; b) ⊆ hι,o(a); hι,o(b). The fact that hι,o(a; b) ⊇ hι,o(a); hι,o(b) follows from +(x, y), (y, z) ∈ ⊤ι,o then x, z ∈ Xι,o and we have (x, z) ∈ ⊤ι,o. Thus hι,o preserves +composition. +We have hι,o(a) ̸= hι,o(b) as (ι, o) ∈ ⊤ι,o. The operation → is preserved by +hι,o as for all (x, y) ∈ ⊤ι,o it holds (x, y) ∈ h(a) ⇐⇒ (x, y) ∈ hι,o(a). Finally, +|Xι,o| ≤ |X|. Thus we conclude that h{ι, o} is a homomorphism for S that +discriminates the pair a ̸≤ b. +Now let us pick for every a ̸≤ b a δ(a, b) = (ι, o) such that (ι, o) ∈ ⊤ is a +discriminator pair for a ̸≤ b and let ˙∪ denote a disjoint union. A mapping +h′ : S → ℘ +� ˙� +a̸≤b∈S⊤δ(a,b) +� +h′(c) = ˙� +a̸≤b∈Shδ(a,b)(c) +still represents ; , → correctly, discriminates all pairs a ̸≤ b (i.e. is injective), +which makes it a representation. Furthermore, h′(0) = ∅ and the size of its base +is bounded by |S|2|X|. +⊓⊔ +11 + +Theorem 37. The (finite) representation decision problem for implication semi- +groups is undecidable. +Proof. We show this by proving that B ∈ BSG is representable if and only if +its ⟨→, ; ⟩-reduct S is representable. The left to right implication is trivial as +a → b ∈ S is term-definable as (−a) + b ∈ B. For the right to left implication, ++, 1 are term-definable in ⟨→, ; ⟩ (see Proposition 5). By Lemma 36 the ⟨→, ; ⟩- +reduct of B is representable if and only if it has a representation where h(0) = ∅. +See how in that representation a → 0 (corresponding to −a + 0 = −a in the +Boolean semigroup) is represented as ⊤ \ a ∪ ∅ = ⊤ \ a. +As the (finite) representation decision problem is undecidable for Boolean +semigroups, we conclude the same for implication semigroups. +⊓⊔ +5 +Problems +In this section we outline some open problems. It follows from the results in +Section 4 that the class of representable implication semigroups is not finitely +axiomatisable, nor does it have the finite representation property, i.e. not ev- +ery finite representable structure in the class is finitely representable. However, +another decision problem of interest is posed below. +Problem 38. Is membership in the equational theory generated by the class of +representable implication semigroups decidable? +The reader can see that if we add the bottom element 0 to the signature, the +undecidability follows from the undecidability of the equational theory of the +Boolean semigroups as described in [7]. This is because all negations of terms +−t can be rewritten as t → 0 and all joins t + t′ as (t → t′) → t′ where t, t′ are +terms. +The problem remains open for the class of representable implication semi- +groups without the bottom element. One of the possible ways to prove undecid- +ability is by using discriminator terms, defined below. +Definition 39. A discriminator term d(a, b, c) is a term defined in terms of +elements of algebra a, b, c such that for all representable algebras d(a, b, c) = c if +a = b and a otherwise. +Although the existence of discriminator terms is not a guarantee for the +undecidability of the equational theory membership decision problem, it is an +interesting open question in its own right. +Problem 40. Is it possible to define a discriminator term in the language of +implication semigroups? +It is well known that subreducts of representable relation algebras form qua- +sivarieties. As such, the class of implication semigroups can be characterised +by quasiequations. However, some open questions about the equational theory +generated by the class of representable implication semigroups are listed below. +12 + +Problem 41. Is the class of representable implication semigroups a variety? +Problem 42. Is the equational theory generated by the class of representable +implication semigroups finitely axiomatisable? +We continue by looking at the alternative interpretations of → operation +for binary relations. An interesting example, as mentioned in the introduction +section is that of a weakening relation defined below. +Definition 43. Let P = (X, ≤) be a poset. R ⊆ X × X is a weakening relation +if and only if ≤; R; ≤ ⊆ R. +In the context of the weakening relation algebras as described in [4], the → +operation can be given in first order terms as +R → S = {(x, y) | ∀x′, y′ : ((x′ ≤ x ∧ y ≤ y′ ∧ (x′, y′) ∈ R) ⇒ (x′, y′) ∈ S)} +where R, S are weakening relations over a poset P = (X, ≤). +This interpretation of the → operation gives rise to the class of representable +weakening implication semigroups, for which the following properties remain +open. +Problem 44. Is the (finite) representation decision problem decidable for the +class of representable weakening implication semigroups? Is the class finitely +axiomatisable and does it have the finite representation property? +Problem 45. Is the class of representable weakening implication semigroups a +(discriminator) variety? Is the equational theory generated by the class finitely +axiomatisable/decidable? +Finally, we note that it can be checked that all results presented in this paper +can be generalised to the dual operation ← by presenting dual axioms for the +class of implication algebras and defining negation as 0 ← a. +References +1. Abott, J.C.: Implicational algebras. Bulletin math´ematique de la Soci´et´e des Sci- +ences Math´ematiques de la R´epublique Socialiste de Roumanie 11 (59)(1), 3–23 +(1967), http://www.jstor.org/stable/43679502 +2. Diego, +A.: +Sobre +´algebras +de +Hilbert. +Notas +de +l´ogica +matem´atica, +Instituto +de +Matem´atica, +Universidad +Nacional +del +Sur +(1965), +https://books.google.co.uk/books?id=AfcSAQAAMAAJ +3. Evans, T.: Embeddability and the word problem. Journal of the London Mathe- +matical Society 1(1), 76–80 (1953) +4. Galatos, N., Jipsen, P.: The structure of generalized bi-algebras and weakening +relation algebras. Algebra universalis 81(3), 1–35 (2020) +5. Galatos, N., Jipsen, P.: Weakening relation algebras and fl2-algebras. In: Interna- +tional Conference on Relational and Algebraic Methods in Computer Science. pp. +117–133. Springer (2020) +13 + +6. Hirsch, R., Hodkinson, I.: Relation algebras by games. Elsevier (2002) +7. Hirsch, R., Hodkinson, I., Jackson, M.: Undecidability of algebras of binary rela- +tions. In: Hajnal Andr´eka and Istv´an N´emeti on Unity of Science, pp. 267–287. +Springer (2021) +8. Hirsch, R., Jackson, M.: Undecidability of representability as binary relations. The +Journal of Symbolic Logic 77(4), 1211–1244 (2012) +9. J´onsson, B., Tsinakis, C.: Relation algebras as residuated boolean algebras. algebra +universalis 30, 469–478 (1993) +10. L¨owenheim, L.: ¨Uber m¨oglichkeiten im relativkalk¨ul. Mathematische Annalen 76, +447–470 (1915) +11. Neuzerling, M.: Undecidability of representability for lattice-ordered semigroups +and ordered complemented semigroups. Algebra universalis 76(4), 431–443 (2016) +12. Rasiowa, H.: An Algebraic Approach to Non-Classical Logics. Amsterdam, Nether- +lands: Warszawa, Pwn - Polish Scientific Publishers (1974) +13. Rogozin, +D.: +The +finite +representation +property +for +some +reducts +of +re- +lation +algebras +(2020). +https://doi.org/10.48550/ARXIV.2007.13079, +https://arxiv.org/abs/2007.13079 +14. Tarski, A., Givant, S.R.: A Formalization of Set Theory Without Variables. Amer- +ican Mathematical Soc. (1987) +14 + diff --git a/V9E0T4oBgHgl3EQfVgDO/content/tmp_files/load_file.txt b/V9E0T4oBgHgl3EQfVgDO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c041f8d8afac36485b4421ece76ea3cc130e6c2 --- /dev/null +++ b/V9E0T4oBgHgl3EQfVgDO/content/tmp_files/load_file.txt @@ -0,0 +1,473 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf,len=472 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='02266v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='LO] 5 Jan 2023 Implication Algebras and Implication Semigroups of Binary Relations⋆ Andrew Lewis-Smith1[0000−0001−9020−4055] and Jaˇs ˇSemrl2[0000−0001−7440−8867] 1 King’s College London andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='lewis-smith@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='uk http://eecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='qmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='uk/profiles/lewis-smithandrewstephen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='html 2 UCL (University College London) j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='semrl@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='uk http://www0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='uk/staff/jsemrl/ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Representable implication algebras are known to be axioma- tised by a finite number of equations (making the representation and finite representation problems decidable here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We show that this also holds in the context of unary (and binary) relations and present a Stone- style representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We then show that the (finite) representa- tion decision problem is undecidable for implication semigroups, in stark contrast with implication algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Keywords: Implication Algebras · Implication Semigroups · Representabil- ity as Binary Relations 1 Introduction The variety of implication algebras, so-named by Abbott [1] and studied by Ra- siowa [12], Diego [2], and their students, forms the algebraic semantics of the implicational fragment of classical propositional logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' These are Boolean alge- bras restricted to one operation (→) and a constant (⊤ or 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The variety of Relation algebras, alias residuated Boolean algebras with an additional involu- tion operator (x⌣ or ‘converse of x’, with x understood as a relation), forms the algebraic semantics of the calculus of relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By a classical result of Korselt at- tributed in [10], the variety of relation algebras corresponds to the three-variable fragment of classical first-order logic, permitting a study of mathematical logic, particularly set theory [14], via a quantifier-free equational theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proofs in this theory consist of simple manipulations of identities, similar to proofs in abstract algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This situation contrasts with proofs of standard mathematical logic (or set theory) which can involve complex alternations of quantifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Meanwhile, implication algebras (having one operation, classical implication) yield an alge- braic analysis of the entailment relation between propositions in classical logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Although informed by different motivations, a certain elegance recommends the study of relation and implication algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⋆ This work was supported by the Engineering and Physical Sciences Research Council EP/S021566/1 The present paper considers a fragment of the signature of relation algebras we call implication semigroups based on adjoining a semigroup operation (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' relational composition, to the implication algebras of Abbott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' When the car- rier set of this structure is a set of binary relations, we obtain the fragment of relation algebras consisting of (S, →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' relation algebras with signature re- stricted to implication and composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' There are good reasons to examine this signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For one, it has not been well-explored: practically speaking, most al- gebraic structures considered in algebraic logic are residuated lattices, groups, or at least monoids – this can be noticed already in a standard definition of relation algebras, as residuated lattices [9] – where the implication and monoidal oper- ations interact via residuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Algebras featuring implication and semigroup operations fall out of the mainstream substructural logic literature as the alge- bra at hand lacks the interdefinability present even in the case of a residuated monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For algebraic logic (particularly relation algebras) the question of whether an algebra has a finite representation looms large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' One typically asks whether a given logical system of interest is not just consistent but has finite models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' models we can inspect within finite time or employing finitely many resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The present paper demonstrates the (finite) representation problem for implica- tion semigroups is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Our results are curious for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' First, implication semigroups represent, in a sense, a limiting case of substructural logics of implication for which the question of decidability of finite representa- tions, to our knowledge, has not been raised, and certainly not approached from the angle considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This suggests a track of further research in what one might call substructural relation algebras, exploring the effects of weakening the Boolean base in relation algebra into other algebras of residuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This is al- ready a current area of research by Peter Jipsen and Nikolaos Galatos [5] [4], and has been broached from another angle in [13], where the signature consid- ered there bears two residuals and a semigroup operation and is in fact a model of the famed Lambek Calculus (thus connecting that algebra to the base sys- tem for infinitely many substructural logics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Second, our results contribute to a research programme seeking a better grasp of the consequences for relation algebras when operating in a restricted signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We are particularly motivated to understand the effect on representability when moving to subsignatures of the standard presentation of a relation algebra [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 2 Preliminaries In this section we present the definitions of the algebraic structures and operators for binary relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We begin by defining Abbott’s implication algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' An implication algebra3 A is a pair (A, →), with A a set and → a binary operation on A satisfying the following properties: 3 Also known as Tarski algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 2 (i) (a → b) → a = a (Contraction) (ii) (a → b) → b = (b → a) → a (Quasi-commutativity) (iii) a → (b → c) = b → (a → c) (Exchange) Trivially, because the class of implication algebras is equationally definable, it forms a variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We shall refer to this class as IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Abbott shows a neat property about these in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 2 (Abbott).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let A = (A, →) be an implication algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We can implicitly define a constant 1 as a → a such that b → 1 = 1 and 1 → b = b, for all b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This also gives us Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For an implication algebra (A, →), we can define a partial order as a ≤ b ⇔ (a → b) = 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let a + b = (a → b) → b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' It is commutative by quasi-commutativity, idempotent by contraction, has 1 as the top by Proposition 2, and can be shown associative (see [1][Theorem 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' So a + b = b forms a partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Observe that if a + b = b then a → b = a → ((a → b) → b) = (a → b) → (a → b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If a → b = 1 then (a → b) → b = 1 → b = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let ⊤ ⊆ X × X be a binary relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Define A(⊤) = (℘(⊤), →) where → is interpreted as proper Boolean implication defined below a → b = (⊤ \\ a) ∪ b One can check that A(⊤) ∈ IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Although ⊤ is conventionally an arbitrary maximal relation, this is not the only possible interpretation of the → operation for binary relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We say that the implication operator is absolute if we require ⊤ = X × X, else we say that it is relative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We say that A ∈ IA is representable if and only if it embeds into A(⊤) for some ⊤ ⊆ X ×X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The embedding (usually denoted h) is called a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If A embeds into A(⊤) and ⊤ is over a finite base X, then we say A is also finitely representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Another standard presentation of implication algebras is A = (A, →, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' How- ever, the constant 1 can be defined as a → a, for any a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Furthermore, the quasi- commutativity axiom is a consequence of the fact that (a → b) → b is equivalent to the Boolean join of a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let A = (A, →) ∈ IA be representable via h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Then h((a → b) → b) = h(a) ∪ h(b) and h(1) = h(a → a) = ⊤, for any a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Since 1 = a → a and h is a representation we get h(1) = h(a → a) = h(a) → h(a) = (⊤ \\ h(a)) ∪ h(a) = ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By h being a representation, DeMorgan’s law, and a ∩ ⊤ = a we also have h((a → b) → b) = (h(a) → h(b)) → h(b) = ⊤\\((⊤\\h(a))∪b)∪h(b) = (h(a)∩(⊤\\ h(b)))∪h(b) = (h(a)∪h(b))∩((⊤\\h(b))∪h(b)) = (h(a)∪h(b))∩⊤ = h(a)∪h(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ 3 We now direct our attention to what happens when we add a semigroup operation (;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=') to the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' An implication semigroup S is a tuple (S, →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ), with a carrier set S and →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' binary operations on S where (i) (S, →) is an implication algebra (ii) (S, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) is a semigroup (iii) ((a → b) → b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' c = (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' c → b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' c) → b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' c (Left quasi-additivity) (iv) c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ((a → b) → b) = (c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' a → c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b) → c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b (Right quasi-additivity) The class of implication semigroups will be called ISG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Similarly to IA we also examine structures where the carrier set is a set of binary relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let ⊤ ⊆ X × X be a transitive binary relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Define S(⊤) = (℘(⊤), →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) where → is interpreted as proper Boolean implication and ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' as proper relational composition defined as a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b = {(x, z) | ∃y ∈ X : (x, y) ∈ a, (y, z) ∈ b} Again checking S(⊤) ∈ ISG is relatively straightforward, note that they are closed under composition due to the transitivity of ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Similarly to IA, S ∈ ISG is (finitely) representable if it embeds into S(⊤) for some transitive ⊤ (over a finite base).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 3 Basic Theory, Stone Representation, and Decidability for Implication Algebras We now present the basic theory of implication algebras, the implicational frag- ment of the implication semigroups discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We first consider the more general positive implication algebras, subsuming the impli- cation algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This culminates in a representation theorem for implication algebras, informing our construction in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 4 The axiomatics here are largely in [1] and [12] with some corrections and modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Their presentations of the implication algebras are quite different, Abbott preferring an equational presentation where Rasiowa utilises a quasiequa- tional definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 8 (Rasiowa 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' A positive implication algebra5 (Postive IA) is a pair (A, →, 1)6, a set A and → satisfying: 4 The representation result for implication algebras appears to have been known to Diego [2], perhaps Abbott [1], but the proof is given in full by Rasiowa in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' It was probably known to several others throughout different traditions of algebraic logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 5 Also known as a Hilbert algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 6 With this axiomatisation we cannot omit 1 from the signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Alternatively, 1 could be replaced with a → a and an extra axiom added as a → a = b → b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 4 (P1) a → (b → a) = 1 (P2) (a → (b → c)) → ((a → b) → (a → c)) = 1 (P3) if a → b = 1 and b → a = 1 then a = b (P4) a → 1 = 1 Without proof, we state the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For proofs, refer to [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 9 (Rasiowa 2(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' In any positive implication algebra, the fol- lowing condition is fulfilled: if a → b = 1 and a = 1, then b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Also, if a = 1, then b → a = 1 for any b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 10 (Rasiowa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For any positive IA A, for all a, b ∈ A, we can define a partial order ≤ on A as a ≤ b ⇐⇒ a → b = 1 and 1 = c → c for all maximal c in the poset (A, ≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 11 (Rasiowa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The following hold in any positive implication algebra: (1) If a ≤ b → c then b ≤ a → c (2) a ≤ (a → b) → b (3) 1 → a = a (4) If b ≤ c, then a → b ≤ a → c (5) If a ≤ b then b → c ≤ a → c (6) a → (b → c) = b → (a → c) Proposition 12 (Distributivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' In any (positive) implication algebra A = (A, →, 1), we have a → (b → c) = (a → b) → (a → c) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We have b ≤ a → b by (P1) and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Applying Proposi- tion 11(5)(6), we get (a → b) → (a → c) ≤ b → (a → c) = a → (b → c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' So, a → (b → c) = (a → b) → (a → c) follows from (P2) and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The proof that distributivity holds in implication algebras is found in [1, Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ We now show that the class of implication algebras lies below the class of pos- itive implication algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Although the following proposition is not in Abbott or Rasiowa, it is latent in the published results concerning implicative, positive implication, and implication algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Any implication algebra (A, →) is a positive implication alge- bra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' (P1) follows from the exchange axiom and Proposition 2, more specifically a → (b → a) = b → (a → a) = b → 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For (P2) follows from Proposition 12 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For (P3) see that by Proposition 3 we have the anti-symmetry for the partial order in implication algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Finally (P4) follows directly from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ 5 Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Any positive implication algebra (A, →) satisfying (a → b) → a = a for all a, b ∈ A is an implication algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' To show the other direction, let (A, →, 1) be a positive implication algebra satisfying (a → b) → a = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The first axiom of implication algebras (a → b) → a = a we have already assumed adjoined to the algebra, and the third axiom, a → (b → c) = b → (a → c), is found in Proposition 11(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' To show the second axiom: (a → b) → b = (b → a) → a, we note a → b ≤ 1 = (b → b) = (b → a) → (b → b) = b → ((b → a) → b) by Proposition 10 and Proposition 11(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By Proposition 11(1) we have b ≤ (a → b) → ((b → a) → b) and thus by Proposition 11(1) and (3) we get (a → b) → b ≤ ((a → b) → (a → b)) → ((b → a) → b) = 1 → ((b → a) → b) = ((b → a) → b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By a completely analogous argument, (a → b) → b ≤ (b → a) → a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Hence (a → b) → b = (b → a) → a as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ In anticipation of the Stone-style representation theorem, we define some required notions like that of an implicative filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 15 (Abbott).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' An implicative filter of a (positive) implication alge- bra A = (A, →) is a subset F ⊆ A such that: (i) 1 ∈ F (ii) if a ∈ F and a → b ∈ F then b ∈ F Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We say that an implicative filter F is proper if F ̸= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We say that a proper implicative filter is irreducible if it is not the intersection of two proper implicative filters distinct from it, or formally: F is irreducible if for any two proper implicative filters F1, F2 such that F = F1 ∩ F2, either F = F1 or F = F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Finally, a proper implicative filter F is said to be prime if a + b ∈ F (or equivalently (a → b) → b ∈ F) implies that either a ∈ F or b ∈ F, for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The proof of the Stone-like Representation theorem follows the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For proofs, refer to [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 17 (Rasiowa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 8 If in any (positive) implication algebra A = (A, →) one of the following conditions is satisfied for all a, b, c ∈ A: (F1) (a → (b → c)) → a → b) → (a → c)) = 1 7 Note that the contraction identity is not provable from the axioms (P1)–(P4), a counterexample can be found using Mace4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 8 Rasiowa states this result for implicative algebras, the weakest algebra she considers in her text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Since all implication algebras are positive implication algebras, and all positive implication algebras are implicative algebras, we can specialise her result for the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 6 (F2) (a → b) → (a → (b → c)) → (a → c)) = 1 then for every implicative filter F in A and for every a ∈ A, the set Fa∗ = {x ∈ A : a → x ∈ F} is an implicative filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If, moreover, for all a, b ∈ A : a → (b → a) = 1, then Fa∗ is the least implicative filter containing F and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 18 (Rasiowa 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If (A, →) is a (positive) implication algebra, then for every implicative filter F and for every element a ∈ A the set Fa∗ = {x ∈ A : a → x ∈ F} is the least implicative filter containing F and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proposition 19 (Rasiowa 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' An implicative filter in an implication algebra is prime if and only if it is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Lemma 20 (Rasiowa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If F0 is an implicative filter in an implicative al- gebra A such that a0 /∈ F0 for some a0 ∈ A then there exists an irreducible implicative filter G such that F0 ⊂ G and a0 /∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Immediately, by Lemma 20 and Proposition 19 we have: Corollary 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If F is an implicative filter in an implicative algebra A such that a /∈ F for some a ∈ A then there exists a prime implicative filter G such that F ⊂ G and a /∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This next corollary we prove, as it is not found in any of the literature cited above and is required for the representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Corollary 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let F be an implicative filter of an implication algebra A = (A, →) such that a → b /∈ F for some a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Then there exists a prime implicative filter G : F ⊆ G such that a ∈ G and b /∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let Fa∗ be the implicative filter generated by the filter F and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Suppose that a → b /∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If b ∈ Fa∗, then we have a → b ∈ F by the definition of Fa∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This contradicts our assumption that a → b /∈ F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' hence b /∈ Fa∗, and applying Corollary 21 for Fa∗ and b we have a prime filter G such that Fa∗ ⊆ G and b /∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Clearly, a ∈ G and F ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ We have then, as an immediate corollary from Corollary 22 and Proposi- tion 19, the following: Corollary 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let F be an implicative filter of an implication algebra A = (A, →) such that a → b /∈ F for some a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Then there exists an irreducible implicative filter G : F ⊆ G such that a ∈ G and b /∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Finally, the culminating representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Rasiowa presents this for irreducible implicative filters [12], which given her equivalence result, one can also state using prime implicative filters, or maximal implicative filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Theorem 24 (Rasiowa 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For any implication algebra A = (A, →), there is a monomorphism h from A to (℘(X), →) of an arbitrary space X with |X| ≥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 7 From this it follows that every implication algebra is isomorphic to an impli- cation algebra of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Since the focus of the present paper is on representations, we note a corollary from this last result [1,2,12]: Corollary 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For any implication algebra A, if A is finite, then A has a finite representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let A be a finite implication algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Then by Theorem 24 A is monomor- phic to the algebra A′ under h, where A′ = (℘(X), →) , an implication algebra of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Now if |X| = A then |℘(X)| = 2|A|, and thus finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' That means h(A), the subalgebra of A′ under h, is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' So we have h(A) is a finite implication algebra (induced by h and A), and hence h is a finite representation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ Now, the focus of the rest of the paper revolves around the (finite) repre- sentation decision problem for implication semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' In the case of IA, this is defined as follows: Definition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The (finite) representation decision problem for implication al- gebras is a decision problem that takes an implication algebra with a (finite) carrier set as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The algebra is a yes instance if and only if it is (finitely) representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Closing this section, we note: Corollary 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' IA is finitely axiomatisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Corollary 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The (finite) representation problem for IA is decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 4 Undecidability Results for Implication Semigroups In this section we build on results from [8,11,7] to show undecidability of some decision problems for .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We begin by defining the representation and the finite representation decision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The (finite) representation decision problem for implication semi- groups is a decision problem that takes an implication semigroup with a finite carrier set as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The semigroup is a yes instance if and only if it is (finitely) representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' As we mention in Section 2, whether a structure is representable, also depends on our interpretation of the constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Here we show that the (finite) decision problem is undecidable in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='1 Representation Problem with Absolute Implication We begin by examining the case with absolute implication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' we require ⊤ = X × X for some (finite) base X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 8 Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' An implication monoid M = (M, 1′, →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) is an algebra where (M, →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) is an implication semigroup and 1′ is the monoidal identity for ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='. For some transitive and reflexive ⊤ ⊆ X × X, we define M(⊤) = (℘(⊤), 1′, →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) where →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' are proper relational implication and composition respectively and 1′ is the proper relational identity for X defined as 1′ = {(x, x) | x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' In [8, Section 4] a construction of a Boolean monoid from a square cancellative partial group G is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Its implication monoid reduct is denoted M(G) = (M, 1′, →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By [8, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='1, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='2] M(G) is representable (over a finite base) if and only if G embeds into a (finite) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' From the fact that both the group and the finite group embedding problems are undecidable [3] for finite structures it follows that the (finite) representation decision problem is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus if we prove that the ISG reduct of M(G) is (finitely) representable if and only if M(G) is representable, we have shown that the (finite) representability is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The right to left implication is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' But we must examine the case where we relax the requirement where we represent 1′ as the true relational identity, and show that this is still sufficient for the structure to remain (finitely) representable with 1′ taken as the true relational identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Suppose we have an embedding h from M(G) to S(⊤), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' an injective mapping that preserves →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', but not necessarily 1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If (x, y) ∈ h(1′) then (y, x) ∈ h(1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Suppose (y, x) ̸∈ h(1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' That means that (y, x) ∈ h(1′ → 0) = h(1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By composition of (x, y) ∈ h(1′) and (y, x) ∈ h(1′) we get that (x, x) ∈ h(1′) and by composing that with (x, y) ∈ h(1′) we have that (x, y) ∈ h(1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' As (x, y) ∈ h(1′) and (x, y) ∈ h(1′) = h(1′ → 0), we also have (x, y) ∈ h(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By a series of compositions we also get that (y, x) ∈ h(0) and because 0 ≤ 1′ we also get (y, x) ∈ h(1′) and we’ve reached a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' h(1′) is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By Lemma 31 we have that h(1′) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Furthermore, since all (x, x) ∈ h(⊤) there must exist a z witnessing 1′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊤ = ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus (x, z) ∈ h(1′) and (z, x) ∈ h(1′) and we compose that to get (x, x) ∈ h(1′), so h(1′) is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Finally, as 1′ = 1′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 1′ we also have that h(1′) is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For all x, x′, y, y′ ∈ X where (x′, x), (y, y′) ∈ h(1′) we have for all a ∈ M(G) that (x, y) ∈ h(a) ⇔ (x′, y′) ∈ h(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If (x, y) ∈ h(a) we have (x, y′) ∈ h(a) by (x′, x), (y, y′) ∈ h(1′) and the composition of 1′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 1′ = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By Lemma 31, we also have (x, x′), (y′, y) ∈ h(1′) so similarly if (x, y′) ∈ h(a) then (x, y) ∈ h(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The (finite) representation decision problem for ISG is undecid- able when → is interpreted as absolute implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' As h(1′) is an equivalence relation by Lemma 32, so we can define h′ : M(G) → X/h(1′) where h′(a) = {([x]h(1′), [y]h(1′)) | (x, y) ∈ h(a)} and show that h′ is indeed an embedding of M(G) into M(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By Lemma 33, we know that if (x, y) ∈ h(a) then for any x′ ∈ [x]h(1′), y′ ∈ [y]h(1′) we have (x′, y′) ∈ h(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Take any a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Then there exists (x, y) ∈ h(a) \\ h(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' From this follows ([x]h(1′), [y]h(1′)) ∈ h′(a) and if it were the case that ([x]h(1′), [y]h(1′)) ∈ h′(b) that would mean that there exist some (x′, y′) ∈ h(b) with (x, x′) ∈ h(1′) and (y′, y) ∈ h(1′) and that would also means that (x, y) ∈ h(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus h′ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Every composition is witnessed by the equivalence class of the witness for the composition in h and if (x, y) ∈ h(a) and (y′, z) ∈ h(b) with y′ ∈ [y]h(1′) we also have (y, y′) ∈ h(1′) and thus we have the composition (x, z) ∈ h(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 1′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b) = h(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus h′ represents ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Finally 1′ is represented correctly as a pair of equivalence classes is in h′(1′) if and only if they are the same equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus we have shown that if we have an embedding of M(G) into S(X × X) then we also have an embedding of M(G) into M(X′ ×X′) where X′ = X/h(1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Furthermore if X is finite, so is X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Trivially if M(G) embeds into M(X × X) it also embeds into S(X × X) via the same embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This, together with the results presented in [8] shows that the (finite) representation decision problem for ISG is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='2 Representation Problem with Relative Implication Now we show the same result for relative implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' A Boolean semigroup is a tuple B = (B, 0, 1, −, +, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) is an algebraic structure where S is a carrier set (i) (B, 0, 1, −, +) is a Boolean algebra (ii) (B, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) is a semigroup (iii) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' is additive over + (iv) 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' a = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 0 = 0 Similarly to , we denote the class of Boolean semigroups BSG and we say that a Boolean semigroup is representable if and only for some transitive ⊤ ⊆ X × X it embeds into B(⊤) = (℘(X × X), ∅, ⊤, −, +, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) where −a is interpreted as proper Boolean negation ⊤ \\ a, + is interpreted as proper Boolean join ∪ and ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' is interpreted as proper relational composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The (finite) representation problem for Boolean semigroups is defined analo- gous to that for implication semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' [7, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='2] shows that the repre- sentation problem for Boolean semigroups is undecidable and [11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='5] shows that the finite representation problem for Boolean semigroups is undecid- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' From this we show that the (finite) representation problem for implication semigroups is also undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 10 Note that the above results require an operation · to be defined in the signa- ture, but much like ⊤ in IA, · is term definable for BSG as a · b = −(−a + −b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let S = (S, →, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ) be an implication semigroup that contains some element 0 such that for all a ∈ S we have 0 ≤ a and 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' a = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If S is representable via some representation h then there exists a representation h′ of S where h′(0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If h is defined over a finite base, so is h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let h(S) be the proper structure defined by h for some ⊤ ⊆ X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' As h is a representation, there exists for every pair a ̸≤ b ∈ S a discriminator pair (ι, o) ∈ ⊤ such that (ι, o) ∈ h(a) \\ h(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Define Xι,o as Xι,o = � x ∈ X | � x = ι ∨ (ι, x) ∈ ⊤ � ∧ � y = o ∨ (y, o) ∈ ⊤ �� ⊤ι,o as ⊤ ∩ (Xι,o × Xι,o), and a mapping hι,o : S → S(⊤ι,o) where hι,o(a) = h(a) ∩ ⊤ι,o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' First observe that hι,o(0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Suppose that there was a pair (x, y) ∈ hι,o(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If x = ι, we have (ι, y) ∈ hι,o(0), else (ι, x) ∈ h(1) = ⊤ and thus (ι, y) ∈ h(0) since 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 0 = 0 and h preserves composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Similarly if y = o we get (ι, o) ∈ h(0), else by composing (ι, y) ∈ h(0) with (y, z) ∈ h(1) we get (ι, o) ∈ h(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Since b ≥ 0 that would mean (ι, o) ∈ h(b) that contradicts the fact that (ι, o) is a discriminator pair for a ̸≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Now let us check that hι,o preserves composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Suppose (x, y) ∈ hι,o(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This means that there exists z ∈ X such that (x, z) ∈ h(a) and (y, z) ∈ h(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' If x = ι, we trivially have (x, ι) ∈ h(1) = ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Else, by composing (ι, x) ∈ h(1) and (x, y) ∈ h(a) we get (ι, y) ∈ h(1) = ⊤ as 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' a ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Similarly (y, o) ∈ ⊤ and thus y ∈ Xι,o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus we have (x, z) ∈ hι,o(a), (y, z) ∈ hι,o(b) and we have shown hι,o(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b) ⊆ hι,o(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' hι,o(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The fact that hι,o(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' b) ⊇ hι,o(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' hι,o(b) follows from (x, y), (y, z) ∈ ⊤ι,o then x, z ∈ Xι,o and we have (x, z) ∈ ⊤ι,o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus hι,o preserves composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We have hι,o(a) ̸= hι,o(b) as (ι, o) ∈ ⊤ι,o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The operation → is preserved by hι,o as for all (x, y) ∈ ⊤ι,o it holds (x, y) ∈ h(a) ⇐⇒ (x, y) ∈ hι,o(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Finally, |Xι,o| ≤ |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Thus we conclude that h{ι, o} is a homomorphism for S that discriminates the pair a ̸≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Now let us pick for every a ̸≤ b a δ(a, b) = (ι, o) such that (ι, o) ∈ ⊤ is a discriminator pair for a ̸≤ b and let ˙∪ denote a disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' A mapping h′ : S → ℘ � ˙� a̸≤b∈S⊤δ(a,b) � h′(c) = ˙� a̸≤b∈Shδ(a,b)(c) still represents ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' , → correctly, discriminates all pairs a ̸≤ b (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' is injective), which makes it a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Furthermore, h′(0) = ∅ and the size of its base is bounded by |S|2|X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ 11 Theorem 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The (finite) representation decision problem for implication semi- groups is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We show this by proving that B ∈ BSG is representable if and only if its ⟨→, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⟩-reduct S is representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The left to right implication is trivial as a → b ∈ S is term-definable as (−a) + b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' For the right to left implication, +, 1 are term-definable in ⟨→, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⟩ (see Proposition 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' By Lemma 36 the ⟨→, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⟩- reduct of B is representable if and only if it has a representation where h(0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' See how in that representation a → 0 (corresponding to −a + 0 = −a in the Boolean semigroup) is represented as ⊤ \\ a ∪ ∅ = ⊤ \\ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' As the (finite) representation decision problem is undecidable for Boolean semigroups, we conclude the same for implication semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ⊓⊔ 5 Problems In this section we outline some open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' It follows from the results in Section 4 that the class of representable implication semigroups is not finitely axiomatisable, nor does it have the finite representation property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' not ev- ery finite representable structure in the class is finitely representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' However, another decision problem of interest is posed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Problem 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is membership in the equational theory generated by the class of representable implication semigroups decidable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The reader can see that if we add the bottom element 0 to the signature, the undecidability follows from the undecidability of the equational theory of the Boolean semigroups as described in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This is because all negations of terms −t can be rewritten as t → 0 and all joins t + t′ as (t → t′) → t′ where t, t′ are terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The problem remains open for the class of representable implication semi- groups without the bottom element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' One of the possible ways to prove undecid- ability is by using discriminator terms, defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' A discriminator term d(a, b, c) is a term defined in terms of elements of algebra a, b, c such that for all representable algebras d(a, b, c) = c if a = b and a otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Although the existence of discriminator terms is not a guarantee for the undecidability of the equational theory membership decision problem, it is an interesting open question in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Problem 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is it possible to define a discriminator term in the language of implication semigroups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' It is well known that subreducts of representable relation algebras form qua- sivarieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' As such, the class of implication semigroups can be characterised by quasiequations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' However, some open questions about the equational theory generated by the class of representable implication semigroups are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 12 Problem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is the class of representable implication semigroups a variety?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Problem 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is the equational theory generated by the class of representable implication semigroups finitely axiomatisable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' We continue by looking at the alternative interpretations of → operation for binary relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' An interesting example, as mentioned in the introduction section is that of a weakening relation defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Definition 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Let P = (X, ≤) be a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' R ⊆ X × X is a weakening relation if and only if ≤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' ≤ ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' In the context of the weakening relation algebras as described in [4], the → operation can be given in first order terms as R → S = {(x, y) | ∀x′, y′ : ((x′ ≤ x ∧ y ≤ y′ ∧ (x′, y′) ∈ R) ⇒ (x′, y′) ∈ S)} where R, S are weakening relations over a poset P = (X, ≤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' This interpretation of the → operation gives rise to the class of representable weakening implication semigroups, for which the following properties remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Problem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is the (finite) representation decision problem decidable for the class of representable weakening implication semigroups?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is the class finitely axiomatisable and does it have the finite representation property?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Problem 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is the class of representable weakening implication semigroups a (discriminator) variety?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Is the equational theory generated by the class finitely axiomatisable/decidable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Finally, we note that it can be checked that all results presented in this paper can be generalised to the dual operation ← by presenting dual axioms for the class of implication algebras and defining negation as 0 ← a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Abott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Implicational algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Bulletin math´ematique de la Soci´et´e des Sci- ences Math´ematiques de la R´epublique Socialiste de Roumanie 11 (59)(1), 3–23 (1967), http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='org/stable/43679502 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Diego, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Sobre ´algebras de Hilbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Notas de l´ogica matem´atica, Instituto de Matem´atica, Universidad Nacional del Sur (1965), https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='uk/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='id=AfcSAQAAMAAJ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Evans, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Embeddability and the word problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Journal of the London Mathe- matical Society 1(1), 76–80 (1953) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Galatos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Jipsen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': The structure of generalized bi-algebras and weakening relation algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Algebra universalis 81(3), 1–35 (2020) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Galatos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Jipsen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Weakening relation algebras and fl2-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' In: Interna- tional Conference on Relational and Algebraic Methods in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 117–133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Springer (2020) 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Hirsch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Hodkinson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Relation algebras by games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Elsevier (2002) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Hirsch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Hodkinson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Jackson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Undecidability of algebras of binary rela- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' In: Hajnal Andr´eka and Istv´an N´emeti on Unity of Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' 267–287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Springer (2021) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Hirsch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Jackson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Undecidability of representability as binary relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' The Journal of Symbolic Logic 77(4), 1211–1244 (2012) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' J´onsson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Tsinakis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Relation algebras as residuated boolean algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' algebra universalis 30, 469–478 (1993) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' L¨owenheim, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': ¨Uber m¨oglichkeiten im relativkalk¨ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Mathematische Annalen 76, 447–470 (1915) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Neuzerling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Algebra universalis 76(4), 431–443 (2016) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Rasiowa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': An Algebraic Approach to Non-Classical Logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Amsterdam, Nether- lands: Warszawa, Pwn - Polish Scientific Publishers (1974) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Rogozin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=': The finite representation property for some reducts of re- lation algebras (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='13079, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='org/abs/2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='13079 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Tarski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=', Givant, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' : A Formalization of Set Theory Without Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' Amer- ican Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} +page_content=' (1987) 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E0T4oBgHgl3EQfVgDO/content/2301.02266v1.pdf'} diff --git a/VNAzT4oBgHgl3EQfX_zt/content/tmp_files/2301.01329v1.pdf.txt b/VNAzT4oBgHgl3EQfX_zt/content/tmp_files/2301.01329v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d050bf8cef9ab550b199cdbd583df91bea659dc --- /dev/null +++ b/VNAzT4oBgHgl3EQfX_zt/content/tmp_files/2301.01329v1.pdf.txt @@ -0,0 +1,553 @@ +arXiv:2301.01329v1 [gr-qc] 3 Jan 2023 +Stellar Instability from Parametric Resonance +Rodrigo Maier∗, +Departamento de F´ısica Te´orica, Instituto de F´ısica, +Universidade do Estado do Rio de Janeiro, +Rua S˜ao Francisco Xavier 524, Maracan˜a, +CEP20550-900, Rio de Janeiro, Brazil +(Dated: January 5, 2023) +In this paper we examine the stability of stellar configurations in which the interior solution is +described by a closed FLRW geometry sourced with a charged pressureless fluid and radiation. An +interacting vacuum component and a conformally coupled massive scalar field are also included. +Given a simple factor for the energy transfer between the pressureless fluid and the vacuum com- +ponent we obtain bounded interior oscillatory solutions. We show that in proper domains of the +parameter space the interior dynamics is highly unstable so that the break of the KAM tori leads to +a disruptive ejection of mass. For such configurations the interior solution asymptotically matches +an exterior Reissner-Nordstr¨om-de Sitter spacetime. +I. +INTRODUCTION +The issue of an interacting dark energy in deep IR as in high UV has been a subject of interest over the last years. +In fact, from a cosmological point of view it has been shown that apart from relieving some cosmological tensions of +observational data[1]-[6], an interacting vacuum component may give rise to nonsingular models[7]. In the framework +of black hole physics, it has been shown that Yukawa black holes[8] or nonsingular Reissner-Nordstr¨om-de Sitter +spacetimes[9] may be obtained if one considers proper interacting vacuum components. In this context, a question +which naturally arises is what would be the consequences of assuming an interacting dark energy in gravitational +collapse processes which may generate stable/unstable stars. +The problem of the gravitational collapse in General Relativity has been object of several important works. In the +realm of black hole formation, the seminal paper due to Oppenheimer and Snyder[10] furnished an interior solution for +a Schwarzschild spacetime assuming the collapse of a spherically symmetric cloud of nonrelativistic and pressureless +particles. As an extension of this model, Vaidya made the inclusion of radiation in the exterior spacetime[11]. On +the other hand, excluding the presence of radiadion, Misner and Sharp made important progress considering the +gravitational collapse of a matter distribution more realistic than dust[12]. In a sequel of their work, a simplified heat- +transfer process was introduced engendering an outward flux of neutrinos [13]. Further analysis due to Chan et al. +(see [14] and references therein) have studied the case of anisotropic gravitational collapse models. Recently, a proper +examination of the stability of neutron stars with a more realistic equation of state was performed in [15]. In this +context, it is well known that the stellar structure in hydrostatic equilibrium is governed by the Tolman-Oppenheimer- +Volkoff (TOV) equations. For the case of a non-perfect fluid, TOV equations were extended[16] in order to include +pressure anisotropy. +In this framework of stellar structure and evolution, two typical behaviours have deserved +attention in the last decades. It is understood that internal mechanical forces, thermal instabilities or turbulent +motions may drive oscillating internal waves which depend on the star interior properties[17]. The propagation of +such waves produces an oscillating power spectrum of the modes which may furnish important information about the +stellar structure[18]. On the other hand, luminous stellar explosions regarded as supernovae (SNe) refer to the final +stage of massive stars in which the progenitor object collapses either to a neutron star, a black hole or is completely +destroyed. Although the observational behaviour of these events is well understood[19–21], a proper explanation about +the mechanisms that trigger SNe ejection of mass remains uncertain. In this paper we propose a simple inceptive +model in which a conformally coupled massive scalar field may account to such a behaviour. +We organize the paper as follows. In Section 2 we present the interior dynamics of a Friedmann star in which +the matter content is given by a charged pressureless fluid and radiation. We show how bounded interior oscillatory +solutions may be obtained once an interacting vacuum component and a conformally coupled massive scalar field +are also assumed. In Section 3 we discuss the exterior spacetime which asymptotically corresponds to a Reissner- +Nordstr¨om-de Sitter geometry for proper configurations. Finally, in Section 4 we leave our final remarks. +∗ rodrigo.maier@uerj.br + +2 +II. +INTERIOR DYNAMICS +We start by considering the Einstein field equations +Gµν = κ2(Tµν − VIgµν) +(1) +where Gµν is the Einstein tensor and κ2 ≡ 8πG. The energy-momentum tensor Tµν is constructed assuming that the +matter content of the model is given by a charged dust fluid, radiation and a conformally coupled massive scalar field. +That is: +Tµν = (d)Tµν + (γ)Tµν + (φ)Tµν, +(2) +where (d)Tµν and (γ)Tµν stand for the energy-momentum tensors of the charged dust fluid and radiation, respectively. +The former can be written[22] as +(d)Tµν = ρduµuν + σMµν, +(3) +where σ is a negative coupling constant (σ ∝ −1/4π) and +Mµν = F +α +µ +Fνα − 1 +4gµνFαβF αβ, +(4) +with Fµν ≡ ∇νAµ − ∇µAν as the Faraday tensor. The radiation component, on the other hand, reads +(γ)Tµν = ργ +3 (4uµuν + gµν). +(5) +Taking into account the lagrangian for a conformally coupled massive scalar field +Lφ = −1 +2 +� +φαφβgαβ + m2φ2 + 1 +6Rφ2� +, +(6) +its respective energy-momentum tensor is given by +(φ)Tµν = φ,µφ,ν + Lφgµν + 1 +6 +� +□(φ2)gµν + Rµνφ2 − (φ2),µ,ν +� +. +(7) +Finally, we denote by VI a vacuum component which interacts with the charged dust fluid. +Such interaction is +described by an energy-momentum 4-vector Qν so that the Bianchi identities furnish +∇µ(d)T µ +ν = Qν = ∇νVI. +(8) +Let us now consider a FLRW interior geometry in comoving coordinates (r, θ, ϕ) given by +ds2 = −dt2 + a2(t) +� +dr2 +1 − kr2 + r2(dθ2 + sin2 θdϕ2) +� +, +(9) +where t is the time coordinate, a(t) the scale factor and k the 3-curvature. +From the conservation equations +∇µ(γ)T µ +ν = 0 we then obtain +ργ = Eγ +a4 , +(10) +where Eγ is a positive constant of integration. On the other hand, the equation of motion for an homogeneous scalar +field reads +∇µ(φ)T µ +ν = 0 → ¨φ + 3H ˙φ + +� k +a2 + 2H2 + ˙H + m2� +φ = 0. +(11) +In the case of spherical symmetry, the only independent nonvanishing component of Fµν is Ftr = F(t, r). Therefore, +the Einstein field equations (1) can be written as +H2 + k +a2 = κ2 +3 +� +ρd + ργ + VI + σ(1 − kr2)F 2(t, r) +2a2 ++ +˙φ2 +2 + H ˙φφ + φ2 +2 +� +H2 + k +a2 + m2�� +, +(12) +˙H + 3H2 +2 ++ +k +2a2 = κ2 +2 +� +VI − ργ +3 + σ(1 − kr2)F 2(t, r) +2a2 +− +˙φ2 +6 + 1 +2(H2 + m2)φ2 + 1 +3 +� +¨φφ + 2Hφ ˙φ + +� +˙H + +k +2a2 +� +φ2�� +. +(13) + +3 +Imposing homogeneous energy densities together with an homogeneous vacuum component, we end up with the +condition +F(t, r) = +N(t) +√ +1 − kr2 . +(14) +Substituting (3) in (8) we then obtain +∇µ +(d)T µ +ν = ∇µ(ρduµuν) + σ∇µM µ +ν = Qν. +(15) +At this stage one may assume that Qν = Qν + JαF +α +ν +where Jα = ǫuα is a 4-current with ǫ being the density of +electric charge. Employing the Maxwell equations we then obtain +σ∇µM µ +ν = σ∇µ(F µ +α)F +α +ν += JαF +α +ν +, +(16) +Making ν = t in (16) we obtain +N(t) = N0 +a , +(17) +where N0 is a constant. On the other hand, for ν = r, equation (16) furnishes +ǫ(t, r) = 2σN0 +�√ +1 − kr2 +ra3 +� +. +(18) +As the physical radius R of the matter distribution is proportional to the scale factor a for a constant comoving radius +r, from the above we note that ǫ scales as R−3, as one should expect. Nevertheless, at a first glance one might identify +a problem in the above charge density profile since it diverges as r → 0. However, given the spherical symmetry of +such matter distribution one should expect that the overall charge should be spread out only in a small neighbourhood +of the surface. In a more realistic model this interior solution could be interpreted as a thin Friedmann layer in a +small neighbourhood of the surface to be matched with a metric which describes a more involved stellar core – an +issue to be addressed in a future work. +To proceed, in order to assure that the interior matter distribution bounces when a minimum 3-volume is reached, +we shall now assume that the energy-momentum 4-vector Qν has the following covariant prescription[7, 9] +Qµ = 4 +3(V0 − VI)(∇αuα)uµ. +(19) +In the above, V0 is a positive constant. Substituting (19) in (15) we then obtain +∇µ(ρduµuν) = ∇µVI = 4 +3(VI − V0)(∇αuα)uµ. +(20) +A straightforward integration of the differential equations (20) furnishes +VI = V0 + λ +a4 , +(21) +ρd = Ed +a3 − 4λ +a4 . +(22) +where λ and Ed are positive constants of integration. Therefore, Einstein equations (12) and (13) read +H2 + k +a2 = κ2 +3 +�Ed +a3 + +�2Eγ + N 2 +0 σ − λ +2a4 +� ++ V0 + +˙φ2 +2 + H ˙φφ + φ2 +2 +� +H2 + k +a2 + m2�� +, +(23) +˙H + 3H2 +2 ++ +k +2a2 = κ2 +2 +� +V0 − +�2Eγ − 3N 2 +0σ − 6λ +6a4 +� +− +˙φ2 +6 + 1 +2(H2 + m2)φ2 + 1 +3 +� +¨φφ + 2Hφ ˙φ + +� +˙H + +k +2a2 +� +φ2�� +. +(24) +It is now useful to rewrite the first integral (23) and the equation of motion of the scalar field (11) in terms of the +so-called conformal time dη = a−1dt together with the rescaling ψ ≡ κaφ. In this case, the Friedmann equation (23) +for a closed metric reads +3a′2 + W(a) = κ2(Eγ − 3V1) + 1 +2[ψ′2 + (1 + m2a2)ψ2], +(25) + +4 +FIG. 1: The potential W (a) for a closed model with k = 1. In the above we have fixed κ = 1 and Ed = 1.5. It is numerically +shown that the potential W (a) has two local extrema for V0 = 2.0 (black curve) and V0 = 2.5 (gray curve). For V0 = 8k3/(κ3Ed)2 +(dashed curve) – the upper limit for V0 – there are no extrema for the potential. Fixing Eγ = 0.01, V1 = 0.052, pψ0 = 0 and +ψ0 = 0.15 we obtain E = −0.13475 (red curve above). +where primes denote derivatives with respect to conformal time, V1 ≡ λ − σN 2 +0 /6 and +W(a) = 3a2 − κ2� +V0a4 + Eda +� +. +(26) +The equation of motion of the scalar field ψ, on the other hand is given by +ψ′′ + (1 + m2a2)ψ = 0. +(27) +We are now in a position to define a dynamical system equivalent to equations (25)–(27): +ψ′ = −pψ, +(28) +a′ = pa/6, +(29) +p′ +ψ = (1 + m2a2)ψ, +(30) +p′ +a = −dW +da + m2aψ2. +(31) +In the above, pψ and pa are the canonical momenta connected to the scalar field ψ and the scale factor a, respectively. +In fact, with the above definitions it is easy to show that (25) turns into a Hamiltonian first integral given by +H = p2 +a +12 + W(a) − κ2(Eγ − 3V1) − 1 +2[p2 +ψ + (1 + m2a2)ψ2] = 0. +(32) +For m = 0 the dynamical system (28)–(31) is separable, hence integrable. In fact, from equations (28) and (30) we +have a first integral E0ψ = (p2 +ψ + ψ2)/2 which is a constant of motion. It can be shown that the potential W(a) has at +most two local extrema (one local minimum a− and one local maximum a+) for a > 0 – as long as V0 < 8k3/(κ3Ed)2. +Considering the surfaces with energy E = κ2(Eγ − 3V1) + E0ψ < 0 so that W(a−) < E < W(a+) we see that the +region 0 < a < a+ is foliated by 2-tori S1 × S1 which are the topological product of periodic orbits of the separable +sectors (a, pa) and (ψ, pψ). Such 2-tori trap the dynamics in a finite region of the phase space and E is a conserved +quantity for those orbits. In the sector (ψ, pψ) orbits have frequency νψ = 1/2π while in the sector (a, pa) +1 +νa += 2 +� β2 +β1 +� +3 +E − W(a)da. +(33) +Here, β1 and β2 are the two smaller real roots of E − W(a). In Fig. 1 we show several plots of W(a). +A relevant question which now arises is whether such tori “survive” once integrability is broken due to a nonvanishing +mass m for the scalar field. In fact, assuming sufficiently small initial conditions (ψ0, pψ0), equation (27) may be +rewritten as +ψ′′ + (1 + m2a2 +0(η))ψ = 0, +(34) + +5 +FIG. 2: The behaviour of the scale factor a(η). For η ≃ 343 the scale factor diverges triggering a disruptive ejection of mass. +where a0(η) is the background solution for the scale factor of the integrable dynamics with m = 0. Defining ˜νψ as the +frequency in the sector (ψ, pψ) given by (34), a resonant behaviour will occur when the ratio R ≡ νa/˜νψ is a rational +number. Expanding a0(η) in (34), one can show that +˜νψ ≃ 1 +2π +� +1 + 1 +2 +� +m(β1 + β2) +2 +�2 +− 1 +8 +� +m(β1 + β2) +2 +�4� +. +(35) +However, as the dynamics evolves the amplitude of the scalar field may grow so that the solution of the integrable case +a0(η) is no longer a good approximation to be introduced in (34). This process may lead the dynamics into a more +unstable behavior, with the amplification of the resonance and the break of the KAM tori[23–25]. To analytically +show this behavior, one may expand the non-integrable term of (32) in the action-angle variables (Θψ = ˜νψη, Jψ, Θa = +νaη, Ja). That is, +1 +2m2a2 +0(η)ψ2(η) = 1 +2m2J (0) +a +J (0) +ψ +� +n +[cn cos(2nπΘa)] cos(4πΘψ) +where ψ(η) is an approximate solution of (34) and cn are constant coefficients. The superior indexes in Ja and Jψ +denote that these are the action variables for the integrable case. The Hamilton equation for Ja can then be integrated +furnishing in its first approximation +Ja ≃ 1 +2m2J (0) +a +J (0) +ψ +� +n +cn +2πn˜νψ +�cos(2πnΘa − 4πΘψ) +νa/˜νψ − 2/n ++ cos(2πnΘa + 4πΘψ) +νa/˜νψ + 2/n +� +. +From the above we see that the dominant resonance terms are those for which +νa +˜νψ +≃ 2 +n. +(36) +When such resonances occur one can eventually obtain a loss of stability so that a → +∞ triggering a disruptive +ejection of mass. +To illustrate the above mentioned behaviour, let us consider the proper domain of the parameters of Fig. 1 with +V0 = 2.0 (black curve). We also fix the initial conditions pψ0 = 0, ψ0 = 0.15 together with Eγ = 0.01, V1 = 0.052 so +that E = −0.13475. The initial condition for the scale factor, a0 ≃ 0.117, is obtained from the first positive root of +W(a) − E. For n = 3, from (33), (35) and (36) we obtain m ≃ 6.39. Feeding the Hamiltonian constraint (32) with +such parameters and initial condition we obtain the remaining initial condition pa0. Evolving the dynamical system +imposing that the hamiltonian constraint is conserved, one can numerically show that the scale factor diverges as +η ≃ 343 triggering a disruptive ejection of mass. In Fig. 2 we illustrate this behaviour. +It is worth mentioning that there is a whole domain in the parametric space (V1, m) in which this unstable behaviour +is manifest. In Fig. 3 we illustrate some examples of such domains for n = 2 (top left panel), n = 3 (top right panel) +and n = 4 (bottom panel). Apart from V1, m and pa0, we used the same parameters and initial conditions considered +in Fig. 2. In Fig. 3 each dark solid line was obtained from our analytical procedure due to (33), (35) and (36) in +order to find the respective resonances. It can be numerically shown that the dynamics is highly unstable once one + +6 +FIG. 3: Resonance domains for n = 2 (top left panel), n = 3 (top right panel) and n = 4 (bottom panel). Here we have fixed +κ = 1, V0 = 2.0, Ed = 1.5, Eγ = 0.01 and initial conditions pψ0 = 0, ψ0 = 0.15 and a0 = 0.117. In each dark solid line – which +were obtained from our analytical procedure due to (33), (35) and (36) – the dynamics is highly unstable. There domain below +such lines – shaded areas – is also resonant so that after a finite amount of time the scale factor diverges triggering a disruptive +ejection of mass. Below the gray lines we restore the domain of parametric stability analogous to that of the integrable case. +considers the parameters/initial conditions connected to these lines. There is also a whole domain below such lines +– shaded areas – were the resonance mechanism is manifest so that after a finite amount of time the scale factor +diverges triggering a disruptive ejection of mass. Below the gray lines we restore the domain of parametric stability +analogous to that of the integrable case. It is also worth noting that one may obtain resonant configurations above +the dark solid lines. However, as our approximation is no longer valid for large masses – above the dark solid lines – +one may safely regard the resonance domain as the shaded portions together with the black and gray lines. +III. +THE EXTERIOR SPACETIME +We now consider the matching of the interior geometry with the exterior spacetime. To this end let us assume that +(¯t, ¯r, ¯θ, ¯φ) are new coordinates defined by +¯t := χ(Γ(t, ¯r)), +¯r = ar, +¯θ = θ, +¯ϕ = ϕ. +(37) +The standard procedure to match the interior geometry with the exterior metric can be found in [9, 10]. Following +the similar notation, we impose that the matching should be performed at the surface r = γ = constant, so that +g¯t¯t = − 1 +g¯r¯r +��� +r=γ ≡ − +� +1 − 2GM +¯r ++ β2 +q +¯r2 − κ2 +3 ¯r2� +V0 + L(η) +���� +r=γ +� +, +where +L(η) = +1 +2κ2a4 +� +p2 +ψ + (1 + m2a2)ψ2� +, +(38) +M = 4πγ3Ed/3, β2 +q = q2G/4πǫ0 and q is the overall charge of the matter distribution[9]. At this stage, it is important +to draw the reader’s attention to a word to note. Let us consider the internal oscillating bounded behaviour of the + +7 +FIG. 4: The behaviour of L(η) as a function of the conformal time. Here we see that L(η) vanishes as a → +∞ at η ≃ 343. +matter distribution – with η < 343 as illustrated in Fig. 2, for instance. During this period the internal oscillating +charged matter is responsible for an ejection of radiation making the exterior spacetime stationary. In fact, such +external radiation is needed in order to support the interior dynamics of the scalar field – through the L(η) function. +However, once the scale factor diverges – for η ≃ 343, for example – there is a disruptive ejection of mass and L(η) +vanishes as a → +∞. Such behaviour is illustrated in Fig. 4. In this sense, the interior solution asymptotically +matches an exterior geometry given by the Reissner-Nordstr¨om-de Sitter spacetime and the exterior metric reads +ds2 = −F(¯r)d¯t2 + +1 +F(¯r)d¯r2 + ¯r2(d¯θ2 + sin2 ¯θdφ2), +(39) +where +F(¯r) ≡ +� +1 − 2GM +¯r ++ β2 +q +¯r2 − κ2V0 +3 +¯r2� +. +(40) +It is worth to note from (40) that V0 plays the same role of a cosmological constant Λ. Therefore, assuming that κ2V0 +is sufficiently small it can be easily seen from (32) that d2a/dt2 ≃ 0 as a → +∞. Therefore, in the asymptotic regime +the ejection of radiation completely ceases making the exterior spacetime static as one should expect. +IV. +FINAL REMARKS +In this paper we propose a first analysis in which stellar stability may be connected to a conformally coupled +massive scalar field. In order to assure that the matter distribution bounces when a minimum 3-volume is reached, +we assume that the internal pressureless matter interacts with vacuum component only through a covariant energy +exchange[7, 9]. In this case, bounded interior oscillatory solutions are obtained. It is worth noting that the dynamics +presented in this paper exhibit similar patterns as several bouncing cosmologies[23, 24]. In this sense, the interaction +assumed in this paper plays just an effective role in order to make the dynamics nonsingular. Similar results – i.e. the +break of the KAM tori leading to a disruptive ejection of mass – should be obtained for different bouncing models. +The obtention of the exterior stationary solution is a rather involved task which we intend to study in a further +publication. As mentioned above, such an exterior stationary metric is needed to support the interior scalar field +dynamics. For the case of stable configurations, the perpetual bounded oscillating interior spacetime could in principle +be matched with some sort of Vaidya spacetime[26]. For the case of unstable configurations on the other hand, the +same procedure could be performed considering a Vaidya layer before its extension to the Reissner-Nordst¨om-de Sitter +exterior solution[27]. +As a future perspective we also intend to consider the results of the present paper in order to furnish more realistic +scenarios in which the oscillating scale factor may account for stellar internal waves together with a disruptive ejection +of mass. The first step in this direction is a full examination of the resonance domains furnishing constrains of more +realistic parameters such as stellar masses. Another issue to be tackled is how the results shown in this paper fit in +several models as neutrino heating, thermonuclear burning and magnetohydrodynamic instabilities which account to +mass ejection in SNe (see [28] and references therein). + +8 +References +[1] V. Salvatelli et al., Phys. Rev. Lett., 113, 181301 (2014). +[2] Y. Wang et al., Phys. Rev. D, 92, 103005 (2015). +[3] G.-B. Zhao, et al., Nature Astronomy, 1, 627 (2017). +[4] J. Sol`a, A. G´omez-Valent, J. de Cruz P´erez, Int. J. of Mod. Phys., A32, 1730014 (2017). +[5] E. Di Valentino, A. Melchiorri, O. Mena, Phys. Rev. D, 96, 043503 (2017). +[6] S. Kumar, R. C. Nunes, Phys. Rev., D96, 103511 (2017). +[7] M. Bruni, R. Maier and D. Wands, Phys. Rev. D 105, no.6, 063532 (2022). +[8] R. Maier, Class. Quant. Grav. 39, 155008 (2022). +[9] R. Maier, Int. J. Mod. Phys. D 29, no.14, 2043023 (2020). +[10] J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455-459 (1939). +[11] P. Vaidya, Proc. Natl. Inst. Sci. India A 33, 264 (1951). +[12] C. W. Misner and D. H. Sharp, Phys. Rev. 136, B571-B576 (1964). +[13] C. W. Misner, Phys. Rev. 137, B1360-B1364 (1965). +[14] R. Chan et al., Monthly Notices of the Royal Astronomical Society, 265(3), 533–544 (1993). +[15] J. M. Z. Pretel and M. F. A. da Silva, Mon. Not. Roy. Astron. Soc. 495, 5027 (2020). +[16] R. Sharma and S. D. Maharaj, Mon. Not. Roy. Astron. Soc. 375, 1265-1268 (2007). +[17] R. Samadi, K. Belkacem and T. Sonoi, EAS Publications Series, Vol. 73–74, 111–191 (2015). +[18] R. A. Garcia, EAS Publications Series, Vol. 73–74, 193–259 (2015). +[19] C. Inserra, Nature Astron. 3, no.8, 697-705 (2019). +[20] S. W. Jha, K. Maguire and M. Sullivan, Nature Astron. 3, no.8, 706-716 (2019). +[21] M. Modjaz, C. P. Gutierrez and I. Arcavi, Nature Astron. 3, no.8, 717-724 (2019). +[22] P. A. Vickers, Ann. Inst. Henri Poincar´e, Vol. XVIII, n2, p. 137-146 (1973). +[23] R. Maier, I. D. Soares and E. V. Tonini, Phys. Rev. D 79, 023522 (2009). +[24] R. Maier, Class. Quant. Grav. 30, 115011 (2013). +[25] V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, 1989). +[26] V. A. Berezin et al., J. Exp. Theor. Phys. 124, no.3, 446-458 (2017). +[27] F. Fayos, M Merc`e Mart´ın-Prats, Jos´e M. M. Senovilla, Class. Quant. Grav 12(10):2565 (1995). +[28] H. T. Janka, Ann. Rev. Nucl. Part. Sci. 62, 407-451 (2012). + diff --git a/VNAzT4oBgHgl3EQfX_zt/content/tmp_files/load_file.txt b/VNAzT4oBgHgl3EQfX_zt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..604940f8e8d6cbc1bb73670e49ab9bd7bb546fea --- /dev/null +++ b/VNAzT4oBgHgl3EQfX_zt/content/tmp_files/load_file.txt @@ -0,0 +1,357 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf,len=356 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='01329v1 [gr-qc] 3 Jan 2023 Stellar Instability from Parametric Resonance Rodrigo Maier∗, Departamento de F´ısica Te´orica, Instituto de F´ısica, Universidade do Estado do Rio de Janeiro, Rua S˜ao Francisco Xavier 524, Maracan˜a, CEP20550-900, Rio de Janeiro, Brazil (Dated: January 5, 2023) In this paper we examine the stability of stellar configurations in which the interior solution is described by a closed FLRW geometry sourced with a charged pressureless fluid and radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' An interacting vacuum component and a conformally coupled massive scalar field are also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Given a simple factor for the energy transfer between the pressureless fluid and the vacuum com- ponent we obtain bounded interior oscillatory solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' We show that in proper domains of the parameter space the interior dynamics is highly unstable so that the break of the KAM tori leads to a disruptive ejection of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' For such configurations the interior solution asymptotically matches an exterior Reissner-Nordstr¨om-de Sitter spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' INTRODUCTION The issue of an interacting dark energy in deep IR as in high UV has been a subject of interest over the last years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In fact, from a cosmological point of view it has been shown that apart from relieving some cosmological tensions of observational data[1]-[6], an interacting vacuum component may give rise to nonsingular models[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In the framework of black hole physics, it has been shown that Yukawa black holes[8] or nonsingular Reissner-Nordstr¨om-de Sitter spacetimes[9] may be obtained if one considers proper interacting vacuum components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this context, a question which naturally arises is what would be the consequences of assuming an interacting dark energy in gravitational collapse processes which may generate stable/unstable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The problem of the gravitational collapse in General Relativity has been object of several important works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In the realm of black hole formation, the seminal paper due to Oppenheimer and Snyder[10] furnished an interior solution for a Schwarzschild spacetime assuming the collapse of a spherically symmetric cloud of nonrelativistic and pressureless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' As an extension of this model, Vaidya made the inclusion of radiation in the exterior spacetime[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' On the other hand, excluding the presence of radiadion, Misner and Sharp made important progress considering the gravitational collapse of a matter distribution more realistic than dust[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In a sequel of their work, a simplified heat- transfer process was introduced engendering an outward flux of neutrinos [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Further analysis due to Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (see [14] and references therein) have studied the case of anisotropic gravitational collapse models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Recently, a proper examination of the stability of neutron stars with a more realistic equation of state was performed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this context, it is well known that the stellar structure in hydrostatic equilibrium is governed by the Tolman-Oppenheimer- Volkoff (TOV) equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' For the case of a non-perfect fluid, TOV equations were extended[16] in order to include pressure anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this framework of stellar structure and evolution, two typical behaviours have deserved attention in the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' It is understood that internal mechanical forces, thermal instabilities or turbulent motions may drive oscillating internal waves which depend on the star interior properties[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The propagation of such waves produces an oscillating power spectrum of the modes which may furnish important information about the stellar structure[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' On the other hand, luminous stellar explosions regarded as supernovae (SNe) refer to the final stage of massive stars in which the progenitor object collapses either to a neutron star, a black hole or is completely destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Although the observational behaviour of these events is well understood[19–21], a proper explanation about the mechanisms that trigger SNe ejection of mass remains uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this paper we propose a simple inceptive model in which a conformally coupled massive scalar field may account to such a behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' We organize the paper as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In Section 2 we present the interior dynamics of a Friedmann star in which the matter content is given by a charged pressureless fluid and radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' We show how bounded interior oscillatory solutions may be obtained once an interacting vacuum component and a conformally coupled massive scalar field are also assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In Section 3 we discuss the exterior spacetime which asymptotically corresponds to a Reissner- Nordstr¨om-de Sitter geometry for proper configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Finally, in Section 4 we leave our final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' ∗ rodrigo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='maier@uerj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='br 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' INTERIOR DYNAMICS We start by considering the Einstein field equations Gµν = κ2(Tµν − VIgµν) (1) where Gµν is the Einstein tensor and κ2 ≡ 8πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The energy-momentum tensor Tµν is constructed assuming that the matter content of the model is given by a charged dust fluid, radiation and a conformally coupled massive scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' That is: Tµν = (d)Tµν + (γ)Tµν + (φ)Tµν, (2) where (d)Tµν and (γ)Tµν stand for the energy-momentum tensors of the charged dust fluid and radiation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The former can be written[22] as (d)Tµν = ρduµuν + σMµν, (3) where σ is a negative coupling constant (σ ∝ −1/4π) and Mµν = F α µ Fνα − 1 4gµνFαβF αβ, (4) with Fµν ≡ ∇νAµ − ∇µAν as the Faraday tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The radiation component, on the other hand, reads (γ)Tµν = ργ 3 (4uµuν + gµν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (5) Taking into account the lagrangian for a conformally coupled massive scalar field Lφ = −1 2 � φαφβgαβ + m2φ2 + 1 6Rφ2� , (6) its respective energy-momentum tensor is given by (φ)Tµν = φ,µφ,ν + Lφgµν + 1 6 � □(φ2)gµν + Rµνφ2 − (φ2),µ,ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (7) Finally, we denote by VI a vacuum component which interacts with the charged dust fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Such interaction is described by an energy-momentum 4-vector Qν so that the Bianchi identities furnish ∇µ(d)T µ ν = Qν = ∇νVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (8) Let us now consider a FLRW interior geometry in comoving coordinates (r, θ, ϕ) given by ds2 = −dt2 + a2(t) � dr2 1 − kr2 + r2(dθ2 + sin2 θdϕ2) � , (9) where t is the time coordinate, a(t) the scale factor and k the 3-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' From the conservation equations ∇µ(γ)T µ ν = 0 we then obtain ργ = Eγ a4 , (10) where Eγ is a positive constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' On the other hand, the equation of motion for an homogeneous scalar field reads ∇µ(φ)T µ ν = 0 → ¨φ + 3H ˙φ + � k a2 + 2H2 + ˙H + m2� φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (11) In the case of spherical symmetry, the only independent nonvanishing component of Fµν is Ftr = F(t, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Therefore, the Einstein field equations (1) can be written as H2 + k a2 = κ2 3 � ρd + ργ + VI + σ(1 − kr2)F 2(t, r) 2a2 + ˙φ2 2 + H ˙φφ + φ2 2 � H2 + k a2 + m2�� , (12) ˙H + 3H2 2 + k 2a2 = κ2 2 � VI − ργ 3 + σ(1 − kr2)F 2(t, r) 2a2 − ˙φ2 6 + 1 2(H2 + m2)φ2 + 1 3 � ¨φφ + 2Hφ ˙φ + � ˙H + k 2a2 � φ2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (13) 3 Imposing homogeneous energy densities together with an homogeneous vacuum component, we end up with the condition F(t, r) = N(t) √ 1 − kr2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (14) Substituting (3) in (8) we then obtain ∇µ (d)T µ ν = ∇µ(ρduµuν) + σ∇µM µ ν = Qν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (15) At this stage one may assume that Qν = Qν + JαF α ν where Jα = ǫuα is a 4-current with ǫ being the density of electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Employing the Maxwell equations we then obtain σ∇µM µ ν = σ∇µ(F µ α)F α ν = JαF α ν , (16) Making ν = t in (16) we obtain N(t) = N0 a , (17) where N0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' On the other hand, for ν = r, equation (16) furnishes ǫ(t, r) = 2σN0 �√ 1 − kr2 ra3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (18) As the physical radius R of the matter distribution is proportional to the scale factor a for a constant comoving radius r, from the above we note that ǫ scales as R−3, as one should expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Nevertheless, at a first glance one might identify a problem in the above charge density profile since it diverges as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' However, given the spherical symmetry of such matter distribution one should expect that the overall charge should be spread out only in a small neighbourhood of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In a more realistic model this interior solution could be interpreted as a thin Friedmann layer in a small neighbourhood of the surface to be matched with a metric which describes a more involved stellar core – an issue to be addressed in a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' To proceed, in order to assure that the interior matter distribution bounces when a minimum 3-volume is reached, we shall now assume that the energy-momentum 4-vector Qν has the following covariant prescription[7, 9] Qµ = 4 3(V0 − VI)(∇αuα)uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (19) In the above, V0 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Substituting (19) in (15) we then obtain ∇µ(ρduµuν) = ∇µVI = 4 3(VI − V0)(∇αuα)uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (20) A straightforward integration of the differential equations (20) furnishes VI = V0 + λ a4 , (21) ρd = Ed a3 − 4λ a4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (22) where λ and Ed are positive constants of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Therefore, Einstein equations (12) and (13) read H2 + k a2 = κ2 3 �Ed a3 + �2Eγ + N 2 0 σ − λ 2a4 � + V0 + ˙φ2 2 + H ˙φφ + φ2 2 � H2 + k a2 + m2�� , (23) ˙H + 3H2 2 + k 2a2 = κ2 2 � V0 − �2Eγ − 3N 2 0σ − 6λ 6a4 � − ˙φ2 6 + 1 2(H2 + m2)φ2 + 1 3 � ¨φφ + 2Hφ ˙φ + � ˙H + k 2a2 � φ2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (24) It is now useful to rewrite the first integral (23) and the equation of motion of the scalar field (11) in terms of the so-called conformal time dη = a−1dt together with the rescaling ψ ≡ κaφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this case, the Friedmann equation (23) for a closed metric reads 3a′2 + W(a) = κ2(Eγ − 3V1) + 1 2[ψ′2 + (1 + m2a2)ψ2], (25) 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 1: The potential W (a) for a closed model with k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In the above we have fixed κ = 1 and Ed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' It is numerically shown that the potential W (a) has two local extrema for V0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='0 (black curve) and V0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='5 (gray curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' For V0 = 8k3/(κ3Ed)2 (dashed curve) – the upper limit for V0 – there are no extrema for the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Fixing Eγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='01, V1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='052, pψ0 = 0 and ψ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='15 we obtain E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='13475 (red curve above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' where primes denote derivatives with respect to conformal time, V1 ≡ λ − σN 2 0 /6 and W(a) = 3a2 − κ2� V0a4 + Eda � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (26) The equation of motion of the scalar field ψ, on the other hand is given by ψ′′ + (1 + m2a2)ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (27) We are now in a position to define a dynamical system equivalent to equations (25)–(27): ψ′ = −pψ, (28) a′ = pa/6, (29) p′ ψ = (1 + m2a2)ψ, (30) p′ a = −dW da + m2aψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (31) In the above, pψ and pa are the canonical momenta connected to the scalar field ψ and the scale factor a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In fact, with the above definitions it is easy to show that (25) turns into a Hamiltonian first integral given by H = p2 a 12 + W(a) − κ2(Eγ − 3V1) − 1 2[p2 ψ + (1 + m2a2)ψ2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (32) For m = 0 the dynamical system (28)–(31) is separable, hence integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In fact, from equations (28) and (30) we have a first integral E0ψ = (p2 ψ + ψ2)/2 which is a constant of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' It can be shown that the potential W(a) has at most two local extrema (one local minimum a− and one local maximum a+) for a > 0 – as long as V0 < 8k3/(κ3Ed)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Considering the surfaces with energy E = κ2(Eγ − 3V1) + E0ψ < 0 so that W(a−) < E < W(a+) we see that the region 0 < a < a+ is foliated by 2-tori S1 × S1 which are the topological product of periodic orbits of the separable sectors (a, pa) and (ψ, pψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Such 2-tori trap the dynamics in a finite region of the phase space and E is a conserved quantity for those orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In the sector (ψ, pψ) orbits have frequency νψ = 1/2π while in the sector (a, pa) 1 νa = 2 � β2 β1 � 3 E − W(a)da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (33) Here, β1 and β2 are the two smaller real roots of E − W(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 1 we show several plots of W(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' A relevant question which now arises is whether such tori “survive” once integrability is broken due to a nonvanishing mass m for the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In fact, assuming sufficiently small initial conditions (ψ0, pψ0), equation (27) may be rewritten as ψ′′ + (1 + m2a2 0(η))ψ = 0, (34) 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 2: The behaviour of the scale factor a(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' For η ≃ 343 the scale factor diverges triggering a disruptive ejection of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' where a0(η) is the background solution for the scale factor of the integrable dynamics with m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Defining ˜νψ as the frequency in the sector (ψ, pψ) given by (34), a resonant behaviour will occur when the ratio R ≡ νa/˜νψ is a rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Expanding a0(η) in (34), one can show that ˜νψ ≃ 1 2π � 1 + 1 2 � m(β1 + β2) 2 �2 − 1 8 � m(β1 + β2) 2 �4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (35) However, as the dynamics evolves the amplitude of the scalar field may grow so that the solution of the integrable case a0(η) is no longer a good approximation to be introduced in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' This process may lead the dynamics into a more unstable behavior, with the amplification of the resonance and the break of the KAM tori[23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' To analytically show this behavior, one may expand the non-integrable term of (32) in the action-angle variables (Θψ = ˜νψη, Jψ, Θa = νaη, Ja).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' That is, 1 2m2a2 0(η)ψ2(η) = 1 2m2J (0) a J (0) ψ � n [cn cos(2nπΘa)] cos(4πΘψ) where ψ(η) is an approximate solution of (34) and cn are constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The superior indexes in Ja and Jψ denote that these are the action variables for the integrable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The Hamilton equation for Ja can then be integrated furnishing in its first approximation Ja ≃ 1 2m2J (0) a J (0) ψ � n cn 2πn˜νψ �cos(2πnΘa − 4πΘψ) νa/˜νψ − 2/n + cos(2πnΘa + 4πΘψ) νa/˜νψ + 2/n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' From the above we see that the dominant resonance terms are those for which νa ˜νψ ≃ 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (36) When such resonances occur one can eventually obtain a loss of stability so that a → +∞ triggering a disruptive ejection of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' To illustrate the above mentioned behaviour, let us consider the proper domain of the parameters of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 1 with V0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='0 (black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' We also fix the initial conditions pψ0 = 0, ψ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='15 together with Eγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='01, V1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='052 so that E = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='13475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The initial condition for the scale factor, a0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='117, is obtained from the first positive root of W(a) − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' For n = 3, from (33), (35) and (36) we obtain m ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Feeding the Hamiltonian constraint (32) with such parameters and initial condition we obtain the remaining initial condition pa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Evolving the dynamical system imposing that the hamiltonian constraint is conserved, one can numerically show that the scale factor diverges as η ≃ 343 triggering a disruptive ejection of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 2 we illustrate this behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' It is worth mentioning that there is a whole domain in the parametric space (V1, m) in which this unstable behaviour is manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 3 we illustrate some examples of such domains for n = 2 (top left panel), n = 3 (top right panel) and n = 4 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Apart from V1, m and pa0, we used the same parameters and initial conditions considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 3 each dark solid line was obtained from our analytical procedure due to (33), (35) and (36) in order to find the respective resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' It can be numerically shown that the dynamics is highly unstable once one 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 3: Resonance domains for n = 2 (top left panel), n = 3 (top right panel) and n = 4 (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Here we have fixed κ = 1, V0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='0, Ed = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='5, Eγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='01 and initial conditions pψ0 = 0, ψ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='15 and a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In each dark solid line – which were obtained from our analytical procedure due to (33), (35) and (36) – the dynamics is highly unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' There domain below such lines – shaded areas – is also resonant so that after a finite amount of time the scale factor diverges triggering a disruptive ejection of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Below the gray lines we restore the domain of parametric stability analogous to that of the integrable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' considers the parameters/initial conditions connected to these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' There is also a whole domain below such lines – shaded areas – were the resonance mechanism is manifest so that after a finite amount of time the scale factor diverges triggering a disruptive ejection of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Below the gray lines we restore the domain of parametric stability analogous to that of the integrable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' It is also worth noting that one may obtain resonant configurations above the dark solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' However, as our approximation is no longer valid for large masses – above the dark solid lines – one may safely regard the resonance domain as the shaded portions together with the black and gray lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' THE EXTERIOR SPACETIME We now consider the matching of the interior geometry with the exterior spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' To this end let us assume that (¯t, ¯r, ¯θ, ¯φ) are new coordinates defined by ¯t := χ(Γ(t, ¯r)), ¯r = ar, ¯θ = θ, ¯ϕ = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (37) The standard procedure to match the interior geometry with the exterior metric can be found in [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Following the similar notation, we impose that the matching should be performed at the surface r = γ = constant, so that g¯t¯t = − 1 g¯r¯r ��� r=γ ≡ − � 1 − 2GM ¯r + β2 q ¯r2 − κ2 3 ¯r2� V0 + L(η) ���� r=γ � , where L(η) = 1 2κ2a4 � p2 ψ + (1 + m2a2)ψ2� , (38) M = 4πγ3Ed/3, β2 q = q2G/4πǫ0 and q is the overall charge of the matter distribution[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' At this stage, it is important to draw the reader’s attention to a word to note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Let us consider the internal oscillating bounded behaviour of the 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 4: The behaviour of L(η) as a function of the conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Here we see that L(η) vanishes as a → +∞ at η ≃ 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' matter distribution – with η < 343 as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 2, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' During this period the internal oscillating charged matter is responsible for an ejection of radiation making the exterior spacetime stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In fact, such external radiation is needed in order to support the interior dynamics of the scalar field – through the L(η) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' However, once the scale factor diverges – for η ≃ 343, for example – there is a disruptive ejection of mass and L(η) vanishes as a → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Such behaviour is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this sense, the interior solution asymptotically matches an exterior geometry given by the Reissner-Nordstr¨om-de Sitter spacetime and the exterior metric reads ds2 = −F(¯r)d¯t2 + 1 F(¯r)d¯r2 + ¯r2(d¯θ2 + sin2 ¯θdφ2), (39) where F(¯r) ≡ � 1 − 2GM ¯r + β2 q ¯r2 − κ2V0 3 ¯r2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' (40) It is worth to note from (40) that V0 plays the same role of a cosmological constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Therefore, assuming that κ2V0 is sufficiently small it can be easily seen from (32) that d2a/dt2 ≃ 0 as a → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Therefore, in the asymptotic regime the ejection of radiation completely ceases making the exterior spacetime static as one should expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' FINAL REMARKS In this paper we propose a first analysis in which stellar stability may be connected to a conformally coupled massive scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In order to assure that the matter distribution bounces when a minimum 3-volume is reached, we assume that the internal pressureless matter interacts with vacuum component only through a covariant energy exchange[7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this case, bounded interior oscillatory solutions are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' It is worth noting that the dynamics presented in this paper exhibit similar patterns as several bouncing cosmologies[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' In this sense, the interaction assumed in this paper plays just an effective role in order to make the dynamics nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Similar results – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' the break of the KAM tori leading to a disruptive ejection of mass – should be obtained for different bouncing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The obtention of the exterior stationary solution is a rather involved task which we intend to study in a further publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' As mentioned above, such an exterior stationary metric is needed to support the interior scalar field dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' For the case of stable configurations, the perpetual bounded oscillating interior spacetime could in principle be matched with some sort of Vaidya spacetime[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' For the case of unstable configurations on the other hand, the same procedure could be performed considering a Vaidya layer before its extension to the Reissner-Nordst¨om-de Sitter exterior solution[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' As a future perspective we also intend to consider the results of the present paper in order to furnish more realistic scenarios in which the oscillating scale factor may account for stellar internal waves together with a disruptive ejection of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' The first step in this direction is a full examination of the resonance domains furnishing constrains of more realistic parameters such as stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Another issue to be tackled is how the results shown in this paper fit in several models as neutrino heating, thermonuclear burning and magnetohydrodynamic instabilities which account to mass ejection in SNe (see [28] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 8 References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Salvatelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', 113, 181301 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' D, 92, 103005 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Zhao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', Nature Astronomy, 1, 627 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Sol`a, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' G´omez-Valent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' de Cruz P´erez, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' of Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', A32, 1730014 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Di Valentino, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Melchiorri, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Mena, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' D, 96, 043503 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Kumar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Nunes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', D96, 103511 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Bruni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Maier and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Wands, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' D 105, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='6, 063532 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Maier, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 39, 155008 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Maier, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' D 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='14, 2043023 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Oppenheimer and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Snyder, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 56, 455-459 (1939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Vaidya, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' India A 33, 264 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Misner and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Sharp, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 136, B571-B576 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Misner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 137, B1360-B1364 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', Monthly Notices of the Royal Astronomical Society, 265(3), 533–544 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Pretel and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' da Silva, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 495, 5027 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Sharma and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Maharaj, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 375, 1265-1268 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Samadi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Belkacem and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Sonoi, EAS Publications Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 73–74, 111–191 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Garcia, EAS Publications Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 73–74, 193–259 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Inserra, Nature Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='8, 697-705 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Jha, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Maguire and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Sullivan, Nature Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='8, 706-716 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Modjaz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Gutierrez and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Arcavi, Nature Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='8, 717-724 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [22] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Vickers, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Henri Poincar´e, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' XVIII, n2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 137-146 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Maier, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Soares and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Tonini, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' D 79, 023522 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Maier, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 30, 115011 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Arnold, Mathematical Methods of Classical Mechanics (Springer, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [26] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Berezin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 124, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content='3, 446-458 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Fayos, M Merc`e Mart´ın-Prats, Jos´e M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Senovilla, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Grav 12(10):2565 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Janka, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} +page_content=' 62, 407-451 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf'} diff --git a/VtFLT4oBgHgl3EQfSS81/content/tmp_files/2301.12040v1.pdf.txt b/VtFLT4oBgHgl3EQfSS81/content/tmp_files/2301.12040v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2299e19f6783c6249b3ef2456f8bab784cd77e25 --- /dev/null +++ b/VtFLT4oBgHgl3EQfSS81/content/tmp_files/2301.12040v1.pdf.txt @@ -0,0 +1,2437 @@ +ProtST: Multi-Modality Learning of Protein Sequences +and Biomedical Texts +Minghao Xu * † 1 2 Xinyu Yuan * 1 2 Santiago Miret 3 Jian Tang 1 4 5 +Abstract +Current protein language models (PLMs) learn +protein representations mainly based on their se- +quences, thereby well capturing co-evolutionary +information, but they are unable to explicitly ac- +quire protein functions, which is the end goal +of protein representation learning. Fortunately, +for many proteins, their textual property descrip- +tions are available, where their various functions +are also described. Motivated by this fact, we +first build the ProtDescribe dataset to augment +protein sequences with text descriptions of their +functions and other important properties. Based +on this dataset, we propose the ProtST frame- +work to enhance Protein Sequence pre-training +and understanding by biomedical Texts. During +pre-training, we design three types of tasks, i.e., +unimodal mask prediction, multimodal represen- +tation alignment and multimodal mask prediction, +to enhance a PLM with protein property informa- +tion with different granularities and, at the same +time, preserve the PLM’s original representation +power. On downstream tasks, ProtST enables +both supervised learning and zero-shot predic- +tion. We verify the superiority of ProtST-induced +PLMs over previous ones on diverse representa- +tion learning benchmarks. Under the zero-shot +setting, we show the effectiveness of ProtST on +zero-shot protein classification, and ProtST also +enables functional protein retrieval from a large- +scale database without any function annotation. +1. Introduction +Proteins serve as the mainstay governing diverse biological +processes and life itself, inducing important applications in +drug discovery (Teague, 2003) and healthcare (Organization +*Equal technical contribution. +†Project lead. +1Mila - +Qu´ebec AI Institute 2Universit´e de Montr´eal 3Intel Labs 4HEC +Montr´eal 5CIFAR AI Research Chair. Correspondence to: Ming- +hao Xu , Santiago Miret , Jian Tang . +Preprint. +& University, 2007). Recent studies have proven the great +promise of machine learning methods in predicting pro- +tein structures (Jumper et al., 2021; Baek et al., 2021) and +functionality (Meier et al., 2021; Gligorijevi´c et al., 2021). +Among these methods, protein language models (PLMs) (El- +naggar et al., 2020; Rives et al., 2021; Lin et al., 2022) +pre-trained on large-scale protein sequence corpus succeed +in acquiring powerful protein representations, which boost +protein structure and function prediction (Xu et al., 2022b). +Most existing PLMs (Elnaggar et al., 2020; Lu et al., 2020; +Rives et al., 2021; Lin et al., 2022) learn protein representa- +tions based only on their sequences, which can well capture +co-evolutionary information but cannot explicitly acquire +protein functions and other important properties like their +subcellular locations. Acquiring such function and property +information is actually the end goal of protein representation +learning. Fortunately, for many proteins, we can get access +to their textual property descriptions in which their diverse +functions are also described. This fact motivates us to study +protein sequence representation learning enriched with di- +verse protein properties described by biomedical texts. +To our best knowledge, OntoProtein (Zhang et al., 2022a) is +the only existing PLM that explicitly captures protein prop- +erties. However, it learns a closed set of properties over a +fixed biological knowledge graph and thus can hardly gener- +alize to unknown properties of new proteins. In comparison, +by modeling textual protein property descriptions, we can +flexibly model the generalization from known properties to +unknown ones based on the semantic correlation of their +text descriptions, as shown by our zero-shot experiments +(Secs. 4.3 and 4.4). +To attain biomedical text enhanced protein sequence repre- +sentation learning, we first build the ProtDescribe dataset, +a paired dataset of protein sequences and textual property +descriptions. We resort to the Swiss-Prot database (Bairoch +& Apweiler, 2000) for high-quality protein annotations and +construct each protein’s property description with the se- +lected annotations of it. ProtDescribe incorporates the in- +formation of protein names, protein functions, subcellular +locations and protein families, and these properties are de- +scribed by biomedical texts with rich expressions. +Based on this dataset, we propose the ProtST framework +arXiv:2301.12040v1 [q-bio.BM] 28 Jan 2023 + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +to enhance protein sequence pre-training and understand- +ing by biomedical texts. During ProtST pre-training, to +preserve the beneficial representation power of a conven- +tional PLM on capturing co-evolutionary information, we +adopt the Unimodal Mask Prediction task for masked pro- +tein modeling. On such basis, two multimodal pre-training +tasks are designed to inject different granularities of perti- +nent protein property information into a PLM: Multimodal +Representation Alignment injects integrated and general +property information into the PLM, in which a biomedi- +cal language model is used to extract structured text rep- +resentations of different property descriptions, and protein +sequence representations are aligned to the corresponding +text representations; Multimodal Mask Prediction models +the fine-grained dependencies between residues in a pro- +tein sequence and property-descriptive words in its property +description, in which a fusion module is employed to de- +rive multimodal representations of residues and words, and, +based on these fused multimodal representations, masked +residues and words are predicted. For downstream applica- +tions, ProtST can conduct supervised learning with only the +PLM and can also perform zero-shot prediction based on +the aligned representation space of protein sequences and +text descriptions. +We investigate the PLMs trained under ProtST by represen- +tation learning and zero-shot prediction. For representation +learning, we verify their superior performance over previous +PLMs on 11 standard benchmarks for protein localization +prediction, fitness landscape prediction and protein function +annotation (Sec. 4.2). For zero-shot protein classification, +ProtST-induced zero-shot classifiers show better data effi- +ciency against various few-shot classifiers (Sec. 4.3.2), and +are proven to be able to enhance the performance of super- +vised learning models via ensemble (Sec. 4.3.3). For zero- +shot text-to-protein retrieval, we verify the effectiveness of +ProtST on retrieving functional proteins from a large-scale +database without any function annotation (Sec. 4.4). +2. Preliminaries +2.1. Problem Definition +In the pre-training phase, we study the problem of learning +informative protein sequence representations guided by the +proteins’ associated biomedical text descriptions. In this +problem, a protein P = (S, T) is represented by an amino +acid sequence S = [s1, s2, · · · , sn] with n amino acids +(a.k.a., residues) and a text description T = [t1, t2, · · · , tm] +with m word tokens. Given a pre-training dataset with N +proteins P = {P1, P2, · · · , PN}, our goal is to extract effec- +tive protein representations by fully utilizing the information +from their sequences and descriptions. The extracted protein +representations are expected to boost various downstream +tasks by supervised learning or zero-shot prediction. +2.2. Protein Language Models +Protein language models (PLMs) (Elnaggar et al., 2020; +Rives et al., 2021; Meier et al., 2021; Lin et al., 2022) pre- +trained on large-scale protein sequence corpus have shown +impressive results on protein function (Meier et al., 2021) +and structure (Lin et al., 2022) prediction. PLMs are com- +monly trained by masked protein modeling, in which partial +residues are masked at input and predicted based on the +context. In this work, we select three state-of-the-art PLMs, +ProtBert (Elnaggar et al., 2020), ESM-1b (Rives et al., 2021) +and ESM-2 (Lin et al., 2022), as baselines and seek to en- +hance their representation power by modeling biomedical +texts at the same time as protein sequence modeling. +2.3. Biomedical Language Models +Compared to the texts from general domains like newswire +and Web, biomedical texts differ a lot in terms of vocabulary +and expressions. To tackle such differences, language mod- +els specific to the biomedical domain (Beltagy et al., 2019; +Lee et al., 2020; Gu et al., 2021) are actively studied. In this +work, we employ a performant biomedical language model, +PubMedBERT (Gu et al., 2021), to represent the biomedical +text descriptions of proteins. +3. Method +In this section, we first motivate the proposed ProtST frame- +work and present its general picture in Sec. 3.1, and then +elucidate the design of pre-training tasks in Sec. 3.2, fol- +lowed by discussing the connections with and advantages +over previous works in Sec. 3.3. +3.1. Motivation and Overview +Motivation: Existing PLMs (Elnaggar et al., 2020; Lu et al., +2020; Rives et al., 2021; Lin et al., 2022) learn protein rep- +resentations primarily based on their sequences, which can +well capture co-evolutionary information but cannot explic- +itly acquire various protein properties like protein functions +and subcellular locations. By acquiring such property in- +formation, the effectiveness of a PLM can be further im- +proved, considering that the protein properties studied in +pre-training and downstream tasks can correlate with each +other (Bhardwaj & Lu, 2005). +To gain such improvement, we curate the ProtDescribe +dataset that augments protein sequences with text descrip- +tions of their diverse properties (see Sec. 4.1 for details). By +injecting such property information into protein sequence +representations, we aim at (1) a PLM that is more effective +than previous ones on various downstream tasks under su- +pervised learning, and (2) it can further enable zero-shot +prediction through the generalization of text descriptions + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +PROTEIN NAME: [MASK] myristoylated +protein 053R. PROTEIN FUNCTION: May +play a critical role in virion [MASK]. +Essential for virus replication in [MASK] … +MGAA[MASK]SINT[MASK]NITKAY +AKIMTTMVT[MASK]QDITADQSQV +F[MASK]IDHVKGDVVIKGDVFTQM +LVINLASLMKAIAT[MASK]SAQDQ… +Fusion +Module +Protein +LM +Biomedical +LM +𝑳𝑴𝑴𝑷 +𝑺 +𝑳𝑴𝑴𝑷 +𝑻 +𝑳𝑮𝑪 +𝑳𝑴𝑷𝑴 +Protein = (AA sequence , Text description) +(a) Multimodal Pre-training +Protein +LM +(b) Downstream Supervised Learning +Cell membrane +Cytoplasm +. . . +Extracellular +MFKKFTREDVHSRSKVKSSIQRTLKA +KLVKQYPKIEDVIDELIPKKSQIELIKC +EDKIQLYSVDGEVLFFQKFDELIPSLK +LVHKFPEAYPTVQVDRGAIKFVLSG… +(c) Zero-shot Protein Classification +(d) Zero-shot Text-to-Protein Retrieval +𝒛𝑺"𝒛𝟏 +𝑻 𝒛𝑺"𝒛𝟐 +𝑻 +𝒛𝑺"𝒛𝑲 +𝑻 +𝒛𝑺 +𝒛𝟏 +𝑻 +𝒛𝟐 +𝑻 +𝒛𝑲 +𝑻 +A protein locating at Cytoplasm +. . . +. . . +𝒛𝑻"𝒛𝟏 +𝑺 𝒛𝑻"𝒛𝟐 +𝑺 +𝒛𝑻"𝒛𝑵 +𝑺 +𝒛𝑻 +𝒛𝟏 +𝑺 +𝒛𝟐 +𝑺 +𝒛𝑵 +𝑺 +. . . +. . . +Biomedical +LM +Protein +LM +A protein locating at Cell membrane +A protein locating at Cytoplasm +A protein locating at Extracellular +... +MFKKFTREDVHSRSKVKSSIQ +RTLKAKLVKQYPKIEDVIDELIP +KKSQIELIKCEDKIQLYSVDG… +Biomedical +LM +Protein +LM +FUNCTION: Binding to a heme, +a compound of iron complexed in +a porphyrin (tetrapyrrole) ring. +MAKQLQARRLDGIDYNPWV … +GHMFGSINLASSLSVDAPGL… +MAKKSNSKKSTPVSTPSKEK… +... +𝑺𝟐 +𝑺𝟏 +𝑺𝑵 +. . . +Retrieval ranks +Figure 1: Graphical illustration of ProtST framework. (a) A protein language model (PLM) is first pre-trained along with +a biomedical language model (BLM) and a fusion module to jointly model protein sequences and biomedical texts. (b) After +this multi-modal pre-training, the PLM can be used individually for supervised learning on downstream tasks. (c) The +couple of pre-trained PLM and BLM can perform zero-shot protein classification using only label descriptions. (d) The +paired PLM and BLM can also retrieve functional proteins from a large-scale database without any function annotation. +between known protein properties and unknown ones. +ProtST Framework: To attain these goals, we first perform +multi-modal pre-training of sequences and texts and then +apply the pre-trained model to three types of downstream +applications (framework overview is shown in Fig. 1): +• Multimodal Pre-training: +Given the ProtDescribe +dataset, we train a PLM together with a biomedical +language model (BLM) and a fusion module to model +the paired protein sequences and text descriptions. We +consider three kinds of pre-training tasks, i.e., unimodal +mask prediction, multimodal representation alignment +and multimodal mask prediction, to capture the protein +property information with different granularities and +also preserve the PLM’s original representation power. +• Downstream Supervised Learning: After such pre- +training, the PLM is enriched by the useful property +information within biomedical texts. For downstream +tasks with labeled proteins, we can employ the PLM +individually to solve the tasks by supervised learning. +• Zero-shot Protein Classification: When a protein clas- +sification task occurs without any labeled data, ProtST +enables zero-shot classification. Specifically, the classi- +fication result can be determined by the representation +similarity comparison between the query protein and +all labels, thanks to the aligned representation space of +protein sequences and label descriptions. +• Zero-shot Text-to-Protein Retrieval: Based on the +aligned representation space, ProtST also allows us to +retrieve functional proteins from a large-scale database +by using only the text descriptions of protein functions, +in which no function annotation is required. +3.2. Pre-training Tasks: Joint Modeling of Protein +Sequences and Biomedical Texts +During ProtST pre-training, we aim to learn informative pro- +tein sequence representations guided by biomedical texts. +To start this process with decent representations of protein +sequences and biomedical texts, we use pre-trained PLM +(i.e., ProtBert (Elnaggar et al., 2020), ESM-1b (Rives et al., +2021) or ESM-2 (Lin et al., 2022)) and pre-trained BLM +(i.e., PubMedBERT (Gu et al., 2021)) for initialization. Dur- +ing training, we tune the parameters of PLM and freeze +those of BLM, since the pre-trained BLM is sufficient for +extracting semantically meaningful representations from +biomedical texts, and it is computationally expensive to +tune both PLM and BLM simultaneously. ProtST involves +the following pre-training tasks for representation learning. +Unimodal Mask Prediction: The PLM for initialization +is pre-trained by masked protein modeling (MPM), i.e., +predicting masked residues based on the protein sequence +context. This task can capture co-evolutionary information +by modeling residue type dependency. To preserve such +unimodal information when injecting the cross-modality +information from biomedical texts, we keep an MPM loss +function LMPM for ProtST pre-training. Specifically, for +each protein sequence, we randomly mask 15% residue +tokens and predict each masked token based on its contextu- +alized representation extracted by the PLM, where LMPM +is formulated as a cross-entropy loss to measure the cost. +Multimodal Representation Alignment: The biomedical +text representations learned by a pre-trained BLM can well +reflect the semantics of the texts (Jin et al., 2019; Gu et al., +2021). Therefore, when given protein property descrip- + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +tions, the BLM can extract semantically meaningful text +representations of proteins. Thanks to this capability, by +aligning protein sequence representations to their associated +text representations, we can naturally inject protein property +information into sequence representations. +To realize such alignment, we perform contrastive learning +between protein sequences and their text descriptions. Given +a batch of M proteins {Pi = (Si, Ti)}M +i=1, we use the PLM +to extract protein sequence representations {zS +i }M +i=1 and the +BLM to derive text description representations {zT +i }M +i=1. A +standard InfoNCE loss (Oord et al., 2018) LGC is defined +to maximize the representation similarity between corre- +sponding sequences and texts and minimize the similarity +between negative pairs: +LGC = − 1 +2M +M +� +i=1 +� +log +exp(zS +i · zT +i /τ) +�M +j=1 exp(zS +i · zT +j /τ) ++ log +exp(zS +i · zT +i /τ) +�M +j=1 exp(zS +j · zT +i /τ) +� +, +(1) +where, under multi-GPU data parallelism, we gather whole- +batch samples separated on different GPUs to form negative +pairs and thus term the loss LGC as a global contrastive +(GC) loss following the convention (Singh et al., 2022), and +τ denotes a learnable temperature parameter. +Multimodal Mask Prediction: Although the general de- +pendency between the whole protein sequences and full +text descriptions can be well modeled by LGC, LGC alone +does not capture the dependency between the residues in +a protein sequence and the words in its text description. +Such fine-grained cross-modality interdependency is actu- +ally ubiquitous. For example, a soluble protein (descriptive +words) always co-occurs with charged and polar surface +residues (Capaldi & Vanderkooi, 1972); high thermostabil- +ity (descriptive words) and high amounts of hydrophobic +residues are correlated with each other (Kumar et al., 2000), +etc. To capture such interdependency, we propose a novel +pre-training task that encourages the model to recover the +corrupted protein sequence (or text description) based on +the information from both modalities. +Specifically, given a protein sequence S = [s1, s2, · · · , sn] +and its corresponding text description T = [t1, t2, · · · , tm], +we first randomly mask 15% residues in the protein se- +quence and 15% words in the text description. Upon the +corrupted inputs, we employ the PLM to extract residue +representations ZS = [zs +1, zs +2, · · · , zs +n] and utilize the BLM +to extract word representations ZT = [zt +1, zt +2, · · · , zt +m]. A +fusion module with both self- and cross-attention is then +used to model the interdependency between residues and +words, in which each residue and word updates its repre- +sentation by attending to all the tokens along both protein +sequence and text description (we state the detailed architec- +ture in Appendix A). The fusion module produces the fused +residue representations ˜ZS = [˜zs +1, ˜zs +2, · · · , ˜zs +n] and the fused +word representations ˜ZT = [˜zt +1, ˜zt +2, · · · , ˜zt +m], in which each +residue/word representation combines the information from +both modalities. Based on ˜ZS and ˜ZT , we perform multi- +modal mask prediction (MMP) to recover masked residues +and words, where a cross-entropy loss LS +MMP measures the +cost on protein sequence, and another cross-entropy loss +LT +MMP measures the cost on text description, inducing the +overall MMP loss LMMP = LS +MMP + LT +MMP. +Overall Pre-training Objective: During the pre-training +process, we seek to minimize the loss functions of all pre- +training tasks simultaneously: +min +θ +LMPM + LGC + LMMP, +(2) +where θ denotes all learnable parameters including those of +the PLM, the fusion module and all projection/prediction +heads. We state the detailed architectures of these modules +in Appendix A. +3.3. Discussion +Now we discuss the connections of our method with previ- +ous works and emphasize its advantages. +Advantages over Self-Supervised PLMs: Previous self- +supervised PLMs (Elnaggar et al., 2020; Rives et al., 2021; +Lin et al., 2022) and the proposed ProtST-induced ones +can both capture co-evolutionary information hidden in pro- +tein sequences by masked protein modeling. On this basis, +ProtST-induced PLMs further utilize the supervision from +textual protein property descriptions, and they are guided +to acquire whole-protein properties by multimodal repre- +sentation alignment and acquire residue-level properties by +multimodal mask prediction. +Advantages over OntoProtein (Zhang et al., 2022a): +Similar to our approach, OntoProtein also seeks to enhance +a self-supervised PLM by involving protein property infor- +mation. In comparison, ProtST could be more effective +mainly in two aspects. (1) Diversity of considered prop- +erties: OntoProtein retrieves Gene Ontology terms (Zhang +et al., 2022a) to cover protein functions and locations; be- +sides these two kinds of properties, ProtST additionally +includes protein names and families which are useful to +indicate protein structural and functional similarity (Murzin +et al., 1995). (2) Property modeling manner: OntoProtein +learns a closed set of protein properties under the context of +a fixed biological knowledge graph, which limits its ability +to generalize to unknown properties of new proteins, while +ProtST can flexibly model such generalization based on the +semantic correlation of text descriptions between known and +unknown properties, leading to decent zero-shot prediction +capability (studied in Secs. 4.3 and 4.4). + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Table 1: Statistics of the ProtDescribe dataset. +Field +Name +Function +Location +Family +#Covered samples +553,052 +460,936 +350,929 +512,276 +Coverage +100% +83.3% +63.5% +92.6% +4. Experiments +4.1. Pre-training Setups +Pre-training Dataset: To inject protein property informa- +tion into PLMs, we build the ProtDescribe dataset with +553,052 aligned pairs of protein sequence and property de- +scription. Specifically, we employ the Swiss-Prot (Bairoch +& Apweiler, 2000) database to provide annotations of var- +ious protein properties, in which we select four property +fields: (1) “Protein Name” gives the full protein name rec- +ommended by the UniProt consortium (Consortium, 2019); +(2) “Function” depicts diverse functions owned by a protein; +(3) “Subcellular Location” describes the location and topol- +ogy of a mature protein in the cell; (4) “Similarity” provides +information about the protein families that a protein belongs +to. A complete property description is formed by concate- +nating these four fields in order, where missing fields are +skipped (see Appendix B.1 for the detailed concatenation +scheme and examples). Tab. 1 presents the statistics of how +each field covers the whole dataset. +Protein Language Models: We seek to enhance three rep- +resentative PLMs, i.e., ProtBert (Elnaggar et al., 2020), +ESM-1b (Rives et al., 2021) and ESM-2 (Lin et al., 2022), +by tuning their weights through the proposed ProtST pre- +training. We name the PLMs after this pre-training phase as +ProtST-ProtBert, ProtST-ESM-1b and ProtST-ESM-2. +For ProtBert, we employ the ProtBert-BFD version which is +trained on the BFD database (Steinegger & S¨oding, 2018). +For ESM-2, we adopt the ESM-2-650M model so as to fairly +compare with ESM-1b under the same model size. +Biomedical Language Models: By default, we utilize the +PubMedBERT-abs (Gu et al., 2021) trained on PubMed ab- +stracts to extract representations of protein property descrip- +tions. We study another model version, PubMedBERT-full +trained with additional full-text articles, in Appendix D.2. +Training Configurations: An Adam optimizer (Kingma +& Ba, 2014) (learning rate: 1.0 × 10−5, weight decay: 0) +is used to train the whole model for 20 epochs on 4 Tesla +V100 GPUs. More settings are introduced in Appendix B.1. +4.2. Representation Learning +4.2.1. EXPERIMENTAL SETUPS +Downstream Benchmark Tasks. We adopt 11 benchmark +tasks within three task types (the “Abbr.” below denotes the +abbreviated task name in Tab. 2 and 3): +• Protein Localization Prediction seeks to predict the +subcellular locations of proteins. We consider two such +problems from DeepLoc (Almagro Armenteros et al., +2017), the subcellular localization prediction (Abbr., +Sub) with 10 location categories and the binary localiza- +tion prediction (Abbr., Bin) with 2 location categories. +We follow the official dataset splits. +• Fitness Landscape Prediction aims to predict the ef- +fect of residue mutations on protein fitness. We em- +ploy the β-lactamase (Abbr., β-lac) landscape from +PEER (Xu et al., 2022b), the AAV and Thermostability +(Abbr., Thermo) landscapes from FLIP (Dallago et al., +2021), and the Fluorescence (Abbr., Flu) and Stability +(Abbr., Sta) landscapes from TAPE (Rao et al., 2019). +For AAV, we use the “two vs many” dataset splits; for +Thermostability, we adopt the “human cell” splits; we +follow the only default splits on all other tasks. +• Protein Function Annotation seeks to annotate a pro- +tein with multiple functional labels simultaneously. We +employ two standard benchmarks proposed by Deep- +FRI (Gligorijevi´c et al., 2021), i.e., Enzyme Commis- +sion (EC) number prediction and Gene Ontology (GO) +term prediction. The GO benchmark is split into three +branches to predict molecular function (Abbr., GO-MF), +biological process (Abbr., GO-BP) and cellular compo- +nent (Abbr., GO-CC). Following Zhang et al. (2022b), +we use the dataset splits under 95% sequence identity +cutoff for both benchmarks. +Baselines: We adopt four protein sequence encoders trained +from scratch, i.e., CNN (Shanehsazzadeh et al., 2020), +ResNet (Rao et al., 2019), LSTM (Rao et al., 2019) and +Transformer (Rao et al., 2019), as naive baselines. We focus +on comparing with four performant PLMs, i.e., ProtBert (El- +naggar et al., 2020), OntoProtein (Zhang et al., 2022a), +ESM-1b (Rives et al., 2021) and ESM-2 (Lin et al., 2022). +Training and Evaluation: We train with an Adam opti- +mizer for 100 epochs on localization and fitness predic- +tion tasks and for 50 epochs on function annotation tasks. +For localization and fitness prediction, all PLMs are evalu- +ated under both fix-encoder learning and full-model tuning +settings, and only full-model tuning is used for PLMs on +function annotation, since it is hard to solve the multiple +binary classification problems on EC and GO with fixed +protein representations. More training details are stated in +Appendix B.2. +For all models on all tasks, we select the checkpoint for +evaluation based on the validation set performance, and all +results are reported on the seed 0. We measure the classifica- +tion accuracy for localization prediction and the Spearman’s +ρ for fitness prediction. Following Gligorijevi´c et al. (2021), +function annotation tasks are measured by AUPR and Fmax +whose detailed definitions are in Appendix B.2. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Table 2: Benchmark results on protein localization and +fitness landscape prediction. We use three color scales of +blue to denote the first, second and third best performance. +Abbr., Loc.: Localization; pred.: prediction; Acc: accuracy. +Model +Loc. pred. (Acc%) +Fitness pred. (Spearman’s ρ) +Bin +Sub +β-lac +AAV +Thermo +Flu +Sta +Mean ρ +Protein sequence encoders trained from scratch +CNN +82.67 +58.73 +0.781 +0.746 +0.494 +0.682 +0.637 +0.668 +ResNet +78.99 +52.30 +0.152 +0.739 +0.528 +0.636 +0.126 +0.436 +LSTM +88.11 +62.98 +0.139 +0.125 +0.564 +0.494 +0.533 +0.371 +Transformer +75.74 +56.02 +0.261 +0.681 +0.545 +0.643 +0.649 +0.556 +PLMs w/ fix-encoder learning +ProtBert +81.54 +59.44 +0.616 +0.209 +0.562 +0.339 +0.697 +0.485 +OntoProtein +84.87 +68.34 +0.471 +0.217 +0.605 +0.432 +0.688 +0.483 +ESM-1b +91.61 +79.82 +0.528 +0.454 +0.674 +0.430 +0.750 +0.567 +ESM-2 +91.32 +80.84 +0.559 +0.374 +0.677 +0.456 +0.746 +0.562 +ProtST-ProtBert +92.29 +78.49 +0.569 +0.219 +0.621 +0.376 +0.719 +0.501 +ProtST-ESM-1b +92.87 +82.00 +0.578 +0.460 +0.680 +0.523 +0.766 +0.601 +ProtST-ESM-2 +92.52 +83.39 +0.565 +0.398 +0.681 +0.499 +0.776 +0.584 +PLMs w/ full-model tuning +ProtBert +91.32 +76.53 +0.731 +0.794 +0.660 +0.679 +0.771 +0.727 +OntoProtein +92.47 +77.59 +0.757 +0.791 +0.662 +0.630 +0.731 +0.714 +ESM-1b +92.40 +78.13 +0.839 +0.821 +0.669 +0.679 +0.694 +0.740 +ESM-2 +91.72 +78.67 +0.867 +0.817 +0.672 +0.677 +0.718 +0.750 +ProtST-ProtBert +91.78 +78.71 +0.863 +0.804 +0.673 +0.679 +0.745 +0.753 +ProtST-ESM-1b +92.35 +78.73 +0.895 +0.850 +0.681 +0.682 +0.751 +0.772 +ProtST-ESM-2 +92.52 +80.22 +0.879 +0.825 +0.682 +0.682 +0.738 +0.761 +4.2.2. EXPERIMENTAL RESULTS +We report the benchmark results on localization and fitness +prediction in Tab. 2 and report function annotation results +in Tab. 3. Based on the benchmark results, we have the +following observations: +ProtST-induced PLMs clearly outperform the vanilla +PLMs. It is observed that: (1) ProtST-ProtBert outperforms +the vanilla ProtBert on 21 out of 24 benchmark metrics +(including both fix-encoder learning and full-model tuning +ones); (2) ProtST-ESM-1b surpasses the vanilla ESM-1b +on 22 out of 24 benchmark metrics; (3) ProtST-ESM-2 +outperforms the vanilla ESM-2 on all 24 benchmark met- +rics. These results demonstrate that ProtST pre-training is +generally beneficial to different PLMs, which boosts their +performance on diverse downstream tasks. +ProtST-ProtBert performs consistently better than On- +toProtein under fair comparison. ProtST-ProtBert and +OntoProtein can be fairly compared with each other, since +they both adopt ProtBert as the initial PLM. ProtST-ProtBert +surpasses OntoProtein on 22 out of 24 benchmark metrics, +which verifies the superiority of the proposed pre-training +dataset and pre-training tasks. +ProtST-ESM-1b performs best on fitness prediction, +and ProtST-ESM-2 performs best on localization pre- +diction and function annotation. We can observe that: (1) +ProtST-ESM-1b achieves the best performance on 4 out +of 6 benchmark metrics for fitness prediction; (2) ProtST- +ESM-2 obtains the highest localization prediction accuracy +on average, and it performs best on 7 out of 8 benchmark +metrics for function annotation. We therefore recommend +these two PLMs as new state-of-the-arts. +Table 3: Benchmark results on protein function annotation. +We use three color scales of blue to denote the first, second +and third best performance. +Model +EC +GO-BP +GO-MF +GO-CC +AUPR +Fmax +AUPR +Fmax +AUPR +Fmax +AUPR +Fmax +Protein sequence encoders trained from scratch +CNN +0.540 +0.545 +0.165 +0.244 +0.380 +0.354 +0.261 +0.387 +ResNet +0.137 +0.187 +0.166 +0.280 +0.281 +0.267 +0.266 +0.403 +LSTM +0.032 +0.082 +0.130 +0.248 +0.100 +0.166 +0.150 +0.320 +Transformer +0.187 +0.219 +0.135 +0.257 +0.172 +0.240 +0.170 +0.380 +PLMs w/ full-model tuning +ProtBert +0.859 +0.838 +0.188 +0.279 +0.464 +0.456 +0.234 +0.408 +OntoProtein +0.854 +0.841 +0.284 +0.436 +0.603 +0.631 +0.300 +0.441 +ESM-1b +0.884 +0.869 +0.332 +0.452 +0.630 +0.659 +0.324 +0.477 +ESM-2 +0.888 +0.874 +0.340 +0.472 +0.643 +0.662 +0.350 +0.472 +ProtST-ProtBert +0.876 +0.856 +0.286 +0.440 +0.615 +0.648 +0.314 +0.449 +ProtST-ESM-1b +0.894 +0.878 +0.328 +0.480 +0.644 +0.661 +0.364 +0.488 +ProtST-ESM-2 +0.898 +0.878 +0.342 +0.482 +0.647 +0.668 +0.364 +0.487 +4.3. Zero-shot Protein Classification +4.3.1. EXPERIMENTAL SETUPS +Zero-shot Protein Classification based on Aligned Rep- +resentation Space: A ProtST-induced PLM naturally al- +lows zero-shot protein classification, thanks to its aligned +representation space of protein sequences and text descrip- +tions. Given the sequence S of a query protein and the +label descriptions {Ti}K +i=1 of all K classes, we employ the +PLM to extract protein representation zS and use the jointly +learned BLM to extract label representations {zT +i }K +i=1. We +then derive classification logits {yi}K +i=1 by comparing the +dot product similarity between protein and label represen- +tations: yi = zS · zT +i /τ (i = 1, · · · , K), which follows the +formula of InfoNCE loss in Eq. (1). Softmax is performed +upon these logits to derive classification probabilities. +Benchmark Tasks: In this part of experiments, we adopt +two protein classification tasks as benchmarks: (1) the sub- +cellular localization prediction task which is same as the +one introduced in Sec. 4.2.1; (2) the reaction classification +task proposed by Hermosilla et al. (2020) which reformu- +lates the EC number prediction task introduced in Sec. 4.2.1 +as a classification task with 384 reaction classes. We follow +the official dataset splits for both tasks. +Prompt Engineering: To extract discriminative label rep- +resentations, we have tried three types of prompt templates +to describe protein function/location labels. (1) Name only: +a label is described only by the name of a function or loca- +tion (e.g., “Cytoplasm”); (2) Natural language: the name +is embedded into a natural language template (e.g., “A pro- +tein locating at Cytoplasm”); (3) Pre-training template: the +name is embedded into the template used during ProtST +pre-training (e.g., “SUBCELLULAR LOCATION: Cyto- +plasm”). The pre-training template is empirically verified +to be more effective than other two templates, and thus it is +used across all experiments of this section. The comparisons +among these templates are provided in Appendix B.3. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +(a) Subcellular localization prediction +(b) Reaction classification +Figure 2: Zero-shot ProtST-ESM-1b outperforms few- +shot classifiers. The horizontal line with a red star denotes +the zero-shot performance of ProtST-ESM-1b. All few-shot +results are averaged over seeds 0, 1, 2, 3 and 4, and gray +intervals denote standard deviations. +4.3.2. DATA EFFICIENCY OF ZERO-SHOT CLASSIFIER +Baselines: We study the data efficiency of zero-shot ProtST- +ESM-1b by comparing it with n-shot classifiers (n ⩾ 1) +which employ n training samples per class for prediction. +We adopt four baselines: (1) the ProtST-ESM-1b with su- +pervised fine-tuning, (2) the ESM-1b with supervised fine- +tuning, (3) the nonparametric ProtST-ESM-1b classifier, and +(4) the nonparametric ESM-1b classifier. We follow Khan- +delwal et al. (2019) to design the nonparametric classifiers +that well fit the few-shot setting, detailed in Appendix B.3. +Results: For subcellular localization prediction (Fig. 2(a)), +the zero-shot ProtST-ESM-1b matches the performance of +3-shot supervised ProtST-ESM-1b and the performance of +5-shot supervised ESM-1b, and the zero-shot classifier out- +performs two 7-shot nonparametric classifiers. For reaction +classification (Fig. 2(b)), the zero-shot ProtST-ESM-1b sur- +passes the 1-shot performance of supervised and nonpara- +metric ProtST-ESM-1b, and it aligns the 2-shot performance +of supervised and nonparametric ESM-1b. These results +demonstrate the data efficiency of ProtST-induced zero-shot +classifiers. In particular, they can be helpful in the down- +stream tasks with limited or even no labeled proteins by +making educated predictions using only label descriptions. +4.3.3. ENHANCING SUPERVISED LEARNING WITH +ZERO-SHOT CLASSIFIER +Ensemble of Supervised Learning Model and Zero-shot +Classifier: We study how zero-shot ProtST-ESM-1b can +boost supervised learning models via ensemble. Specifically, +we combine the classification logits produced by a super- +vised learning model and the zero-shot classification logits +as below: {yk = ysup +k ++ α yzero +k +}K +k=1 (K is the number of +classes), where α controls the contribution of the zero-shot +classifier. Empirically, we set α as the ratio of the zero-shot +classifier’s validation set performance over the validation +performance of the supervised learning model. +Baselines: We employ ProtST-ESM-1b and ESM-1b with +supervised fine-tuning on downstream tasks as baselines. +(a) Subcellular localization prediction +(b) Reaction classification +Figure 3: Zero-shot ProtST-ESM-1b enhances few-shot +classifiers’ performance via ensemble. The horizontal +line with a red star denotes the zero-shot performance of +ProtST-ESM-1b. All few-shot results are averaged over +seeds 0, 1, 2, 3 and 4, and gray intervals denote standard +deviations. +Table 4: Zero-shot ProtST-ESM-1b enhances full-shot +classifiers’ performance via ensemble. Abbr., loc.: local- +ization; Acc: accuracy. +Model +Subcellular loc. (Acc%) +Reaction (Acc%) +ProtST-ESM-1b +82.00 +86.73 +[Ensemble] ProtST-ESM-1b +82.37 +87.14 +ESM-1b +79.82 +80.54 +[Ensemble] ESM-1b +80.20 +83.03 +We consider fine-tuning under both the few-shot setting and +the full-shot setting (i.e., trained with all training samples). +Results: According to Fig. 3 and Tab. 4, we can observe that +zero-shot ProtST-ESM-1b succeeds in enhancing the perfor- +mance of all few-shot and full-shot baselines on both bench- +marks. These results verify that ProtST-induced zero-shot +classifiers are useful tools to enhance supervised learning +models, which is realized by refining decision boundaries. +4.4. Zero-shot Text-to-Protein Retrieval +Zero-shot Text-to-Protein Retriever: +Based on the +protein-text aligned representation space, ProtST enables us +to retrieve functional proteins from a large-scale database +without any function annotation. To be specific, the PLM is +first employed to extract the representations {zS +i }N +i=1 of all +proteins in the database. During the retrieval process, given +the text description (i.e., prompt) T of a protein function, +the BLM is used to extract its representation zT , and all pro- +teins are then ranked based on their representation similarity +{ϵi = zS +i · zT }N +i=1 with the prompt. +Experimental Setups: We use ProtST-ESM-1b to retrieve +the Gene Ontology (GO) dataset introduced in Sec. 4.2.1. +We build each prompt by adding the “FUNCTION:” prefix +before the molecular function definition from GO. +Results: In Fig. 4, we visualize the top-4 retrieved candi- +dates of heme binders. We present the text prompt, the dock- +ing result of each candidate binding with heme (AutoDock +Vina (Trott & Olson, 2010) is used for docking), the binding +affinity predicted by AutoDock Vina (the lower the better), + +60 +50 +% +40 +Accuracy ( +30 +++t +ProtST-ESM-1b +20 +ESM-1b +[Nonparam] ProtST-ESM-1b +10 +[Nonparam] ESM-1b +0 +1 +2 +3 +4 +5 +# Training samples per class60 +50 +40 +Accuracy (%) +30 +ProtST-ESM-1b +20 +ttt +ESM-1b +[Nonparam] ProtST-ESM-1b +10- +[Nonparam] ESM-1b +01 +2 +3 +4 +5 +6 +7 +# Training samples per class50 +(%) +40 +Accuracy ( +30 +[Ensemblel ProtST-ESM-1b +20 +t+t +ProtST-ESM-1b +[Ensemble] ESM-1b +10 +ESM-1b +0 +1 +2 +3 +4 +5 +# Training samples per class60 +50- +(%) +Accuracy ( +40 +30 +[Ensemblel ProtST-ESM-1b +ProtST-ESM-1b +20 +[Ensemblel ESM-1b +ESM-1b +01 +2 +3 +4 +5 +6 +7 +# Training samples per classProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Prompt - FUNCTION: Binding to a heme, a compound composed of iron complexed in a porphyrin (tetrapyrrole) ring. +(1st) 2N91-A: +• +Affinity: -7.3 (kcal/mol) +• +GO-MF label: Bind +(2nd) 1YHU-A: +• +Affinity: -7.9 (kcal/mol) +• +GO-MF label: Bind +(3rd) 5B3I-A: +• +Affinity: -8.1 (kcal/mol) +• +GO-MF label: Bind +(4th) 5VPR-A: +• +Affinity: -7.4 (kcal/mol) +• +GO-MF label: Non-bind +Figure 4: Zero-shot text-to-protein retrieval of heme binders based on ProtST-ESM-1b. +Table 5: Ablation study of pre-training losses on ProtST-ESM- +1b. Abbr., Loc.: localization prediction; Fit.: fitness prediction; +Func.: function annotation; Fix-enc.: fix-encoder learning; Full-m.: +full-model tuning. Blue denotes the largest decay. +Config +Loc. (mean Acc%) +Fit. (mean ρ) +Func. (mean Fmax) +Fix-enc. +Full-m. +Fix-enc. +Full-m. +Full loss +87.44 +85.54 +0.601 +0.772 +0.627 +w/o LMPM +87.40(↓0.05%) 85.12(↓0.49%) +0.593(↓1.33%) 0.766(↓0.78%) +0.625(↓0.32%) +w/o LGC +86.34(↓1.26%) 85.21(↓0.39%) +0.579(↓3.66%) 0.758(↓1.81%) +0.613(↓2.23%) +w/o LMMP +87.41(↓0.03%) 84.97(↓0.67%) +0.588(↓2.16%) 0.751(↓2.72%) +0.615(↓1.91%) +and the GO molecular function labels of heme binding. We +can observe that the top-3 candidates are annotated as heme +binders by GO, and the 4th candidate owns decent bind- +ing affinity though annotated as non-binding (only 0.54% +proteins are annotated as heme binders in the GO dataset). +These results verify the effectiveness of ProtST-ESM-1b on +retrieving heme binders. We provide more case studies in +Appendix C. Other visualization results are in Appendix E. +4.5. Ablation Study +Effect of Pre-training Losses: Tab. 5 reports the averaged +performance of ProtST-ESM-1b by using full or partial +pre-training losses (per-task results are in Appendix D.1). +By removing any of three pre-training losses, performance +decay occurs on all three types of tasks. Such phenomenon +verifies the necessity of each ProtST pre-training loss, where +LGC and LMMP inject different granularities of protein +property information into a PLM, and LMPM preserves the +PLM’s original representation power. +Effect of PLM: According to the results in Tabs. 2 and 3, +we can observe that the strength of a ProtST-induced PLM +correlates with the strength of its initial PLM. To be specific, +the better performance of ESM-1b and ESM-2 over ProtBert +is inherited by their ProtST-induced variants. +5. Related Work +Protein Representation Learning: Learning effective pro- +tein representations is of great importance for machine learn- +ing guided protein understanding. Existing works learn +protein representations in two ways: (1) Sequence-based +methods model protein sequences on evolutionary scale (El- +naggar et al., 2020; Rives et al., 2021; Lin et al., 2022) or +on individual protein families (Meier et al., 2021; Biswas +et al., 2021); (2) Structure-based methods seek to represent +different levels of protein structures including residue-level +structures (Gligorijevi´c et al., 2021; Zhang et al., 2022b; +Xu et al., 2022a), all-atom structures (Jing et al., 2020) and +protein surfaces (Gainza et al., 2020). Our work aims to +enhance protein sequence representation learning by using +textual protein property descriptions. +Multimodal Representation Learning: +It has been +broadly studied how to learn better image (Radford et al., +2021; Singh et al., 2022), video (Luo et al., 2020; Xu et al., +2021), speech (Chung et al., 2020; Qian et al., 2021) and +molecule (Edwards et al., 2021; Liu et al., 2022) representa- +tions by incorporating text supervision, while such study is +lacked for proteins. OntoProtein (Zhang et al., 2022a) learns +protein representations under the context of a knowledge +graph, while it investigates less the effect of biomedical +texts. Our work takes the initiative of enhancing protein +sequence representation learning by biomedical texts. +6. Conclusions and Future Work +In this work, we propose the ProtST framework to study +how textual protein property descriptions can boost protein +sequence pre-training and understanding. We build the Prot- +Describe dataset that aligns protein sequences with their +diverse property descriptions. ProtST pre-training injects +the property information with different granularities into a +protein language model (PLM). The ProtST-induced PLMs +are verified to be generally effective on various downstream +applications including supervised learning, zero-shot pro- +tein classification and zero-shot text-to-protein retrieval. +In the future, we will extend the ProtDescribe dataset by +incorporating protein structures and study biomedical text +enhanced protein structure representation learning. Also, we +will go beyond text-to-protein retrieval towards text-guided +controllable protein design. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +References +Almagro Armenteros, J. J., Sønderby, C. K., Sønderby, S. K., +Nielsen, H., and Winther, O. Deeploc: prediction of pro- +tein subcellular localization using deep learning. Bioin- +formatics, 33(21):3387–3395, 2017. +Baek, M., DiMaio, F., Anishchenko, I., Dauparas, J., +Ovchinnikov, S., Lee, G. R., Wang, J., Cong, Q., Kinch, +L. N., Schaeffer, R. D., et al. Accurate prediction of pro- +tein structures and interactions using a three-track neural +network. Science, 373(6557):871–876, 2021. +Bairoch, A. and Apweiler, R. The swiss-prot protein se- +quence database and its supplement trembl in 2000. Nu- +cleic acids research, 28(1):45–48, 2000. +Beltagy, I., Lo, K., and Cohan, A. +Scibert: +A pre- +trained language model for scientific text. arXiv preprint +arXiv:1903.10676, 2019. +Bhardwaj, N. and Lu, H. Correlation between gene ex- +pression profiles and protein–protein interactions within +and across genomes. Bioinformatics, 21(11):2730–2738, +2005. +Biswas, S., Khimulya, G., Alley, E. C., Esvelt, K. M., and +Church, G. M. Low-n protein engineering with data- +efficient deep learning. Nature methods, 18(4):389–396, +2021. +Capaldi, R. A. and Vanderkooi, G. The low polarity of +many membrane proteins. Proceedings of the National +Academy of Sciences, 69(4):930–932, 1972. +Chang, A., Jeske, L., Ulbrich, S., Hofmann, J., Koblitz, +J., Schomburg, I., Neumann-Schaal, M., Jahn, D., and +Schomburg, D. Brenda, the elixir core data resource in +2021: new developments and updates. Nucleic Acids +Research, 49(D1):D498–D508, 2021. +Chen, T., Kornblith, S., Norouzi, M., and Hinton, G. A +simple framework for contrastive learning of visual rep- +resentations. In International conference on machine +learning, pp. 1597–1607. PMLR, 2020. +Chung, Y.-A., Zhu, C., and Zeng, M. +Splat: Speech- +language joint pre-training for spoken language under- +standing. arXiv preprint arXiv:2010.02295, 2020. +Consortium, U. Uniprot: a worldwide hub of protein knowl- +edge. Nucleic acids research, 47(D1):D506–D515, 2019. +Dallago, C., Mou, J., Johnston, K. E., Wittmann, B. J., Bhat- +tacharya, N., Goldman, S., Madani, A., and Yang, K. K. +Flip: Benchmark tasks in fitness landscape inference for +proteins. bioRxiv, 2021. +Edwards, C., Zhai, C., and Ji, H. Text2mol: Cross-modal +molecule retrieval with natural language queries. In Pro- +ceedings of the 2021 Conference on Empirical Methods +in Natural Language Processing, pp. 595–607, 2021. +Elnaggar, A., Heinzinger, M., Dallago, C., Rihawi, G., +Wang, Y., Jones, L., Gibbs, T., Feher, T., Angerer, C., +Steinegger, M., et al. Prottrans: towards cracking the lan- +guage of life’s code through self-supervised deep learn- +ing and high performance computing. arXiv preprint +arXiv:2007.06225, 2020. +Gainza, P., Sverrisson, F., Monti, F., Rodola, E., Boscaini, +D., Bronstein, M., and Correia, B. Deciphering interac- +tion fingerprints from protein molecular surfaces using +geometric deep learning. Nature Methods, 17(2):184– +192, 2020. +Gligorijevi´c, V., Renfrew, P. D., Kosciolek, T., Leman, J. K., +Berenberg, D., Vatanen, T., Chandler, C., Taylor, B. C., +Fisk, I. M., Vlamakis, H., et al. Structure-based protein +function prediction using graph convolutional networks. +Nature communications, 12(1):1–14, 2021. +Gu, Y., Tinn, R., Cheng, H., Lucas, M., Usuyama, N., Liu, +X., Naumann, T., Gao, J., and Poon, H. Domain-specific +language model pretraining for biomedical natural lan- +guage processing. ACM Transactions on Computing for +Healthcare (HEALTH), 3(1):1–23, 2021. +He, K., Fan, H., Wu, Y., Xie, S., and Girshick, R. Mo- +mentum contrast for unsupervised visual representation +learning. In Proceedings of the IEEE/CVF conference on +computer vision and pattern recognition, pp. 9729–9738, +2020. +Hermosilla, P., Sch¨afer, M., Lang, M., Fackelmann, G., +V´azquez, P. P., Kozl´ıkov´a, B., Krone, M., Ritschel, T., and +Ropinski, T. Intrinsic-extrinsic convolution and pooling +for learning on 3d protein structures. +arXiv preprint +arXiv:2007.06252, 2020. +Jin, Q., Dhingra, B., Cohen, W. W., and Lu, X. Probing +biomedical embeddings from language models. arXiv +preprint arXiv:1904.02181, 2019. +Jing, B., Eismann, S., Suriana, P., Townshend, R. J., and +Dror, R. Learning from protein structure with geometric +vector perceptrons. arXiv preprint arXiv:2009.01411, +2020. +Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., +Ronneberger, O., Tunyasuvunakool, K., Bates, R., ˇZ´ıdek, +A., Potapenko, A., et al. Highly accurate protein structure +prediction with alphafold. Nature, 596(7873):583–589, +2021. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Khandelwal, U., Levy, O., Jurafsky, D., Zettlemoyer, L., +and Lewis, M. Generalization through memorization: +Nearest neighbor language models. +arXiv preprint +arXiv:1911.00172, 2019. +Kingma, D. P. and Ba, J. Adam: A method for stochastic +optimization. arXiv preprint arXiv:1412.6980, 2014. +Kumar, S., Tsai, C.-J., and Nussinov, R. Factors enhancing +protein thermostability. Protein engineering, 13(3):179– +191, 2000. +Lee, J., Yoon, W., Kim, S., Kim, D., Kim, S., So, C. H., +and Kang, J. Biobert: a pre-trained biomedical language +representation model for biomedical text mining. Bioin- +formatics, 36(4):1234–1240, 2020. +Lin, Z., Akin, H., Rao, R., Hie, B., Zhu, Z., Lu, W., dos +Santos Costa, A., Fazel-Zarandi, M., Sercu, T., Candido, +S., et al. Language models of protein sequences at the +scale of evolution enable accurate structure prediction. +bioRxiv, 2022. +Liu, S., Nie, W., Wang, C., Lu, J., Qiao, Z., Liu, L., Tang, +J., Xiao, C., and Anandkumar, A. Multi-modal molecule +structure-text model for text-based retrieval and editing. +arXiv preprint arXiv:2212.10789, 2022. +Lu, A. X., Zhang, H., Ghassemi, M., and Moses, A. M. Self- +supervised contrastive learning of protein representations +by mutual information maximization. BioRxiv, 2020. +Luo, H., Ji, L., Shi, B., Huang, H., Duan, N., Li, T., Li, J., +Bharti, T., and Zhou, M. Univl: A unified video and lan- +guage pre-training model for multimodal understanding +and generation. arXiv preprint arXiv:2002.06353, 2020. +Meier, J., Rao, R., Verkuil, R., Liu, J., Sercu, T., and Rives, +A. Language models enable zero-shot prediction of the +effects of mutations on protein function. bioRxiv, 2021. +Murzin, A. G., Brenner, S. E., Hubbard, T., and Chothia, C. +Scop: a structural classification of proteins database for +the investigation of sequences and structures. Journal of +molecular biology, 247(4):536–540, 1995. +Oord, A. v. d., Li, Y., and Vinyals, O. Representation learn- +ing with contrastive predictive coding. arXiv preprint +arXiv:1807.03748, 2018. +Organization, W. H. and University, U. N. Protein and +amino acid requirements in human nutrition, volume 935. +World Health Organization, 2007. +Qian, Y., Bianv, X., Shi, Y., Kanda, N., Shen, L., Xiao, +Z., and Zeng, M. +Speech-language pre-training for +end-to-end spoken language understanding. In ICASSP +2021-2021 IEEE International Conference on Acoustics, +Speech and Signal Processing (ICASSP), pp. 7458–7462. +IEEE, 2021. +Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G., +Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., +et al. Learning transferable visual models from natural +language supervision. In International Conference on +Machine Learning, pp. 8748–8763. PMLR, 2021. +Rao, R., Bhattacharya, N., Thomas, N., Duan, Y., Chen, P., +Canny, J., Abbeel, P., and Song, Y. Evaluating protein +transfer learning with tape. Advances in neural informa- +tion processing systems, 32, 2019. +Rives, A., Meier, J., Sercu, T., Goyal, S., Lin, Z., Liu, J., +Guo, D., Ott, M., Zitnick, C. L., Ma, J., et al. Biological +structure and function emerge from scaling unsupervised +learning to 250 million protein sequences. Proceedings +of the National Academy of Sciences, 118(15), 2021. +Shanehsazzadeh, A., Belanger, D., and Dohan, D. Is trans- +fer learning necessary for protein landscape prediction? +arXiv preprint arXiv:2011.03443, 2020. +Singh, A., Hu, R., Goswami, V., Couairon, G., Galuba, +W., Rohrbach, M., and Kiela, D. Flava: A foundational +language and vision alignment model. In Proceedings +of the IEEE/CVF Conference on Computer Vision and +Pattern Recognition, pp. 15638–15650, 2022. +Steinegger, M. and S¨oding, J. Clustering huge protein se- +quence sets in linear time. Nature communications, 9(1): +1–8, 2018. +Teague, S. J. Implications of protein flexibility for drug +discovery. Nature reviews Drug discovery, 2(7):527–541, +2003. +Trott, O. and Olson, A. J. Autodock vina: improving the +speed and accuracy of docking with a new scoring func- +tion, efficient optimization, and multithreading. Journal +of computational chemistry, 31(2):455–461, 2010. +Van der Maaten, L. and Hinton, G. Visualizing data using +t-sne. Journal of machine learning research, 9(11), 2008. +Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, +L., Gomez, A. N., Kaiser, Ł., and Polosukhin, I. At- +tention is all you need. Advances in neural information +processing systems, 30, 2017. +Xu, H., Ghosh, G., Huang, P.-Y., Okhonko, D., Aghajanyan, +A., Metze, F., Zettlemoyer, L., and Feichtenhofer, C. +Videoclip: Contrastive pre-training for zero-shot video- +text understanding. arXiv preprint arXiv:2109.14084, +2021. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Xu, M., Guo, Y., Xu, Y., Tang, J., Chen, X., and Tian, Y. Eu- +rnet: Efficient multi-range relational modeling of spatial +multi-relational data. arXiv preprint arXiv:2211.12941, +2022a. +Xu, M., Zhang, Z., Lu, J., Zhu, Z., Zhang, Y., Ma, C., Liu, +R., and Tang, J. Peer: A comprehensive and multi-task +benchmark for protein sequence understanding. arXiv +preprint arXiv:2206.02096, 2022b. +Zhang, N., Bi, Z., Liang, X., Cheng, S., Hong, H., Deng, +S., Lian, J., Zhang, Q., and Chen, H. Ontoprotein: Pro- +tein pretraining with gene ontology embedding. arXiv +preprint arXiv:2201.11147, 2022a. +Zhang, Z., Xu, M., Jamasb, A., Chenthamarakshan, V., +Lozano, A., Das, P., and Tang, J. Protein representa- +tion learning by geometric structure pretraining. arXiv +preprint arXiv:2203.06125, 2022b. +Zhu, Z., Shi, C., Zhang, Z., Liu, S., Xu, M., Yuan, X., +Zhang, Y., Chen, J., Cai, H., Lu, J., et al. Torchdrug: A +powerful and flexible machine learning platform for drug +discovery. arXiv preprint arXiv:2202.08320, 2022. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +A. Model Architecture for Pre-training +Fusion Module: The fusion module extracts multimodal +representations from the unimodal representations of protein +sequence and text description. As shown in Fig. 5, each fu- +sion layer of this module receives a sequence of residue rep- +resentations ZS = [zs +1, zs +2, · · · , zs +n] ∈ Rn×d and a sequence +of word representations ZT = [zt +1, zt +2, · · · , zt +m] ∈ Rm×d +(d denotes the hidden dimension), and the layer updates +each residue/word representation by attending to all residues +and all words. Specifically, two sets of projection matrices +(W S +q , W S +k , W S +v ) and (W T +q , W T +k , W T +v ) are respectively used +to derive the queries, keys and values for protein sequence +and text description as below (each projection matrix is in +Rd×d): +QS = ZSW S +q , +KS = ZSW S +k , +V S = ZSW S +v , +(3) +QT = ZT W T +q , +KT = ZT W T +k , +V T = ZT W T +v , (4) +where QS, KS, V S ∈ Rn×d are the queries, keys and val- +ues for protein sequence, and QT , KT , V T ∈ Rm×d are the +queries, keys and values for text description. Multi-head +self- and cross-attention are then applied to update each +residue and word representation as below: +˜ZS = 1 +2 +� +MHA(QS, KS, V S) + MHA(QS, KT , V T ) +� +, +(5) +˜ZT = 1 +2 +� +MHA(QT , KT , V T ) + MHA(QT , KS, V S) +� +, (6) +where ˜ZS ∈ Rn×d and ˜ZT ∈ Rm×d are the updated +residue and word representations, and MHA(·, ·, ·) denotes +the multi-head attention operation (Vaswani et al., 2017). +In our implementation, each fusion layer contains 8 attention +heads, and we equip the fusion module with a single fusion +layer so as to restrict the capacity of fusion module and +facilitate the representation power of PLM. Upon the fused +residue and word representations produced by the fusion +module, multimodal mask prediction is performed. +Projection Head for Multimodal Representation Align- +ment: Following SimCLR (Chen et al., 2020), we use a +two-layer MLP (with ReLU nonlinearity in between) to +project the protein sequence representation extracted by the +PLM, and another two-layer nonlinear MLP is employed to +project the text description representation extracted by the +BLM. The projected sequence and text representations are +then used to compute the global contrastive loss defined in +Eq. (1). +Prediction Head for Masked Protein Modeling (MPM): +Based on the residue representations extracted by the PLM, +we utilize a two-layer MLP (with ReLU nonlinearity in +between) to predict the type of each residue token masked +at input. +Figure 5: Architecture of the fusion layer. This layer fuses +the protein representation and the text representation by +querying over them with self-attention and cross-attention. +Prediction Head for Multimodal Mask Prediction +(MMP): Upon the fused residue representations output from +the fusion module, a two-layer MLP (with ReLU nonlinear- +ity in between) is used to predict the type of each residue +token masked at input protein sequence. Upon the fused +word representations produced by the fusion module, an- +other two-layer nonlinear MLP is employed to predict each +word token masked at input text description. +B. More Experimental Setups +B.1. More Pre-training Setups +Pre-training Data Curation: We add prefixes to denote +annotations from different fields, i.e., “PROTEIN NAME” +for the protein name field, “FUNCTION” for the protein +function field, “SUBCELLULAR LOCATION” for the sub- +cellular location field, and “SIMILARITY” for the protein +family field. The complete protein property description is +formed by concatenating all annotations of the protein in the +order of (1) protein name, (2) protein function, (3) subcel- +lular location, and (4) protein family. In Tab. 6, we present +several property descriptions coupled with the Swiss-Prot +entry names of their corresponding proteins. +Training Configurations: We list the training configura- +tions of three ProtST-induced PLMs in Tab. 7. In general, an +Adam optimizer with the constant learning rate of 1.0×10−5 +is used to train the model for 20 epochs on 4 Tesla V100 +GPUs, where ProtST-ProtBert adopts the batch size of 16 (4 +proteins per GPU), and ProtST-ESM-1b and ProtST-ESM-2 +adopt the batch size of 12 (3 proteins per GPU). Since the +PLM is pre-trained, we set its learning rate as 1.0 × 10−6, +i.e., one tenth of other modules. The weights of PubMed- + +Multi-head +Multi-head +Attention +Attention +Qs +Ks +sA +Ks +Vs +zs +zs +ZT +KT +VT +os +Multi-head +Multi-head +Attention +Attention +Self-Attention +Cross-AttentionProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Table 6: Examples of property descriptions in the ProtDescribe dataset. We index each description with the Swiss-Prot entry +name of its corresponding protein. +Entry +name +Description +14336 ORYSJ +PROTEIN NAME: 14-3-3-like protein GF14-F. FUNCTION: Is associated with a DNA binding complex +that binds to the G box, a well-characterized cis-acting DNA regulatory element found in plant genes. +SUBCELLULAR LOCATION: Cytoplasm. Nucleus. SIMILARITY: Belongs to the 14-3-3 family. +053R FRG3G +PROTEIN NAME: Putative myristoylated protein 053R. FUNCTION: May play a critical role in virion +formation. Essential for virus replication in vitro. SUBCELLULAR LOCATION: Host membrane; Multi-pass +membrane protein. +1A16 ORYSJ +PROTEIN NAME: 1-aminocyclopropane-1-carboxylate synthase 6. FUNCTION: Catalyzes the formation of +1-aminocyclopropane-1-carboxylate, a direct precursor of ethylene in higher plants (By similarity). Required +for the regulation of starch grain size in endosperm. SUBCELLULAR LOCATION: Plastid, amyloplast +membrane. Note=Localizes to the amyloplast membrane surrounding starch grains in endosperm, pollen, and +pericarp. SIMILARITY: Belongs to the class-I pyridoxal-phosphate-dependent aminotransferase family. +17KD RICPR +PROTEIN NAME: 17 kDa surface antigen. SUBCELLULAR LOCATION: Cell outer membrane; Lipid- +anchor. SIMILARITY: Belongs to the rickettsiale 17 kDa surface antigen family. +1A1D CYBSA +PROTEIN NAME: 1-aminocyclopropane-1-carboxylate deaminase. FUNCTION: Catalyzes a cyclopropane +ring-opening reaction, the irreversible conversion of 1-aminocyclopropane-1-carboxylate (ACC) to ammonia +and alpha-ketobutyrate. SIMILARITY: Belongs to the ACC deaminase/D-cysteine desulfhydrase family. +1AP1 BRAOT +PROTEIN NAME: Floral homeotic protein APETALA 1-1. FUNCTION: Transcription factor that promotes +early floral meristem identity in synergy with LEAFY. Displays a redundant function with CAULIFLOWER +in the up-regulation of LEAFY. Required subsequently for the transition of an inflorescence meristem into a +floral meristem, and for the normal development of sepals and petals in flowers. Regulates positively B class +homeotic proteins (By similarity). SUBCELLULAR LOCATION: Nucleus. +Table 7: ProtST pre-training configurations. Abbr., lr.: learn- +ing rate; bs.: batch size. +Model +optimizer +lr. +bs. +#epochs +train time +ProtST-ProtBert +Adam +1.0 × 10−5 +16 +20 +117h 10min +ProtST-ESM-1b +Adam +1.0 × 10−5 +12 +20 +205h 36min +ProtST-ESM-2 +Adam +1.0 × 10−5 +12 +20 +206h 12min +BERT are frozen along the whole process. To reduce the +memory cost, we truncate the protein sequences that have +more than 450 residues to the length of 450, where the +truncation starts from a random residue before the last 450 +ones. Following MoCo (He et al., 2020), we initialize the +temperature parameter τ in Eq. (1) as 0.07 and optimize it +along the training process. +B.2. More Representation Learning Setups +Architecture of Prediction Heads: Following the default +settings in TorchDrug (Zhu et al., 2022), the prediction of +each task is performed by a two-layer MLP with ReLU +nonlinearity in between. To be specific, given the protein +representation, the MLP head is used to predict classification +logits for localization prediction, regression score for fitness +prediction and per-function classification logits for function +Table 8: Configurations of fix-encoder learning and full- +model tuning on three task types. Abbr., lr.: learning rate; +bs.: batch size; MSE: mean squared error; CE: cross en- +tropy; BCE: binary cross entropy. +Task +optimizer +lr. +bs. +#epochs +loss +fix-encoder learning +Localization +Adam +5.0 × 10−5 +128 +100 +CE +Fitness +Adam +5.0 × 10−5 +128 +100 +MSE +full-model tuning +Localization +Adam +2.0 × 10−4 +12 +100 +CE +Fitness +Adam +2.0 × 10−4 +24 +100 +MSE +Annotation +Adam +1.0 × 10−4 +8 +50 +BCE +annotation. +Training Configurations: In Tab. 8, we present the de- +tailed configurations of fix-encoder learning and full-model +tuning on three task types, which mainly follows the config- +urations used in PEER benchmark (Xu et al., 2022b). For +full-model tuning, the learning rate of the PLM is set as one +tenth of the value in Tab. 8. The protein sequence encoders +trained from scratch do not use smaller learning rates. All +experiments are conducted on 4 Tesla V100 GPUs. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Table 9: Zero-shot protein classification performance under dif- +ferent prompt templates. Abbr., Acc: accuracy; loc.: localization. +Prompt template +Label +Subcellular loc. (Acc%) +Reaction (Acc%) +Name only +Name +25.68 +25.27 +Natural language +Name +36.24 +26.93 +Pre-training template +Name +43.49 +29.85 +Pre-training template +Description +29.90 +21.91 +Evaluation Metrics: The protein function annotation tasks +are measured by AUPR and Fmax. We clarify their defini- +tions as below: +(1) AUPR denotes the pair-centric area under precision- +recall curve. It computes the average precision scores for all +protein-function pairs, which is exactly the micro-average +precision score for the multiple binary classification prob- +lem. +(2) Fmax denotes the protein-centric maximum F-score. +Given a decision threshold t ∈ [0, 1], it first calculates the +precision and recall for each protein: +precisioni(t) = +� +f 1[f ∈ Pi(t) ∩ Ti] +� +f 1[f ∈ Pi(t)] +, +(7) +recalli(t) = +� +f 1[f ∈ Pi(t) ∩ Ti] +� +f 1[f ∈ Ti] +, +(8) +where f denotes a functional term of EC or GO, Ti is the +set collecting all experimentally determined functions for +protein i, Pi(t) denotes the predicted functions for protein i +whose scores are at least t, and 1[·] represents the indicator +function. The precision and recall are then averaged over all +proteins: +precision(t) = +1 +M(t) +� +i +precisioni(t), +(9) +recall(t) = 1 +N +� +i +recalli(t), +(10) +where N is the total number of proteins, and M(t) denotes +the number of proteins that contain at least one prediction +larger than t, i.e., |Pi(t)| > 0. +Finally, the Fmax score is computed as the maximum value +of F-measure over all thresholds: +Fmax = max +t +�2 · precision(t) · recall(t) +precision(t) + recall(t) +� +. +(11) +B.3. More Zero-shot Protein Classification Setups +Prompt Engineering for Subcellular Localization Pre- +diction: +Based +on +the +information +provided +by +DeepLoc (Almagro Armenteros et al., 2017), we consider +two label formats, the name of each subcellular location +(i.e., the “Location” field in the Tab. 1 of DeepLoc paper) +and the description of each location (i.e., the “Sublocations” +field in the Tab. 1 of DeepLoc paper). We further embed the +labels into three prompt templates: (1) Name only: only the +label itself is used; (2) Natural language: the label is em- +bedded into the template “A protein locating at {label}.”; +(3) Pre-training template: the label is embedded into the +template “SUBCELLULAR LOCATION: {label}”. +According to the results in Tab. 9, we can observe that the +pre-training template clearly outperforms other two tem- +plates on the subcellular localization prediction task, which +mainly owes to the alignment of text format across pre- +training and zero-shot prediction. It is shown that represent- +ing the labels with location names leads to better perfor- +mance than using location descriptions, since the location +names better fit the biomedical text distribution that the +BLM is trained on. Based on these results, we represent the +labels with the location names coupled with the pre-training +prompt template on this task. +Prompt Engineering for Reaction Classification: Same +as subcellular localization prediction, we also use two sets of +label notations for reaction classification, i.e., the name and +the description. (1) The name refers to the composition of +the enzyme class name and its alternative names, allowing +unambiguous identification of each enzyme class. (2) The +description further adds the scientific comments that discuss +each class of enzymes in depth, which are extracted from +scientific articles published by the International Union of +Biochemistry and Molecular Biology (IUBMB). We retrieve +all the information from Chang et al. (2021). +We embed such label information into three prompt tem- +plates: (1) Name only: the concatenation of the name +and alternative names of an enzyme class, i.e., “{Name} +{AlterNames}”; (2) Natural Language: the label is incor- +porated into a natural-language-like template “A {Name} +enzyme. This enzyme is also known as {AlterNames}.”; +(3) Pre-training template: the label is merged into the tem- +plate used for pre-training, i.e., “FUNCTION: {Name} +{AlterNames}” (scientific comments “{Comments}” +are appended after the names if the description is used). +According to Tab. 9, the pre-training template performs the +best on the reaction classification task, mainly thanks to the +consistent format of text descriptions between pre-training +and zero-shot prediction. Injecting detailed scientific com- +ments does not bring further benefits to the zero-shot per- +formance. Therefore, we represent each enzyme class with +its name and alternative names along with the pre-training +prompt template for this task. +Nonparametric Few-shot Classifier: We adopt the non- +parametric classifier proposed by Khandelwal et al. (2019) +as baseline. Specifically, given n-shot K-class training sam- + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +Table 10: Ablation study of pre-training losses on localization +and fitness prediction. Abbr., Loc.: Localization; pred.: prediction; +Acc: accuracy. Gray denotes the performance decay. +Model +Loc. pred. (Acc%) +Fitness pred. (Spearman’s ρ) +Bin +Sub +β-lac +AAV +Thermo +Flu +Sta +Mean ρ +Fix-encoder learning +ProtST-ESM-1b +92.87 +82.00 +0.578 +0.460 +0.680 +0.523 +0.766 +0.601 +ProtST-ESM-1b (w/o LMPM) +92.52 +82.28 +0.558 +0.475 +0.680 +0.522 +0.730 +0.593 +ProtST-ESM-1b (w/o LGC) +92.12 +80.55 +0.560 +0.448 +0.684 +0.467 +0.738 +0.579 +ProtST-ESM-1b (w/o LMMP) +92.81 +82.00 +0.544 +0.479 +0.681 +0.504 +0.731 +0.588 +Full-model tuning +ProtST-ESM-1b +92.35 +78.73 +0.895 +0.850 +0.681 +0.682 +0.751 +0.772 +ProtST-ESM-1b (w/o LMPM) +92.64 +77.59 +0.894 +0.842 +0.681 +0.685 +0.726 +0.766 +ProtST-ESM-1b (w/o LGC) +91.67 +78.75 +0.891 +0.798 +0.674 +0.686 +0.741 +0.758 +ProtST-ESM-1b (w/o LMMP) +91.90 +78.03 +0.902 +0.804 +0.677 +0.678 +0.696 +0.751 +Table 11: Ablation study of pre-training losses on function anno- +tation. Gray denotes the performance decay. +Model +EC +GO-BP +GO-MF +GO-CC +AUPR +Fmax +AUPR +Fmax +AUPR +Fmax +AUPR +Fmax +Full-model tuning +ProtST-ESM-1b +0.894 +0.878 +0.328 +0.480 +0.644 +0.661 +0.364 +0.488 +ProtST-ESM-1b (w/o LMPM) +0.898 +0.873 +0.324 +0.483 +0.642 +0.660 +0.350 +0.482 +ProtST-ESM-1b (w/o LGC) +0.894 +0.870 +0.322 +0.463 +0.638 +0.656 +0.327 +0.462 +ProtST-ESM-1b (w/o LMMP) +0.890 +0.871 +0.328 +0.456 +0.635 +0.659 +0.340 +0.473 +ples {{(Sk +i , yk +i = k)}n +i=1}K +k=1 composed of pairs of protein +sequence and label, we employ the PLM to extract the rep- +resentations {{zk +i }n +i=1}K +k=1 of all protein sequences. When +a test protein S′ comes, the nonparametric classifier first +extracts its representation z′ via the PLM and then derives +its classification logits {y′ +k}K +k=1 by computing its represen- +tation similarity with each training protein: +y′ +k = +n +� +i=1 +exp +� +−||z′ − zk +i ||2 +2 +� +, +k = 1, · · · , K. +(12) +Softmax is performed upon these logits to derive classifi- +cation probabilities. Such a classifier predicts based on the +relations between test sample and training samples, which +well fits the few-shot setting. In our experiments, the non- +parametric classifier based on ESM-1b and the one based +on ProtST-ESM-1b serve as two baselines for zero-shot +classifiers. +C. More Zero-shot Text-to-Protein Retrieval +Results +In Fig. 10, we study four more sets of text-to-protein re- +trieval of ligand binders based on ProtST-ESM-1b. For each +study, we visualize the text prompt and the top-4 retrieved +candidates. For each candidate, we present the docking +result of it binding with the ligand, the binding affinity +and its GO molecular function label of binding with the +ligand, where AutoDock Vina (Trott & Olson, 2010) is +used to estimate docking pose and binding affinity. It is +observed that, among the top-4 candidates, ProtST-ESM-1b +succeeds in retrieving 3 GO-annotated ATP binders (only +3.99% proteins are annotated as ATP binders in GO), 3 GO- +annotated GTP binders (only 1.18% proteins are annotated +Table 12: Ablation study of BLM on localization and fitness +prediction. ProtST-ESM-1b serves as the base model. Abbr., Loc.: +Localization; pred.: prediction; Acc: accuracy. +BLM +Loc. pred. (Acc%) +Fitness pred. (Spearman’s ρ) +Bin +Sub +β-lac +AAV +Thermo +Flu +Sta +Mean ρ +Fix-encoder learning +PubMedBERT-abs +92.87 +82.00 +0.578 +0.460 +0.680 +0.523 +0.766 +0.601 +PubMedBERT-full +93.04 +82.28 +0.548 +0.458 +0.682 +0.507 +0.744 +0.588 +Full-model tuning +PubMedBERT-abs +92.35 +78.73 +0.895 +0.850 +0.681 +0.682 +0.751 +0.772 +PubMedBERT-full +92.87 +78.77 +0.899 +0.785 +0.672 +0.680 +0.722 +0.752 +Table 13: Ablation study of BLM on function annotation. ProtST- +ESM-1b serves as the base model. +BLM +EC +GO-BP +GO-MF +GO-CC +AUPR +Fmax +AUPR +Fmax +AUPR +Fmax +AUPR +Fmax +Full-model tuning +PubMedBERT-abs +0.894 +0.878 +0.328 +0.480 +0.644 +0.661 +0.364 +0.488 +PubMedBERT-full +0.905 +0.878 +0.323 +0.475 +0.630 +0.652 +0.374 +0.485 +as GTP binders in GO), 2 GO-annotated P5P binders (only +0.17% proteins are annotated as P5P binders in GO), and +2 GO-annotated NAD+ binders (only 0.05% proteins are +annotated as NAD+ binders in GO). The rest candidates +annotated as non-binding also own decent binding affinity, +e.g., the better binding affinity of protein 2AKA-B (with- +out ATP binder annotation) against protein 6EAC-A (with +ATP binder annotation), the better binding affinity of protein +5DHG-A (without NAD+ binder annotation) against protein +3GFB-A (with NAD+ binder annotation), etc. These results +demonstrate the general effectiveness of ProtST-ESM-1b on +retrieving the binders of diverse ligands. In the future work, +we will study how ProtST enables zero-shot text-to-protein +retrieval of other types of functional proteins, e.g., antigen +binders, toxic substance binders, transcription factors, etc. +D. More Ablation Study +D.1. Ablation Study of Pre-training Losses +In Tabs. 10 and 11, we report the performance of ProtST- +ESM-1b on all benchmark tasks by using full or partial +pre-training losses. It can be observed that: (1) removing +the loss LMPM leads to performance decay on 16 out of +24 benchmark metrics; (2) removing the loss LGC leads to +decay on 20 out of 24 benchmark metrics; (3) removing the +loss LMMP diminishes model performance on 19 out of 24 +benchmark metrics. Therefore, all pre-training losses are +necessary to maximize the effectiveness of a ProtST-induced +PLM, where LGC and LMMP inject different granularities +of protein property information into a PLM, and LMPM +preserves the PLM’s original representation power. +D.2. Ablation Study of Biomedical Language Model +PubMedBERT owns two versions: (1) the PubMedBERT- +abs trained by using only PubMed abstracts, and (2) the + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +A +R +N +D +C +E +Q +G +H +I +L +K +M +F +P +S +T +W +Y +V +Hydrophobic (aromatic) +Hydrophobic (aliphatic) +Positive +Negative +Polar neutral +Special cases +Small (<130 Dalton) +Medium +Big (>150 Dalton) +Figure 6: Amino acid representations learned by the linear layer +for unimodal mask prediction (ProtST-ESM-1b is used). +A +R +N +D +C +E +Q +G +H +I +L +K +M +F +P +S +T +W +Y +V +Hydrophobic (aromatic) +Hydrophobic (aliphatic) +Positive +Negative +Polar neutral +Special cases +Small (<130 Dalton) +Medium +Big (>150 Dalton) +Figure 7: Amino acid representations learned by the linear layer +for multimodal mask prediction (ProtST-ESM-1b is used). +PubMedBERT-full trained by using additional PubMed Cen- +tral full-text articles. In this experiment, we compare the +effectiveness of these two models by respectively using +them as the BLM of ProtST-ESM-1b. +Tabs. 12 and 13 report the performance comparison of these +two models on all benchmark tasks. We can observe that: +(1) PubMedBERT-full outperforms PubMedBERT-abs on +all four benchmark metrics of localization prediction; (2) +PubMedBERT-abs performs better than PubMedBERT-full +on 10 out of 12 benchmark metrics of fitness prediction; +(3) PubMedBERT-abs outperforms PubMedBERT-full on 5 +out of 8 benchmark metrics of function annotation. There- +fore, PubMedBERT-full does not show superiority over +PubMedBERT-abs in ProtST pre-training, which owes to +the fact that the protein property descriptions in the ProtDe- +scribe dataset are more like abstracts than full-text articles. +Membrane-bound +Soluble +Figure 8: Visualization of protein representations on the binary +localization prediction dataset (ProtST-ESM-1b is used). +Cell membrane +Cytoplasm +Endoplasmic reticulum (ER) +Golgi apparatus +Lysosome/Vacuole +Mitochondrion +Nucleus +Peroxisome +Plastid +Extracellular +Figure 9: Visualization of protein representations on the subcellu- +lar localization prediction dataset (ProtST-ESM-1b is used). +E. More Visualization +Well-trained PLMs should have the capacity to extract struc- +tural, functional, and even evolutionary features of proteins. +As a result, the learned representations in PLMs are ex- +pected to have certain intrinsic organization patterns in the +embedding space to capture these protein characteristics. To +demonstrate the effectiveness of ProtST-ESM-1b, we use +t-SNE (Van der Maaten & Hinton, 2008) to visualize such +information at different scales from amino acid decomposi- +tions to protein functional properties. +Biophysical Properties of Amino Acids: It is known that +the biophysical properties of amino acids, such as hydropho- +bicity, aromaticity and charge, highly influence the bio- +logical structures of proteins and therefore their biological +functions as well. To investigate if ProtST-ESM-1b captures +such intrinsic features, we apply t-SNE to the two linear +layers used for unimodal mask prediction and multimodal + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +mask prediction. As shown in Figs. 6 and 7, hydrophobic +and polar residues exhibit clear distinct clusterings, even to +the level of aliphatic v.s. aromatic. The clustering is also +coherent in terms of the charge and size of the amino acids. +Biological and Biochemical Properties of Proteins: As +introduced in Sec. 4.1, our proposed ProtDescribe dataset +provides ProtST-ESM-1b with direct access to knowledge +like protein subcellular localizations, which refers to a spe- +cific region within a cell where the proteins can be found. +For a protein, such locations can influence its activity and +interaction with other molecules, thus helping the PLMs to +better capture the biological and biomedical protein func- +tions. To validate this assumption, we adopt the datasets +used in two protein localization prediction tasks, i.e., the +subcellular localization prediction and the binary localiza- +tion prediction. With t-SNE, we project protein representa- +tions to the 2-dimensional space for these two benchmark +datasets. In Figs. 8 and 9, certain clustering patterns of +different cellular locations are observed. + +ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts +(1st) 6C6B-A: +• +Affinity: -8.7 (kcal/mol) +• +GO-MF label: Bind +(2nd) 6EAC-A: +• +Affinity: -8.2 (kcal/mol) +• +GO-MF label: Bind +(3rd) 2AKA-B: +• +Affinity: -8.4 (kcal/mol) +• +GO-MF label: Non-bind +(4th) 1YID-B: +• +Affinity: -7.8 (kcal/mol) +• +GO-MF label: Bind +(a) Prompt - FUNCTION: Binding to ATP, adenosine 5'-triphosphate, a universally important coenzyme and enzyme regulator. +(b) Prompt - FUNCTION: Binding to GTP, guanosine triphosphate. +(1st) 5C1S-A: +• +Affinity: -7.5 (kcal/mol) +• +GO-MF label: Bind +(2nd) 2CVH-A: +• +Affinity: -7.5 (kcal/mol) +• +GO-MF label: Non-bind +(3rd) 4DHE-A: +• +Affinity: -6.8 (kcal/mol) +• +GO-MF label: Bind +(4th) 5HXB-X: +• +Affinity: -6.4 (kcal/mol) +• +GO-MF label: Bind +(c) Prompt - FUNCTION: Binding to pyridoxal 5' phosphate, the biologically active form of vitamin B6. +(1st) 4ILS-A: +• +Affinity: -5.8 (kcal/mol) +• +GO-MF label: Bind +(2nd) 5LL2-A: +• +Affinity: -5.3 (kcal/mol) +• +GO-MF label: Bind +(3rd) 1EHI-A: +• +Affinity: -5.6 (kcal/mol) +• +GO-MF label: Non-bind +(4th) 3AG6-A: +• +Affinity: -5.8 (kcal/mol) +• +GO-MF label: Non-bind +(d) Prompt - FUNCTION: Binding to the oxidized form, NAD, of nicotinamide adenine dinucleotide, +a coenzyme involved in many redox and biosynthetic reactions. +(1st) 5DHG-A: +• +Affinity: -8.6 (kcal/mol) +• +GO-MF label: Non-bind +(2nd) 3GFB-A: +• +Affinity: -7.9 (kcal/mol) +• +GO-MF label: Bind +(3rd) 5OXU-A: +• +Affinity: -9.7 (kcal/mol) +• +GO-MF label: Non-bind +(4th) 3GGO-A: +• +Affinity: -7.2 (kcal/mol) +• +GO-MF label: Bind +Figure 10: Zero-shot text-to-protein retrieval of (a) ATP binders, (b) GTP binders, (c) P5P binders, and (d) NAD+ binders +based on ProtST-ESM-1b. + diff --git a/VtFLT4oBgHgl3EQfSS81/content/tmp_files/load_file.txt b/VtFLT4oBgHgl3EQfSS81/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..465f2ac5c126abd5f35f8670744b1a8ed0512dbd --- /dev/null +++ b/VtFLT4oBgHgl3EQfSS81/content/tmp_files/load_file.txt @@ -0,0 +1,1823 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf,len=1822 +page_content='ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts Minghao Xu * † 1 2 Xinyu Yuan * 1 2 Santiago Miret 3 Jian Tang 1 4 5 Abstract Current protein language models (PLMs) learn protein representations mainly based on their se- quences, thereby well capturing co-evolutionary information, but they are unable to explicitly ac- quire protein functions, which is the end goal of protein representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' Fortunately, for many proteins, their textual property descrip- tions are available, where their various functions are also described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' Motivated by this fact, we first build the ProtDescribe dataset to augment protein sequences with text descriptions of their functions and other important properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' Based on this dataset, we propose the ProtST frame- work to enhance Protein Sequence pre-training and understanding by biomedical Texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' During pre-training, we design three types of tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=', unimodal mask prediction, multimodal represen- tation alignment and multimodal mask prediction, to enhance a PLM with protein property informa- tion with different granularities and, at the same time, preserve the PLM’s original representation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' On downstream tasks, ProtST enables both supervised learning and zero-shot predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' We verify the superiority of ProtST-induced PLMs over previous ones on diverse representa- tion learning benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' Under the zero-shot setting, we show the effectiveness of ProtST on zero-shot protein classification, and ProtST also enables functional protein retrieval from a large- scale database without any function annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' Introduction Proteins serve as the mainstay governing diverse biological processes and life itself, inducing important applications in drug discovery (Teague, 2003) and healthcare (Organization Equal technical contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' †Project lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' 1Mila - Qu´ebec AI Institute 2Universit´e de Montr´eal 3Intel Labs 4HEC Montr´eal 5CIFAR AI Research Chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFLT4oBgHgl3EQfSS81/content/2301.12040v1.pdf'} +page_content=' Correspondence to: Ming- hao Xu 0). +The following result is obvious. +Lemma 2.2 (see [5, Theorem (14.5), Chapter V]). Let (F, h) be a Hermitian holomorphic +vector bundle over X, and let E, G be two holomorphic subbundles of F such that F = E⊕G +and E is orthogonal to G, then the curvature of these bundles satisfies +ΘF = ΘE ⊕ ΘG. +One of the main ingredients in our argument of the main results is the following result of +Berndtsson. +Lemma 2.3 ([3, (3.1)]). If Ω and ϕ satisfy the conditions in Theorem 1.1 and ϕ is strictly +plurisubharmonic, then for any smooth sections u1, · · · , un of the trivial bundle E, we have +� +j,l +(ΘE +jluj, ul) ≥ +� +j,l +� +D +H(ϕ)jlujule−ϕdλz +where +H(ϕ)jl := ϕjl − +� +α,β +ϕαβϕjαϕlβ, +where (ϕαβ)m×m is the inverse matrix of (ϕαβ)m×m. + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +11 +In the above Lemma, j, l = 1, · · · , n represent the indices of the components of t = +(t1, · · · , tn), α, β = 1, · · · , m represent the indices of the components of z = (z1, · · · , zm), +ϕj,l = +∂2ϕ +∂tj∂¯tl and ϕjα, ϕαβ are given in the same way. +2.3. Optimal L2-estimate condition and curvature positivity. +We first recall a fundamental result about the L2-estimate of ¯∂ for a Hermitian holomorphic +vector bundle with Nakano positive curvature, which is due to H¨ormander and Demailly. +Lemma 2.4 (see [5, Theorem (4.5), Chapter VIII]). Let X be a complete K¨ahler mani- +fold, with a K¨ahler metric ω which is not necessarily complete. Let (E, h) be a Hermitian +vector bundle of rank r over X, and assume that the curvature operator B := [iΘE,h, Λω] +is semi-positive definite everywhere on Λp,qT ∗ +X ⊗ E, for some q ≥ 1. Then for any form +g ∈ L2(X, Λp,qT ∗ +X ⊗ E) satisfying ¯∂g = 0 and +� +X⟨B−1g, g⟩dVω < +∞, there exists f ∈ +L2(X, Λp,q−1T ∗ +X ⊗ E) such that ¯∂f = g and +� +X +|f|2dVω ≤ +� +X +⟨B−1g, g⟩dVω. +The following result of Deng-Ning-Wang-Zhou shows that the converse of the above Lemma +also holds, and hence gives an equivalent integral form characterization of the curvature +positivity of Hermitian holomorphic vector bundles. +Lemma 2.5 ([8, Theorem 1.1]). Let U ⊂ Cn be a bounded domain, (E, h) be a Hermitian +holomorphic vector bundle over U with smooth Hermitian metric h, and θ ∈ C0(U, ∧1,1T ∗ +U ⊗ +End(E)) with θ∗ = θ. +If for any strictly plurisubharmonic function ψ on U and f ∈ +C∞ +c (U, ∧n,1T ∗ +U ⊗ E) with ¯∂f = 0 and i∂ ¯∂ψ ⊗ IdE + θ > 0 on supp(f), there is a measurable +section u of ∧n,0T ∗ +U ⊗ E on U, satisfying ¯∂u = f and +(2.2) +� +U +|u|2 +he−ψdλz ≤ +� +U +⟨B−1 +i∂ ¯∂ψ,θf, f⟩hdλz, +provided that the right hand side is finite, then iΘE,h ≥ θ in the sense of Nakano, where +ω = i �n +j=1 dzj ∧ d¯zj and +Bi∂ ¯∂ψ,θ = [i∂ ¯∂ψ ⊗ IdE + θ, Λω]. +The above Lemma is a modified version of Theorem 1.1 in [8] (please see [8, Remark 1.2].) +3. The proof of Theorem 1.10 +We first give the proof in the case that Ω is a product domain. +Lemma 3.1. Let Ω := U × D ⊂ Cn +t × Cm +z be a bounded domain, D be a (connected) pseu- +doconvex circular domain containing the origin. We assume that ϕ is a C2 plurisubharmonic +function defined on some neighborhood of Ω and is S1-invariant with respect to z. Let k ≥ 0 + +12 +F. DENG, J. HU, AND X. QIN +and Ek +t be the space of homogenous polynomials on Cm of degree k, with inner product ht +given by +ht(f, g) = +� +D +f¯ge−ϕtdλz, f, g ∈ Ek +t . +We set Ek = ∪t∈UEk +t and view it as a (trivial) holomorphic vector bundle over U in a natural +way. Let R, M > 0 satisfy +sup{∥z∥; z ∈ D} ≤ R, sup{|ϕ(t, z)|; (t, z) ∈ Ω} ≤ M. +If there exist 0 < r < s such that Br,s := {z ∈ Cm|r ≤ ∥z∥ ≤ s} ⊂ D and ϕ is strictly +plurisubharmonic on U ×Br,s, then the curvature of the Hermitian holomorphic vector bundle +(Ek, h) satisfies: +� +j,l +(Θ(Ek,h) +jl +uj, ul) ≥ δ +� +j +h(uj, uj) +for any sections u1, · · · , un of Ek, where δ > 0 is a constant depending on R, M, r, s and the +complex Hessian of ϕ on U × Br,s. +Proof. For any ǫ > 0, let ϕǫ := ϕ + ǫ(|t|2 + |z|2) and denote the complex Hessian matrix +of ϕǫ as +� +(ϕǫ)jl +(ϕǫ)jα +(ϕǫ)βl +(ϕǫ)βα +� +, +where j, l = 1, · · · , n represent the indices of the components of t = (t1, · · · , tn), α, β = +1, · · · , m represent the indices of the components of z = (z1, · · · , zm). Then ϕǫ is strictly +plurisubharmonic on Ω. We consider the Hermitian metric hǫ on Ek given by: +hǫ +t(f, g) = +� +D +f¯ge−(ϕǫ)tdλz, f, g ∈ Ek +t . +Let E be the trivial vector bundle over U as in Theorem 1.1. Then Ek is a holomorphic +subbundle of E. Since D is a circular domain containing the origin, any f ∈ O(D) can be +represented as a series +f = ++∞ +� +j=0 +fj +that is convergent locally uniformly on D, where each fj is a homogenous polynomial of +degree j. For any S1-invariant continuous bounded function ψ on D, and any homogenous +polynomials gj, gl of degree j and l respectively, we have +� +D +gj¯gle−ψ = 0 +whenever j ̸= l. It follows that, for any t ∈ U, an element f in the orthogonal complement +(Ek)⊥ +t of Ek +t in Et has the form +f = +� +j≥0,j̸=k +fk, + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +13 +where each fj is a homogeneous polynomial of degree j. Hence (Ek)⊥ +t as a vector space is +independent of the choice of the weight function ϕ and is also a holomorphic subbundle of +E. +We now fix an arbitrary t0 ∈ U. By Lemma 2.2 and Lemma 2.3, for any u1, · · · , un of Ek +t0, +we have +(3.1) +� +j,l +(Θ(Ek,hǫ) +jl +uj, ul) ≥ +� +D +� +j,l +H(ϕǫ)jl(t0, z)ujule−(ϕǫ)t0dλz. +where H(ϕǫ) is a Hermitian matrix defined as in Lemma 2.3. +If we write +� +ϕjl +ϕjα +ϕβl +ϕβα +� += +� +A +B +C +F +� +, +then we have H(ϕ) = A − BF −1C provided that F is nonsigular, and +� +H(ϕ) +0 +∗ +F +� += +� +A − BF −1C +0 +∗ +F +� += +� +I +−BF −1 +0 +I +� � +A +B +C +F +� � +I +0 +−(BF −1)∗ +I +� +. +It follows that H(ϕ) is positively definite if ϕ is strictly plurisubharmonic. So we have +(3.2) +� +j,l +(Θ(Ek,hǫ) +jl +uj, ul)|t0 ≥ +� +Br,s +� +j,l +H(ϕǫ)jl(t0, z)ujule−(ϕǫ)t0dλz. +By assumption and by continuity, there is a constant δ0 > 0 such that +� +j,l +H(ϕ)jl(t0, z)ujul ≥ δ0 +� +j +|uj|2 +for z ∈ Br,s. On the other hand, it is clear that +H(ϕǫ)(t0, z) = H(ϕ)(t0, z) + oǫ(1) +on Br,s, where oǫ(1) represents functions on Br,s that converge to 0 uniformly as ǫ → 0. It +follows that +� +j,l +(Θ(Ek,hǫ) +jl +uj, ul)|t0 ≥ δ0 +� +Br,s +� +j +(1 + oǫ(1))|uj|2e−(ϕǫ)t0dλz. +Since hǫ converges to h in the sense of C2 as ǫ → 0+, Θ(Ek,hǫ) converges to Θ(Ek,h) as ǫ → 0+. +We thus have +� +j,l +(Θ(Ek,h) +jl +uj, ul) ≥ δ0 +� +Br,s +� +j +|uj|2e−ϕt0dλz. +Note that uj are homogenous polynomials of degree k, D is bounded, and ϕ(t0, z) is bounded +on D, there exists a constant δ > 0, which is independent of uj, such that +δ0 +� +Br,s +� +j +|uj|2e−ϕt0dλz ≥ δ +� +D +� +j +|uj|2e−ϕt0dλz. + +14 +F. DENG, J. HU, AND X. QIN +It follows that +� +j,l +(Θ(Ek,h) +jl +uj, ul) ≥ δ +� +D +� +j +|uj|2e−ϕt0dλ. +□ +We shall deduce Theorem 1.10 from Lemma 3.1 and Lemma 2.5. +Theorem 3.2 (=Theorem 1.10). Let Ω ⊂ U × Cm be a family of bounded domains over U +that admits a plurisubharmonic defining function, and ϕ be a C2 plurisubharmonic function +defined on some neighborhood of Ω in U × Cm. We assume that all fibers Ωt (t ∈ U) are +(connected) circular domains in Cm containing the origin and ϕ(t, z) is S1- invariant with +respect to z. Let k ≥ 0 and Ek +t be the space of homogenous polynomials on Cm of degree k, +with inner product ht given by +ht(f, g) = +� +Ωt +f¯ge−ϕtdλz, f, g ∈ Ek +t . +We set Ek = ∪t∈UEk +t and view it as a (trivial) holomorphic vector bundle over U in the +natural way. If there exist 0 < r < s such that Br,s := {z ∈ Cm|r ≤ ∥z∥ ≤ s} ⊂ Ωt for all +t ∈ U and ϕ is strictly plurisubharmonic on U × Br,s, then the curvature of the holomorphic +Hermitian vector bundle (Ek, h) is strictly positive in the sense of Nakano. +Proof. Let ρ(t, z) be a plurisubharmonic defining function of Ω, by averaging, we may assume +that ρ is S1-invariant with respect to z. For any fixed t0 ∈ U and 0 < h << 1, let D = +{(t0, z) ∈ U × Cn|ρ(t0, z) ≤ h}. Then there exists a neighborhood U′ of t0 in U such that ρ +and ϕ are defined on some neighborhood of the closure of U′ × D and p−1(U′) ∩ Ω ⊂ U′ × D, +where p : Cn × Cm → Cn is the natural projection. Since the result to be proved is local in +nature with respect to t, we may assume that U = U′, then we have Ω ⊂ ˜Ω := U × D. +For any positive integer N, let +ϕN = ϕ + N max( 1 +N2 , 1 +N2 ){0, ρ − 1 +N }, +where max( 1 +N2 , 1 +N2 ){0, ρ} is the regularized max function defined as in Lemma 2.1. For N >> +1, ϕN is equal to ϕ on Ω. Applying Lemma 3.1 to ˜Ω and ϕN, we get a constant δ > 0 such +that +� +(Θ(Ek,hN) +jl +uj, ul) ≥ δ +� � +D +|uj|2e−ϕN. +for any sections u1, · · · , un of Ek, where the metric hN on Ek is given by +hN +t (f, g) = +� +D +f¯ge−(ϕN)tdλz, f, g ∈ Ek +t . +In other words, if we take +θ = iδ +� +j +dtj ∧ d¯tj ⊗ IdEk ∈ C0(U, ∧1,1T ∗ +U ⊗ End(Ek)), + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +15 +then we have iΘ(Ek,hN) ≥Nak θ. +We want to apply Lemma 2.5 to prove that iΘ(Ek,h) ≥Nak θ. The main idea is as follows. +From the above curvature estimate and the L2-estimate of ¯∂, we know that (Ek, hN) satisfy +the L2-estimate condition presented in Lemma 2.5. As N → ∞, we have hN → h and one +can see that (E, h) also satisfies the L2-estimate condition. Then it follows from Lemma 2.5 +that the curvature of (E, h) satisfies iΘ(Ek,h) ≥Nak θ. The detail of the argument is as follows. +Let ψ(t) be a strictly plurisubharmonic function on U, and f ∈ C∞ +c (U, ∧n,1T ∗ +U ⊗Ek) satisfies +¯∂f = 0 and +� +U +< B−1 +i∂ ¯∂ψ,θf, f >h e−ψdλt < +∞, +where ω = i �n +j=1 dtj ∧ d¯tj and Bi∂ ¯∂ψ,θ is given as in Lemma 2.5. Then there exists M > 0 +such that +� +U +< B−1 +i∂ ¯∂ψ,θf, f >hN e−ψdλt ≤ M, ∀N. +By Lemma 2.4, there are measurable sections uN of ∧n,0T ∗ +U ⊗ Ek on U, such that ¯∂uN = f +and +� +U +|uN|2 +hNe−ψdλt ≤ +� +U +< B−1 +i∂ ¯∂ψ,θf, f >hN e−ψdλt ≤ M. +Since ϕN and ϕ are equal on Ω, we have +� +U +|uN|2 +he−ψdλt ≤ +� +U +|uN|2 +hNe−ψdλt ≤ M +for all N ≥ 1. In particular, {uN} is a bounded sequence in the Hilbert space H of square +integrable sections of ∧n,0T ∗ +U ⊗ Ek on U with weight e−ψ. Hence there is a subsequence of +{uN}, assumed to be {uN} itself without loss of generality, that converges weakly in H to +some u. Note that we also have ¯∂u = f in the sense of distribution. On one hand, we have +� +U +|u|2 +he−ψdλt ≤ lim sup +N→∞ +� +U +|uN|2 +he−ψdλt, +and on the other hand, we have +lim +N→∞ +� +U +< B−1 +i∂ ¯∂ψ,θf, f >hN e−ψdλt = +� +U +< B−1 +i∂ ¯∂ψ,θf, f >h e−ψdλt +by Lebesgue’s dominated convergence theorem. So we get +� +U +|u|2 +he−ψdλt ≤ +� +U +< B−1 +i∂ ¯∂ψ,θf, f >h e−ψdλt. +It follows from Lemma 2.5 that iΘ(Ek,h) ≥Nak θ. +□ + +16 +F. DENG, J. HU, AND X. QIN +4. The proof of Theorem 1.2 and Theorem 1.3 +The difficulty of Theorem 1.2 compared with Theorem 1.10 is that the weight function does +not have strict plurisubharmonicity. We will use the strict psedoconvexity of the domain to +get the Nakano positivity. In the proof of Theorem 1.2, in addition to using Berndtsson’s +estimate of curvature (Lemma 2.3) and Deng-Ning-Wang-Zhou’s integral characterization of +the Nakano positivity of Hermitian vector bundles (Lemma 2.5), an important role is also +played by the simple observation that the integral +� r +0 Ne−Nh(x)dx has a uniform positive limit +as N → ∞ for all r > 0 and all smooth function h with h(0) = 0 and h′(0) ≤ 1. +We first give a Lemma. +Lemma 4.1. Let Ω be a bounded domain in Rn with C2- boundary. For any 0 < r << 1, +let +Ωr := {x ∈ Rn\Ω|d(x, ∂Ω) < r}. +Then there exists a constant c > 0 such that +� +Ωr +hdx1 ∧ · · · ∧ dxn ≥ c +� +∂Ω +dS +� r +0 +h(ζ + tnζ)dt +for any positive integrable functions h on Ωr, where nζ is the outward unit normal of ∂Ω at +ζ and dS is the volume form on ∂Ω. +Proof. We can choose r0 > 0 such that the map +f : ∂Ω × [0, r0) → Ωr0; (ζ, t) �→ ζ + tnζ +is a diffeomorphism. Let µ = dS ∧ dt be the product measure on ∂Ω × [0, r0) → Ωr0 and µ0 +be the Lebesgue measure on Ωr0. Then there is a continuous positive function σ on Ωr0 such +that µ0 = σ · f∗µ on Ωr0. For any 0 < r < r0, taking c = min{σ(x)|x ∈ Ωr}, then c > 0 and +µ0 ≥ cf∗µ. From it the lemma follows. +□ +Theorem 4.2 (=Theorem 1.2). Let Ω ⊂ U × Cm be a strictly pseudoconvex family of +bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some +neighborhood of Ω in U × Cm. We assume that all fibers Ωt (t ∈ U) are (connected) circular +domains in Cm containing the origin and ϕ(t, z) is S1- invariant with respect to z. Let k ≥ 0 +and Ek +t be the space of homogenous polynomials on Cm of degree k, with inner product ht +given by +ht(f, g) = +� +Ωt +f¯ge−ϕtdλz, f, g ∈ Ek +t . +We set Ek = ∪t∈UEk +t and view it as a (trivial) holomorphic vector bundle over U in a natural +way. +Then the curvature of the holomorphic Hermitian vector bundle (Ek, h) is strictly +positive in the sense of Nakano. + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +17 +Proof. Since Ω is strictly pseudoconvex with C2 boundary, there is a defining function ρ +that is strictly plurisubharmonic on some neighborhood ˜Ω of Ω in U × Cm and S1 invariant +with respect to z. +For any fixed t0 ∈ U, we can take a neighborhood U′ of t0 in U and a pseudoconvex circular +domain D ⊂ Cm such that p−1(U′) ∩ Ω ⊂ U′ × D ⊂ ˜Ω. Since the conclusion to be proved is +local in nature on t, we may assume that U = U′. We denote U × D by Ω′. +For N ∈ Z+, we set +ϕN = ϕ + N max( 1 +N3 , 1 +N3 ){0, ρ}, +which is a C2 plurisubharmonic function defined on ˜Ω and is S1- invariant with respect to +z, where max( 1 +N3 , 1 +N3 ){0, ρ} is the regularized max function defined as in Lemma 2.1. For any +ǫ > 0, define +ϕN,ǫ := ϕ + N max( 1 +N3 , 1 +N3 ){0, ρ} + ǫ|t|2 + ǫ|z|2. +Let hN,ǫ be the Hermitian metric on Ek given by +hN,ǫ +t += +� +D +f¯ge−(ϕN,ǫ)tdλz, f, g ∈ Et. +By Lemma 2.3, we know for any u1, · · · , un of Ek +t0 that +� +j,l +(Θ(Ek,hN,ǫ) +jl +uj, ul) ≥ +� +D +� +j,l +H(ϕN,ǫ)jl(t0, z)ujule−(ϕN,ǫ)t0dλ, +where H(ϕN,ǫ)jl is defined as in Lemma 2.3. +For 0 < r << 1, as in Lemma 4.1, we set +Ωt0,r = {z ∈ Cm\Ωt0|d(z, ∂Ωt0) < r} +and set ΩN +t0,r = Ωt0,r\Ωt0,1/N2 for N > 0. We now fix such an r such that Ωt0,r ⊂ D. Note +that max( 1 +N3 , 1 +N3 ){0, ρ} = ρ on ΩN +t0,r for all N. +Note that +H(ϕ + Nρ + ǫ|t|2 + ǫ|z|2) = NH(ρ + (ϕ + ǫ|t|2 + ǫ|z|2)/N), +we have +H(ϕ + Nρ + ǫ|t|2 + ǫ|z|2) ≥ N +2 H(ρ) + +18 +F. DENG, J. HU, AND X. QIN +on ΩN +t0,r for N sufficiently large. Combining with Lemma 4.1, we can see there exist constants +δ0, δ1 > 0 such that +� +j,l +(Θ(Ek,hN,ǫ) +jl +uj, ul) +≥ +� +D +� +H(ϕN,ǫ)jl(t0, z)ujule−(ϕN,ǫ)t0dλz +≥ +� +ΩN +t0,r +� +H(ϕN,ǫ)jl(t0, z)ujule−(ϕN,ǫ)t0dλz +≥δ0 +� +ΩN +t0,r +N +� +|uj|2e−Nρdλz +≥δ1 +� +ζ∈∂Ωt0 +dS +� r +1/N2 +� +N|uj(ζ + τnζ)|2e−Nρ(ζ+τnζ)dτ +≥δ1 +� +ζ∈∂Ωt0 +dS +� +inf +1/N2≤τ≤r |uj(ζ + τnζ)|2 +� r +1/N2 +� +Ne−Nρ(ζ+τnζ)dτ +≥δ1 +� +ζ∈∂Ωt0 +dS +� +inf +0≤τ≤r |uj(ζ + τnζ)|2 +� r +1/N2 Ne−NTτdτ, +where nζ is the unit outward normal of ∂Ωt0 at ζ, dS is the volume form on ∂Ωt0, and T > 0 +is a constant such that ρ(ζ + τnζ) ≤ Tτ for all ζ ∈ ∂Ωt0 and 0 ≤ τ ≤ r. We now need the +obvious but important fact that limN→∞ +� r +1/N2 Ne−NTτdτ = +1 +T > 0. We then get from the +above calculation that +� +j,l +(Θ(Ek,hN,ǫ) +jl +uj, ul) ≥ δ2 +� � +∂Ωt0 +inf +0≤τ≤r |uj(ζ + τnζ)|2dS +for some constant δ2 > 0 and for N sufficiently large. Let ǫ → 0, and denote hN,0 by hN, we +get +� +j,l +(Θ(Ek,hN) +jl +uj, ul) ≥ δ2 +� � +∂Ωt0 +inf +0≤τ≤r |uj(ζ + τnζ)|2dS +(4.1) +for N sufficiently large. +For u ∈ Ek +t0, we need to control its norm +∥u∥hN +t0 = +� +D +|u|2e−(ϕN)t0dλz +in terms of the integral +� +∂Ωt0 inf0≤τ≤r |u(ζ + τnζ)|2dS, where ϕN = ϕN,0. +Let Q = {u ∈ Ek +t0; ∥u∥2 +hN = 1}. Note that functions in Q are homogenous polynomials of +degree k and Ωt0 contains the origin, we can choose a constant M > 0 and a large ball B +with D ⊂ B such that +� +B |u|2dλz ≤ M for all u ∈ Q. By Cauchy’s inequality for holomorphic + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +19 +functions, there is a constant C > 0 such that |du2| < C on D for all u ∈ Q. It follows that +(4.2) +inf +0≤τ≤r |u(ζ + τnζ)|2 ≥ |u(ζ)|2 − rC +for all ζ ∈ ∂Ωt0 and for all u ∈ Q. +We now move to prove that we can choose r and a constant δ3 > 0 such that +� +∂Ωt0 +inf +0≤τ≤r |u(ζ + τnζ)|2dS ≥ δ3 +for all u ∈ Q. +By the maximum principle and continuity, we can take ζ′ ∈ ∂Ωt0 such +that |u| takes its maximum on Ωt0 at ζ′. +Again, since functions in Q are homogenous +polynomials of degree k and Ωt0 contains the origin, we can choose a constant C1 > 0 such +that +� +Ωt0 |u|2dλz ≥ C1 for all u ∈ Q. It follows that +|u(ζ′)|2 ≥ C1 +|Ωt0|, +where |Ωt0| is the Lebesgue measure of Ωt0. +Again by Cauchy’s inequality, if choosing 0 < r < +C1 +2C|Ωt0|, we get +|u(ζ)|2 ≥ |u(ζ′)|2 − Cr ≥ +C1 +2|Ωt0| +for all u ∈ Q and for all ζ ∈ ∂Ωt0 with |ζ − ζ′| < r. It follows that +� +∂Ωt0 +inf +0≤τ≤r |u(ζ + τnζ)|2dS +≥ +� +B(ζ′,r)∩∂Ωt0 +inf +0≤τ≤r |u(ζ + τnζ)|2dS +≥ +� +B(ζ′,r)∩∂Ωt0 +(|u|2 − rC)dS +≥ C1 +2|Ωt0||B(ζ′, r) ∩ ∂Ωt0|, +where B(ζ′, r) is the ball in Cm with center ζ′ and radius r. Note that ∂Ωt0 is compact and +the function +σ : ∂Ωt0 −→ R : ζ → |B(ζ, r) ∩ ∂Ωt0| +is continuous and positive, we have +δ3 := inf +ζ∈∂Dr |B(ζ, r) ∩ ∂Ωt0| > 0. +So we get +� +∂Ωt0 +inf +0≤τ≤r |u(ζ + τnζ)|2dS ≥ δ3. + +20 +F. DENG, J. HU, AND X. QIN +By (4.1), for N sufficiently large, we have +(4.3) +� +j,l +(Θ(Ek,hN) +jl +uj, ul) ≥ δ +� +j +∥uj∥2 +hN +t0 +for any tuple u1, · · · , un ∈ Ek +t0. Just as the last step in the proof of Theorem 1.10, we can +derive from (4.3) and Lemma 2.5 that +� +j,l +(Θ(Ek,h) +jl +uj, ul) ≥ δ +� +j +∥uj∥2 +ht0 +for any tuple u1, · · · , un ∈ Ek +t0. In particular, the curvature of (Ek, h) is strictly positive in +the sense of Nakano. +□ +Similar results holds for a strictly pseudoconvex family of Reinhardt domains. +Theorem 4.3 (=Theorem 1.3). Let Ω ⊂ U × Cm be a strictly pseudoconvex family of +bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some +neighborhood of Ω in U ×Cm. We assume that all fibers Ωt (t ∈ U) are (connected) Reinhardt +domains in Cm and ϕ(t, z) is T m invariant with respect to z. Then for any nonnegative +integers k1, · · · , km, the function ψ(t) defined by +e−ψ(t) = +� +Ωt +|zk1 +1 · · · zkm +m |2e−ϕtdλz +is a strictly plurisubharmonic function on U. +Proof. Since the proof is almost the same as the proof of Theorem 1.2, we just give a +sketch of it. +For any nonnegative integers k1, · · · , km, we consider the 1-dimensional vector space +Ek1,··· ,km +t += Czk1 +1 · · · zkm +m , +with inner product ht given by +ht(f, g) = +� +Ωt +f¯ge−ϕtdλz, f, g ∈ Ek1,··· ,km +t +. +We set Ek1,··· ,km = ∪t∈UEk1,··· ,km +t +and view it as a holomorphic line bundle over U in the +natural way. +Since the conclusion to be proved is local in nature with respect to t ∈ U, we may assume +there is a bounded pseudoconvex Reinhardt domain D ⊂ Cn such that Ω ⊂ Ω′ := U ×D and +ϕ and ρ are defined on some neighborhood of Ω′. +Note that +� +D +zk1 +1 · · · zkm +m zl1 +1 · · · zlm +m e−ϕtdλz = 0 +for any nonnegative integers k1, · · · , km and l1, · · · , lm with kj ̸= lj for some 1 ≤ j ≤ m. So +by Lemma 2.2 the curvature of (Ek1,··· ,km, h) is the restriction of the curvature of (E, h′) on + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +21 +Ek1,··· ,km, where (E, h′) represents the vector bundle given in Theorem 1.1 with Ω replaced +by Ω′. +With the above discussions at hand, the remaining of the proof of the theorem can go ahead +following the same way as in the proof of Theorem 1.2, and we omit the details here. +□ +5. Some consequences of Theorem 1.2 and Theorem 1.3 +We now discuss some consequences of Theorem 1.2 and Theorem 1.3. +5.1. Consequences in complex analysis. +We prove Corollary 1.4 and Corollary 1.5 in this subsection. +Corollary 5.1 (=Corollary 1.4). Let Ω ⊂ U × Cm be a strictly pseudoconvex family of +bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some +neighborhood of Ω in U × Cm that satisfy the conditions in Theorem 1.2 or Theorem 1.3. +For t ∈ U, let K(t, z) be the weighted Bergman kernel of Ωt with weight ϕt. Then ln K(t, z) +is a strictly plurisubharmonic function on Ω. +The proof is provided in the following discussion, which indeed gives us more information. +We assume Ω and ϕ satisfies the conditions in Theorem 1.2, and the remaining case can +be proved in the same way. +Note that ln K(t, z) is strictly plurisubharmonic with respect to z, it is enough to prove +that for any (t0, z0) ∈ Ω and any local holomorphic map ξ(t) : B → Cm defined on some small +neighborhood B of t0 with ξ(t0) = z0, the function ln K(t, ξ(t)) is strictly plurisubharonic as +a function on B (the reason is that any non-vertical tangent vector of Ω at (t0, z0) lies in the +image of dξ(t0)) for some such a map ξ). +Let Ek +t be the space with inner product defined as in Theorem 1.2, and let uk +1, · · · , uk +mk be +an orthogonal normal basis of Ek +t . We set +Kk(t, z) = +mk +� +j=1 +|uj(z)|2, +then it is clear that +(5.1) +K(t, z) = +∞ +� +k=0 +Kk(t, z). +Let p : Ω → U be the natural projection. Then the pull back +( ˜Ek, ˜h) := (p∗Ek, p∗h) +of the bundle (Ek, h) on U is a Hermitian holomorphic vector bundle over Ω whose curvature +is semi-positive in the sense of Nakano. + +22 +F. DENG, J. HU, AND X. QIN +Let F = Ω × C be the trivial line bundle on Ω and denote by e the canonical frame of F +on Ω. Then we have a natural vector bundle morphism σk : ˜Ek → L given by +f �→ (t, z, f(z)) ∈ F +for f ∈ ˜Ek +(t,z) = Ek +t . Let +Ωk = {(t, z) ∈ Ω|Kk(t, z) ̸= 0}, +or equivalently, (t, z) ∈ Ωk if and only if f(z) ̸= 0 for some homogenous polynomial f on Cm +of degree k. Then σk is a surjective bundle morphism from ˜Ek|Ωk to F|Ωk. One can see that +the quotient metric, say hk on F|Ωk induced from this morphism is given by +∥e∥2 +hk = +1 +Kk(t, z) = e− ln Kk(t,z). +Since the curvature of ( ˜Ek, ˜h) is semi-positive in the sense of Nakano, and note the curvature +increasing property under taking quotient metric [5, a) in Proposition (6.10)], we know the +curvature of (F|Ωk, hk) is semi-positive, which implies that ln Kk(t, z) is plurisubharmonic on +Ω. +For any given (t0, z0) ∈ Ω, and any holomorphic map ξ(t) : B → Cm defined on some small +neighborhood B of t0 with ξ(t0) = z0, we denote by +Γ = {(t, ξ(t))|t ∈ B} ⊂ Ω +the graph of ξ. Then ( ˜E0, ˜h)|Γ is a (trivial) Hermitian line bundle over Γ whose curvature is +strictly positive, since p|Γ : Γ → B is a biholomoprhic map. Note also that σ0 : ˜E0 → L is an +isomorphism of vector bundles, it follows that ln K0(t, ξ(t)) is strictly plurisubharmonic on Γ, +and hence is strictly plurisubharmonic as a function of t. By (5.1), we know that ln K(t, ξ(t)) +is strictly plurisubharmonic as a function of t. Hence ln K(t, z) is strictly plurisubharmonic +on Ω. The proof of the above corollary is complete. +In fact, by the same argument, one can show, for any nonnegative integer k, that ”the +relative log character Bergman kernel” ln Kk(t, z) is plurisubharmonic on Ω and is strictly +plurisubharmonic on Ωk. +Corollary 5.2 (=Corollary 1.5). Let Ω ⊂ U × Cm be a strictly pseudoconvex family of +domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood +of Ω in U × Cm. +(1) If Ω and ϕ satisfy the conditions in Theorem 1.2 or Theorem 1.3, then the function +˜ϕ defined by +e− ˜ϕ(t) = +� +Ωt +e−ϕ(t,z)dλz +is a strictly plurisubharmonic function on U. + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +23 +(2) If all fibers Ωt are tube domains of the form Xt + iRm withXt bounded, and ϕ(t, z) +does not depend on the imaginary part of z, then the function ˜ϕ defined by +e− ˜ϕ(t) = +� +Xt +e−ϕ(t,Rez)dλRez +is a strictly plurisubharmonic function on U. +Proof. It is clear that (1) is equivalent to the curvature strict positivity of (E0, h) in +Theorem 1.2 or Theorem 1.3. We now give the proof of (2). +Let us consider the map +f : Ω → Cn +t × Cm +w +(t1, · · · , tn, z1, · · · , zm) �→ (t1, · · · , tn, ez1, · · · , ezm), +then Ω∗ := f(Ω) ⊂ Cn +t × Cm +w is a strictly pseudoconvex family of Reinhardt domains over U. +Note that +ψ(t, w) := ϕ(t, ln |w1|, · · · , ln |wm|) + 2(ln |w1| + · · · + ln |wm|) +is a C2 and plurisubharmonic function defined on some neighborhood of the closure of Ω∗ in +U × Cm, applying (1) to Ω∗ and ψ, we see that the function ˜ϕ defined by +e− ˜ϕ(t) = +� +Xt +e−ϕ(t,Rez)dλRez = +1 +(2π)n +� +Ω∗ +t +e−ψ(t,w)dλw +is a strictly plurisubharmonic function on U. +□ +5.2. Consequences in convex analysis. +The bridge connecting strictly convex families of bounded domains in Rm and strictly +pseudoconvex families of tube domains in Cm is indicated in the proof of the following +corollary. +Corollary 5.3 (=Corollary 1.7). Let D ⊂ U0 × Rm be a strictly convex family of bounded +domains over a domain U0 ⊂ Rn and ϕ be a C2 convex function defined on some neighborhood +of the closure of D in U0 × Rm. Then the function ˜ϕ defined by +e− ˜ϕ(t) = +� +Dt +e−ϕ(t,x)dλx +is a strictly convex function on U0. +Proof. We first complexify U0 to U = U0 × iRn +l with complex coordinate τ = t + il, +then U is a domain in Cn +τ . We secondly complexify Rm to Rm +x + iRm +y = Cm +z , with complex +coordinate z = x + iy. Then +Ω = D + iRn+m = {(τ, z) ∈ Cn × Cm|(Reτ, Rez) ∈ D} + +24 +F. DENG, J. HU, AND X. QIN +is a strictly pseudoconvex family of tube domains over U. For τ ∈ U, Ωτ is a tube domain of +the form Ωτ = Dτ + iRm, where Dτ ⊂ Rm can be naturally identified with DReτ. By setting +ψ(τ, z) = ϕ(Reτ, Rez), +we extend ϕ to a C2 plurisubharmonic function ψ on some neighborhood of Ω in U × Cm, +such that ψ(τ, z) is independent of the imaginary part of τ, z. By (2) in Corollary 1.5, the +function ˜ψ defined by +e− ˜ψ(τ) = +� +Dτ +e−ψ(τ,Rez)dλRez +is a strictly plurisubharmonic function on U. It is clear that ˜ψ(τ) is independent of the +imaginary part of τ and ˜ψ|U0 = ˜ϕ, thus ˜ϕ is a strictly convex function on U0. +□ +5.3. Curvature negativity of determinant line bundle. +We now explain the meaning of Corollary 1.9 and give its proof. +Let π : E → X be a holomorphic vector bundle of rank m over a complex manifold X +equipped with a smooth Finsler metric h. By definition, h is a continuous function h : E → R +such that h ≥ 0, h(λv) = |λ|h(v) for λ ∈ C and v ∈ E, and h is smooth on E\ZE, where +ZE ⊂ E is the zero section of E. Recall that (E, h) is defined to be strictly negatively curved +if ln h is strictly plurisubharmonic on E\ZE. +We now define the Hermitian metric deth induced from h on the determinant line bundle +detE = ΛmE of E via the measure µ on Et with µ(Bt) = 1 for t ∈ X, where +Bt = {v ∈ Et|h(v) ≤ 1}. +A more explicit description of det h in terms of local frame is as follows. Let e1, · · · , em be a +holomorphic local frame of E over some open set U ⊂ X. We get a local trivialization of E +over U: +φ : E|U → U × Cm, (t, z1v1 + · · · + zrvm) �→ (t, z1, · · · , zm). +Then e := e1 ∧ · · · ∧ em is a local frame of detE over U, whose norm with respect to det h is +given by +∥e(t)∥2 +det h = +1 +µ0(φt(Bt)), +where µ0 is the Lebesgue measure on Cm. +By Corollary 1.8, we know that − ln µ0(φt(Bt)) is a strictly plurisubharmonic function on +U provided that h is strictly negatively curved. Note that the curvature of (detE, deth) on +U is given by i∂ ¯∂ ln µ0(φt(Bt)), so the curvature of the induced Hermitian metric deth on +detE is strictly negative. We thus get +Corollary 5.4 (=Corollary 1.9). Let π : E → X be a holomorphic vector bundle over a +complex manifold X equipped with a smooth Finsler metric h. If (E, h) is strictly negatively +curved, then the curvature of the induced Hermitian metric deth on detE is strictly negative. + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +25 +6. Deduce Theorem 1.12 from Theorem 1.2 or Theorem 1.10 +In this section, we discuss the relation of Theorem 1.12 with Theorem 1.2 or Theorem 1.10. +We show that Theorem 1.12 can be deduced from Theorem 1.2 or Theorem 1.10. For this +consideration, the symmetric structure appearing in Theorem 1.2 or Theorem 1.10 plays an +indispensable role. +6.1. Basic properties of ample vector bundles. +This subsection recalls some well known basic knowledge about ample vector bundles. +Let π : E → X be a holomorphic vector bundle over a compact complex manifold X. For +each x ∈ X, we denote by Ex the fiber of E over x and denote by E∗ +x its dual. Let P(E∗ +x) be +the projective space of E∗ +x, which is the space of one-dimensional complex linear subspaces +of E∗ +x with the natural complex structure, and let OP(E∗x)(1) be the dual of the tautological +line bundle over P(E∗ +x). Then +P(E∗) := ∪x∈XP(E∗ +x) +is a complex manifold that can be naturally realized as a holomorphic fiber bundle over X +with P(E∗ +x) as fibers, and +OP(E∗x)(1) := ∪x∈XOP(E∗x)(1) +can be naturally realized as a holomorphic line bundle over P(E∗) whose restriction to P(E∗ +x) +is just OP(E∗x)(1). By definition, E is called an ample vector bundle if OP(E∗)(1) is an ample +line bundle over P(E∗). +We now assume that E is ample. +Then there is a Hermitian metric h on OP(E∗)(−1) +whose curvature is negative. Let ρ : OP(E∗)(−1) → R≥0 be the length function associated +to h, namely ρ(v) = +� +h(v, v) for v ∈ OP(E∗)(−1). +Then ρ is strictly plurisubharmonic +on OP(E∗)(−1)\ZOP(E∗)(−1), where ZOP(E∗)(−1) is the zero section of OP(E∗)(−1), viewed as a +submanifold of OP(E∗)(−1). +Note that OP(E∗)(−1) can be viewed as the blow up of E∗ along its zero section ZE∗, with +ZOP(E∗)(−1) as the exceptional divisor, we can naturally identify E∗\ZE∗ with OP(E∗)(−1)\ZOP(E∗)(−1). +Through this identification, we can view ρ as a function on E∗, with ρ|ZE∗ ≡ 0. +In conclusion, we get a plurisubharmonic function ρ on E∗, which is strictly plurisubhar- +monic on E∗\ZE∗ and invariant under the natural S1 action on E∗. In other words, ρ is a +smooth Finsler metric on E∗ whose curvature is strictly negative. +6.2. Some linear algebra. +We present some knowledge about linear algebra that is needed in the proof of Theorem +1.12. +Let V be a complex vector space with complex dimension m and V ∗ be its dual space. Let +P(V ) be the set of all polynomials on V , and Pk(V ) be the space of homogeneous polynomials + +26 +F. DENG, J. HU, AND X. QIN +of degree k on V . Then +P(V ) = +� +k≥0 +Pk(V ). +We have P0(V ) = C and P1(V ) = V ∗. In an obvious manner, we can naturally identify +Pk(V ) with SkV , the k-th symmetric product of V ∗. +We realize the circle group S1 as the space of complex numbers with unit norm. Then S1 +acts on V via scalar product. This induces an action of S1 on P(V ) as follows: +α · f(v) = f(αv), +where f ∈ P(V ), v ∈ V , α ∈ S1. Then Pk(V ) are character subspaces of P(V ) associated to +this action, namely, for k ≥ 0 we have +Pk(V ) = {f ∈ P(V ) +��α · f = αkf, ∀α ∈ S1}. +The cotangent bundle of V is +T ∗V = V × V ∗. +It follows that the canonical bundle of V is +KV = V ⊗ det V ∗, +where det V ∗ = ∧mV ∗. +We now consider the coordinate representation of KV . Let u1, · · · , um be a basis of V +and u∗ +1, · · · , u∗ +m be the associated dual basis of V ∗. Then u∗ +1 ∧ · · · ∧ u∗ +m is a basis of det V ∗. +Consider linear isomorphism: +V −→ Cn : z1u1 + · · · + zmum �→ (z1, · · · , zm), +then u∗ +1 ∧ · · · ∧ u∗ +m corresponds to dz1 ∧ · · · ∧ dzm, a basis of det (Cm)∗. +Let Ω ⊂ V be an S1 invariant domain containing 0, then we have the following identification +(6.1) +H0(Ω, KΩ) = O(Ω, det V ∗), +where O(Ω, det V ∗) is the space of holomorphic mappings from Ω to det V ∗. The action of +S1 on O(Ω, det V ∗) is given as follows: +α · f(x) = f(αx), α ∈ S1. +Under coordinate form, if we identify V with Cm as above and view Ω as a domain in Cm, +then we have the following identification +(6.2) +H0(Ω, KΩ) ∼= {f(z1, · · · , zm)dz1 ∧ · · · ∧ dzm|f ∈ O(Ω)}, +and the action of S1 on H0(Ω, KΩ) is realized as +α · (f(z1, · · · , zm)dz1 ∧ · · · ∧ dzm) = f(αz1, · · · , αzm)dz1 ∧ · · · ∧ dzm. +It is clear that the action of S1 on the Hilbert space +A2(Ω) = {f ∈ H0(Ω, KΩ) : ||f|| < +∞} + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +27 +is unitary, where +||f||2 = +� +Ω +cmf ∧ ¯f, +with cm = im2 +2m is set to make the form cmf ∧ ¯f real and nonnegative. +For any k ≥ 0, let +(6.3) +P′ +k(Ω) = {f ∈ A2(Ω)| α · f = αkf, ∀α ∈ S1}, +then +(6.4) +P +′ +k(Ω) = {f(z1, · · · , zm)dz1 ∧ · · · ∧ dzm| f ∈ Pk(Cm)}. +It follows that +(6.5) +P +′ +k(Ω) = Pk(V ) ⊗ det V ∗ = Sk V ∗ ⊗ det V ∗. +6.3. The proof of Theorem 1.12. +Let π : E → X be an ample holomorphic vector bundle of rank m over a compact complex +manifold X of dimension n. Let E∗ be the dual bundle of E and let ZE∗ be the zero section +of E∗, viewed naturally as a submanifold E∗. From §6.1, we know that E∗ admits a smooth +Finsler metric ρ : E∗ → R≥0 whose curvature is strictly negative. +Let Ω = {v ∈ E∗|ρ(v) ≤ 1}, then Ω is an S1 invariant bounded domain in E∗ whose +boundary is strictly pseudoconvex. +As usual, we denote Ω ∩ E∗ +t by Ωt for t ∈ X. Note +that Ωt is an S1-invariant domain in E∗ +t containing the origin. By (6.1), we can canonically +identify H0(Ωt, KΩt) with O(Ωt, det Et). For k ≥ 0, if we define P′ +k(Ωt) as in (6.3), we have +P′ +k(Ωt) = SkEt ⊗ det Et from (6.5). +Let ϕ be an S1-invariant smooth plurisubharonic function defined on some neighborhood +of the closure Ω of Ω in E∗. On P′ +k(Ωt), we can define a Hermitian inner product ht by setting +∥f∥2 +ht = +� +Ωt +cmf ∧ ¯fe−ϕt, f ∈ P′ +k(Ωt), +where ϕt is the restriction of ϕ on Ωt. In this way, we get a Hermitian metric h on SkE⊗det E. +Our propose is to deduce from Theorem 1.2 or Theorem 1.10 that the curvature of the +Hermitian vector bundle (SkE ⊗ det E, h) over X is strictly positive in the sense of Nakano, +for suitable choice of ϕ (indeed for all such ϕ), and hence get new proofs of Theorem 1.12. +The argument goes as follows. Let (U, t1, · · · , tn) be a local coordinate on X, and e1, · · · , em +be a holomorphic local frame of E∗ over U. Then we get an isomorphism σ : π−1(U) −→ +U × Cm given by +(t, z1e1 + · · · + zmem) �→ (t1, · · · , tn, z1, · · · , zm), +where π : E∗ → X is the bundle map. This isomorphism realizes Ω ∩ π−1(U) as a strictly +pseudoconvex family of bounded domains over U whose fibers σ(Ωt) ⊂ Cm are circular + +28 +F. DENG, J. HU, AND X. QIN +domains containing the origin. By (6.4), for t ∈ U, via σ we can identify P +′ +k(Ωt) with the +space +{f(z1, · · · , zm)dz1 ∧ · · · dzm|f ∈ Pk(Cm)}, +with the Hermitian inner product ht given by +∥f(z1, · · · , zm)dz1 ∧ · · · dzm∥2 +ht = +� +σ(Ωt) +|f|2e−ϕt◦σ−1dλz. +It follows from Theorem 1.2 that the curvature of (SkE ⊗ det E, h) is strictly positive in the +sense of Nakano, and hence we get Theorem 1.12. +In a similar way, we can deduce Theorem 1.12 from Theorem 1.10 by choosing ϕ = +max1/4,1/4{1/4, ρ}. (see Lemma 2.1 for the definition of the regularized maximum function). + +STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES +29 +References +[1] Berndtsson B. Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions. +Mathematische Annalen, 312(1998), No.4, 785-792. +[2] Berndtsson B. Subharmonicity properties of the Bergman kernel and some other functions associated to +pseudoconvex domains. Ann. Inst. Fourier (Grenoble) 56 (2006), No.6, 1633-1662. +[3] Berndtsson B. Curvature of vector bundles associated to holomorphic fibrations. Annals of Mathematics, +2009, 169(2):531-560. +[4] Berndtsson B. Strict and nonstrict positivity of direct image bundles. Math. Z. 269 (2011), No.3-4, 1201- +1218. +[5] Demailly JP. Complex analytic and differential geometry. Electric book, available in the author’s home- +page. +[6] Demailly JP. , Skoda H. Relations entre les notions de positivit´es de P. A. Griffiths et de S. Nakano pour +les fibr´es vectoriels. S´eminaire Pierre Lelong-Henri Skoda (Analyse), (1978/79), 304-309, Lecture Notes +in Math. 822, Springer-Verlag, New York, 1980. +[7] Deng F., Hu J., Jiang W. Curvature positivity of invariant direct images of Hermitian vector bundles. +Annali di Matematica Pura ed Applicata, 2022. +[8] Deng F., Ning J., Wang Z., Zhou X. Positivity of holomorphic vector bundles in terms of Lp-properties +of ¯∂. arXiv:2001.01762, to appear in Math. Ann. . +[9] Deng F., Jiang W., Qin X. Deduce some results in convex analysis form complex analysis and vice versa, +to appear. +[10] Deng F., Zhang H., Zhou X. Positivity of character subbundles and minimum principle for noncompact +group actions. Math. Z. 286 (2017), No. 1-2, 431-442. +[11] Grauert, H., On Levis problem and the imbedding of real-analytic manifolds. Ann. of Math. (2) 68 +(1958) 460-472. +[12] Grauert H. ¨Uber Modifikationen und exzeptionelle analytische Mengen. Mathematische Annalen, 1962, +146(4):331-368. +[13] Griffiths PA. Hermitian differential geometry, Chern classes, and positive vector bundles. Global Anal- +ysis, 1969. +[14] Liu Z., Yang H., Zhou X., On the Multiplier Submodule Sheaves Associated to Singular Nakano Semi- +positiveMetrics, https://arxiv.org/abs/2111.13452, 2021 +[15] Pr´ekopa A. On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 (1973), 335-343. +[16] Umemura H. Some results in the theory of vector bundles. Nagoya Math. J. 52 (1973), 97-128. +Fusheng Deng: +School of Mathematical Sciences, University of Chinese Academy of +Sciences, Beijing 100049, P. R. China +Email address: fshdeng@ucas.ac.cn +Jinjin Hu: School of Mathematical Sciences, University of Chinese Academy of Sciences, +Beijing 100049, P. R. China +Email address: hujinjin21@mails.ucas.ac.cn +Xiangsen Qin: +School of Mathematical Sciences, University of Chinese Academy of Sci- +ences, Beijing 100049, P. R. China +Email address: qinxiangsen@amss.ac.cn + diff --git a/W9AyT4oBgHgl3EQfWPem/content/tmp_files/load_file.txt b/W9AyT4oBgHgl3EQfWPem/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..14addacfc1224693d0828ecb09fdb49639e3a390 --- /dev/null +++ b/W9AyT4oBgHgl3EQfWPem/content/tmp_files/load_file.txt @@ -0,0 +1,890 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf,len=889 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='00160v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='CV] 31 Dec 2022 CURVATURE STRICT POSITIVITY OF DIRECT IMAGE BUNDLES ASSOCIATED TO PSEUDOCONVEX FAMILIES OF DOMAINS FUSHENG DENG, JINJIN HU, AND XIANGSEN QIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We consider the curvature strict positivity of the direct image bundle associated to a pseudoconvex family of bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded circular do- mains or Reinhardut domains are strictly positive in the sense of Nakano, even if the weight functions are not strictly plurisubharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' This result gives a new geometric insight about the property of strict pseudoconvexity, and has some applications in complex analysis and convex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We investigate that the main result implies a remarkable result of Berndts- son which states that, for an ample vector bundle E over a compact complex manifold X and any k ≥ 0, the bundle SkE ⊗ det E admits a Hermitian metric whose curvature is strictly positive in the sense of Nakano, where SkE is the k-th symmetric product of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The two main ingredients in the argument of the main theorems are Berndtsson’s estimate of the lower bound of curvature of direct image bundles and Deng-Ning-Wang-Zhou’s characteriza- tion of the curvature Nakano positivity of Hermitian vector bundles in terms of L2-estimate of ¯∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Introduction 2 Acknowledgements 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Preliminaries 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Regular maximum of plurisubharmonic functions 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Curvature positivity of Hermitian holomorphic vector bundles 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Optimal L2-estimate condition and curvature positivity 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Some consequences of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Consequences in complex analysis 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Consequences in convex analysis 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Curvature negativity of determinant line bundle 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Basic properties of ample vector bundles 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Some linear algebra 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 27 1 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN References 29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Introduction Let U ⊂ Cn and D ⊂ Cm be pseudoconvex bounded domains, and let ϕ be a smooth plurisubharmonic function defined on some (open) neighborhood of the closure of Ω := U ×D in Cn × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For t ∈ U, we define the Hilbert space Et = {f ∈ O(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' ∥f∥2 t := � D |f|2e−ϕtdλz < ∞}, where O(D) is the space of holomorphic functions on D, ϕt(z) = ϕ(t, z), and dλz is the Lebesgue measure on Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' When t varies in U, Et is invariant as a vector space, but the inner product defined by the above norm varies if ϕ is not constant with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Set E = ∪t∈UEt and take π : E → U by setting π(Et) = {t}, then E is a holomorphic vector bundle (of infinite rank) over U with a Hermitian metric h given by ht(f, g) = � D f¯ge−ϕtdλz, f, g ∈ Et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' A fundamental result of Berndtsson is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 ([3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' With the above notations and assumptions, the curvature of the Hermitian vector bundle (E, h) is semi-positive in the sense of Nakano, and is strictly positive in the sense of Nakano if ϕ is strictly plurisubharmnic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Our main purpose is to study strict positivity of curvature of direct image bundles defined in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1, we see that the strict positivity of the curvature of (E, h) comes from the strict plurisubharmonicity of the weight function ϕ, which can be viewed as the strict curvature positivity of the trivial line bundle over Ω with Hermitian metric given by e−ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In the present work, we show that the strict positivity of the curvature can come from a completely different source, namely, the strict pseudoconvexity of the total space of the family of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' To state the main result, we first introduce some notions and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Denote by p : Cn × Cm → Cn the natural projection, and for a set A ⊂ Cn × Cm, we denote p−1(t) ∩ A by At, which is called the fiber of A over t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Of course we can view At as a family of subsets in Cm depending on the parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (1) A family of domains of dimension m over a domain U ⊂ Cn is a domain Ω ⊂ U ×Cm such that p(Ω) = U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' such a family is called a family of bounded domains if all fibers Ωt ⊂ Cm (t ∈ U) are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 3 (2) A family of domains Ω over U has Ck (k ≥ 1) boundary if there exists a Ck function ρ(t, z) defined on some neighborhood of the closure Ω of Ω in U × Cm, such that Ω = {(t, z) ∈ U × Cn|ρ(t, z) < 0} and d(ρ|∂Ωt) ̸= 0 for all t ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Such a function ρ is called a defining function of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (3) A family of domains Ω ⊂ U ×Cm over U is called pseudoconvx if Ω is a pseudoconvex domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (4) A family of domains Ω ⊂ U ×Cm with C2-boundary is said to have plurisubharmonic defining function if it admits a defining function that is plurisubharmonic on some neighborhood of Ω in U×Cm, and is called strictly pseudoconvex if it admits a defining function that is strictly plurisubharmonic on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If U is pseudoconvex and Ω has a plurisubharmonic defining function, then Ω is a pseudo- convex family of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' But the opposite is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We consider pseudoconvex families of bounded domains with certain symmetries, namely, pseudoconvex families of bounded domains whose fibers are circular domains or Reinhardt domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Recall that a domain D ⊂ Cm is called a circular domain if it is invariant under the action of S1 on Cm given by eiθ · (z1, · · · , zm) = (eiθz1, · · · , eiθzm), θ ∈ R, and is called a Reinhardt domain if it is invariant under the action of the torus group T m on Cm given by (eiθ1, · · · , eiθm) · (z1, · · · , zm) = (eiθ1z1, · · · , eiθmzm), θi ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that all fibers Ωt (t ∈ U) are (connected) circular domains in Cm containing the origin and ϕ(t, z) is S1- invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let k ≥ 0 and Ek t be the space of homogenous polynomials on Cm of degree k, with inner product ht given by ht(f, g) = � Ωt f¯ge−ϕtdλz, f, g ∈ Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We set Ek = ∪t∈UEk t and view it as a (trivial) holomorphic vector bundle over U in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then the curvature of the holomorphic Hermitian vector bundle (Ek, h) is strictly positive in the sense of Nakano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Similar results holds for a strictly pseudoconvex family of Reinhardt domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U ×Cm be a strictly pseudoconvex family of bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U ×Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that all fibers Ωt (t ∈ U) are (connected) Reinhardt domains in Cm and 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN ϕ(t, z) is T m- invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then for any nonnegative integers k1, · · · , km, the function ψ(t) defined by e−ψ(t) = � Ωt |zk1 1 · · · zkm m |2e−ϕtdλz is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In fact, in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, if we assume all fibers Ωt has no intersection with any coordinate axis, then k1, · · · , km can be taken to be any integers (not necessarily nonnegative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now discuss the relation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If we assume that both Ω is a product domain as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 and ϕ is strictly plurisubharmonic, then, as observed in [10], the conclusions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 can be deduced from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1, with the help of some basic group representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' On the other hand, as we will see in the proofs, if one of the above two assumptions is dropped, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 essentially go beyond Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Here the key point we want to emphasize about Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 is that strict pseudoconvexity of the family Ω encodes the curvature strict positivity of the (character) direct image bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It is also interesting to compare Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 with the main result in [4], where Berndtsson shows that the curvature of the direct image bundle of the relative canonical bundle twisted with a Hermitian line bundle associated to a K¨ahler family of compact man- ifolds is strictly positive in the sense of Nakano, provided that the related Kodaira-Spencer map is nondegenerate and the curvature of the line bundle is strictly positive along fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Berndtsson also gives counterexamples to this result if one of the two conditions is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In connection to Berndtsson’s result, one may imagine from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 that strict pseudoconvexity of the family Ω implicitly implies nontrivial deformation of the fibers and certain curvature positivity along fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' From this point of view, it seems that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 provide a very deep new geometric insight about strict psedo- convexity in complex analysis and complex geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It seems that more profound potential relations of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 with Berndtsson’s result deserves further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' On the other hand, we conjecture that certain appropriate form of the converse of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 ( or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3) holds, namely, the curvature strict positivity of the direct images implies the strict pseudoconvexity of the family Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We want to point out that the symmetry involved in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 does not play essential role, and it is mainly used to avoid considering the whole space of L2- holomorphic functions on Ωt as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 and bundles of infinite rank without local trivialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' As mentioned above, the key role is played by the strict psedoconvexity of the family Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' On the other hand, as we will see, the S1-symmetry is indispensable when we apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 to the study of ample vector bundles over projective manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now discuss some consequences of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 5 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm that satisfy the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For t ∈ U, let K(t, z) be the weighted Bergman kernel of Ωt with weight ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then ln K(t, z) is a strictly plurisubharmonic function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We believe that Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4 holds for an arbitrary strictly pseudoconvex family of bounded domains, without symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' But we will not discuss this topic further in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (1) If Ω and ϕ satisfy the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, then the function ˜ϕ defined by e− ˜ϕ(t) = � Ωt e−ϕ(t,z)dλz is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (2) If all fibers Ωt are tube domains of the form Xt + iRm with Xt bounded, and ϕ(t, z) does not depend on the imaginary part of z, then the function ˜ϕ defined by e− ˜ϕ(t) = � Xt e−ϕ(t,Rez)dλRez is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Taking ϕ ≡ 0 in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5, we get Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of domains over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (1) If Ω satisfies the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, then the function given by t �→ − ln |Ωt| is a strictly plurisubharmonic function on U, where |Ωt| is the Lebesgue measure of Ωt ⊂ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (2) If all fibers Ωt are tube domains of the form Xt + iRm with Xt bounded, then the function given by t �→ − ln |Xt| is a strictly plurisubharmonic function on U, where |Xt| is the Lebesgue measure of Xt ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The plurisubharmonicity of ln K(t, z) in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4 was proved in [2], and the plurisub- harmonicity of the functions considered in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='6 were proved in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The contribution here is on the strict plurisubharmonicity of those functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We will explain that the above corollaries imply some parallel results in convex analysis, following a general principle given in [9] that connecting convex analysis and complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN In a similar way as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1, we define a strictly convex family of domains in Rm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let U0 ⊂ Rn be a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By definition, a strictly convex family of domains over U0 is a convex domain D ⊂ U0 × Rm such that p0(D) = U0, and there exists a C2 strictly convex function ρ0(t, x) on some neighborhood ˜D of D in U0 × Rm such that D = {(t, x) ∈ ˜D|ρ0(t, x) < 0} and d(ρ0|∂Dt) ̸= 0 for all t ∈ U0, where p0 : Rn × Rm → Rn is the natural projection and Dt = p−1 0 (t) ∩ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The function ρ0 is called a defining function of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Here a C2 function is called strictly convex if its Hessian is positively definite everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let D ⊂ U0 × Rm be a strictly convex family of bounded domains over a domain U0 ⊂ Rn and ϕ be a C2 convex function defined on some neighborhood of the closure of D in U0 × Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then the function ˜ϕ defined by e− ˜ϕ(t) = � Dt e−ϕ(t,x)dλx is a strictly convex function on U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The convexity of ˜ϕ in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='7 can be deduced from the Pr´ekopa’s theorem [15], but here we are interested in the strict convexity of ˜ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Taking ϕ ≡ 0 in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='7, we get a stronger form of the classical Brunn-Minkowski inequality in convex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let D ⊂ U0 × Rm be a strictly convex family of bounded domains over U0 ⊂ Rn, then the function given by t �→ − ln |Dt| is a strictly convex function on U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 has a direct application to vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let π : E → X be a holomorphic vector bundle of rank m over a complex manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By definition, a smooth Finsler metric on E is a continuous function h : E → R such that h ≥ 0, h(λv) = |λ|h(v) for λ ∈ C and v ∈ E, and h is smooth on E\\ZE, where ZE ⊂ E is the zero section of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We call (E, h) is strictly negatively curved if ln h is strictly plurisubharmonic on E\\ZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (Note that if h is a smooth Hermitian metric, then (E, h) is strictly negatively curved if and only if its curvature is strictly negative in the sense of Griffiths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=') Given a smooth Finsler metric h on E, we can define an induced Hermitian metric deth on the determinant line bundle detE = ΛmE of E via the measure µ on Et with µ({v ∈ Et|h(v) ≤ 1}) = 1 (see §5 for details), t ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (In fact, the definition still works even if h is just a singular Finsler metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=') STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 7 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let π : E → X be a holomorphic vector bundle over a complex manifold X equipped with a smooth Finsler metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If (E, h) is strictly negatively curved, then the curvature of the induced Hermitian metric deth on detE is strictly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Motivated by the study of ample vector bundles, we also establish a result about the curvature strict positivity of invariant direct images from another perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a family of bounded domains over U that admits a plurisubharmonic defining function, and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that all fibers Ωt (t ∈ U) are (connected) circular domains in Cm containing the origin and ϕ(t, z) is S1 invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let k ≥ 0 and Ek t be the space of homogenous polynomials on Cm of degree k, with inner product ht given by ht(f, g) = � Ωt f¯ge−ϕtdλz, f, g ∈ Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We set Ek = ∪t∈UEk t and view it as a (trivial) holomorphic vector bundle over U in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If there exists 0 < r < s such that Br,s := {z ∈ Cm|r ≤ ∥z∥ ≤ s} ⊂ Ωt for all t ∈ U and ϕ is strictly plurisubharmonic on U × Br,s, then the curvature of the holomorphic Hermitian vector bundle (Ek, h) is strictly positive in the sense of Nakano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For Reinhardt domains, we have a similar result which is stronger in form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a family of bounded domains over U that admits a plurisubharmonic defining function, and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that all fibers Ωt (t ∈ U) are (connected) Reinhardt domains in Cm and ϕ(t, z) is T m- invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If ϕ is strictly plurisubharmnic on some open subset O in Ω such that p(O) = U, then for any nonnegative integers k1, · · · , km, the function ψ(t) defined by e−ψ(t) = � Ωt |zk1 1 · · · zkm m |2e−ϕtdλz is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We can also deduce some consequences from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='11 that are parallel to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We leave the details to the readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It is helpful to compare Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' While the strict positivity of the curvature in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 comes from the strict pesudoconvexity of Ω, it seems that the strict positivity of the curvature in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 essentially comes from the strict plurisub- harmonicity of the weight function on certain subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In connection to this, we do not know whether Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 still holds if Ω is just assumed to be a pseudoconvex family (may without plurisubharmonic defining function), with other conditions unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The second part of the paper aims to establish the connection of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 to the study of ample vector bundles (see §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 for definition) over projective manifolds, 8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN which is indeed one of the original motivations for us to consider Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By Kodaira’s embedding theorem, one can show that a holomorphic vector bundle over a compact complex manifold must be ample if it is Griffiths positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', admits a Hermitian metric with positive curvature in the sense of Griffiths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In 1969, Griffiths conjectured that the converse is true, namely, such a vector bundle is Griffiths positive if it is ample [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' This conjecture is known as Griffiths conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Griffiths conjecture is known to be true if the base space is a Riemannian surface [16], but is still widely open otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Along a related direction, Demailly and Skoda proved in 1980 [6] that E ⊗det E is Nakano positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', admits a Hermitian metric with positive curvature in the sense of Nakano) if E is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In 2009, Berndtsson proved the following remarkable result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 ([3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 and the remark following it]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If E is an ample vector bundle over a compact complex manifold X, then SkE ⊗ det E is Nakano positive for all k ≥ 0, where SkE denotes the k-th symmetric product of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We explain briefly how to deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let E be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12, then one can easily see that there is a smooth strictly negatively curved Finsler metric h : E∗ → R on the dual bundle E∗ of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω = {v ∈ E∗|h(v) < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Bn be a coordinate ball in X and identify E∗|Bn with Bn × Cm via a local trivialization, where m is the rank of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then Ω ∩ E∗|Bn ⊂ Bn × Cm is a strictly pseudoconvex family of bounded circular domains over Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' As in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, we get a Hermitian vector bundle (Ek, h) over Bn whose curvature is strictly positive in the sense of Nakano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then the point is that Ek can be canonically identified with (SkE ⊗ detE)|Bn, and the metric h is indeed invariant under transformation of local frame of E, and hence is a global Hermitian metric on (SkE ⊗ detE)|Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In this way, we see that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 is a direct consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We can also deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that, our argument is different from that in [3] and we do not need to consider the projectivization P(E) of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We hope that our method can throw new light on the study of ample vector bundles and Griffiths conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We should point out that, to see from a bounded domain Ω ⊂ E∗ the whole structure of E, we have to consider the natural S1-action on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' So as mentioned above the symmetric structure of Ω in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 is indispensable in their application to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We end the introduction by presenting the main ideas of proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The two main ingredients in the proofs of the theorems are Berndtsson’s curvature estimate in [3] and Deng-Ning-Wang-Zhou’s integral characterization of the Nakano positivity of Hermitian vector bundles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' More precisely, we first consider product domains and use Berndtsson’s estimate to get a positive lower bound of the curvature, and then take STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 9 a limit to come back to the metric in the non-product domain case and use Deng-Ning- Wang-Zhou’s result to show that the curvature of the limit metric also has the same lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The idea in the first step was motivated by Berdtsson’s proof of a complex version of the Prekopa’s theorem for non-product domains [1], and the idea in the second step was applied by Liu-Yang-Zhou to solve a problem of Lempert via Deng-Ning-Wang-Zhou’s result mentioned above [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In the proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, one key observation is that a piece of area near the boundary of Ω, no matter how small it is, can produce a positive lower bound of the curvature of the concerned vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10, and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='11 are possible to be generalized to holomorphic vector bundles on more general spaces with general compact group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' However, to keep the main idea transparent, we do not touch such general abstract setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The remaining of the paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' After presenting some necessary pre- liminaries in §2, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='11 is almost the same as the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10, so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 in §4 and deduce the corollaries of them in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In the final section §6, we connect Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 with the study of ample vector bundles and deduce from them Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The first author thanks Professor Jiafu Ning, Zhiwei Wang, and Xi- angyu Zhou for helpful discussions on related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' This research is supported by Na- tional Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 2021YFA1003100), NSFC grants (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 11871451, 12071310), and the Fundamental Research Funds for the Central Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Preliminaries In this section, we collect some knowledge that are needed in our discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Regular maximum of plurisubharmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let ψ ∈ C∞(R) be a nonnegative even function, which is supported on [−1, 1] and satisfies � R ψ(h)dh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 (see [5, Lemma (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='18), Chapter I]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any η := (η1, η2) ∈ (0, +∞) × (0, +∞), the function maxη : R2 → R defined as (t1, t2) �→ � R2 max{t1 + h1, t2 + h2} 1 η1η2 ψ(h1 η1 )ψ(h2 η2 )dh1dh2 possesses the following properties (i) maxη{t1, t2} is non decreasing in all variables, smooth and convex on R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (ii) max{t1, t2} ≤ maxη{t1, t2} ≤ max{t1 + η1, t2 + η2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (iii) If u1, u2 are plurisubharmonic functions, then maxη{u1, u2} is also plurisubharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Curvature positivity of Hermitian holomorphic vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let X be a complex manifold of complex dimension n, and (E, h) be a Hermitian holomorphic vector bundle over X of rank r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let D be the (1, 0)-part of the Chern connection of (E, h), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1) Θ(E,h) := [D, ¯∂] = D ¯∂ + ¯∂D be the Chern curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Over a coordinate chart (Ω, (t1, · · · , tn)) ⊂ X, we have ∂tj(u, v) = (Dtju, v) + (u, ¯∂tjv), ∀u, v ∈ Γ(X, E), where ∂tj := ∂ ∂tj and ¯∂tj := ∂ ∂¯tj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The Chern curvature is Θ(E,h) = � Θ(E,h) jk dtj ∧ d¯tk, where these coefficients are the commutators Θ(E,h) jk := [Dtj, ¯∂tk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The curvature of (E, h) is said to be positive (or strictly positive) in the sense of Nakano if for any nonzero n-tuple (u1, · · · , un) of sections of E � (Θ(E,h) jk uj, uk) ≥ 0 (or > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The following result is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 (see [5, Theorem (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5), Chapter V]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let (F, h) be a Hermitian holomorphic vector bundle over X, and let E, G be two holomorphic subbundles of F such that F = E⊕G and E is orthogonal to G, then the curvature of these bundles satisfies ΘF = ΘE ⊕ ΘG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' One of the main ingredients in our argument of the main results is the following result of Berndtsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 ([3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If Ω and ϕ satisfy the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 and ϕ is strictly plurisubharmonic, then for any smooth sections u1, · · · , un of the trivial bundle E, we have � j,l (ΘE jluj, ul) ≥ � j,l � D H(ϕ)jlujule−ϕdλz where H(ϕ)jl := ϕjl − � α,β ϕαβϕjαϕlβ, where (ϕαβ)m×m is the inverse matrix of (ϕαβ)m×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 11 In the above Lemma, j, l = 1, · · · , n represent the indices of the components of t = (t1, · · · , tn), α, β = 1, · · · , m represent the indices of the components of z = (z1, · · · , zm), ϕj,l = ∂2ϕ ∂tj∂¯tl and ϕjα, ϕαβ are given in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Optimal L2-estimate condition and curvature positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We first recall a fundamental result about the L2-estimate of ¯∂ for a Hermitian holomorphic vector bundle with Nakano positive curvature, which is due to H¨ormander and Demailly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4 (see [5, Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5), Chapter VIII]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let X be a complete K¨ahler mani- fold, with a K¨ahler metric ω which is not necessarily complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let (E, h) be a Hermitian vector bundle of rank r over X, and assume that the curvature operator B := [iΘE,h, Λω] is semi-positive definite everywhere on Λp,qT ∗ X ⊗ E, for some q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then for any form g ∈ L2(X, Λp,qT ∗ X ⊗ E) satisfying ¯∂g = 0 and � X⟨B−1g, g⟩dVω < +∞, there exists f ∈ L2(X, Λp,q−1T ∗ X ⊗ E) such that ¯∂f = g and � X |f|2dVω ≤ � X ⟨B−1g, g⟩dVω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The following result of Deng-Ning-Wang-Zhou shows that the converse of the above Lemma also holds, and hence gives an equivalent integral form characterization of the curvature positivity of Hermitian holomorphic vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5 ([8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let U ⊂ Cn be a bounded domain, (E, h) be a Hermitian holomorphic vector bundle over U with smooth Hermitian metric h, and θ ∈ C0(U, ∧1,1T ∗ U ⊗ End(E)) with θ∗ = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If for any strictly plurisubharmonic function ψ on U and f ∈ C∞ c (U, ∧n,1T ∗ U ⊗ E) with ¯∂f = 0 and i∂ ¯∂ψ ⊗ IdE + θ > 0 on supp(f), there is a measurable section u of ∧n,0T ∗ U ⊗ E on U, satisfying ¯∂u = f and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2) � U |u|2 he−ψdλz ≤ � U ⟨B−1 i∂ ¯∂ψ,θf, f⟩hdλz, provided that the right hand side is finite, then iΘE,h ≥ θ in the sense of Nakano, where ω = i �n j=1 dzj ∧ d¯zj and Bi∂ ¯∂ψ,θ = [i∂ ¯∂ψ ⊗ IdE + θ, Λω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The above Lemma is a modified version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 in [8] (please see [8, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 We first give the proof in the case that Ω is a product domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω := U × D ⊂ Cn t × Cm z be a bounded domain, D be a (connected) pseu- doconvex circular domain containing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that ϕ is a C2 plurisubharmonic function defined on some neighborhood of Ω and is S1-invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let k ≥ 0 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN and Ek t be the space of homogenous polynomials on Cm of degree k, with inner product ht given by ht(f, g) = � D f¯ge−ϕtdλz, f, g ∈ Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We set Ek = ∪t∈UEk t and view it as a (trivial) holomorphic vector bundle over U in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let R, M > 0 satisfy sup{∥z∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' z ∈ D} ≤ R, sup{|ϕ(t, z)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (t, z) ∈ Ω} ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If there exist 0 < r < s such that Br,s := {z ∈ Cm|r ≤ ∥z∥ ≤ s} ⊂ D and ϕ is strictly plurisubharmonic on U ×Br,s, then the curvature of the Hermitian holomorphic vector bundle (Ek, h) satisfies: � j,l (Θ(Ek,h) jl uj, ul) ≥ δ � j h(uj, uj) for any sections u1, · · · , un of Ek, where δ > 0 is a constant depending on R, M, r, s and the complex Hessian of ϕ on U × Br,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any ǫ > 0, let ϕǫ := ϕ + ǫ(|t|2 + |z|2) and denote the complex Hessian matrix of ϕǫ as � (ϕǫ)jl (ϕǫ)jα (ϕǫ)βl (ϕǫ)βα � , where j, l = 1, · · · , n represent the indices of the components of t = (t1, · · · , tn), α, β = 1, · · · , m represent the indices of the components of z = (z1, · · · , zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then ϕǫ is strictly plurisubharmonic on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We consider the Hermitian metric hǫ on Ek given by: hǫ t(f, g) = � D f¯ge−(ϕǫ)tdλz, f, g ∈ Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let E be the trivial vector bundle over U as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then Ek is a holomorphic subbundle of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since D is a circular domain containing the origin, any f ∈ O(D) can be represented as a series f = +∞ � j=0 fj that is convergent locally uniformly on D, where each fj is a homogenous polynomial of degree j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any S1-invariant continuous bounded function ψ on D, and any homogenous polynomials gj, gl of degree j and l respectively, we have � D gj¯gle−ψ = 0 whenever j ̸= l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that, for any t ∈ U, an element f in the orthogonal complement (Ek)⊥ t of Ek t in Et has the form f = � j≥0,j̸=k fk, STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 13 where each fj is a homogeneous polynomial of degree j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Hence (Ek)⊥ t as a vector space is independent of the choice of the weight function ϕ and is also a holomorphic subbundle of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now fix an arbitrary t0 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, for any u1, · · · , un of Ek t0, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1) � j,l (Θ(Ek,hǫ) jl uj, ul) ≥ � D � j,l H(ϕǫ)jl(t0, z)ujule−(ϕǫ)t0dλz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' where H(ϕǫ) is a Hermitian matrix defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If we write � ϕjl ϕjα ϕβl ϕβα � = � A B C F � , then we have H(ϕ) = A − BF −1C provided that F is nonsigular, and � H(ϕ) 0 ∗ F � = � A − BF −1C 0 ∗ F � = � I −BF −1 0 I � � A B C F � � I 0 −(BF −1)∗ I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that H(ϕ) is positively definite if ϕ is strictly plurisubharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' So we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2) � j,l (Θ(Ek,hǫ) jl uj, ul)|t0 ≥ � Br,s � j,l H(ϕǫ)jl(t0, z)ujule−(ϕǫ)t0dλz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By assumption and by continuity, there is a constant δ0 > 0 such that � j,l H(ϕ)jl(t0, z)ujul ≥ δ0 � j |uj|2 for z ∈ Br,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' On the other hand, it is clear that H(ϕǫ)(t0, z) = H(ϕ)(t0, z) + oǫ(1) on Br,s, where oǫ(1) represents functions on Br,s that converge to 0 uniformly as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that � j,l (Θ(Ek,hǫ) jl uj, ul)|t0 ≥ δ0 � Br,s � j (1 + oǫ(1))|uj|2e−(ϕǫ)t0dλz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since hǫ converges to h in the sense of C2 as ǫ → 0+, Θ(Ek,hǫ) converges to Θ(Ek,h) as ǫ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We thus have � j,l (Θ(Ek,h) jl uj, ul) ≥ δ0 � Br,s � j |uj|2e−ϕt0dλz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that uj are homogenous polynomials of degree k, D is bounded, and ϕ(t0, z) is bounded on D, there exists a constant δ > 0, which is independent of uj, such that δ0 � Br,s � j |uj|2e−ϕt0dλz ≥ δ � D � j |uj|2e−ϕt0dλz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN It follows that � j,l (Θ(Ek,h) jl uj, ul) ≥ δ � D � j |uj|2e−ϕt0dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' □ We shall deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 (=Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a family of bounded domains over U that admits a plurisubharmonic defining function, and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that all fibers Ωt (t ∈ U) are (connected) circular domains in Cm containing the origin and ϕ(t, z) is S1- invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let k ≥ 0 and Ek t be the space of homogenous polynomials on Cm of degree k, with inner product ht given by ht(f, g) = � Ωt f¯ge−ϕtdλz, f, g ∈ Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We set Ek = ∪t∈UEk t and view it as a (trivial) holomorphic vector bundle over U in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If there exist 0 < r < s such that Br,s := {z ∈ Cm|r ≤ ∥z∥ ≤ s} ⊂ Ωt for all t ∈ U and ϕ is strictly plurisubharmonic on U × Br,s, then the curvature of the holomorphic Hermitian vector bundle (Ek, h) is strictly positive in the sense of Nakano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let ρ(t, z) be a plurisubharmonic defining function of Ω, by averaging, we may assume that ρ is S1-invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any fixed t0 ∈ U and 0 < h << 1, let D = {(t0, z) ∈ U × Cn|ρ(t0, z) ≤ h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then there exists a neighborhood U′ of t0 in U such that ρ and ϕ are defined on some neighborhood of the closure of U′ × D and p−1(U′) ∩ Ω ⊂ U′ × D, where p : Cn × Cm → Cn is the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since the result to be proved is local in nature with respect to t, we may assume that U = U′, then we have Ω ⊂ ˜Ω := U × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any positive integer N, let ϕN = ϕ + N max( 1 N2 , 1 N2 ){0, ρ − 1 N }, where max( 1 N2 , 1 N2 ){0, ρ} is the regularized max function defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For N >> 1, ϕN is equal to ϕ on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 to ˜Ω and ϕN, we get a constant δ > 0 such that � (Θ(Ek,hN) jl uj, ul) ≥ δ � � D |uj|2e−ϕN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' for any sections u1, · · · , un of Ek, where the metric hN on Ek is given by hN t (f, g) = � D f¯ge−(ϕN)tdλz, f, g ∈ Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In other words, if we take θ = iδ � j dtj ∧ d¯tj ⊗ IdEk ∈ C0(U, ∧1,1T ∗ U ⊗ End(Ek)), STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 15 then we have iΘ(Ek,hN) ≥Nak θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We want to apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5 to prove that iΘ(Ek,h) ≥Nak θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The main idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' From the above curvature estimate and the L2-estimate of ¯∂, we know that (Ek, hN) satisfy the L2-estimate condition presented in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' As N → ∞, we have hN → h and one can see that (E, h) also satisfies the L2-estimate condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5 that the curvature of (E, h) satisfies iΘ(Ek,h) ≥Nak θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The detail of the argument is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let ψ(t) be a strictly plurisubharmonic function on U, and f ∈ C∞ c (U, ∧n,1T ∗ U ⊗Ek) satisfies ¯∂f = 0 and � U < B−1 i∂ ¯∂ψ,θf, f >h e−ψdλt < +∞, where ω = i �n j=1 dtj ∧ d¯tj and Bi∂ ¯∂ψ,θ is given as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then there exists M > 0 such that � U < B−1 i∂ ¯∂ψ,θf, f >hN e−ψdλt ≤ M, ∀N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4, there are measurable sections uN of ∧n,0T ∗ U ⊗ Ek on U, such that ¯∂uN = f and � U |uN|2 hNe−ψdλt ≤ � U < B−1 i∂ ¯∂ψ,θf, f >hN e−ψdλt ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since ϕN and ϕ are equal on Ω, we have � U |uN|2 he−ψdλt ≤ � U |uN|2 hNe−ψdλt ≤ M for all N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In particular, {uN} is a bounded sequence in the Hilbert space H of square integrable sections of ∧n,0T ∗ U ⊗ Ek on U with weight e−ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Hence there is a subsequence of {uN}, assumed to be {uN} itself without loss of generality, that converges weakly in H to some u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that we also have ¯∂u = f in the sense of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' On one hand, we have � U |u|2 he−ψdλt ≤ lim sup N→∞ � U |uN|2 he−ψdλt, and on the other hand, we have lim N→∞ � U < B−1 i∂ ¯∂ψ,θf, f >hN e−ψdλt = � U < B−1 i∂ ¯∂ψ,θf, f >h e−ψdλt by Lebesgue’s dominated convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' So we get � U |u|2 he−ψdλt ≤ � U < B−1 i∂ ¯∂ψ,θf, f >h e−ψdλt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5 that iΘ(Ek,h) ≥Nak θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' □ 16 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 The difficulty of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 compared with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 is that the weight function does not have strict plurisubharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We will use the strict psedoconvexity of the domain to get the Nakano positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, in addition to using Berndtsson’s estimate of curvature (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3) and Deng-Ning-Wang-Zhou’s integral characterization of the Nakano positivity of Hermitian vector bundles (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5), an important role is also played by the simple observation that the integral � r 0 Ne−Nh(x)dx has a uniform positive limit as N → ∞ for all r > 0 and all smooth function h with h(0) = 0 and h′(0) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We first give a Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω be a bounded domain in Rn with C2- boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any 0 < r << 1, let Ωr := {x ∈ Rn\\Ω|d(x, ∂Ω) < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then there exists a constant c > 0 such that � Ωr hdx1 ∧ · · · ∧ dxn ≥ c � ∂Ω dS � r 0 h(ζ + tnζ)dt for any positive integrable functions h on Ωr, where nζ is the outward unit normal of ∂Ω at ζ and dS is the volume form on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We can choose r0 > 0 such that the map f : ∂Ω × [0, r0) → Ωr0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (ζ, t) �→ ζ + tnζ is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let µ = dS ∧ dt be the product measure on ∂Ω × [0, r0) → Ωr0 and µ0 be the Lebesgue measure on Ωr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then there is a continuous positive function σ on Ωr0 such that µ0 = σ · f∗µ on Ωr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any 0 < r < r0, taking c = min{σ(x)|x ∈ Ωr}, then c > 0 and µ0 ≥ cf∗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' From it the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 (=Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that all fibers Ωt (t ∈ U) are (connected) circular domains in Cm containing the origin and ϕ(t, z) is S1- invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let k ≥ 0 and Ek t be the space of homogenous polynomials on Cm of degree k, with inner product ht given by ht(f, g) = � Ωt f¯ge−ϕtdλz, f, g ∈ Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We set Ek = ∪t∈UEk t and view it as a (trivial) holomorphic vector bundle over U in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then the curvature of the holomorphic Hermitian vector bundle (Ek, h) is strictly positive in the sense of Nakano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since Ω is strictly pseudoconvex with C2 boundary, there is a defining function ρ that is strictly plurisubharmonic on some neighborhood ˜Ω of Ω in U × Cm and S1 invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any fixed t0 ∈ U, we can take a neighborhood U′ of t0 in U and a pseudoconvex circular domain D ⊂ Cm such that p−1(U′) ∩ Ω ⊂ U′ × D ⊂ ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since the conclusion to be proved is local in nature on t, we may assume that U = U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We denote U × D by Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For N ∈ Z+, we set ϕN = ϕ + N max( 1 N3 , 1 N3 ){0, ρ}, which is a C2 plurisubharmonic function defined on ˜Ω and is S1- invariant with respect to z, where max( 1 N3 , 1 N3 ){0, ρ} is the regularized max function defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any ǫ > 0, define ϕN,ǫ := ϕ + N max( 1 N3 , 1 N3 ){0, ρ} + ǫ|t|2 + ǫ|z|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let hN,ǫ be the Hermitian metric on Ek given by hN,ǫ t = � D f¯ge−(ϕN,ǫ)tdλz, f, g ∈ Et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, we know for any u1, · · · , un of Ek t0 that � j,l (Θ(Ek,hN,ǫ) jl uj, ul) ≥ � D � j,l H(ϕN,ǫ)jl(t0, z)ujule−(ϕN,ǫ)t0dλ, where H(ϕN,ǫ)jl is defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For 0 < r << 1, as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1, we set Ωt0,r = {z ∈ Cm\\Ωt0|d(z, ∂Ωt0) < r} and set ΩN t0,r = Ωt0,r\\Ωt0,1/N2 for N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now fix such an r such that Ωt0,r ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that max( 1 N3 , 1 N3 ){0, ρ} = ρ on ΩN t0,r for all N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that H(ϕ + Nρ + ǫ|t|2 + ǫ|z|2) = NH(ρ + (ϕ + ǫ|t|2 + ǫ|z|2)/N), we have H(ϕ + Nρ + ǫ|t|2 + ǫ|z|2) ≥ N 2 H(ρ) 18 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN on ΩN t0,r for N sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Combining with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' we can see there exist constants δ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' δ1 > 0 such that � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='l (Θ(Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='hN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ǫ) jl uj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' ul) ≥ � D � H(ϕN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ǫ)jl(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' z)ujule−(ϕN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ǫ)t0dλz ≥ � ΩN t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='r � H(ϕN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ǫ)jl(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' z)ujule−(ϕN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ǫ)t0dλz ≥δ0 � ΩN t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='r N � |uj|2e−Nρdλz ≥δ1 � ζ∈∂Ωt0 dS � r 1/N2 � N|uj(ζ + τnζ)|2e−Nρ(ζ+τnζ)dτ ≥δ1 � ζ∈∂Ωt0 dS � inf 1/N2≤τ≤r |uj(ζ + τnζ)|2 � r 1/N2 � Ne−Nρ(ζ+τnζ)dτ ≥δ1 � ζ∈∂Ωt0 dS � inf 0≤τ≤r |uj(ζ + τnζ)|2 � r 1/N2 Ne−NTτdτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' where nζ is the unit outward normal of ∂Ωt0 at ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' dS is the volume form on ∂Ωt0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' and T > 0 is a constant such that ρ(ζ + τnζ) ≤ Tτ for all ζ ∈ ∂Ωt0 and 0 ≤ τ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now need the obvious but important fact that limN→∞ � r 1/N2 Ne−NTτdτ = 1 T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We then get from the above calculation that � j,l (Θ(Ek,hN,ǫ) jl uj, ul) ≥ δ2 � � ∂Ωt0 inf 0≤τ≤r |uj(ζ + τnζ)|2dS for some constant δ2 > 0 and for N sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let ǫ → 0, and denote hN,0 by hN, we get � j,l (Θ(Ek,hN) jl uj, ul) ≥ δ2 � � ∂Ωt0 inf 0≤τ≤r |uj(ζ + τnζ)|2dS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1) for N sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For u ∈ Ek t0, we need to control its norm ∥u∥hN t0 = � D |u|2e−(ϕN)t0dλz in terms of the integral � ∂Ωt0 inf0≤τ≤r |u(ζ + τnζ)|2dS, where ϕN = ϕN,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Q = {u ∈ Ek t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' ∥u∥2 hN = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that functions in Q are homogenous polynomials of degree k and Ωt0 contains the origin, we can choose a constant M > 0 and a large ball B with D ⊂ B such that � B |u|2dλz ≤ M for all u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By Cauchy’s inequality for holomorphic STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 19 functions, there is a constant C > 0 such that |du2| < C on D for all u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2) inf 0≤τ≤r |u(ζ + τnζ)|2 ≥ |u(ζ)|2 − rC for all ζ ∈ ∂Ωt0 and for all u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now move to prove that we can choose r and a constant δ3 > 0 such that � ∂Ωt0 inf 0≤τ≤r |u(ζ + τnζ)|2dS ≥ δ3 for all u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By the maximum principle and continuity, we can take ζ′ ∈ ∂Ωt0 such that |u| takes its maximum on Ωt0 at ζ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Again, since functions in Q are homogenous polynomials of degree k and Ωt0 contains the origin, we can choose a constant C1 > 0 such that � Ωt0 |u|2dλz ≥ C1 for all u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that |u(ζ′)|2 ≥ C1 |Ωt0|, where |Ωt0| is the Lebesgue measure of Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Again by Cauchy’s inequality, if choosing 0 < r < C1 2C|Ωt0|, we get |u(ζ)|2 ≥ |u(ζ′)|2 − Cr ≥ C1 2|Ωt0| for all u ∈ Q and for all ζ ∈ ∂Ωt0 with |ζ − ζ′| < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that � ∂Ωt0 inf 0≤τ≤r |u(ζ + τnζ)|2dS ≥ � B(ζ′,r)∩∂Ωt0 inf 0≤τ≤r |u(ζ + τnζ)|2dS ≥ � B(ζ′,r)∩∂Ωt0 (|u|2 − rC)dS ≥ C1 2|Ωt0||B(ζ′, r) ∩ ∂Ωt0|, where B(ζ′, r) is the ball in Cm with center ζ′ and radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that ∂Ωt0 is compact and the function σ : ∂Ωt0 −→ R : ζ → |B(ζ, r) ∩ ∂Ωt0| is continuous and positive, we have δ3 := inf ζ∈∂Dr |B(ζ, r) ∩ ∂Ωt0| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' So we get � ∂Ωt0 inf 0≤τ≤r |u(ζ + τnζ)|2dS ≥ δ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 20 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1), for N sufficiently large, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3) � j,l (Θ(Ek,hN) jl uj, ul) ≥ δ � j ∥uj∥2 hN t0 for any tuple u1, · · · , un ∈ Ek t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Just as the last step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10, we can derive from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5 that � j,l (Θ(Ek,h) jl uj, ul) ≥ δ � j ∥uj∥2 ht0 for any tuple u1, · · · , un ∈ Ek t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In particular, the curvature of (Ek, h) is strictly positive in the sense of Nakano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' □ Similar results holds for a strictly pseudoconvex family of Reinhardt domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 (=Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U ×Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume that all fibers Ωt (t ∈ U) are (connected) Reinhardt domains in Cm and ϕ(t, z) is T m invariant with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then for any nonnegative integers k1, · · · , km, the function ψ(t) defined by e−ψ(t) = � Ωt |zk1 1 · · · zkm m |2e−ϕtdλz is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since the proof is almost the same as the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, we just give a sketch of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any nonnegative integers k1, · · · , km, we consider the 1-dimensional vector space Ek1,··· ,km t = Czk1 1 · · · zkm m , with inner product ht given by ht(f, g) = � Ωt f¯ge−ϕtdλz, f, g ∈ Ek1,··· ,km t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We set Ek1,··· ,km = ∪t∈UEk1,··· ,km t and view it as a holomorphic line bundle over U in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since the conclusion to be proved is local in nature with respect to t ∈ U, we may assume there is a bounded pseudoconvex Reinhardt domain D ⊂ Cn such that Ω ⊂ Ω′ := U ×D and ϕ and ρ are defined on some neighborhood of Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that � D zk1 1 · · · zkm m zl1 1 · · · zlm m e−ϕtdλz = 0 for any nonnegative integers k1, · · · , km and l1, · · · , lm with kj ̸= lj for some 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' So by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 the curvature of (Ek1,··· ,km, h) is the restriction of the curvature of (E, h′) on STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 21 Ek1,··· ,km, where (E, h′) represents the vector bundle given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 with Ω replaced by Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' With the above discussions at hand, the remaining of the proof of the theorem can go ahead following the same way as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, and we omit the details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Some consequences of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 We now discuss some consequences of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Consequences in complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5 in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 (=Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of bounded domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm that satisfy the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For t ∈ U, let K(t, z) be the weighted Bergman kernel of Ωt with weight ϕt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then ln K(t, z) is a strictly plurisubharmonic function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof is provided in the following discussion, which indeed gives us more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We assume Ω and ϕ satisfies the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, and the remaining case can be proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that ln K(t, z) is strictly plurisubharmonic with respect to z, it is enough to prove that for any (t0, z0) ∈ Ω and any local holomorphic map ξ(t) : B → Cm defined on some small neighborhood B of t0 with ξ(t0) = z0, the function ln K(t, ξ(t)) is strictly plurisubharonic as a function on B (the reason is that any non-vertical tangent vector of Ω at (t0, z0) lies in the image of dξ(t0)) for some such a map ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ek t be the space with inner product defined as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2, and let uk 1, · · · , uk mk be an orthogonal normal basis of Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We set Kk(t, z) = mk � j=1 |uj(z)|2, then it is clear that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1) K(t, z) = ∞ � k=0 Kk(t, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let p : Ω → U be the natural projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then the pull back ( ˜Ek, ˜h) := (p∗Ek, p∗h) of the bundle (Ek, h) on U is a Hermitian holomorphic vector bundle over Ω whose curvature is semi-positive in the sense of Nakano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 22 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN Let F = Ω × C be the trivial line bundle on Ω and denote by e the canonical frame of F on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then we have a natural vector bundle morphism σk : ˜Ek → L given by f �→ (t, z, f(z)) ∈ F for f ∈ ˜Ek (t,z) = Ek t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ωk = {(t, z) ∈ Ω|Kk(t, z) ̸= 0}, or equivalently, (t, z) ∈ Ωk if and only if f(z) ̸= 0 for some homogenous polynomial f on Cm of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then σk is a surjective bundle morphism from ˜Ek|Ωk to F|Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' One can see that the quotient metric, say hk on F|Ωk induced from this morphism is given by ∥e∥2 hk = 1 Kk(t, z) = e− ln Kk(t,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Since the curvature of ( ˜Ek, ˜h) is semi-positive in the sense of Nakano, and note the curvature increasing property under taking quotient metric [5, a) in Proposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10)], we know the curvature of (F|Ωk, hk) is semi-positive, which implies that ln Kk(t, z) is plurisubharmonic on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any given (t0, z0) ∈ Ω, and any holomorphic map ξ(t) : B → Cm defined on some small neighborhood B of t0 with ξ(t0) = z0, we denote by Γ = {(t, ξ(t))|t ∈ B} ⊂ Ω the graph of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then ( ˜E0, ˜h)|Γ is a (trivial) Hermitian line bundle over Γ whose curvature is strictly positive, since p|Γ : Γ → B is a biholomoprhic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note also that σ0 : ˜E0 → L is an isomorphism of vector bundles, it follows that ln K0(t, ξ(t)) is strictly plurisubharmonic on Γ, and hence is strictly plurisubharmonic as a function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1), we know that ln K(t, ξ(t)) is strictly plurisubharmonic as a function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Hence ln K(t, z) is strictly plurisubharmonic on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of the above corollary is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In fact, by the same argument, one can show, for any nonnegative integer k, that ”the relative log character Bergman kernel” ln Kk(t, z) is plurisubharmonic on Ω and is strictly plurisubharmonic on Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 (=Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ U × Cm be a strictly pseudoconvex family of domains over U ⊂ Cn and ϕ be a C2 plurisubharmonic function defined on some neighborhood of Ω in U × Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (1) If Ω and ϕ satisfy the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3, then the function ˜ϕ defined by e− ˜ϕ(t) = � Ωt e−ϕ(t,z)dλz is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 23 (2) If all fibers Ωt are tube domains of the form Xt + iRm withXt bounded, and ϕ(t, z) does not depend on the imaginary part of z, then the function ˜ϕ defined by e− ˜ϕ(t) = � Xt e−ϕ(t,Rez)dλRez is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It is clear that (1) is equivalent to the curvature strict positivity of (E0, h) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now give the proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let us consider the map f : Ω → Cn t × Cm w (t1, · · · , tn, z1, · · · , zm) �→ (t1, · · · , tn, ez1, · · · , ezm), then Ω∗ := f(Ω) ⊂ Cn t × Cm w is a strictly pseudoconvex family of Reinhardt domains over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that ψ(t, w) := ϕ(t, ln |w1|, · · · , ln |wm|) + 2(ln |w1| + · · · + ln |wm|) is a C2 and plurisubharmonic function defined on some neighborhood of the closure of Ω∗ in U × Cm, applying (1) to Ω∗ and ψ, we see that the function ˜ϕ defined by e− ˜ϕ(t) = � Xt e−ϕ(t,Rez)dλRez = 1 (2π)n � Ω∗ t e−ψ(t,w)dλw is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Consequences in convex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The bridge connecting strictly convex families of bounded domains in Rm and strictly pseudoconvex families of tube domains in Cm is indicated in the proof of the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3 (=Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let D ⊂ U0 × Rm be a strictly convex family of bounded domains over a domain U0 ⊂ Rn and ϕ be a C2 convex function defined on some neighborhood of the closure of D in U0 × Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then the function ˜ϕ defined by e− ˜ϕ(t) = � Dt e−ϕ(t,x)dλx is a strictly convex function on U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We first complexify U0 to U = U0 × iRn l with complex coordinate τ = t + il, then U is a domain in Cn τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We secondly complexify Rm to Rm x + iRm y = Cm z , with complex coordinate z = x + iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then Ω = D + iRn+m = {(τ, z) ∈ Cn × Cm|(Reτ, Rez) ∈ D} 24 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN is a strictly pseudoconvex family of tube domains over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For τ ∈ U, Ωτ is a tube domain of the form Ωτ = Dτ + iRm, where Dτ ⊂ Rm can be naturally identified with DReτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By setting ψ(τ, z) = ϕ(Reτ, Rez), we extend ϕ to a C2 plurisubharmonic function ψ on some neighborhood of Ω in U × Cm, such that ψ(τ, z) is independent of the imaginary part of τ, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By (2) in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5, the function ˜ψ defined by e− ˜ψ(τ) = � Dτ e−ψ(τ,Rez)dλRez is a strictly plurisubharmonic function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It is clear that ˜ψ(τ) is independent of the imaginary part of τ and ˜ψ|U0 = ˜ϕ, thus ˜ϕ is a strictly convex function on U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Curvature negativity of determinant line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now explain the meaning of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='9 and give its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let π : E → X be a holomorphic vector bundle of rank m over a complex manifold X equipped with a smooth Finsler metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By definition, h is a continuous function h : E → R such that h ≥ 0, h(λv) = |λ|h(v) for λ ∈ C and v ∈ E, and h is smooth on E\\ZE, where ZE ⊂ E is the zero section of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Recall that (E, h) is defined to be strictly negatively curved if ln h is strictly plurisubharmonic on E\\ZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now define the Hermitian metric deth induced from h on the determinant line bundle detE = ΛmE of E via the measure µ on Et with µ(Bt) = 1 for t ∈ X, where Bt = {v ∈ Et|h(v) ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' A more explicit description of det h in terms of local frame is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let e1, · · · , em be a holomorphic local frame of E over some open set U ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We get a local trivialization of E over U: φ : E|U → U × Cm, (t, z1v1 + · · · + zrvm) �→ (t, z1, · · · , zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then e := e1 ∧ · · · ∧ em is a local frame of detE over U, whose norm with respect to det h is given by ∥e(t)∥2 det h = 1 µ0(φt(Bt)), where µ0 is the Lebesgue measure on Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='8, we know that − ln µ0(φt(Bt)) is a strictly plurisubharmonic function on U provided that h is strictly negatively curved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that the curvature of (detE, deth) on U is given by i∂ ¯∂ ln µ0(φt(Bt)), so the curvature of the induced Hermitian metric deth on detE is strictly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We thus get Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4 (=Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let π : E → X be a holomorphic vector bundle over a complex manifold X equipped with a smooth Finsler metric h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' If (E, h) is strictly negatively curved, then the curvature of the induced Hermitian metric deth on detE is strictly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 In this section, we discuss the relation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We show that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 can be deduced from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For this consideration, the symmetric structure appearing in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 plays an indispensable role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Basic properties of ample vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' This subsection recalls some well known basic knowledge about ample vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let π : E → X be a holomorphic vector bundle over a compact complex manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For each x ∈ X, we denote by Ex the fiber of E over x and denote by E∗ x its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let P(E∗ x) be the projective space of E∗ x, which is the space of one-dimensional complex linear subspaces of E∗ x with the natural complex structure, and let OP(E∗x)(1) be the dual of the tautological line bundle over P(E∗ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then P(E∗) := ∪x∈XP(E∗ x) is a complex manifold that can be naturally realized as a holomorphic fiber bundle over X with P(E∗ x) as fibers, and OP(E∗x)(1) := ∪x∈XOP(E∗x)(1) can be naturally realized as a holomorphic line bundle over P(E∗) whose restriction to P(E∗ x) is just OP(E∗x)(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By definition, E is called an ample vector bundle if OP(E∗)(1) is an ample line bundle over P(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now assume that E is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then there is a Hermitian metric h on OP(E∗)(−1) whose curvature is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let ρ : OP(E∗)(−1) → R≥0 be the length function associated to h, namely ρ(v) = � h(v, v) for v ∈ OP(E∗)(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then ρ is strictly plurisubharmonic on OP(E∗)(−1)\\ZOP(E∗)(−1), where ZOP(E∗)(−1) is the zero section of OP(E∗)(−1), viewed as a submanifold of OP(E∗)(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that OP(E∗)(−1) can be viewed as the blow up of E∗ along its zero section ZE∗, with ZOP(E∗)(−1) as the exceptional divisor, we can naturally identify E∗\\ZE∗ with OP(E∗)(−1)\\ZOP(E∗)(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Through this identification, we can view ρ as a function on E∗, with ρ|ZE∗ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In conclusion, we get a plurisubharmonic function ρ on E∗, which is strictly plurisubhar- monic on E∗\\ZE∗ and invariant under the natural S1 action on E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In other words, ρ is a smooth Finsler metric on E∗ whose curvature is strictly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Some linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We present some knowledge about linear algebra that is needed in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let V be a complex vector space with complex dimension m and V ∗ be its dual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let P(V ) be the set of all polynomials on V , and Pk(V ) be the space of homogeneous polynomials 26 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN of degree k on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then P(V ) = � k≥0 Pk(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We have P0(V ) = C and P1(V ) = V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In an obvious manner, we can naturally identify Pk(V ) with SkV , the k-th symmetric product of V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We realize the circle group S1 as the space of complex numbers with unit norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then S1 acts on V via scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' This induces an action of S1 on P(V ) as follows: α · f(v) = f(αv), where f ∈ P(V ), v ∈ V , α ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then Pk(V ) are character subspaces of P(V ) associated to this action, namely, for k ≥ 0 we have Pk(V ) = {f ∈ P(V ) ��α · f = αkf, ∀α ∈ S1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The cotangent bundle of V is T ∗V = V × V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that the canonical bundle of V is KV = V ⊗ det V ∗, where det V ∗ = ∧mV ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' We now consider the coordinate representation of KV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let u1, · · · , um be a basis of V and u∗ 1, · · · , u∗ m be the associated dual basis of V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then u∗ 1 ∧ · · · ∧ u∗ m is a basis of det V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Consider linear isomorphism: V −→ Cn : z1u1 + · · · + zmum �→ (z1, · · · , zm), then u∗ 1 ∧ · · · ∧ u∗ m corresponds to dz1 ∧ · · · ∧ dzm, a basis of det (Cm)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω ⊂ V be an S1 invariant domain containing 0, then we have the following identification (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1) H0(Ω, KΩ) = O(Ω, det V ∗), where O(Ω, det V ∗) is the space of holomorphic mappings from Ω to det V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The action of S1 on O(Ω, det V ∗) is given as follows: α · f(x) = f(αx), α ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Under coordinate form, if we identify V with Cm as above and view Ω as a domain in Cm, then we have the following identification (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2) H0(Ω, KΩ) ∼= {f(z1, · · · , zm)dz1 ∧ · · · ∧ dzm|f ∈ O(Ω)}, and the action of S1 on H0(Ω, KΩ) is realized as α · (f(z1, · · · , zm)dz1 ∧ · · · ∧ dzm) = f(αz1, · · · , αzm)dz1 ∧ · · · ∧ dzm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It is clear that the action of S1 on the Hilbert space A2(Ω) = {f ∈ H0(Ω, KΩ) : ||f|| < +∞} STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 27 is unitary, where ||f||2 = � Ω cmf ∧ ¯f, with cm = im2 2m is set to make the form cmf ∧ ¯f real and nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For any k ≥ 0, let (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3) P′ k(Ω) = {f ∈ A2(Ω)| α · f = αkf, ∀α ∈ S1}, then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4) P ′ k(Ω) = {f(z1, · · · , zm)dz1 ∧ · · · ∧ dzm| f ∈ Pk(Cm)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5) P ′ k(Ω) = Pk(V ) ⊗ det V ∗ = Sk V ∗ ⊗ det V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let π : E → X be an ample holomorphic vector bundle of rank m over a compact complex manifold X of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let E∗ be the dual bundle of E and let ZE∗ be the zero section of E∗, viewed naturally as a submanifold E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' From §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1, we know that E∗ admits a smooth Finsler metric ρ : E∗ → R≥0 whose curvature is strictly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let Ω = {v ∈ E∗|ρ(v) ≤ 1}, then Ω is an S1 invariant bounded domain in E∗ whose boundary is strictly pseudoconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' As usual, we denote Ω ∩ E∗ t by Ωt for t ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Note that Ωt is an S1-invariant domain in E∗ t containing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1), we can canonically identify H0(Ωt, KΩt) with O(Ωt, det Et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' For k ≥ 0, if we define P′ k(Ωt) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3), we have P′ k(Ωt) = SkEt ⊗ det Et from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let ϕ be an S1-invariant smooth plurisubharonic function defined on some neighborhood of the closure Ω of Ω in E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' On P′ k(Ωt), we can define a Hermitian inner product ht by setting ∥f∥2 ht = � Ωt cmf ∧ ¯fe−ϕt, f ∈ P′ k(Ωt), where ϕt is the restriction of ϕ on Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In this way, we get a Hermitian metric h on SkE⊗det E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Our propose is to deduce from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 or Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 that the curvature of the Hermitian vector bundle (SkE ⊗ det E, h) over X is strictly positive in the sense of Nakano, for suitable choice of ϕ (indeed for all such ϕ), and hence get new proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' The argument goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Let (U, t1, · · · , tn) be a local coordinate on X, and e1, · · · , em be a holomorphic local frame of E∗ over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Then we get an isomorphism σ : π−1(U) −→ U × Cm given by (t, z1e1 + · · · + zmem) �→ (t1, · · · , tn, z1, · · · , zm), where π : E∗ → X is the bundle map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' This isomorphism realizes Ω ∩ π−1(U) as a strictly pseudoconvex family of bounded domains over U whose fibers σ(Ωt) ⊂ Cm are circular 28 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' DENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' HU, AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' QIN domains containing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4), for t ∈ U, via σ we can identify P ′ k(Ωt) with the space {f(z1, · · · , zm)dz1 ∧ · · · dzm|f ∈ Pk(Cm)}, with the Hermitian inner product ht given by ∥f(z1, · · · , zm)dz1 ∧ · · · dzm∥2 ht = � σ(Ωt) |f|2e−ϕt◦σ−1dλz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' It follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='2 that the curvature of (SkE ⊗ det E, h) is strictly positive in the sense of Nakano, and hence we get Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' In a similar way, we can deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='12 from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='10 by choosing ϕ = max1/4,1/4{1/4, ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='1 for the definition of the regularized maximum function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' STRICT CURVATURE POSITIVITY OF DIRECT IMAGE BUNDLES 29 References [1] Berndtsson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Mathematische Annalen, 312(1998), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='4, 785-792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [2] Berndtsson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Fourier (Grenoble) 56 (2006), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='6, 1633-1662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [3] Berndtsson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Curvature of vector bundles associated to holomorphic fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Annals of Mathematics, 2009, 169(2):531-560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [4] Berndtsson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Strict and nonstrict positivity of direct image bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 269 (2011), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='3-4, 1201- 1218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [5] Demailly JP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Complex analytic and differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Electric book, available in the author’s home- page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [6] Demailly JP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' , Skoda H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Relations entre les notions de positivit´es de P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Griffiths et de S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Nakano pour les fibr´es vectoriels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' S´eminaire Pierre Lelong-Henri Skoda (Analyse), (1978/79), 304-309, Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 822, Springer-Verlag, New York, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [7] Deng F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Hu J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Jiang W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Curvature positivity of invariant direct images of Hermitian vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Annali di Matematica Pura ed Applicata, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [8] Deng F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Ning J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Wang Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Zhou X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Positivity of holomorphic vector bundles in terms of Lp-properties of ¯∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='01762, to appear in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [9] Deng F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Jiang W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Qin X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Deduce some results in convex analysis form complex analysis and vice versa, to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [10] Deng F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Zhang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Zhou X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Positivity of character subbundles and minimum principle for noncompact group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 286 (2017), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 1-2, 431-442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [11] Grauert, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', On Levis problem and the imbedding of real-analytic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (2) 68 (1958) 460-472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [12] Grauert H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' ¨Uber Modifikationen und exzeptionelle analytische Mengen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Mathematische Annalen, 1962, 146(4):331-368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [13] Griffiths PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Hermitian differential geometry, Chern classes, and positive vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Global Anal- ysis, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [14] Liu Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Yang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', Zhou X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=', On the Multiplier Submodule Sheaves Associated to Singular Nakano Semi- positiveMetrics, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='13452, 2021 [15] Pr´ekopa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' On logarithmic concave measures and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' (Szeged) 34 (1973), 335-343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' [16] Umemura H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Some results in the theory of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Nagoya Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' 52 (1973), 97-128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' Fusheng Deng: School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' China Email address: fshdeng@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='cn Jinjin Hu: School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' China Email address: hujinjin21@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='cn Xiangsen Qin: School of Mathematical Sciences, University of Chinese Academy of Sci- ences, Beijing 100049, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content=' China Email address: qinxiangsen@amss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfWPem/content/2301.00160v1.pdf'} +page_content='cn' metadata={'source': 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Artificial Intelligence, Toronto, ON M5G 1M1, Canada +3 Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada +4 Department of Computer Science, University of Victoria, Victoria, BC V8P 5C2, Canada +itamblyn@squareup.com +Abstract +We discuss our simulation tool, fintech-kMC, which is de- +signed to generate synthetic data for machine learning model +development and testing. fintech-kMC is an agent-based +model driven by a kinetic Monte Carlo (a.k.a. continuous time +Monte Carlo) engine which simulates the behaviour of cus- +tomers using an online digital financial platform. The tool +provides an interpretable, reproducible, and realistic way of +generating synthetic data which can be used to validate and +test AI/ML models and pipelines to be used in real-world +customer-facing financial applications. +To appear at AAAI-23 Bridge Program: AI for Financial +Services, Washington D.C., February 7 – 8, 2023 +Introduction +Simulated data is useful in machine learning (ML) work +for developing and testing algorithms in a repeatable and +verifiable way. We note that although historically there +has been some debate about the utility of simulated data +in machine learning applications given the challenges of +sim2real (H¨ofer et al. 2020), irrespective of whether sim- +ulated data is valuable for training production models, they +are extremely valuable in the initial stages of model design, +unit testing, and verification processes. Because it is possible +to have absolute certainty regarding ground truth labels as +well as microscopic control over their data generation con- +ditions, it is possible to remove the initial complexity and +variability of noisy, real-world data, enabling a clean de- +velopment environment to verify the functioning and logic +of machine learning pipelines and models. Simulations are +particularly valuable when the system under study is charac- +terized by large class imbalances, rare events, and significant +label noise. All of these challenging modelling problems fre- +quently appear in fintech machine learning. +This article is organized as follows. We briefly describe +possible approaches to synthetic dataset generation. We then +describe the kinetic Monte Carlo (kMC) algorithm in a gen- +eral way. Next, we explain attributes that define agents in +fintech-kMC, followed by a discussion related to the +treatment of the rate constants which govern their behaviour. +We provide examples of our simulation output, as well as +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +some simple supervised machine learning tests which make +use of this output. We discuss the limitations of our method, +and finally, conclude. Our major contributions are: +• a new agent-based simulation tool, driven by kinetic +Monte Carlo is introduced +• a detailed description of the tool is provided, including +its design, modelling capabilities, and output format +• demonstration of a predictive model trained to detect +“bad actors” is given as a prototypical example of an ML +model that uses our synthetic data +Different types of synthetic data generation protocols ex- +ist. They include simple heuristics (e.g. if an input feature +takes some value X, set the output to label Y ), simula- +tions which can be deterministic (Papageorgiou and Paskov +1999) or stochastic (Wang 2012), and generative mod- +els (Goodfellow et al. 2014; Albergo, Kanwar, and Shanahan +2019) which have been trained from sample data streams +from the true system. We believe that generative models, +while they may be able to best mimic a real-world dataset, +introduce an entirely new set of issues since they themselves +have dependencies on high-quality data pipelines, training +infrastructure, monitoring, etc. A stochastic simulation is an +excellent balance between being realistic enough to find im- +plementation bugs in machine learning workflows yet sim- +ple enough not to introduce them. +kinetic Monte Carlo (kMC) +kMC has a long history of use in the physical sciences for +describing the microscopic time evolution of a wide range of +systems, such as molecules on catalytic surfaces (Stamatakis +and Vlachos 2012), self-assembly and growth (Ghosh et al. +2021), and radiation damage (Voter 2007). The method has +also been successfully applied to the social sciences, where +it has been used to simulate the structure and informa- +tion flow through large online social graphs such as Twit- +ter (Ryczko et al. 2017). +Unlike some other Monte Carlo based simulation tech- +niques (most notably Markov Chain Monte Carlo, MCMC), +kMC is a rejection-free algorithm. An event occurs at ev- +ery simulation step. Additionally, we note that the simu- +lation step size, ∆t, is not fixed. In a hypothetical system +with only a single type of event, occurring at one charac- +teristic rate, r, each step in simulation time will be approxi- +arXiv:2301.01807v1 [cs.LG] 4 Jan 2023 + +Algorithm 1: kinetic Monte Carlo algorithm +Parameters: num agents, maximum time, etc +Output: logfile +1: while simulation time < maximum time do +2: +update rates() +3: +rates := [r1, r2, ..., rn] {array of rates for n events} +4: +cummulative array = cumsum(rates/sum(rates)) +5: +u1 := rand() {Get random number u1 ∈ (0, 1]} +6: +event := binary search(cummulative array, u1) +7: +carry out event(event) +8: +u2 := rand() {Get random number u2 ∈ (0, 1], up- +date simulation time} +9: +simulation time := simulation time −ln(u2)/R +10: +write out event information to logfile +11: end while +mately ∆t = r−1 apart. In a simulation with many possible +events (and rates), the simulation will progress forward in +time according to the total effective rate of the system (e.g. +R−1 = r−1 +1 ++ r−1 +2 ++ ...). In practice, this means that as +the total number of possible events grows, the time between +each successive one decreases. This is exactly what happens +in a real-world system. Consider an early-stage financial +platform with exactly 1 customer. If that customer deposits +money once per week, then after 1 month the transaction log +will contain approximately 4 entries, spaced roughly 1 week +apart. If the number of customers increases by x100 (and +the timing of their deposities is randomly distributed) then +after the same period of one month, we would expect 400 +entries, with the average time between updates decreasing +by x100. kMC simulations mimic this behaviour, and allow +for irregularly spaced events and changing values of ∆t as +the simulation progresses. +We note that in the above example, we assumed that the +rates of client deposits were uncorrelated with each other +and occurred at random times. In general this of course is +not guaranteed to be true, particularly with payments which +are associated with customer paycheques, rent, loan repay- +ments, etc, which tend to occur around specific times of the +month or the year. See Sec. Dynamic rates for further dis- +cussion on how we account for this.. +Disclaimer: in the following discussion, for pedagogical +purposes, we give examples of actions or rules which might +take place on a fintech platform. These are purely hypothet- +ical, and are not intended to describe a recommended set of +policies of a real fintech platform. Such platforms are sub- +ject to a large number of policies and regulations (partic- +ularly with respect to ID requirements, customer age limits, +etc). While fintech-kMC could be adapted to incorporate +those specific requirements as needed, for this discussion we +do not handle this complexity or region-specific limitations. +Agents +Within fintech-kMC, customers are represented as +agents. Currently, we have 2 distinct types of agents which +are included within our model: +• Individual customers: These are the most common type +of agent in typical simulations and represent individual +customers. Within this type, several different archetypes +exist (see Sec. Agent Archetypes). +• Businesses: Representing organizations or businesses, +these agents engage in a restricted set of actions (e.g. they +do not pay rent, repay loan, or buy btc). See +Sec. Available actions for further discussion of agent ac- +tions. Currently we have only a single archetype here. +Individual or business accounts can also be “bad actors” +(e.g. engaged in credit card fraud, consumer scams, etc). +Actions taken by these agents are primarily associated with +positive labels. Subtle behavioural differences between such +agents and regular, healthy, customers are what allow ML +algorithms to make predictions. +Static, Dynamical, and Triggered Rates +fintech-kMC allows for the concurrent use of both static +rates and dynamical rates. An example of a static rate could +be the rate at which new customers join the platform. For the +timescales we typically consider, this rate can be assumed to +be constant. Conversely, dynamical rates can change over +the course of the simulation. Some dynamical rates can +change continuously, while others are activated according to +a pre-established set of rules or when a particular simulation +state has been reached (e.g. an agent successfully has their +ID verified). +Dynamic rates Dynamical rate schedules can be based +on time (e.g. the absolute simulated world-clock time, or a +relative time, such as how many days a customer has been +been a user of the platform). Regular events such as pay- +cheque deposits are also included in this manner. For sched- +uled events such as paycheques, rent, etc, we set the rates of +all affected agents to a very high value on days correspond- +ing to such payments (e.g. 1st day of the month). This means +the probability such an event will be selected is significantly +higher than any other process. As soon as the event occurs, +we reset this rate to 0. Using this approach, it is possible to +naturally incorporate such “guaranteed events” into the kMC +approach. +Other rules which govern dynamical rates can be based +on the current state of the agent. For example, an agent +may purchase bitcoin (BTC) at a frequency of ≈ 1/week, +provided their account balance is above a threshold and +they have passed the id verification process. Similarly, an +agent may make a monthly loan payment via a peer-to- +peer (p2p send) payment until the balance is 0. Allowing +for dynamic and flexible modifications to the rate constants +which determine agent behaviour allows for a realistic ap- +proach to modelling human behaviour. +Available actions Agents interact with one another and +evolve through time via the actions they take within the sim- +ulation. Many such actions exist. +• cash in: money is moved into the account. The range +of possible transfer amounts depends on whether or not +the agent has completed id verification. +• cash out: money is removed from the account. The +range of possible transfer amounts depends on whether +or not the agent has completed id verification. + +• p2p send: money is removed from the account and is +transferred to another agent via a peer-to-peer payment. +• id verification: successful verification of an +agent “unlocks” other actions (specifically btc buy) +and changes the maximum allowed cash in and +cash out amounts. A customer will attempt to do this +action until they are successful, with bad actors having a +lower probability of success. +• btc buy: after an agent has successfully verfied their +ID via the id verification action, they can pur- +chase BTC. This action reduces their cash balance and +increases their BTC balance. +• pay rent: if an agent pays rent, it will do so at a fre- +quency of once every 30 days. +• deposit paycheque: if an agent receives a pay- +cheque, it will be deposited into their account once every +14 days. +• repay loan: loan repayments occur on a 7 day cycle +(at a fixed percentage of the original amount) until the +balance of the loan is zero. +We note that any actions involving money movements +above can only occur if there is sufficient balance to do so +(accounts cannot go negative). +Importantly, to represent the diversity of customer be- +haviour observed on fintech platforms by real customers, the +individual rates each agent operates with are not identical, +but rather are sampled randomly from a statistical distribu- +tion. We also use the concept of agent archetypes. +Agent Archetypes +While no two people are the same, in +modelling and business analytics it is often useful to invoke +the concept of customer archetypes. By understanding the +behaviours and motivations of customers on our platform, +we can segment them into groups which have similar actions +and (likely) motivations. Within fintech-kMC, we make +use of this idea by defining different agent archetypes which +give rise to distinct behaviours. +For +example, +an +agent +archetype +of +a +cypto enthusiast +will +be +assigned +a +rate +for +BTC purchases from the normal distribution N(µ, σ), +where µ = 2 and σ = 1 (in units of days−1), whereas a +crypto skeptic would have that rate set to zero. The +specific values of µ and σ are adjustable input parameters +that the modeller can use to best mimic the details of the +particular fintech platform they wish to simulate. Similarly, +a customer that is a big spender may have an average +cash in amount which is significantly larger than their +peers. Defining the agent archetype determines the distri- +bution of rates and the typical range of money movement +values an agent will use during the simulation. This allows +for the creation of a meaningful diversity of agents within +the dataset. Agents which belong to the same archetype will +not have identical behaviours, but rather their preferences +and propensities will be sampled from the same statistical +distributions. +Output +fintech-kMC provides simulation output in the form +of structured files which should be familiar to practi- +tioners working in the area of datascience and machine +learning. They are CSV files consisting of timestamped +records of all events and include information about the +generating agent, the action that was taken, and any +value associated with it. For example, when an agent +has their ID verified, the following information is re- +ported: time, initiating token, action, value. For example, +this +may +appear +as +2022-09-03 04:50:05.00, +C 83dhpqzz, id verification, True. +A +p2p +currency exchange includes an addtional field corresponding +to the receiving token. For example: 2022-11-14 +04:50:05.00, C b6589f, p2p sent, 2000, +C 83dhpqzz. +Here initiating token is a hash of the agent’s +unique numerical ID. Typically such tokens are used in real- +world systems to uniquely specific customers without re- +vealing any personally identifying information (PII) data. +We use this formatting style to mimic the type of data one +would find in raw log files from a real-world system as +closely as possible. In a similar spirit, we note that a final +dataframe has NULL values that must be accounted for (a +common occurrence in realistic datasets) because different +events can have different numbers of attributes. We inten- +tionally provide the output data in this format where actions +and agents are intermixed so that preprocessing steps used +in the ML pipeline can also be validated and tested. +Testing machine learning models +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Precision +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +ROC curve (area = 0.988) +4 +2 +0 +2 +4 +SHAP value (impact on model output) +time_diff_std_cash_in +time_diff_median_cash_in +time_diff_median_btc_buy +time_diff_median_all +cash_in_value +p2p_sent_ratio +btc_buy_count +time_diff_median_p2p_sent +cash_in_ratio +time_diff_std_all +cash_out_ratio +time_diff_mean_cash_out +cash_out_value +btc_buy_value +time_diff_median_cash_out +time_diff_mean_all +customer_verification_ratio +btc_buy_value_ratio +p2p_sent_value_ratio +cash_out_value_ratio +Low +High +Feature value +Figure 1: a) Precision-Recall and Receiver operating charac- +teristic, ROC (inset) and b) SHAP (Lundberg and Lee 2017) +analysis for our simple example binary classifier model. +Model training and validation data were generated using the +fintech-kMC tool. The full list of features used to train +the model is listed in the online Supplementary Information. +To demonstrate the utility of fintech-kMC with ML +tools, here we show results from a toy experiment to de- +tect and classify bad actors from regular customers. For this +particular simulation, bad actors were configured to have +a lower rate of ID verification success (50% compared to +75% for regular customers) and slightly different peer-to- +peer money transfer behaviours (an average transfer amount +of 5±3 vs 8±3 and a willingness to do such transfers only +when their balance was above 15±3 vs 30±3). Here ± de- +notes the standard deviation of the normal distributions from +which we sample to determine the specific transfer amounts. + +0 +20 +40 +60 +80 +100 +120 +140 +160 +epoch +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +log(loss) +model loss +train +val +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Precision +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +ROC curve (area = 0.965) +Figure 2: a) LSTM loss curves for train and validation sets b) +Precision-Recall and ROC (inset) for the model (evaluated +at epoch denoted by vertical red line). Hyperparameters are +listed in the online Supplementary Information. +We created a population of 1000 agents where 50% were bad +actors and the remaining were regular, healthy customers. +Our simulation covered a time period of ≈ 8 days, which, +given the various rate constants we used, represents approx- +imately 100 actions for each agent. We use the convention +that a positive label is deleterious (e.g. a scam, an account +take-over, or some other activity we wish to identify and pre- +vent). To train a model, we constructed a series of features +listed in the online Supplementary Information. We used a +train-validation-test split of 80%-10%-10% and trained an +XGBoost binary classifier (Chen and Guestrin 2016), Fig. 1, +as well as an LSTM (Hochreiter and Schmidhuber 1997), +Fig. 2, on the data (full model training hyperparameters are +provided in the online Supplementary Information). +Limitations of the method +• Introducing new types of actions requires more soft- +ware implementation. We cannot learn directly from data +streams from our real-world system. +• Rate constant probability distributions in the control file +are set manually by the user. To match them to a partic- +ular dataset, they must be optimized in a separate loop. +Conceivably this could be achieved by automatic differ- +entiation, but such capability is not currently included. +• Large scale network effects are currently not well de- +scribed due to the small number of agents we typically +simulate using the tool (e.g. 1k). In principle, agent-based +kMC can be used to simulate larger “worlds” (e.g. 1- +10M) but for simplicity we have not implemented the +typical optimization or parallelization techniques needed +to achieve such performance here. +• The current implementation is written in python. This is +convenient for prototyping and adding new features, but +results in a tool which is not as performant as it could be. +• In the current implementation, actions can occur at any +time during the simulated day, whereas in real-world data +customers tend to exhibit daily trends. This is not, how- +ever, a fundamental limitation of the methodology. +• Our positive labels are currently assigned at the customer +level, rather than the individual transaction level. We are +currently working to remove this restriction. +• Data produced by the model should not be seen as a +source of data for production models. +Conclusions +fintech-kMC is an agent-based model which can sim- +ulate the behaviour of customers of online digital finan- +cial platforms. The tool implements many actions typical of +such platforms, such as peer-to-peer money movements, ID +verification, and crypto purchases. By using kinetic Monte +Carlo, events occur at realistic timescales and can have +meaningful sequential dependencies. Data produced by the +tool can be used to test and validate machine learning work- +flows in a controllable and repeatable way. +References +Albergo, M. S.; Kanwar, G.; and Shanahan, P. E. 2019. +Flow-based generative models for Markov chain Monte +Carlo in lattice field theory. Phys. Rev. D, 100: 034515. +Chen, T.; and Guestrin, C. 2016. +XGBoost: A Scalable +Tree Boosting System. In Proceedings of the 22nd ACM +SIGKDD International Conference on Knowledge Discov- +ery and Data Mining, KDD ’16, 785–794. New York, +NY, USA: Association for Computing Machinery. +ISBN +9781450342322. +Ghosh, P.; Gupta, N.; Dhankhar, M.; and Ranganathan, M. +2021. Kinetic Monte Carlo simulations of self-organization +of Ge islands on Si(001). Phys. Chem. Chem. Phys., 23: +19022–19031. +Goodfellow, I. J.; Pouget-Abadie, J.; Mirza, M.; Xu, B.; +Warde-Farley, D.; Ozair, S.; Courville, A.; and Bengio, Y. +2014. Generative Adversarial Networks. . +Hochreiter, S.; and Schmidhuber, J. 1997. Long Short-Term +Memory. Neural Computation, 9(8): 1735–1780. +H¨ofer, S.; Bekris, K.; Handa, A.; Gamboa, J. C.; Golemo, F.; +Mozifian, M.; Atkeson, C.; Fox, D.; Goldberg, K.; Leonard, +J.; Liu, C. K.; Peters, J.; Song, S.; Welinder, P.; and White, +M. 2020. Perspectives on Sim2Real Transfer for Robotics: +A Summary of the R:SS 2020 Workshop. . +Lundberg, S.; and Lee, S.-I. 2017. A Unified Approach to +Interpreting Model Predictions. . +Papageorgiou, A.; and Paskov, S. H. 1999. Deterministic +Simulation for Risk Management. The Journal of Portfolio +Management, 25(5): 122–127. +Ryczko, K.; Domurad, A.; Buhagiar, N.; and Tamblyn, I. +2017. Hashkat: large-scale simulations of online social net- +works. Social Network Analysis and Mining, 7(1): 4. +Stamatakis, M.; and Vlachos, D. G. 2012. Unraveling the +Complexity of Catalytic Reactions via Kinetic Monte Carlo +Simulation: Current Status and Frontiers. +ACS Catalysis, +2(12): 2648–2663. +Voter, A. F. 2007. Introduction to the kinetic Monte Carlo +method. In Sickafus, K. E.; Kotomin, E. A.; and Uberu- +aga, B. P., eds., Radiation Effects in Solids, 1–23. Dordrecht: +Springer Netherlands. ISBN 978-1-4020-5295-8. +Wang, H. 2012. Monte Carlo simulation with applications +to finance. CRC Press. + +Supplementary Information +Architexture details of our LSTM and XGBoost models are +presented in Table S1 and Table S2. We also provide a list +of the hand-designed features we used to train our feature- +based ML models (Table S3). +base score +0.5 +booster +gbtree +colsample bylevel +1 +colsample bynode +1 +colsample bytree +1 +gamma +0 +learning rate +0.1 +max delta step +0 +max depth +3 +min child weight +1 +missing +None +n estimators +100 +nthread +1 +objective +binary:logistic +reg alpha +0 +reg lambda +1 +scale pos weight +1 +seed +0 +subsample +1 +verbosity +1 +tree method +hist +Table 1: Parameters used to train our XGBoost (Chen and +Guestrin 2016) model + +Layer +Type +Output units +Dropout rate +Return sequence +0 +Bidirectional LSTM +64 +0.2 +True +1 +Bidirectional LSTM +64 +0.2 +True +2 +Bidirectional LSTM +64 +0.2 +False +3 +Dense +1 +Table 2: Architexture (layer definitions) of our LSTM (Hochreiter and Schmidhuber 1997) neural network model +total events +cash in count +customer verification count +cash out count +p2p sent count +btc buy count +cash in ratio +customer verification ratio +cash out ratio +p2p sent ratio +btc buy ratio +cash in value +customer verification value +cash out value +p2p sent value +btc buy value +cash in value ratio +customer verification value ratio +cash out value ratio +p2p sent value ratio +btc buy value ratio +time diff mean all +time diff median all +time diff std all +time diff mean cash in +time diff median cash in +time diff std cash in +time diff mean customer verification +time diff median customer verification +time diff std customer verification +time diff mean cash out +time diff median cash out +time diff std cash out +time diff mean p2p sent +time diff median p2p sent +time diff std p2p sent +time diff mean btc buy +time diff median btc buy +time diff std btc buy +Table 3: Hand-designed features we created and used as in- +put to our XGBoost model + diff --git a/X9AzT4oBgHgl3EQf1_4_/content/tmp_files/load_file.txt b/X9AzT4oBgHgl3EQf1_4_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f317b073663534ed78f52346adef3fe117f3611 --- /dev/null +++ b/X9AzT4oBgHgl3EQf1_4_/content/tmp_files/load_file.txt @@ -0,0 +1,466 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf,len=465 +page_content='fintech-kMC: Agent based simulations of financial platforms for design and testing of machine learning systems Isaac Tamblyn,1,2,3 Tengkai Yu, 1,4 Ian Benlolo 1,3 1 Cash App, Block, Toronto, ON M5A 1J7, Canada 2 Vector Institute for Artificial Intelligence, Toronto, ON M5G 1M1, Canada 3 Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada 4 Department of Computer Science, University of Victoria, Victoria, BC V8P 5C2, Canada itamblyn@squareup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='com Abstract We discuss our simulation tool, fintech-kMC, which is de- signed to generate synthetic data for machine learning model development and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' fintech-kMC is an agent-based model driven by a kinetic Monte Carlo (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' continuous time Monte Carlo) engine which simulates the behaviour of cus- tomers using an online digital financial platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The tool provides an interpretable, reproducible, and realistic way of generating synthetic data which can be used to validate and test AI/ML models and pipelines to be used in real-world customer-facing financial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' To appear at AAAI-23 Bridge Program: AI for Financial Services, Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=', February 7 – 8, 2023 Introduction Simulated data is useful in machine learning (ML) work for developing and testing algorithms in a repeatable and verifiable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We note that although historically there has been some debate about the utility of simulated data in machine learning applications given the challenges of sim2real (H¨ofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2020), irrespective of whether sim- ulated data is valuable for training production models, they are extremely valuable in the initial stages of model design, unit testing, and verification processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Because it is possible to have absolute certainty regarding ground truth labels as well as microscopic control over their data generation con- ditions, it is possible to remove the initial complexity and variability of noisy, real-world data, enabling a clean de- velopment environment to verify the functioning and logic of machine learning pipelines and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Simulations are particularly valuable when the system under study is charac- terized by large class imbalances, rare events, and significant label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' All of these challenging modelling problems fre- quently appear in fintech machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We briefly describe possible approaches to synthetic dataset generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We then describe the kinetic Monte Carlo (kMC) algorithm in a gen- eral way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Next, we explain attributes that define agents in fintech-kMC, followed by a discussion related to the treatment of the rate constants which govern their behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We provide examples of our simulation output, as well as Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' some simple supervised machine learning tests which make use of this output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We discuss the limitations of our method, and finally, conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Our major contributions are: a new agent-based simulation tool, driven by kinetic Monte Carlo is introduced a detailed description of the tool is provided, including its design, modelling capabilities, and output format demonstration of a predictive model trained to detect “bad actors” is given as a prototypical example of an ML model that uses our synthetic data Different types of synthetic data generation protocols ex- ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' They include simple heuristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' if an input feature takes some value X, set the output to label Y ), simula- tions which can be deterministic (Papageorgiou and Paskov 1999) or stochastic (Wang 2012), and generative mod- els (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Albergo, Kanwar, and Shanahan 2019) which have been trained from sample data streams from the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We believe that generative models, while they may be able to best mimic a real-world dataset, introduce an entirely new set of issues since they themselves have dependencies on high-quality data pipelines, training infrastructure, monitoring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' A stochastic simulation is an excellent balance between being realistic enough to find im- plementation bugs in machine learning workflows yet sim- ple enough not to introduce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' kinetic Monte Carlo (kMC) kMC has a long history of use in the physical sciences for describing the microscopic time evolution of a wide range of systems, such as molecules on catalytic surfaces (Stamatakis and Vlachos 2012), self-assembly and growth (Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2021), and radiation damage (Voter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The method has also been successfully applied to the social sciences, where it has been used to simulate the structure and informa- tion flow through large online social graphs such as Twit- ter (Ryczko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Unlike some other Monte Carlo based simulation tech- niques (most notably Markov Chain Monte Carlo, MCMC), kMC is a rejection-free algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' An event occurs at ev- ery simulation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Additionally, we note that the simu- lation step size, ∆t, is not fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In a hypothetical system with only a single type of event, occurring at one charac- teristic rate, r, each step in simulation time will be approxi- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='01807v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='LG] 4 Jan 2023 Algorithm 1: kinetic Monte Carlo algorithm Parameters: num agents, maximum time, etc Output: logfile 1: while simulation time < maximum time do 2: update rates() 3: rates := [r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=', rn] {array of rates for n events} 4: cummulative array = cumsum(rates/sum(rates)) 5: u1 := rand() {Get random number u1 ∈ (0, 1]} 6: event := binary search(cummulative array, u1) 7: carry out event(event) 8: u2 := rand() {Get random number u2 ∈ (0, 1], up- date simulation time} 9: simulation time := simulation time −ln(u2)/R 10: write out event information to logfile 11: end while mately ∆t = r−1 apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In a simulation with many possible events (and rates), the simulation will progress forward in time according to the total effective rate of the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' R−1 = r−1 1 + r−1 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In practice, this means that as the total number of possible events grows, the time between each successive one decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' This is exactly what happens in a real-world system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Consider an early-stage financial platform with exactly 1 customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' If that customer deposits money once per week, then after 1 month the transaction log will contain approximately 4 entries, spaced roughly 1 week apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' If the number of customers increases by x100 (and the timing of their deposities is randomly distributed) then after the same period of one month, we would expect 400 entries, with the average time between updates decreasing by x100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' kMC simulations mimic this behaviour, and allow for irregularly spaced events and changing values of ∆t as the simulation progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We note that in the above example, we assumed that the rates of client deposits were uncorrelated with each other and occurred at random times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In general this of course is not guaranteed to be true, particularly with payments which are associated with customer paycheques, rent, loan repay- ments, etc, which tend to occur around specific times of the month or the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Dynamic rates for further dis- cussion on how we account for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='. Disclaimer: in the following discussion, for pedagogical purposes, we give examples of actions or rules which might take place on a fintech platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' These are purely hypothet- ical, and are not intended to describe a recommended set of policies of a real fintech platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Such platforms are sub- ject to a large number of policies and regulations (partic- ularly with respect to ID requirements, customer age limits, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' While fintech-kMC could be adapted to incorporate those specific requirements as needed, for this discussion we do not handle this complexity or region-specific limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Agents Within fintech-kMC, customers are represented as agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Currently, we have 2 distinct types of agents which are included within our model: Individual customers: These are the most common type of agent in typical simulations and represent individual customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Within this type, several different archetypes exist (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Agent Archetypes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Businesses: Representing organizations or businesses, these agents engage in a restricted set of actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' they do not pay rent, repay loan, or buy btc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Available actions for further discussion of agent ac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Currently we have only a single archetype here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Individual or business accounts can also be “bad actors” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' engaged in credit card fraud, consumer scams, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Actions taken by these agents are primarily associated with positive labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Subtle behavioural differences between such agents and regular, healthy, customers are what allow ML algorithms to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Static, Dynamical, and Triggered Rates fintech-kMC allows for the concurrent use of both static rates and dynamical rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' An example of a static rate could be the rate at which new customers join the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For the timescales we typically consider, this rate can be assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Conversely, dynamical rates can change over the course of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Some dynamical rates can change continuously, while others are activated according to a pre-established set of rules or when a particular simulation state has been reached (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' an agent successfully has their ID verified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Dynamic rates Dynamical rate schedules can be based on time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' the absolute simulated world-clock time, or a relative time, such as how many days a customer has been been a user of the platform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Regular events such as pay- cheque deposits are also included in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For sched- uled events such as paycheques, rent, etc, we set the rates of all affected agents to a very high value on days correspond- ing to such payments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 1st day of the month).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' This means the probability such an event will be selected is significantly higher than any other process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' As soon as the event occurs, we reset this rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Using this approach, it is possible to naturally incorporate such “guaranteed events” into the kMC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Other rules which govern dynamical rates can be based on the current state of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For example, an agent may purchase bitcoin (BTC) at a frequency of ≈ 1/week, provided their account balance is above a threshold and they have passed the id verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Similarly, an agent may make a monthly loan payment via a peer-to- peer (p2p send) payment until the balance is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Allowing for dynamic and flexible modifications to the rate constants which determine agent behaviour allows for a realistic ap- proach to modelling human behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Available actions Agents interact with one another and evolve through time via the actions they take within the sim- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Many such actions exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' cash in: money is moved into the account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The range of possible transfer amounts depends on whether or not the agent has completed id verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' cash out: money is removed from the account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The range of possible transfer amounts depends on whether or not the agent has completed id verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' p2p send: money is removed from the account and is transferred to another agent via a peer-to-peer payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' id verification: successful verification of an agent “unlocks” other actions (specifically btc buy) and changes the maximum allowed cash in and cash out amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' A customer will attempt to do this action until they are successful, with bad actors having a lower probability of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' btc buy: after an agent has successfully verfied their ID via the id verification action, they can pur- chase BTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' This action reduces their cash balance and increases their BTC balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' pay rent: if an agent pays rent, it will do so at a fre- quency of once every 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' deposit paycheque: if an agent receives a pay- cheque, it will be deposited into their account once every 14 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' repay loan: loan repayments occur on a 7 day cycle (at a fixed percentage of the original amount) until the balance of the loan is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We note that any actions involving money movements above can only occur if there is sufficient balance to do so (accounts cannot go negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Importantly, to represent the diversity of customer be- haviour observed on fintech platforms by real customers, the individual rates each agent operates with are not identical, but rather are sampled randomly from a statistical distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We also use the concept of agent archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Agent Archetypes While no two people are the same, in modelling and business analytics it is often useful to invoke the concept of customer archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' By understanding the behaviours and motivations of customers on our platform, we can segment them into groups which have similar actions and (likely) motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Within fintech-kMC, we make use of this idea by defining different agent archetypes which give rise to distinct behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For example, an agent archetype of a cypto enthusiast will be assigned a rate for BTC purchases from the normal distribution N(µ, σ), where µ = 2 and σ = 1 (in units of days−1), whereas a crypto skeptic would have that rate set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The specific values of µ and σ are adjustable input parameters that the modeller can use to best mimic the details of the particular fintech platform they wish to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Similarly, a customer that is a big spender may have an average cash in amount which is significantly larger than their peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Defining the agent archetype determines the distri- bution of rates and the typical range of money movement values an agent will use during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' This allows for the creation of a meaningful diversity of agents within the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Agents which belong to the same archetype will not have identical behaviours, but rather their preferences and propensities will be sampled from the same statistical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Output fintech-kMC provides simulation output in the form of structured files which should be familiar to practi- tioners working in the area of datascience and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' They are CSV files consisting of timestamped records of all events and include information about the generating agent, the action that was taken, and any value associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For example, when an agent has their ID verified, the following information is re- ported: time, initiating token, action, value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For example, this may appear as 2022-09-03 04:50:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='00, C 83dhpqzz, id verification, True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' A p2p currency exchange includes an addtional field corresponding to the receiving token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For example: 2022-11-14 04:50:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='00, C b6589f, p2p sent, 2000, C 83dhpqzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Here initiating token is a hash of the agent’s unique numerical ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Typically such tokens are used in real- world systems to uniquely specific customers without re- vealing any personally identifying information (PII) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We use this formatting style to mimic the type of data one would find in raw log files from a real-world system as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In a similar spirit, we note that a final dataframe has NULL values that must be accounted for (a common occurrence in realistic datasets) because different events can have different numbers of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We inten- tionally provide the output data in this format where actions and agents are intermixed so that preprocessing steps used in the ML pipeline can also be validated and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Testing machine learning models 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 False Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 True Positive Rate ROC curve (area = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='988) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='SHAP value (impact on model output) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_std_cash_in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_median_cash_in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_median_btc_buy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_median_all ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='cash_in_value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='p2p_sent_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='btc_buy_count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_median_p2p_sent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='cash_in_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_std_all ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='cash_out_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_mean_cash_out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='cash_out_value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='btc_buy_value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_median_cash_out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time_diff_mean_all ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='customer_verification_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='btc_buy_value_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='p2p_sent_value_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='cash_out_value_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='Feature value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='Figure 1: a) Precision-Recall and Receiver operating charac- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='teristic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' ROC (inset) and b) SHAP (Lundberg and Lee 2017) analysis for our simple example binary classifier model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Model training and validation data were generated using the fintech-kMC tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The full list of features used to train the model is listed in the online Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' To demonstrate the utility of fintech-kMC with ML tools, here we show results from a toy experiment to de- tect and classify bad actors from regular customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' For this particular simulation, bad actors were configured to have a lower rate of ID verification success (50% compared to 75% for regular customers) and slightly different peer-to- peer money transfer behaviours (an average transfer amount of 5±3 vs 8±3 and a willingness to do such transfers only when their balance was above 15±3 vs 30±3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Here ± de- notes the standard deviation of the normal distributions from which we sample to determine the specific transfer amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 epoch 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 log(loss) model loss train val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 False Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='0 True Positive Rate ROC curve (area = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='965) Figure 2: a) LSTM loss curves for train and validation sets b) Precision-Recall and ROC (inset) for the model (evaluated at epoch denoted by vertical red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Hyperparameters are listed in the online Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We created a population of 1000 agents where 50% were bad actors and the remaining were regular, healthy customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Our simulation covered a time period of ≈ 8 days, which, given the various rate constants we used, represents approx- imately 100 actions for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We use the convention that a positive label is deleterious (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' a scam, an account take-over, or some other activity we wish to identify and pre- vent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' To train a model, we constructed a series of features listed in the online Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We used a train-validation-test split of 80%-10%-10% and trained an XGBoost binary classifier (Chen and Guestrin 2016), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 1, as well as an LSTM (Hochreiter and Schmidhuber 1997), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2, on the data (full model training hyperparameters are provided in the online Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Limitations of the method Introducing new types of actions requires more soft- ware implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We cannot learn directly from data streams from our real-world system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Rate constant probability distributions in the control file are set manually by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' To match them to a partic- ular dataset, they must be optimized in a separate loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Conceivably this could be achieved by automatic differ- entiation, but such capability is not currently included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Large scale network effects are currently not well de- scribed due to the small number of agents we typically simulate using the tool (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 1k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In principle, agent-based kMC can be used to simulate larger “worlds” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 1- 10M) but for simplicity we have not implemented the typical optimization or parallelization techniques needed to achieve such performance here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The current implementation is written in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' This is convenient for prototyping and adding new features, but results in a tool which is not as performant as it could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In the current implementation, actions can occur at any time during the simulated day, whereas in real-world data customers tend to exhibit daily trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' This is not, how- ever, a fundamental limitation of the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Our positive labels are currently assigned at the customer level, rather than the individual transaction level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We are currently working to remove this restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Data produced by the model should not be seen as a source of data for production models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Conclusions fintech-kMC is an agent-based model which can sim- ulate the behaviour of customers of online digital finan- cial platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The tool implements many actions typical of such platforms, such as peer-to-peer money movements, ID verification, and crypto purchases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' By using kinetic Monte Carlo, events occur at realistic timescales and can have meaningful sequential dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Data produced by the tool can be used to test and validate machine learning work- flows in a controllable and repeatable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' References Albergo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Kanwar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Shanahan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Flow-based generative models for Markov chain Monte Carlo in lattice field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' D, 100: 034515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Guestrin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' XGBoost: A Scalable Tree Boosting System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discov- ery and Data Mining, KDD ’16, 785–794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' ISBN 9781450342322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Ghosh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Gupta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Dhankhar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Ranganathan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Kinetic Monte Carlo simulations of self-organization of Ge islands on Si(001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=', 23: 19022–19031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Pouget-Abadie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Mirza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Warde-Farley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Ozair, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Courville, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Generative Adversarial Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Hochreiter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Schmidhuber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Long Short-Term Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Neural Computation, 9(8): 1735–1780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' H¨ofer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Bekris, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Handa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Gamboa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Golemo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Mozifian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Atkeson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Fox, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Goldberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Leonard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Peters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Welinder, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and White, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Perspectives on Sim2Real Transfer for Robotics: A Summary of the R:SS 2020 Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Lundberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' A Unified Approach to Interpreting Model Predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Papageorgiou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Paskov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Deterministic Simulation for Risk Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' The Journal of Portfolio Management, 25(5): 122–127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Ryczko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Domurad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Buhagiar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Tamblyn, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Hashkat: large-scale simulations of online social net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Social Network Analysis and Mining, 7(1): 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Stamatakis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Vlachos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Unraveling the Complexity of Catalytic Reactions via Kinetic Monte Carlo Simulation: Current Status and Frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' ACS Catalysis, 2(12): 2648–2663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Voter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Introduction to the kinetic Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' In Sickafus, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Kotomin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' and Uberu- aga, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=', Radiation Effects in Solids, 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Dordrecht: Springer Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' ISBN 978-1-4020-5295-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Monte Carlo simulation with applications to finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' CRC Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' Supplementary Information Architexture details of our LSTM and XGBoost models are presented in Table S1 and Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' We also provide a list of the hand-designed features we used to train our feature- based ML models (Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content=' base score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='5 booster gbtree colsample bylevel 1 colsample bynode 1 colsample bytree 1 gamma 0 learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='1 max delta step 0 max depth 3 min child weight 1 missing None n estimators 100 nthread 1 objective binary:logistic reg alpha 0 reg lambda 1 scale pos weight 1 seed 0 subsample 1 verbosity 1 tree method hist Table 1: Parameters used to train our XGBoost (Chen and Guestrin 2016) model Layer Type Output units Dropout rate Return sequence 0 Bidirectional LSTM 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 True 1 Bidirectional LSTM 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 True 2 Bidirectional LSTM 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='False ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='Dense ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='Table 2: Architexture (layer definitions) of our LSTM (Hochreiter and Schmidhuber 1997) neural network model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='total events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='cash in count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='customer verification count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='cash out count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='p2p sent count ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time diff std p2p sent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time diff mean btc buy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time diff median btc buy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='time diff std btc buy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='Table 3: Hand-designed features we created and used as in- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} +page_content='put to our XGBoost model' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AzT4oBgHgl3EQf1_4_/content/2301.01807v1.pdf'} diff --git a/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf b/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1d70b7d254bc777170f6badeabb9dc04b0d34e8e --- /dev/null +++ b/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3988856a3a8944adf1075c505375b8279060e0935800486d0e2fc3ecdd73589e +size 3038143 diff --git a/_9E1T4oBgHgl3EQfVAPl/content/tmp_files/2301.03098v1.pdf.txt b/_9E1T4oBgHgl3EQfVAPl/content/tmp_files/2301.03098v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f96133046841e68b3abf89371daa8bbc97cb82f --- /dev/null +++ b/_9E1T4oBgHgl3EQfVAPl/content/tmp_files/2301.03098v1.pdf.txt @@ -0,0 +1,3583 @@ +Comprehensive Mapping of +Continuous/Switching Circuits in CCM and +DCM to Machine Learning Domain using +Homogeneous Graph Neural Networks + +Ahmed K. Khamis +EE Dept., AASTMT, Alex, Egypt +ECE Dept., University at Albany SUNY +Albany NY, USA +Mohammed Agamy +ECE Dept., University at Albany, SUNY +Albany NY, USA + + +Abstract: This paper proposes a method of transfer- +ring physical continuous and switching/converter circuits +working in continuous conduction mode (CCM) and +discontinuous conduction mode (DCM) to graph repre- +sentation, independent of the connection or the number +of circuit components, so that machine learning (ML) +algorithms and applications can be easily applied. Such +methodology is generalized and is applicable to circuits +with any number of switches, components, sources and +loads, and can be useful in applications such as artificial +intelligence (AI) based circuit design automation, layout +optimization, circuit synthesis and performance moni- +toring and control. The proposed circuit representation +and feature extraction methodology is applied to seven +types of continuous circuits, ranging from second to +fourth order and it is also applied to three of the +most common converters (Buck, Boost, and Buck-boost) +operating in CCM or DCM. A classifier ML task can +easily differentiate between circuit types as well as their +mode of operation, showing classification accuracy of +97.37% in continuous circuits and 100% in switching +circuits. +Index Terms—Electric circuit, Bond Graph, Graph Neu- +ral Networks (GNN), Machine Learning +I. INTRODUCTION +AI algorithms are used to model computationally +complex systems or systems/processes with significant +parameter uncertainties. Modern improvements in com- +putation resources enable the incorporation of AI algo- +rithms in power converter design and control. Complex +nonlinear problems such as thermal and electromagnetic +designs, modeling of layout parasitics and estimation of +component stresses under different operating conditions +are some areas where AI algorithms can significantly +simplify and optimize the design process. [3]–[6]. Power + +A preliminary version of this paper was presented in [1], [2] +electronics applications of ML have focused on control, +component design and maintenance [7]. ML-based sur- +rogate/black box models are used for online prediction +tasks to reduce computational effort, memory and power +used by classical simulation/mathematical-based models +[7]. Design optimization is an additional target as ML +models obtain the optimal target without compromising +other design constraints or trade-offs of design, which is +known in its mathematical formulation as Pareto front +[8]. ML-based circuit design should be able to reflect +circuit component connectivity as well as the effect of +varying the values of these components. In [9] a graph +representation of circuits with a combined feature map +for input and output nodes was proposed. However, it +does not represent details of component types or connec- +tivity, rather it is just a numerical input/output transfer +characteristic of the circuit. Reinforcement Learning +(RL) was introduced in [10] to optimize passive com- +ponent values. An updated version of the RL agent +was presented in [11], where the RL-based optimization +algorithm is used to optimally size transistors. In this +case, based on a given design flow, the RL algorithm +updates the node embedding in Graph Neural Network +(GNN) representation of the circuit to maximize the cost +function. One-hot encoding is used to represent tran- +sistors, in addition to other internal parameters, which +are passed as features to a Graph Convolution Network +(GCN) to extract node embedding. Despite the simplicity +of this approach, it it is incorrect and does not guarantee +a solution in the inverse problem. In other words, in cir- +cuit synthesis/generation problem, there is no guarantee +that the circuit synthesis neural network can transform +the generated graph to a physically realisable circuit. +Existing methods do not provide a systematic way of +circuit feature embedding in GNNs. These models have +several limitations including scalability of circuit size +(number of nodes and/or components), mapping con- + +nectivity and identifying component types within the +circuit. This paper proposed a systematic approach for +electric circuit representation to enable use of ML design +or performance prediction tools. This method has the +benefits of being scalable and topology agnostic. In this +paper the following key contributions are proposed: +• A comparative review of different research attempts +in mapping circuits to ML domain including cir- +cuit representation techniques, feature assignment, +intended task and how components and connections +are represented. +• Proposing three possible circuit representation tech- +niques, listing the advantages and disadvantages +while providing mathematical reasons for technique +selection. +• The circuit representation includes different circuit +element types and circuit connection types, without +indulging the concept with numerical tuning or the +empirical hyper-parameters optimization of ML. +• Proposing a unified (applied to all circuit elements) +node feature assignment algorithm, irrespective of +number of connections present in circuit or circuit +order, while combining the feature maps of the +nodes to generate the feature map for the whole +graph in a GNN. +• Proposing a dataset generation algorithm, that is +easily applicable to the ML task or application, +capturing circuit performance variables of interest +in a standardized data format that can be used in +ML problems. +• A proof of concept classifier problem applied to +variable structure continuous circuits or switching +circuits operating in CCM or DCM is presented. +The target ML task covers a wide range of possible +tasks or even a combination of tasks including re- +gression, classification and clustering tasks, whether +it is supervised or unsupervised tasks. +The proposed mapping approach enables a wide range +of possible ML tasks or a combination of tasks including +regression, classification, clustering, and synthesis of +power electronic converter circuits. +II. PROBLEM DESCRIPTION +Neural networks can construct model from training +data after being processed in order to obtain features +to characterize the built model. In the case of electri- +cal circuits, the process does not have an established +methodology or criteria. Problems with interfacing elec- +tric circuits to ML tools are highlighted in this section, +while different solutions are proposed in next section. +A. Circuit Structure Representation Problem +The main problem faced when circuits are to be fed +to a NN is the fixed size input layer, which has a +defined dimension, invalidating the scalability require- +ment. The workaround proposed in [12] pre-processes +a matrix consisting of multiple vectors representing +circuit components, so that the input to the Convolutional +Neural Network (CNN) is of a fixed size. Eventually, +this workaround added more computational overhead +and increased training time and computational resources. +Moreover, from a circuit standpoint, it is an incomplete +circuit model because it has no explicit representation +of the circuit structure or the dynamic behaviour or +circuit elements interactions. In this paper we lay some +foundations on how the physical properties of an electric +circuit can be mapped to ML space, as follows: +1) Circuit performance is independent of circuit enter- +ing order or elements order variation as long as the +connection is kept invariant (isomorphic circuits). +This makes the circuit representation Permutation +Invariant. +2) Circuit connectivity (series or parallel connections) +and circuit elements values define the circuit per- +formance. +3) Circuits may have any numbers of elements and has +no upper boundary. +4) For circuits of similar input/output response (e.g. +dual circuits [13]), circuit type/connection will be +the identifying factor in each case. +The realization of the last three definitions necessitates +that the ML input layer be independent of the size of +the input dataset. Hence, the representation becomes +Scalable. +B. Dataset Expressiveness Problem +Machine learning algorithms gain knowledge by iter- +ative training. Datasets contain standardized/normalized +data according to the nature of the ML task. Neither +a generalized and confirmed methodology to handle +circuit datasets nor a feature extraction/definition algo- +rithm are defined that independently capture the circuit +topology and the effects of component variation. More +importantly, a clear measure of dataset expressiveness +is absent. Given the circuits in Fig. 2, every class has +identical component count however, their performance is +different and depends on component values, especially +at resonance, and the dataset should indicate that differ- +ence. +C. Neural Network Topology Problem +The physical circuit topology and the influence of +parameter variation on its output variables must be +clearly expressed by the selected NN topology. As an +example, same circuit performance, can be obtained by +using dual components [13]. In [14] a model of similar +purpose employs CNN and takes placement images as +its features. Arguably, Graph Neural Networks (GNN) +are superior in capturing the netlist topology, which is +a graph. Moreover, GNN is more efficient in feature +encoding. For instance, the shape of a transistor can +be represented by two real numbers (width and height) + +in GNN while it requires an array of pixels for CNN. +The spatial features can be easily embraced in GNN by +taking the location coordinates as features, which are +motivations to take the GNN approach. +III. REVIEW OF CIRCUIT REPRESENTATION +TECHNIQUES +This section offers all possible solutions to presented +problems in section II, and highlights the flow of work +and derivations made from initial problem statements +and better explains available solutions by offering de- +tailed comparisons between them. There has been a lot of +attempts to better represent circuits in ML domain, which +are thoroughly explained in this paper. Moreover, the +paper will also highlight why solutions offered are insuf- +ficient, ungeneralizable and empirical solutions, which +either require fixed layout, huge datasets or extensive +training and very complex models. +A. Circuit Representation Methodologies +The main problem is to properly encode circuit prob- +lem into computer interpretable form, which has been +addressed by three modelling techniques, i.e graph the- +ory, Y-Matrix and Bond graph [15]–[17]. A brief is given +on every modelling technique, with an expanded illus- +tration on the one used in this paper, wile a comparison +between the merits and disadvantage of three modelling +techniques are listed in Table II. +1) Graph Theory Representation +Graph theory is a mathematical tool used to model +complex systems in a simplified way. In the field +of power electronics and converters, graph theory has +proved to be a powerful tool for representing and an- +alyzing the complex network of components and their +interactions.There have been numerous studies in the +literature which use graph theory to represent power +electronics and converters [18]. The use of graph theory +to represent power electronics and converters has sev- +eral advantages. Graphs provide a concise and intuitive +way to represent the components and their interactions. +Furthermore, graph algorithms can be used to analyze +the system and identify system faults. However, the +use of graph theory to represent power electronics and +converters also has some limitations. Graphs are lim- +ited in their ability to represent complex systems with +many components, as the number of nodes and edges +increases, the graph becomes cluttered and difficult to +interpret. In addition, graph matrices are usually very +large and computationally intensive, making it difficult +to obtain simulation results in real-time. This can result +in inaccurate or unsatisfactory results [19]. Furthermore, +due to the complex relationship between the different +components in the power system, the graph model may +not be able to accurately represent the real-world system, +leading to incorrect results [20]. Graph theory cannot +account for nonlinearity and non-smoothness. Power +electronic converters are nonlinear systems and their +circuits may contain high-frequency harmonics, which +is difficult to capture using graph theory [21].Finally, +when using graph theory to model a power electronic +converter, the system needs to be linearized, which may +neglect certain important nonlinear effects. This can lead +to incorrect results and further limitations to the accuracy +of the model [20]. +2) Y-Matrix Representation +The admittance matrix is a powerful tool used to +represent power systems and power electronic convert- +ers. This method of representation has been used since +its inception in the 1960s, and continues to be an +efficient and novel way to model electrical systems. The +admittance matrix is a complex quantity that describes +the relationship between the voltage and the current in +an electrical network. It consists of a matrix whose ele- +ments are admittances of electrical components such as +resistors, capacitors, and inductors [22]. This relationship +between the voltage and the current provides a useful +representation for solving electrical circuit problems [?]. +The admittance matrix has been used for many appli- +cations such as transient analysis and stability analysis. +In particular, it has been used to study power systems +[23]. In power system analysis, the admittance matrix +can represent the components in the power system such +as transmission lines, transformers, and loads, which +can be analyzed in both the frequency domain and the +time domain [24]. The advantage of the admittance +matrix is that it is computationally efficient and provides +a concise representation of the wide range of power +system components [25].The admittance matrix has also +been used for analyzing the stability of power electronic +converter systems [26]. Power electronic converters are +devices used to convert AC power to DC power or vice +versa, and they generally consist of power switches, +capacitors, and inductors [27]. Using the admittance ma- +trix, the stability of the power electronic converter can be +accurately analyzed in the frequency and time domains +[28]. This method of representation is relatively old but +can provides for accurate and efficient simulations of +power electronic converters. +In a preliminary attempt of this work, different circuits +were modelled utilizing Y-bus admittance matrix, where +nodes represented buses and admittances serve as node +features, while edges represent whether a connection is +established between nodes. Fig. 1 shows a three and four +element bus systems and its equivalent Y-bus admittance +matrix and the corresponding features. However, this +representation was proven to be non expressive based on +the fact that it is not uniformly scalable, i.e a three and +four element (admittances) systems can both have the +same number of nodes, which in this case is two, hence +losing a very important feature in graph notation. This + + + + + + + +Graph Representation as Two Nodes +with Y-Matrix as node feature + + + + +Fig. 1 – Early attempt of converting circuit to graph by using Y-Matrix + + + +is because the branch elements are lumped together into +a single equivalent admittance making it impossible to +distinguish between different elements. Moreover, with +this representation, the change in node feature values +doesn’t discriminate between whether a new element is +added or component value has changed. +3) Bond Graph Representation +Bond graphs (BG) were proposed as a graphical +language and systematic representation, to overcome +limitations of block diagram models [29]. Using BG, a +circuit can be modeled as bonds during all possible series +and parallel connection permutations and combinations. +Two key model elements were devised the 0 junction +that is used to represent a parallel connection and 1 +junction for series connections [29], [30]. In addition +to electric circuits, this approach can be extended to +mechanical and chemical models as well [31]–[33]. The +BG representation capturing the dynamics of a system is +based on transforming (mapping) system components to +their BG model counterparts. The bond graph analogies +used to describe physical systems in the form of bonds +and paths are listed in Table I. +Bond graphs in opposition to transfer function which +are behavioral models, belong to the class of struc- +tural models. Controllability and structural observability +are applicable to BG, which are structural properties +of models [37]. Moreover, it was proven in [36] that +BGs are structurally identifiable, which allows a unique +set of parameters to associate with given input/output +response. In other words, bidirectional transformation +governs circuit to graph and graph to circuit transforma- +tion and hence, graphs generated from ML algorithms +can be translated into a circuit if they match structural +identifiability criterion. +IV. REVIEW OF NEURAL NETWORK TOPOLOGIES +A. Classical Neural Network Topologies +Linear regression, random forest (RF) and artificial +neural networks (ANN) are classical regression models +used as attempts for regression tasks. For classifica- +tion tasks, support vector machine (SVM), K-Nearest- +Neighbor (KNN) algorithm and RF are used. Convolu- +tional neural network (CNN) and recurrent neural net- +works are extensively used in ML tasks. CNN models are +composed of convolutional layers and other basic blocks +such as non-linear activation functions and down-sample +pooling functions. While CNN is suitable for feature +extraction on grid structure data like 2-D image, RNN is +good at processing sequential data such as text or audio +[38] due to their ability to leverage statistical properties +of the image as euclidean data such as stationarity and +compositionality through local statistics. On the contrary, +non-Euclidean data has no familiar properties as global +parameterization, common system of coordinates, vec- +tor space structure, or shift-invariance. Operations like +convolution that are taken for granted in the Euclidean +case are even not well defined on non-Euclidean domains +[39]. From that prospective, it is necessary to use an +ML topology that can better represent non-euclidean +structures like electric circuits. +B. Graph Neural Networks +GNNs are composed of definite function layers, but +unlike other neural networks, the input is a graph. +Acyclic, cyclic, directed, and undirected graphs can be +processed by GNN as was stated in the first GNN model +in [40]. Scalablity and permutation invariance are unique +properties in GNNs allowing input layer to be variable +while graph node re-ordering will not affect the NN +layer output, which satisfies the requirements needed +for physical circuits representations. RNNs and GNNs, +capable of directly processing graphs with labeled nodes +and edges. An image classification task showed that +GNNs outperforms RNNs, both in terms of accuracy +and error rate [41]. Convolution operation on graphs +is defined by spectral and spatial operations. In [42], +spectral-based GCNs was proposed, which used the +spectral graph theory to develop a new variant of graph +convolutional operation. Graph mutual dependence com- +plexity was solved using non-recursive layers presented +in [43]. Moreover, spatial GCNs have been developed +based on the fact that spectral GCNs are difficult to +extend to large-scale graphs [44]. This makes GNNs +suitable for circuit representation. +1) Graph definition +Graph G is a defined as (V, E) with V the set of +vertices/Nodes equals v1, ..., vN , while set of Edges E +⊑ V × V . Let N and M be the number of vertices and +edges, respectively. Each graph can be represented by an +adjacency matrix A of size N × N : Ai,j = 1 if there +2 +Y2 +1 +Y1 +Y4 +Y3 +G +G +1 +2 + +++-Y2Ya+ Y + Y4Table I – Bond Graph terminologies [34] + +Terminology +Description +Strong Bond +A single bond that causes effort in the 0 junction and flow in the 1 junction Passive Element A one port element that stores +input power as potential energy (C-element), as kinetic energy (I-element) or transforms it into dissipative power (R-element). +Causal BG +A BG is called causally completed or causal if the causal stroke known as causality is added on one end of each bond + +Causal Path +A sequence of bonds with/without a transformer in between having causality at the same end of all bonds or a sequence +of bonds with a gyrator in between, and all the bonds of one side of the gyrator having same end causality while all the bonds +on the other side with causality on opposite end. That means gyrator switches the direction of efforts/flows on one of its side [9]. +A causal path can be a backward or forward or both depending upon the junction structure, elements and causality + +Branch +A branch is a series of junctions having parent-child relationship. +Two differ-ent sequences of junctions can be connected with a common bond or two-port element. +Thus, one of the junction’s sequence acts as parent branch and the other one as child. +Causal Loop +A causal loop is a closed causal path with bonds (of the child branch) either connected +to a similar junction or two different junctions of the parent branch + +Table II – Comparison between different circuit representation techniques + +Method +Representation +Methods +Merits +Drawbacks + + + +Graph Theory + + +Component terminals are nodes. +Circuit Elements are edges. + + +Multi-discipline physics based +modelling technique. +More intuitive graph for human reader. +Converter modelling foundations +(duty cycle, CCM & DCM ..etc) +are missing/never been addressed +No research on graph identifiability +from graph to circuit. +Circuit graph can be defined using +three matrices as shown in [35]. + + + + + + + +Bond Graph + + + + + + +Elements and connections are +nodes with different attributes +Solid foundations on circuits/converter +modelling in CCM & DCM. +BG is a linear transformation and +is mathematically identifiable +as shown in [36]. +Multi-discipline physics based +modelling technique. +Generated graph can be defined with +one Adjacency matrix. +Maintains causality invariance of +the system for any operational mode, +i.e the state vector resulting from +state equation of the system does +not change for any operating mode. + + + + + +Non-intuitive modelling technique. +Added complexity of causality +assignment. +Can yield a bigger graph than +graph theory method. + + +Y admittance +matrix + + +Circuit buses are nodes. +Connections between buses +are edges +Well known methodology for circuit +representation. +Number of circuit sources can’t be +extracted. +System components can be lumped +altogether and information about +element count is lost. + +Used only for power system +representation. +Node count is independent from +number of components. + +is an edge from vertex vi to vertex vj , and Ai,j = 0 +otherwise. Every edge has a set of edge features e +V. REVIEW OF CIRCUIT REPRESENTATION AND +DESIGN USING GNN +In [45] it was shown that the most intuitive way to rep- +resent circuit, netlists or layouts is graph representation. +It was also stated that graph neural networks (GNNs) are +an opportunity replace shallow methods or mathematical +optimization techniques, and Table III shows the state +of the art circuit representation trials. Many research has +utilized GNN in circuit optimizations/classification op- +erations and in many applications like transistor sizing, +capacitor value optimization and many more. In [46], + +[47] , the model leverages reinforcement learning (RL) to +learn the optimal policy for best parameter selection by +rewarding the model for the best Figure of Merits (FOM) +composed of several performance metrics. The circuit is +embedded into a graph whose vertices are components +and edges are wires, while generating a vector for each +transistor and passing the graph to the RL agent. Finally, +the RL agent processes each vertex in the graph and +generates an action vector for each node, then process +the graph with an action vector with the purpose of max- +imizing the reward. [48] proposes a model that solves the +forward and inverse problems. In which, the model maps +a given circuit to the corresponding transfer function and + +vise versa. Inversely, the model utilizes gradient descent +to optimize the circuit parameters to produce a transfer +function. The model leverages the differentiable nature +of the neural network and applying gradient descent +methods to optimize the input parameters of the neural +network. However, the neural network is trained for a +particular circuit topology, and hence cannot be used +for general circuit representation, in addition to the lack +of switching circuit representation. Moreover, [49] pro- +posed a technique for combining the feature maps of the +nodes to generate the feature map for the whole graph in +a GNN. By propagating information from nodes to nodes +representing input and output instead of pooling opera- +tion. The paper represents graphs as a concatenations +of the feature maps of the input and output nodes. In +resonator circuits applications, [49] introduced a model +that learns to simulate electromagnetic properties of +distributed circuits. Circuit were mapped on system level +basis, such that each node refers to a resonator and each +edge refers to the interaction between a pair of resonators +(i.e., the electromagnetic coupling) between a pair of +resonators. This representation does not incorporate the +resonator internal structure or if the system had different +resonators with different characteristics. By propagating +information from nodes to nodes, while representing cir- +cuits as concatenation of input and output node features +instead of pooling operation, regression task is utilized +to obtain predictions about circuit performance. On the +other hand, feature concatenation is not the correct +technique to represent circuit. Feature concatenation is +a numerical representation of circuit inputs and outputs +that properly tuned by minimizing the loss function. +Attempts has been made to include different circuit +topologies and obtain predictions as in [50], where two +circuit types were included in the study: the ladder +circuits and two stage operational amplifier circuits, with +20k training data instances of resistor ladders with 2 to +10 branches with equal distribution weight. The model +is based on DeepGEN architecture and was able to make +predictions on ladder circuits with higher number of +branches. However, the model’s ability to generalize and +applicability to other circuit topologies and types remain +questionable. Moreover, no clue was given on how to +distinguish connection type, and its effect on circuit +performance. Moreover, the representation was limited +to transistors, without the inclusion of other circuit pa- +rameters or elements(Transistor/resistor/voltage sources, +.. etc). Also, no guidelines/rules were given on how to +model circuit elemtents properties like frequency, phase +shift, .. etc. One major drawback in this representation is +the elements with multiple terminals like transistors are +represented as four connected nodes, which can cause +unnecessary excessive computations . In [51], heteroge- +neous GNN were utilized to construct a graph based on a +circuit schematic, where each device (transistor, resistor, +capacitor, etc.) can be mapped into different node and +edge type within the graph. The model target is to predict +net capacitance, which was achieved by mapping con- +nections as nodes with corresponding node information +(i.e. net capacitance), preventing information loss if nets +were represented as graph edges. To complete the struc- +ture, circuits were represented as multi-graphs, where +graphs have two edges with opposing directions, and +are mapped between every net node and the appropri- +ate device nodes corresponding to terminal connections +within the schematic. Despite leveraging heterogeneous +GNNs to differentiate between circuit elements nodes +and netlist nodes, this representation works around the +circuit connection type problem (series or parallel) in +the netlist nodes by assigning four types of connection +signal (Net to transistor gate, transistor gate to net, Net +to transistor drain, and transistor drain to net), resulting +in an over complicated representation that extensively +require more time at training. Physically, connections in +series share the same current and connections in parallel +share the same voltage, which are not shown in multi- +graph heterogeneous graphs. In the area of analog circuit +layout automation, [52] showed a GNN based model that +can identify symmetry constraints in analog circuits That +can be extended to other pairwise constraints. However, +the graph representation of circuits is simplistic as it +treats device instances and device pins as graph nodes, +while edges represents connections between pin nodes of +devices. Eventually, this simplistic representation creates +a problem of isomorphic graphs, which was mitigated by +adding an additional a two-dimensional vector to node +feature to distinguish between whether a node is a device +or a pin, which eventually increases computational cost +at training. Followed by [53] in which circuits was +represented as heterogeneous multi-graphs to the purpose +of modelling active and passive elements for analog and +mixed signal circuits. In this representation, four types +of edges (To transistor (drain), To transistor (source), To +transistor (gate), To passive device) are used to represent +connections between device/circuit elements, which were +represented as nodes. Circuit representation in previous +research can be summarized as: +• All methods for circuit to graph representation are +arbitrary, without any mathematical/scientific base. +• These methods disregards mapping the connection +type and hence is substituted by a significant in- +crease in the number of hidden layers, number of +neurons, training for many epochs, ... etc. +• Other implications of disregarding connection type +in previous methods are the limited scope of the +methodology. Previous methods cannot be applied +to any circuit except what it is intended for. +• All methodologies had deficiency in modelling + +Table III – Review of circuit representation in previous research + + +Node Features +Edge Features +Circuit Representation +Task +Network +type +Circuit components +Connections +(Series/Parallel) + +DC operating points, +One-hot encoding +of simulation step, +Transistor parameters, +Internal capacitances + + +Featureless + + +Every circuit element is represented as node +, where node features define the element +type and DC operating conditions. +No indication was given on connection +representation, or its effect on +analog circuit performance. + + +Learning design policy +for selecting optimal +circuit parameters. + + +RNN+RL + +[46] + + + +One-hot encoding +of element type +Circuit order, +Passive and active +characteristics + + +Featureless + + +GCN+RL + +[47] + + + +[54] +Gate logic level, +Controllability, +Observability + +Featureless +Limited circuit representation in the form +of connected nodes according to +the physical connection. +Determine whether an +observation point should +be added on the output +port or not + +Meta-path ++ GCN + +Subcircuit coordinates, +Center position of the +Subcircuit, Angular +position of the slit. +Position +of the two +subcircuits, +Gap length , +shift +System level representation, where every +subcircuit is represented as a node, +while edges between two nodes represent +distance between two subcircuits. +Electromagnetic +outputs prediction +based on resonators +relative positions + + +GCN +[48] +[49] + + +[55] +Operation type +Bitwidth. +Signal +information +System level representation, +where every node represents a microbench +operation, while edges represent signals. +Operation Delay +Prediction for +FPGA HLS + +GraphSAGE + +[50] +One-hot encoding +of terminal type, +Device parameters. + +Featureless +Edges , +but component terminals +are represented as nodes +No direct indication +of connection +DC output voltage +prediction + +Deep-GEN + + + + +[51] +gate poly length, +number of fingers, +number of fins, +number of copies, +length of resistor, +Capacitors, +number of copies, +net N + + + + +Featureless + + + + +Nodes + + + +No direct indication +of connection + + +Net parasitics +Predictions based +on physical +devices parameters + +GraphSage, +Relation GCN +and Graph +Attention +Networks. + + + + +[52] + + + +One hot encoding +(Device/Pin) +Path based feature + + + + +Featureless +Nodes represent component +terminals and pins. +Components can have +multiple nodes representing +Pins. Pin/Components are +distinguished by node features. +Power/GND are +represented as I/O nodes. + + + +No direct indication +of connection + + + +Binary Classification +of layout symmetry + + + + +GCN + +[53] +Node type, +Geometry, +layer + +Featureless + +Devices and circuit elements + +No direct indication +of connection + +Binary Classification +of layout symmetry +Gated +Recurrent +Unit based +GNN + + + + +[56] +Device type, +Functional +Module, +Current mirror, +Differential pair, +Active load, +Device dimension, +Device location. + +Horizontal and +vertical distance +between pins +Pin metal layer, +Pin length, +Pin type + + + + +Nodes with different types + + + +No direct indication +of connection + + + +Prediction of IC +placement impact on +circuit performance + + + +GAT ++ Pooling +(PEA) + +Proposed + +Element ID, +Normalized Component +Values. + +One for +continuous Circuits, +Duty Cycle for +switching circuits. + + + +Nodes with different types + + +one and zero nodes +for every branch/voltage node + +Different circuit +topologies based +ML tasks +(Classifier, Regression, +Clustering). + + +GCN ++ Pooling + + +common circuit properties like frequency, phase +shift, ... etc. +• Most methodologies mention only elements of in- +terest (Transistors and capacitors), but ignores other +circuit parameters like inductance, resistance, volt- +age source, current source, transformers, ... etc. +• Some methodologies try to simulate the connection +type by adding component terminals as nodes and +define the circuit as a multi-graph heterogeneous + +graph. Despite the added complexity and extensive +computational cost of heterogeneous graphs, This +representation suffers a major disadvantage as dif- +ferent circuit topologies can have the same graph +representations (isomorphic graphs). This problem +is usually addressed by defining another node fea- +ture the define whether a node is a pin or a device +at the expense of added computational cost. +• Some representations omits voltage and current + +C +L +V +C +Se +1 +R +R +I +C +L +V +C +R +Se +R +I +L +V +C +C +R +C +L +V +C +L +R +Se +R +I +I +C +L +V +L +C +R +Se +1 +0 +1 +R +I +I +C +C + + + +Se +1 +0 +R +L C +V +L +R + +I +I +C +C +L +V C +L +C +R +Se +1 +0 +1 +0 R +I +I +C +Circuit Element +Bondgraph Equivalent Element +Voltage Source (V) +Effort Source (Se) +Current Source (I) +Flow Source (Sf) +Resistance (R) +Resistance (R) +Inductance (L) +Inertance (I) +capacitance (C) +Compliance (C) + +Table IV – Circuit to bondgraph equivalent elements + +sources nodes to focus on circuit structure. How- +ever, this is incorrect representation since source +location can change the circuit behavior. +• Some methodologies include one-hot encoding of +device position in circuit along with device type, +which inherently means the node features vector +size per node is linearly proportional to the circuit +size. +VI. PROPOSED CONVERTER CIRCUITS MODELING +FOR MACHINE LEARNING APPLICATIONS +In this section, the proposed formulation of a graph +representation of continuous or switching circuits that +allow the application of ML algorithms to circuit de- +sign and control will be presented. This formulation is +completed in several steps: +1) Bond graph modeling of circuit topology. +2) Generating standardized datasets that capture circuit +topology, input and output circuit variables and +operating conditions. +3) Defining a scalable and permutation invariant NN +structure. +A. Graph Creation Using Bond Graph Modeling +This section explains how to model electric circuit as +a graph for further processing. +1) Continuous circuit presentation as Bond Graph +An electrical circuit consists of five main compo- +nents such as resistors, inductors, capacitors, voltage +source, and current source. The generalized BG ele- +ments and their mathematical relations can describe any +continuous circuit and perform analysis of dynamics of +electrical systems. Zero-junction is assigned for each +distinct voltage node in the circuit where according to +Kirchhoff’s voltage law (KVL)—the algebraic sum of +all voltage drops around a closed circuit is equal to +zero. Additionally, one-junction is assigned for each +element in the circuit, according to Kirchhoff’s current +law (KCL)—the algebraic sum of all electrical currents +entering and leaving a node is equal to zero), taking +into consideration the relative voltage or drops related + + + + + +(a) + +C +Class 2 + +Se +1 +0 +R + + + +I + + +C + + + + + + + +(b) + + + + + + + + + + +(c) +Fig. 2 – Converter circuits to Bondgraphs: (a) Two elements circuits, +(b) Three elements circuits, (c) Four elements circuits + + +2) Switching Circuit Representation as Bond Graph +A study in [57], [58] showed that switches (unidi- +to each element located between two 0-junctions, since +1-junction represents and effort summation point. Fig. +2 shows the bond graph models of seven classes of +resonant circuits of increasing order and Table IV shows +the equivalent notations used in BGs with their circuit +counterparts. +rectional or bidirectional) can be represented in BG +by the concept of Switched Power Junctions (SPJ) and +activated bonds and hence, BG can be used to model +switching circuits. Other switch modelling techniques +including Modulated Transformer (MTF) with Boolean +modulation index m and a resistive element R or the +1 +0 +1 +0 +Class 1 +Class 0 +Four Elements Circuits +Three Elements Circuits +Two Elements Circuits +Class 6 +Class 5 +Class 4 +Class 3 + +Ideal Switch Element method where switch state depends +on the junction to which the switch element is connected, +an energetic connection is established or broken [59], +[60]. A comparative study in [61] shows that the most +convenient method is the SPJ Modelling method as +it does not lead to causality conflicts and leads to a +unified model, like the Modulated Transformer method, +but does not require additional elements (R) to eliminate +algebraic loops. In this paper, the SPJ method will + + + +D +C +S2 B +S1 +D +D +D + + + + +A + + + + + +C + + + +Sf = 0 +Sf = 0 + +D +1s +7 B +S2 + + + + + +A +be used to represent switches. Converter topology and +its function are defined by the location of the energy +storage/resonance elements (L & C) and the type and +order of the switching cell. Simplification of Single Pole +Double Throw switching cell can be in the form of +two Single Pole Single Throw (SPST). Every SPST is +modelled as a 1s-junction with two flow decider bonds. +For the sake of completion, the physical interpretation +of current interruption when the SPST switch is OFF is +represented when one flow decider bond is modelled as +the zero current source (Sf) and the other flow decider +bond is connected to the system. The current source has +a zero value, indicating that current falls to zero when +Fig. 3 – Switching cell and equivalent BG formulation + +Fig. 4 – Circuit with equivalent BG formulation +switch is OFF. D and D¯ are the control signals that +control the junction flows. This is uniformly analogous to +the duty cycle (D) physical concept in converter circuits. +Based on [57], [58], SPST switches combinations can +be modelled using (0s and 1s) junctions. Fig. 3 shows a +switching cell represented as two SPST switches and its +equivalent bond graph representation, the flow decider +bond and the zero value flow sources. Additionally, +switched power junctions are a generalisation of the +already existing zero and one-junction concepts of the +bond graph element set [57]. Thus, the traditional zero +and one-junctions are special cases of the more general +switched power zero and switched power one-junctions. +When converters operates in DCM, the inductor current +reaches zero before switching cycle is over. This paper +utilizes the virtual switch concept to represent converter +operation in DCM mode. As the inductor current reaches +zero, both switches S1 and S2 are in OFF state. This +virtual switch only closes when both switches become +OFF. D1, D2, D3 are mutually exclusive control signal +to control switches operation. The concept of virtual +switch presented in [62] is used to express the converter +operation in DCM. This representation is based on the +fact that inductor current reaches zero in DCM. The +virtual switch shorts the inductor ensuring no current +passes through, while connecting certain circuit nodes +to maintain voltage balance equations during the DCM +time period D3. This representation compatible with the +predefined physical property namely Scalability. +B. Circuits to Graph Representation +The second step is to convert the BG formulation +to a graph representation containing all gathered and +simulated information including circuit types, classes, +nodes, edges, node and edge features. Fig. 4 shows a +continuous circuit represented as graph following BG +formulation, with minor changes in Switching circuits. +Nodes are used to represent circuit element as well as +zero and one junctions. Edges are used to describe circuit +connection between nodes. Node and edge features de- +scribe operating condition of the circuit. In continuous +circuits, edge features are set as one describing 100% +connection between designated nodes. The same notation +is used for switching circuit. Node features are used to +describe element type as well as the element value placed +in circuit. Some switching circuit properties require +special consideration and explained as: +a) Duty Cycle Representation +The duty cycle is a property in every switching circuit +and physically represent the percentage of the connection +existence within switching cycle. Duty cycle is mapped +as a feature of the edges the connects to switching nodes +(0s & 1s nodes). +b) Switching Frequency Representation +The one/zero switching junctions representing switch- +ing cell are connected to zero-valued current source, +interrupting the switch current with frequency equal to +switching frequency. In other words, the zero-valued +current source works as a control source for every switch. +Based on the physical properties of the control source, +including the switching frequency as a property of the +BG control source aligns with the physical properties of +VC = DVA + DVB +0s +D +D +D +1s S1 + +CircuitElementNode Features +EdgeFeatures +CS +Connection NodeFeaturesthe circuit. +c) Switching Pattern Representation +A generalized switching pattern representation is pro- +posed, allowing all types of switching patterns and duty +cycle variations. This adds more flexibility to represent +converters that operate differently when subjected to +different switching patterns, i.e resonant converters oper- +ating with different control modes. The switching pattern +representation is expressed in the control source (flow +source in BG representation) node features. Fig. 6 shows +two cases of switching patterns. In the first case, the +switching is aligned so that the first switching operation +compliments the second one. The current source node + + + + +Circuit +Element + +V → +I → +C → +R → +L → +0 → +1 → +Table VI – Feature matrix assignment +Concatenated Feature Matrix + + +features should indicate the same phase shift reference, +and by default is set to zero. In the second case, where +switch operations are not aligned either at turn on or turn +off, a phase shift φ indicates that delay, and is set the con- +trol source of the delayed switch. Combining the phase +shift information along with duty cycle information, +allows complete representation of the switching patterns +in switch operations. Table V summarizes the switching +pattern modes and their node feature representation. +Table V – General representation of all possible switching patterns as +node features + + +Representation + +Case 1 +*Phase shift is set to φ=0 +*Edge Features represents duty cycle +*Switches which are controlled dependently +are represented with the same phase shift. +Case 2 +*φ Is the phase shift +*Delayed switch include phase shift as node feature + +C. Dataset Generation +Generating a dataset of different circuit topologies, +circuit elements and circuit order is shown in this section. +Also, a proposed technique for storing recorded data in +a general format for any ML task is highlighted. Fig. +7 shows a paradigm for such dataset generation step, +where a circuit netlist is converted to its equivalent bond +graph model. Since BG is a graph notation for modeling +circuits, they inherently have all graph characteristics, +with all requirements of graph definitions like number +of nodes, node types, edge weights and the adjacency +matrix. Finally, BGs are passed to feature assignment +algorithm, where features are assigned to each node in +graph. +1) Feature Assignment +Node features are defined based on circuit element +type and its behavior in circuit using the proposed algo- +rithm. Circuit simulations are used to obtaining features +describing circuit performance such as node voltages and +loop currents. Simulations run for multiple instances at +multiple operating points for all circuits including dif- +ferent component values and circuit conditions. Output +values are normalized to common base to avoid sparsity +of the feature vector, which is referred in Table VI +as ”Normalized Values Vector”. The proposed feature +assignment algorithm is expandable and can include +many circuit features if it is desired to be included in the +dataset. Therefore, the normalized values vector can be +multiple columns listing not only component’s value, but +also different component properties i.e source frequency +in continuous circuits or phase shift in switching circuits. +One main function of feature extraction algorithm is to +define the circuit element types, which are defines the +concept of Element ID. Element ID assigns a binary +code based on circuit element type by utilizing one- +hot encoding [63]. The second main function of feature +assignment algorithm is to concatenate the assigned one- +hot encoded vector with normalized values vector, +forming the feature matrix of the whole graph with +dimension N × din, where N is number of nodes and +din is the dimension of feature vector. +2) Dataset Format +Extracted features and other graph information like +types and number of node, adjacency matrix and edge +features are saved in a unique graph dataframe for- +mat.This unique dataset format features independent +graph dataset of circuits, which allows using this graph +representation in any ML library independent of saved +graph dataset. Since there are many graph ML libraries +like pytorch-Geometric [64], DGL [65], Keras [66] .. etc, +the final step in the algorithm is to process the dataset +to be in a compatible format. Pytorch-Geometric GNN +library was chosen to build the GNN structure. +D. Different Circuit Examples Using Proposed Method- +ology +This section shows some examples from different +areas where the proposed methodology is applicable to +many ML applications. +1) Example 1: Power System +Power systems (PS) area have a lot of research where +ML methodologies has bee applied. Recently GNN +has been on the spotlight for application in PS, and +xx +xx +xx +xx +xx +xx +xx + +Normalized +Value +Element ID +V +I +L +R +C +1 +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 + + +Conveter Circuits +Equivalent CCM Bond Graphs +Equivalent DCM Bond Graphs + + + + +R + +Buck + + + + +R + + +Boost + + + + +R + +Buck-Boost + + +Fig. 5 – Buck, boost and Buck-Boost converters and their equivalent BondGraphs in CCM + + + +Fig. 6 – Switching pattern representation as features + + + +many publications utilizing GNN in power systems have +emerged. A comprehensive overview of GNN appli- +cations such as fault scenario application, time series +prediction, power flow calculation, and data generation +are reviewed in [67]. In [68], [69] the provided network +learns to solve load flow problem on random power grids +whose size range from 10 to 110 buses. A method to +identify the topology of a PS network is proposed in [70] +based on GNN, avoiding errors in Traditional knowledge +graphs in the case of errors or informational conflicts in + +the data. All previously mentioned research empirically +transform the PS network into graph without following +a circuit-laws-consistent formulation. Fig. 9a shows a +PS network example and its graph equivalent with node +features, following the proposed methodology. +2) Example 2: Two-Stage Amplifier +Fig. 9b shows a two-stage amplifier that was used in +[10] as a circuit layout. The equivalent graph represen- +tation proposed in this work was arbitrarily transformed +into a graph by representing every transistor, resistor +and capacitor as nodes connected to each other by +edges, disregarding the original connection or the physi- +cal/electrical consequences of such connections. The Fig. +also shows the proposed graph representation includes +component and connection nodes, in addition to node +features for each node. +E. Graph Convolution Network +NN have many variants like GCN [71], GraphSage +[72], Gated Convolution [73], Transformer convolution +[74] and many more, but the most common is GCN. +GCN was chosen for the following reasons: +• Unique ability to extract latent information from +graph data compared to other GNN structures as +reported in [75]. +• Most practical circuit GNN based applications in +Table III utilize GCN as their main network model +or a part of the model, hence the results from this +study can be fairly compared to previous ones. +Sf +Sf +C +Se +1 +0 +1 +0 +Sf +1 Virtual Switch +I +R +Se GND +Class 5 +Sf + + +1 Virtual Switch +Sf +C +Se +0 +1 +0 +1 +0 +I +Sf +1 +R +Se GND +Class 4 +Sf +Sf +C +1 Virtual Switch +Se +1 +0 +0 +1 +Sf +1 +R +I +Se GND +Class 3 +Sf +Sf +C +Se +1 +0 +1 +0 +I +R +Class 2 +I +I +Sf +C +Se +1 +0 +1 +0 +Sf +1 +R +Se (GND) +Class 1 +Sf +I +I +C +Se +1 +0 +1 +0 +Sf +1 +R +Se (GND) +Class 0 + +S1 +S2 +L +C +L +S2 + +S1 +C +S1 +S2 + + +L +C + +D3 +Di +D2Di +D2DSwitching Circuits + + + + + + + + + + + + + + + + + + + + + + + + + + +Fig. 7 – From circuit to ML Block diagram + + + +(a) + +(b) +Fig. 8 – Equivalent graph with node and edge features in: a) LCC +Continuous circuits, b) Buck converter switching circuit + + +• Simple construction and implementation, which can +be beneficial if implemented as digital twin on a +microcontroller [76] +The selection of GCN as the engine for the proposed +GNN has allowed better focus on other hyperparame- +ters and eventually led to better circuit representation. +GCNs obtain updated features by inspecting neighboring +nodes, and aggregating current node information to other +neighbours through message-passing process then updat- +ing the node state. Eventually, all the nodes in graph +obtain knowledge about self and surrounding neighbor +information. Fig. 10 shows three layer message passing +applied to a single node (node of type 1) of class 1 +circuit. A deeper level of neighbor nodes exploration +and better awareness of self node position can be gained +by adding an additional GCN layer, at the expense +of additional computational effort. Three layer GCN +network is utilized in this paper as a mid point between +exploration depth and computational efficiency. Node +features are repetitively aggregated through the GCN +layers via multiple message passing layers. At the end +of this process, the final node embeddings contain self +and all neighbor information. +Mathematically, this initial embedding function is rep- +resented by equation (1). The aggregation layer has mul- +tiple Graph Convolution Networks (GCN) that performs +multiple message passing leaps to collect information +about neighbouring nodes and keeps updating the latent +dimensional vector with dimension d, which is mathe- +matically represented as in equation (2). +X(0) = E(X) +(1) +S1 +L +S1 +L +V +L +C +S2 +C +R +S1 +S2 +C +R +L +S2 +C +R +R +L +V +C +R +Circuit +Netlist +Computer +Simulations +L +C +C +R +Equivalent Bond +Graph +Circuit Class +assignment +Continous Circuits +Sf +Sf +I +I +C +Sf +C +True Class +1 Virtual Switch +C +Se +1 +0 +0 +Se +1s +0 +1 +0 +1 +R +Sf +1 +Sf +1 +R +GCN +Se +1 +R +s +I +Se GND +Se (GND) +Sf +C +I +I +Sf +C +I +1 Virtual Switch +Sf +C +Global Mean Pooling +Se +1 +0 +1 s +0 +Se +1 +0 +R +Se +0 +1 +0 +1 +0 +Sf +1s +R +R +I +Sf +1 +Linear +Transformation +I +Se (GND) +Sf +Sf +C +Sf +Se GND +Sf +C +Softmax +Se +1s +0 +1s +0 +Se +1 +0 +1 +0 +I +R +Sf +1 Virtual Switch +I +R +Continuous circuits Bond Graph +Switching circuits Bond Graph in CCM +Se GND +Predicted +Class +Switching circuits Bond Graph in DCM +Buck +Boost +Buck-Boost +Weight update +Loss Function + +00000.01.V +0440044.m +Vs +40001440s +Cs +Cp0.00.0,010.51 +0,0,0,0,1,0,0,F +Os +0,0,0,0,0,0,1,V +0.0.0.1.0.0.0. +0,0.0.0,0,1.0.T2. +VgndF +Circuit +Element ID +Vall +Val2 +Element +(One-hot encoding) +Se +1 +0 +0 +0 +0 +0 +xX +xX +Sf> +0 +1 +0 +0 +0 +F +0 +0 +0 +0 +xX +xX +0 +0 +0 +1 +0 +0 +0 +xX +xX +R- +0 +0 +0 +0 +1 +0 +0 +XX +XX +1> +0 +0 +0 +0 +1 +0 +XX +XX +10 +001000 +10 +xX +XXNxd, +RNxds → R1xd +Da +R1xd → RIxC +RI×C →RIxC2 +2 +VDD +T7 +0 +VBp +Vx + + +Cx +0 +1 +Tx +Rx +1 +0 +1 +0 +1 +C +T4 +R + + +0 +1 +0 +T6 +1 +1 +T3 +0 Vin+ +T5 +T2 +0 +0 +Vin- +1 +T1 +0 VBN + + +Third Message +Passing Layer +Second Message +Passing Layer +First Message +Passing Layer + + + +Fig. 10 – Rooted subtree showing message passing applied to node of +of type 1 in the circuit of class 1 in Fig. 2 with three GCN layers + +X(l+1) = σ(Dˆ − 1 AˆDˆ − 1 XlΘl) +(2) + + + + + + + + + + + + + +(a) + +(b) +Fig. 9 – Examples of proposed concept in different applications: a) +Power system example b) 65 nm 2 stage amplifier example. [10] +where Θl is a weight matrix for the l-th neural network +layer and σ is a non-linear activation function like the +ReLU, Aˆ= A + I, where I is the identity matrix and Dˆ +is the diagonal node degree matrix of Aˆ. This allows the +GCN to scale well, because the number of parameters +in the model is not tied to the size of the graph. +F. GCN Time complexity and Graph Scalability Limit +Generally speaking, there are no limitation on the size +of the circuit fed to the ML model (theoretically, the +circuit order can be infinite). However, the computation +time and RAM consumption are the main concerns +when feeding circuit graphs to model, which mainly +depends on how the model was built, the libraries used +to build the model (pytorch or keras or tensorflow +. . . .etc), the layers depth, operating system used, the +model architecture and the output size, . . . etc. From a +GNN designer prospective, Graph circuit for a GNN +input can be represented in two ways: +• sparse: As a list of nodes and a list of edge indices +• +dense: As a list of nodes and an adjacency matrix +For any graph G with N vertices of feature vector +length F and E edges, the sparse version will operate on +the nodes of size N*F and a list of edge indices of size +2*E. The dense representation in contrast will require an +adjacency matrix of size N*N, with node degree of d. +The choice of dense or sparse representation not +only affects the memory usage, but also the calculation +method. Dense and sparse graph tensors require graph +convolutions that operate on dense or sparse inputs (or +alternatively as seen in some implementations convert +between sparse and dense inside the network layer). +Sparse graph tensors would operate on sparse con- +volutions that use sparse operations. Generally, dense +computations would be more expensive but faster than +sparse, because sparse graphs would require processing +of operations in the shape of a list. For simplicity, we +assume the node features at every layer are size- F . As +such, Θl is an F × F matrix. The time complexity of +the convolution operation can be decomposed as: +I +GCN +I +1 +GCN +Se +0 +1 GCN +Se +GCN +C +GCN +0 +0 +GCN +R +GCN +0 +L2 +L1 +L3 +C1 +G1 +G2 +C1 +G1 +1 +0 +1 +G2 +L1 +L2 +L3 + +Voc +HT +R +T += C +65nm +Two-stage Amplifier +0, 0, 0, 0, 0, 1. Il +O, 0. 1. 0, 0, 0. Xa +0, 0, 1, 0, 0, 0, R[0, 0, 0, 1, 0, 0, Ca] +[0, 0, 0, 0, 1, 0, Ea +[0, 0, 1, 0, 0, 0, La] +1. 0. 0, 0, 0, 0, VanormalizedFrequency +Frequency×C +C +normalizedFrequency +resonanceFrequency +normalizedFrequency +C +L +L +• Equation (1) : which is a dense matrix multiplica- +tion between matrices of size N ×Fl and Fl ×Fl+1. +We a.ssumeΣfor all l, Fl = Fl+1 = F . Therefore, this +is O NF 2 . +• Equation (2): which is a multiplication between +ma.trices Σof size N × N and N × F , yielding +O N 2F +time complexity. Hence, the neighbor- +Fig. 12 is the second set of experiments, where edge +weights were set as the normalized frequency, while +nodes that represents capacitive elements were set to +have ( +1 +) as edge feature. Another +experiment is to test whether negative component values +would increase the accuracy, or setting the capacitive +components as 1 . These experiments are reflection from +hood aggregation for each node therefore requires +C +circuit analysis as Xc = + +−j +. However, the +O(dF) work, with a total of O(NdF) = O(EF ). σ(·) +: is the activation function which is an element- +wise function, so its cost is O(N ). +Over L layers, this results in computational time +results shows that negative capacitive element value and +its edge feature as ( +1 +) have negative +effect on the accuracy of the classifier, while setting +capacitive elements as of inverted value( 1 ) had a +com +. plexity of: +Σ +. +O .LNF 2 + LNdF + LN += O LNF 2 + LNdF +Σ +significant training accuracy boost to 91.12%. It is imper- += +ative to modify node features expression for capacitive +O LNF 2+ LEF ) +G. Optimal Node And Edge Features Exploration +To determine the optimal representation of circuit +component values, twelve experiments were performed +on the continuous circuits of Fig. 2 and the results are +shown in Fig. 11 - Fig. 14. The dataset contained 6000 +graphs representing the seven circuit types. 70% of the +dataset was used for training.The data is shuffled before +being applied to the model, and there was no mutual data +between training and testing. Cross entropy loss function +is used in training the model with Adam optimizer +[77] with learning rate of 0.02. Twelve experiments +were conducted in order to obtain conclusions and a +paradigm of how the node and edge features should +represent the circuit parameters. These experiments were +divided into four sets. Each set contains three experi- +ments and a conclusion based on observations from these +experiments. The conditions/modifications applied on the +dataset when fed to the classifier are listed on the left of +each set. The purpose of these experiments is to identify +the effect of different component representations, and +how would that affect the ML task. Figures also show +the classifier problem evolution ranging from three class +to seven class classifier problem, along with physical +circuit elements representation as features. +The purpose of the upcoming experiments is to ex- +plore the highest impact features on task accuracy. How- +ever, since features are hyper-parameters, some result +obtained from edge features may eventually update how +the node features are expressed. In the first set of +experiments shown in Fig. 11, edge features are explored +and the problem is limited to three classes classifier, edge +weights are separately tested as normalized frequency +( circuitF requency ) vs. being set as ones, vs. being the +circuit frequency. This experiment is concluded with the +highest accuracy achieved is when edge weights were set +as normalized frequency and as ones. As frequency can +be included as edge features, it can be tested if capacitive +elements can to be expressed as ( +1 +), +which is the purpose of the second experiment set. +elements. Eventually, circuit graph dataset was modified +to include this change in the the third experiment set. +Also, from the first experiment set, edge features set as +one had the highest accuracy score. The next experiment +aims to explore if the concluded node and edge features +modifications can enhance the accuracy. +In the third experiment set, the highest accuracy of +100% was achieved in training and testing when edge +weights were set to ones and capacitive elements has +node feature values of ( 1 ). The first experiment tested +whether edge feature can be used as a scaling factor +substituted by the node feature. The second one tested +whether edge weights can be set to one, while the third +experiment tested if inductive elements can be set as( 1 ). +From results shown in Fig. 13, it can be concluded +that utilizing edge features for scaling deteriorates the +classification accuracy as well as representing inductive +elements as( 1 ). The optimal edge feature can be defined +to be one, without embedding any circuit characteristics +or parameters. +In the last set of experiments in Fig. 14, all outcomes +and recommendations that was concluded from previous +experiments were taken into consideration, while in- +creasing the classification problem difficulty to four, five +and seven classes classification problem to further verify +the optimal representation. In a four-classes problem, +the classifier scored a training accuracy of 92.3%, while +in five-classes problem the training accuracy score was +95.92%. Lastly, the seven-classes problem resulted in +training accuracy score of 97.37%. The discrepancy of +accuracy scores while using the same feature represen- +tation is due to the change in dataset number of circuits. +The result is a graph of a circuit with connection nodes +and element nodes each has its own features. Nodes are +connected by edges having edge features of one. +VII. CASE STUDY +As a proof of concept, the proposed approach is +applied to map two types of topologies: i) continuous +circuits and ii) switching circuits, to a ML compatible +representation. Seven resonant circuit topologies of cir- + +• Capacitive elements have node + + + +• Three-Class classification Problem + + +With Frequency As Edge Features +Training Accuracy: 0.6898, Testing Accuracy: 0.6906 +(a) + + +Edge Features Set As one +Training Accuracy: 0.8606, Testing Accuracy: 0.8669 +(b) + + +Normalized Frequency as edges +Training Accuracy: 0.7574, Testing Accuracy: 0.7709 +(c) +Conclusion +When edge weights set as normalized frequency and when they are set as ones are of the highest accuracy. + +Fig. 11 – First experiment set. Edge weights are set as: a) Frequency, b) value of one, c)Normalized frequency. + +• Three-Class classification +Problem +• Edge weights represents the +normalized frequency. +• Capacitive elements have +edge weight of +1/norm.frequency. + + +Training Accuracy: 0.722, Testing Accuracy: 0.72 +(a) + + +*Capacitive elements has node feature value of 1/C +Training Accuracy: 0.9112, Testing Accuracy: 0.9175 +(b) + +*Capacitive elements has negative node feature value of +1/C Training Accuracy: 0.9107, Testing Accuracy: 0.9163 +(c) + +Conclusion +• Capacitive elements having edge weights of 1/Frequency has negative effect on accuracy. +• Minor decrease in accuracy when capacitors have negative values. +• Setting 1/C value has a major accuracy boost. + +Fig. 12 – Second experiment set. a) No change in node features, b) Capacitive element representation is 1 , c) Capacitive element representation + +−1 +C +is C + + +• Three-Class classification Problem + +feature value of 1/C. + + +Edge weight is set to 1000 which is +substituted from Node value feature +Training Accuracy: 0.6385, Testing Accuracy: 0.6434 +(a) + + +Edge weights are set as one +Training Accuracy: 1, Testing Accuracy: +1 +(b) + +Edge weights are set as one +Inductive elements have values of 1/L +Training Accuracy: 0.7928, Testing Accuracy: 0.7817 +(c) + +Conclusion +• Setting Inductive elements values of 1/L deteriorates the accuracy. +• Scale representation as edge weights has a negative effect on the accuracy. +• Edge weights set as ones significantly increase the accuracy. + +Fig. 13 – Third experiment set. Edge weights are set as: a) Scaling factor, b) value of one, c) value of one but different inductive element +representation. + + +cuit orders ranging from second to fourth order as shown +in Fig. 2, and three switching circuit topologies in CCM +and DCM shown in Fig. 3 are fed to a classifier to show +the applicability of the proposed methodology to any +ML task. Following the sequence illustrated in Fig. 7 +and same steps presented in this paper and in [1] and +[2], converters are converted to graph form and computer +simulations are used to assign normalized node features +of the generated graph according to section VI-C1. +Steady state simulations are run for multiple instances at + +multiple operating points for all circuits including differ- +ent component values and circuit conditions and circuit +behavior is recorded and stored. The circuit simulation +sampling rate is a measure of the accuracy of the circuit +simulations in the continuous circuit classifier case. In +this case study, a dataset of 6000 graphs with 6000 steady +state simulations have been normalized to a common +base. This helps to ensure that each feature vector is +consistent and not overly sparse. The normalized values +vector is then used to provide a representation of the +. + +1.0 +Traing Dataset +Testing Dataset +0.8 +ccuracy +0.6 +20.4 +0.2 +0.0+ +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +80 +0.6 +0.2 +Traing Dataset +Testing Dataset +0.0 + +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +Traing Dataset +Testing Dataset +0.8 +0.6 +0.2 +0.0 + +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +Traing Dataset +Testing Dataset +0.8 - +2 0.4 +0.2 +0.0 +0 +200 +400 +600 +008 +1000 +1200 +Epoch1.0 +0.8 +0.6 +20.4 +0.2 +Traing Dataset +Testing Dataset +0.0+ +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +0.8 +20.4 +0.2 +Traing Dataset +TestingDataset +0.0 +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +Traing Dataset +Testing Dataset +0.8 +0.4 +0.2 +0.0 + +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +8°0 +0.4 +0.2 +Traing Dataset +Testing Dataset ++00 +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +TraingDataset +Testing Dataset +0.8 +0.4 +0.2 +0.0 +0 +200 +400 +600 +800 +1000 +1200 +Epoch + +• Multi-Class classification +problem +• Edges has weight of One +• Capacitive elements have +node feature value of 1/C. + + +Four-Class classification problem +Training Accuracy: 0.9230, Testing Accuracy: 0.9246 +(a) + + +Five-Class classification problem +Training Accuracy: 0.9592, Testing Accuracy: 0.9611 +(b) + +Seven-Class classification problem +Training Accuracy: 0.9737, Testing Accuracy: 0.9710 +(c) + +Final +Conclusion +Highest classification accuracy obtained when: +• Edge weights are to be set as ones. +• Capacitive elements have node feature value of 1/C. + +Fig. 14 – Fourth experiment set. a) Four-class, b) Five-class, c) Seven-class classification problem. + + + + + + + + + + + + +Fig. 15 – Circuit classifier structure [1] + +circuit simulation data that is accurate and reliable. To +ensure that the sampling rate is accurate, the graphs are +divided into a number of subsets based on circuit class, +and each subset is simulated separately. Each of these +subsets is tested for accuracy, and any discrepancies are +noted and addressed. After all the subsets have been +tested and corrected, the overall sampling rate of the +circuit simulations can be determined. Once the sampling +rate has been determined, the normalized values vector +is concatunated with element ID to complement the +feature vector. Fig. 15 shows a block diagram of the +classifier structure. Three GCN layers are used to get +information about 3rd level neighbors. The classifier +output layer computes a probability score for the class +of each topology. +1) Classifier Problem Formulation +Circuit topologies in graph forms (G) are fed to +the classifier. Each circuit graph has number of nodes +(N) along with their corresponding node features (X) +each has dimension (din). The adjacency matrix (A) +defines connections between each node. The classifier +outputs a probability (Y) of a converter to belong to a +certain class (C). Sub-GCN networks are embedded in +each GCN layer, allowing aggregation processes between +feature vectors in the neighboring nodes. Hyperbolic +tangent (“tanh”) is used as the non-linear activation +function, while being slower than the Rectified Linear +Unit (ReLU) activation function, it helps to avoid the +dying ReLU problem due to the very different values of +both inputs and outputs [78]. The global mean readout +(GM-Read out) layer returns graph level outputs by aver- +aging GCN processed node features. A fully Connected +(FC) linear layer is a score function for each circuit, +while (Softmax) output layer is used to calculate the +probability, in range of [0-1], of each circuit belonging +to a certain class. The Softmax function formula σ() +is stated in equation (10). The classifier uses training +datasets and updates weights or GCN layers and linear +layers by minimizing the cross entropy loss function, +which is shown in equation (11), where: +• M - Number of classes + +• log - The natural log + +• Y - Binary indicator (0 or 1) if class label c is the +correct classification for observation O. + +• p - Predicted probability observation O is of class +C. +A mathematical formulation of the transformations of +the designed classifier is stated as: +Y = classifier(X, A) +(3) +Where +3 X GCN layers + +GCN +Global Mean +Pooling +Linear +Transformation +Softmax +Graphs Representing Circuits +Probability for each Class + +1.0 +0.8 +0.4 +0.2 +Traing Dataset +Testing Dataset +0.0+ +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +80 +Accur +0.4 +0.2 +Traing Dataset +Testing Dataset +0.0+ +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +80 +0.4 +0.2 +Traning Dataset +Testing Dataset ++00 +200 +400 +600 +800 +1000 +1200 +Epoch000000 +RNxdin → RNxd +R1xd RIxC +RIxc +RIxC +RNxdn → R1xdX ∈ R(N )×din +(4) +Y ∈ RC×1 +(5) +No. of correct predictions belonging to specific class += +Total No. of predictions belonging to that class +GCN (k) : RN×din ›→ RN×d, k ∈ +. +Σ +0, 1, .., k − 1 +TP +(12) +(6) +GM − Readout : RN×d ›→ R1×d +(7) +FC : R1×d ›→ R1×C +(8) +Softmax = R1×C ›→ R1×C +(9) +Recall = +TP + FN +No. of correct predictions belonging to specific class += +Total No. of correct predictions in the dataset +(13) +2 × Precision × Recall +where +F 1score = + + +Precision + Recall +(14) +ezi +σ(zi) = ΣK +j=1 +for i = 1, 2, . . . , K +(10) +ezj +Σ +M +Class two and three have F1 scores of 0.87 and 0.89, +respectively. Since F1 score embeds precision and re- +call into one computation, the weighted average of F1 +should be used to compare classifier models, not global +CrossEntropy = − + +2) Results and Analysis + +c=1 +yo,c log(po,c) +(11) +accuracy. The Recall of class 2 is 0.77, indicating a +misclassification occurs. On the other hand, the Recall +score of class three is 1, indicating all class 3 circuits +a) Continuous Circuit Classifier +Training and testing accuracy after 1200 epochs are +shown in Fig 16, scoring 97.37% and 97.10 %, respec- +tively. 70% of the dataset containing 6000 graphs rep- +resenting the seven circuit classes was used for training. +Cross entropy loss function is used in training the model +with Adam optimizer with learning rate of 0.02. Fig. 17 +shows the 2-D embedding of the classifier testing dataset +output. It can be clearly seen that graphs falling in the +same class cluster together. +The confusion matrix shown in Fig. 18 is used +to analyze the classifier behavior and obtain insights +about its functionality. The array gives an insight about +overlaps/errors in class predictions. Other classifier as- +sessment metrics are listed in Table VII, which shows +the precision, recall, F1 and support metrics. The fol- +lowing notations are used to assess binary classifiers +performance, but are also extended to multi-classification +problems. +• Positive: The graph is classified as a member of the +circuit class the classifier is trying to identify. +• Negative: The instance is classified as not being a +member of the class we are trying to identify. +True or false can be added to Positive or negative to +indicate whether the classifier has correctly predicted +the class or misclassified it. Generally, precision is a +measure of true positive instances, which shows how +many of the positive predictions made are correct. Recall +aka sensitivity, is a measure of how many of the positive +cases the classifier correctly predicted with respect to +the over all the positive cases in the data. The F1 +score is the percentage of correct class predictions. A +mathematical formulation of the evaluation metrics are +listed in equations (13-14). +TP +Precision = +TP + FP +were correctly classified. This analysis indicates misclas- +sification of 52 class 2 circuit graphs as class 3, resulting +in a precision measure of 0.8.Additional observations +from confusion matrix, classifier metrics and the 2-D +vector mapping can be summarized as follows: +• Circuits with similar connections are distinctly clas- +sified but the clusters appear close in the 2-D vector +mapping. Classes (four and six) are fourth order +circuits but are dissimilar in physical connection, +hence are mapped in the same vicinity but close. +Similarly are classes (Zero and one), follow the +same principle. On the other hand, classes (two +and three) are second order circuits sharing almost +identical circuit connection, hence are mapped very +close to each other. +• The same concept is applied to circuits with dis- +similar circuit structures, as they are clustered far +from each other in the 2-D map i.e classes 0 and 5. +• The similarity between classes two and three in con- +nection and number of nodes causes 2.63% classifi- +cation inaccuracy. Further tuning of the weights of +the linear layer can improve the classifier selectivity. +b) Switching Circuit Classifier +The trained classifier scored 100% for training and +testing data, when trained for 200 epochs. In Fig. 20, a +2-D output representation of 1800 test dataset graphs are +plotted and colorized according to their predicted class. +Circuits of the same topology are distinctly identified +and clustered together. Further, the operating mode of +each of the circuits (CCM or DCM) is also identified. +The different loci of the 2D plot from every class +is a result of convolution operation taking all graph +properties representing circuits like component values, +type and switches duty cycle and converting it to a +lower dimension (2-D). It is also noted that graphs of +the same converter topology form groups and cluster in + + + + +Fig. 16 – Circuit classifier accuracy +(a) + + + +3 + + + +2 + + + +Fig. 17 – 2-D embeddings of circuit graphs + +Fig. 18 – Confusion matrix for seven classes circuits + + +1 + + + + + +0 + + + + + +1 + + + + + +2 + + + + + +3 + + +X + +(b) +Fig. 20 – a) training and testing data classification accuracy, b) 2D +embedding of the three converters in CCM and DCM after classifica- +tion + + + + + + +1000 + + + + + + + + + + + +800 + + + + + + + + + + + +600 + + + + + + + + + + + +400 + + + + + + + + + + + +200 +Table VII – Continuous circuits classifier assessment metrics + + + + +Buck +CCM + + +Boost +CCM + + +Buck-Boost +CCM + + +Buck +DCM + + +Boost +DCM + + +0 +Buck-Boost +DCM +VIII. DISCUSSION AND FUTURE WORK +Predicted Class + + +Fig. 19 – Confusion matrix for DC-DC converters + +close proximity. +This methodology of circuit representation allows +incorporating ML techniques in many applications, and +can serve the purpose of generating application-specific +circuits. Machine learning and neural network models +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +981 +0 +0 +0 +0 +0 +1013 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1031 +1027 +1015 +1008 +Classes +Buck +Boost +Buck-Boost +Buck-DCM +Boost-DCM +BuckBoost-DCM +Buck Converter +Boost Converter +Buck-Boost Converter +4 +2 +0 +2 +4 +True Class +Buck-Boost +DCM +Boost +DCM +Buck +DCM +Buck-Boost +CCM +Boost +CCM +Buck +CCM +Y +Circuit Class +Precesion +Recall +F1 score +Support +Class 0 +1.00 +1.00 +1.00 +310 +Class 1 +1.00 +1.00 +1.00 +247 +Class 2 +1.00 +0.77 +0.87 +226 +Class 3 +0.80 +1.00 +0.89 +213 +Class 4 +1.00 +1.00 +1.00 +225 +Class 5 +1.00 +1.00 +1.00 +289 +Class 6 +1.00 +1.00 +1.00 +282 +Macro avg +0.97 +0.97 +0.97 +1792 +Weighted avg +0.98 +0.97 +0.97 +1792 +Accuracy + + +0.97 +1792 + + +1.0 +T +0.8 +Accuracy +0.6 +0.4 +0.2 +TraningDataset +Testing +Dataset +0.0 +0 +200 +400 +600 +800 +1000 +1200 +Epoch1.0 +0.8 +Accuracy +0.6 +0.4 - +0.2 +Traning Dataset +Testing D +Dataset +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +EpochClass0 +310 +0 +0 +0 +0 +0 +0 +247 +0 +0 +0 +0 +250 +0 +174 +52 +0 +0 +200 +0 +0 +0 +213 +0 +0 +0 +150 +0 +225 +100 +0 +0 +0 +289 +0 +50 +0 +0 +0 +0 +282 +- 0 +Class0 +Class 1 +Class2 +Class 3 +Class 4 +Class 5 +Class 62.5 +2.0 +1.5 +1.0 +0.5ClassesTable VIII – DC-DC converters classifier assessment metrics +estimation ...etc. +IX. CONCLUSION + + + + + + + + + + +in general are heavily dependent on hyper-parameter +tuning. Several aspects are to be included when circuit +designer incorporate ML model in circuit design like +network depth, number of neuron, activation functions, +pooling layers . . . etc. These uncertainties in ML models +adds more burden when incorporating ML techniques in +circuit design. Eventually, a network update is a must +at some point of the design process, and eventually +designer must fine tweak the ML based design tool. +The proposed method can be applied to a wide range +of applications such as, power electronic converters con- +dition monitoring and prognostics, since the developed +representation maps the circuit structure and thus voltage +stresses at each node and current stresses in each branch +can be evaluated and tied to a component/converter +reliability function. Another application is network struc- +ture and fault detection in large power systems [79]. +Circuit design is another application that fits the pro- +posed methodology, where circuit performance parame- +ters are set, and the GNN model can generate a circuit +topology that meets the input criteria. Moreover, this +study can be further developed to for the purpose of +linking finite element modelling software in AI assisted +design of magnetic components for the purpose of +optimal component values/shape design. Additionally, +the proposed methodology has very high potential in +circuit obfuscation and reverse engineering when it is +required to identify/obscure circuit structure [80]. One +idea works on the circuit side utilizing the GNN capabil- +ity of learning the proper transformation function of the +converter, i.e can obtain a mathematical transformation +of every circuit component and eventually all circuit +behavior. On the application side, the end goals whether +they are gain, current ripples, magnetic design .. etc, +are transformed into a fictitious statistical domain, and +the purpose of the GNN is to generate circuits with +similar statistical domain. This can be beneficial to train +AI to generate application specific converters, which +eventually will help reduce component size, increase +power density, speed and efficiency. This methodology +is also applicable in power system applications such as +network reconstruction and fault detection and load flow +In this paper a graph representation of electric circuits +is proposed. This method enables a dynamically scalable +interface of different circuit aspects including physical +connections, component values and mode of operation, +to the machine learning domain. Applying the circuit +graphs as inputs to a GNN different circuit modeling, +design and optimization tasks can be performed. The +effect of bond graph feature selection, scaling and for- +mulation was also analyzed. Optimal feature represen- +tation results in a more well defined feature matrix +and consequently a more accurate circuit and operating +mode identification. As a proof of concept case studies +of classifiers of continuous and switching circuits were +presented where, the proposed algorithms were proven +to distinctly identify with high accuracy circuit types +based on physical connectivity as well as identifying +their mode of operation based on parameter values and +control variable values. + +REFERENCES +[1] A. K. Khamis and M. Agamy, “Mapping continuous circuit struc- +tures to machine learning space,” in 2022 IEEE 31st International +Symposium on Industrial Electronics (ISIE), 2022, pp. 149–155. +[2] A. K. Khamis and A. M., “Converter circuits to machine learning: +Optimal feature selection,” in 2022 IEEE Energy Conversion +Congress and Exposition (ECCE), 2022, pp. 9–14. +[3] M. Takamiya, K. Miyazaki, H. Obara, T. Sai, K. Wada, +and T. Sakurai, “Power electronics 2.0: Iot-connected and al- +controlled power electronics operating optimally for each user,” +in 2017 29th International Symposium on Power Semiconductor +Devices and IC’s (ISPSD), 2017, pp. 29–32. +[4] X. Li, X. Zhang, F. Lin, and F. Blaabjerg, “Artificial-intelligence- +based design (ai-d) for circuit parameters of power converters,” +IEEE Transactions on Industrial Electronics, 2021. +[5] S. Zhao and H. Wang, “Enabling data-driven condition moni- +toring of power electronic systems with artificial intelligence: +Concepts, tools, and developments,” IEEE Power Electronics +Magazine, vol. 8, no. 1, pp. 18–27, 2021. +[6] S. Peyghami, T. Dragicevic, and F. Blaabjerg, “Intelligent long- +term performance analysis in power electronics systems,” Scien- +tific Reports, vol. 11, no. 1, pp. 1–18, 2021. +[7] S. Zhao, F. Blaabjerg, and H. Wang, “An overview of artificial in- +telligence applications for power electronics,” IEEE Transactions +on Power Electronics, vol. 36, no. 4, pp. 4633–4658, 2021. +[8] B. K. Bose, “Neural network applications in power electronics +and motor drives—an introduction and perspective,” IEEE Trans- +actions on Industrial Electronics, vol. 54, no. 1, pp. 14–33, 2007. +[9] G. Zhang, H. He, and D. Katabi, “Circuit-GNN: Graph neural +networks for distributed circuit design,” in Proceedings of the +36th International Conference on Machine Learning, ser. Pro- +ceedings of Machine Learning Research, vol. 97. PMLR, 09–15 +Jun 2019, pp. 7364–7373. +[10] H. Wang, J. Yang, H. Lee, and S. Han, “Learning to design +circuits,” CoRR, vol. abs/1812.02734, 2018. [Online]. Available: +http://arxiv.org/abs/1812.02734 +[11] H. Wang, K. Wang, J. Yang, L. Shen, N. Sun, H.-S. Lee, and +S. Han, “Gcn-rl circuit designer: Transferable transistor sizing +with graph neural networks and reinforcement learning,” in +Proceedings of the 57th ACM/EDAC/IEEE Design Automation +Conference, ser. DAC ’20. IEEE Press, 2020. +[12] A. Fayyazi, S. Shababi, P. Nuzzo, S. Nazarian, and M. Pedram, +“Deep learning-based circuit recognition using sparse mapping +Circuit Class +Precesion +Recall +F1 score +Support +Buck-CCM +1.00 +1.00 +1.00 +1008 +Boost-CCM +1.00 +1.00 +1.00 +1015 +Buck-Boost-CCM +1.00 +1.00 +1.00 +1027 +Buck-DCM +1.00 +1.00 +1.00 +981 +Boost-DCM +1.00 +1.00 +1.00 +1013 +Buck-Boost-DCM +1.00 +1.00 +1.00 +1031 +Macro avg +1.00 +1.00 +1.00 +6075 +Weighted avg +1.00 +1.00 +1.00 +6075 +Accuracy + + +1.00 +6075 + + +and level-dependent decaying sum circuit representations,” in +2019 Design, Automation Test in Europe Conference Exhibition, +2019, pp. 638–641. +[13] Y. Li and Y. W. Li, “Power converters topological transformation +using dual and isomorphic principles,” IEEE Open Journal of +Power Electronics, vol. 1, pp. 74–87, 2020. +[14] M. Liu, K. Zhu, J. Gu, L. Shen, X. Tang, N. Sun, and D. Z. Pan, +“Towards decrypting the art of analog layout: Placement quality +prediction via transfer learning,” in 2020 Design, Automation +Test in Europe Conference Exhibition (DATE), 2020, pp. 496– +501. +[15] P. C. Breedveld, “Multibond graph elements in physical systems +theory,” Journal of the Franklin Institute, vol. 319, no. 1-2, pp. +1–36, 1985. +[16] F. Harary, “Graph theory and electric networks,” IRE Transac- +tions on Information Theory, vol. 5, no. 5, pp. 95–109, 1959. +[17] Z. X. Han, “Generalized method of analysis of simultaneous +faults in electric power system,” IEEE Transactions on Power +Apparatus and Systems, vol. PAS-101, no. 10, pp. 3933–3942, +1982. +[18] Y. Li, J. Kuprat, Y. Li, and M. Liserre, “Graph-theory-based +derivation, modeling and control of power converter systems,” +IEEE journal of emerging and selected topics in power electron- +ics, pp. 1–1, 2022. +[19] N. Fitzgerald, Graph theory with applications to engineering and +computer science. Univ Sci Books, 2001. +[20] H. Q. Tran and C. Canudas de Wit, “Modeling and control +of power electronic converters with graph theory and hybrid +automata,” IEEE Transactions on Control Systems Technology, +vol. 26, no. 1, pp. 285–296, 2018. +[21] J. E. Diaz, S. Mises, and R. Kevork, “Modelling of power +electronic converters in graph theory,” IEEE Access, vol. 7, pp. +128 285–128 297, 2019. +[22] H. Patel, “Anomalous ringing in circuits and its cancellation,” IET +Circuits, Devices & Systems, vol. 10, no. 1, pp. 33–39, 2016. +[23] P. Kundur, Power system stability and control. McGraw-Hill, +1994. +[24] T. D. Sundar, “Admittance matrix based z-bus formulation for +large-scale power systems,” 2018 International Conference on +Applied and Theoretical Computing and Communication Tech- +nology (iCATccT), pp. 1–6, 2011. +[25] A. M. Kettner and M. Paolone, “On the properties of the +compound nodal admittance matrix of polyphase power systems,” +IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 444– +453, 2018. +[26] H. Saadat, Power system analysis. McGraw-Hill Professional, +2003. +[27] M. Nasir, Power electronic converters: modeling, simulation, and +control using MATLAB/SIMULINK. CRC Press, 2008. +[28] Y. Li, Z. Shuai, X. Liu, Y. Chen, Z. Li, Y. Hong, and Z. J. +Shen, “Stability analysis and location optimization method for +multiconverter power systems based on nodal admittance ma- +trix,” IEEE Journal of Emerging and Selected Topics in Power +Electronics, vol. 9, no. 1, pp. 529–538, 2019. +[29] V. D. Gebben, “Bond Graph Bibliography for 1961–1976,” +Journal of Dynamic Systems, Measurement, and Control, +vol. 99, no. 2, pp. 143–145, 06 1977. [Online]. Available: +https://doi.org/10.1115/1.3427087 +[30] P. J. Gawthrop and G. P. Bevan, “Bond-graph modeling,” IEEE +Control Systems Magazine, vol. 27, no. 2, pp. 24–45, 2007. +[31] Y. Baqqal and M. E. hammoumi, “A generic approach of mod- +elling perturbation of mechanical systems: Concept with bond +graph,” in 2018 IEEE International Conference on Technology +Management, Operations and Decisions (ICTMOD), 2018, pp. +219–223. +[32] K. Sirivadhna, E. F. Richards, and M. D. Anderson, “The +application of bond graphs to electrical machinery and power +engineering,” IEEE Power Engineering Review, vol. PER-3, +no. 5, pp. 35–35, 1983. +[33] P. Gawthrop and E. J. Crampin, “Bond graph representation of +chemical reaction networks,” IEEE Transactions on NanoBio- +science, vol. 17, no. 4, pp. 449–455, 2018. +[34] U. Qasim, A. Rashid, and O. Hasan, “Formalization of bond +graph using higher-order-logic theorem proving,” ISA Transac- +tions, vol. 128, pp. 453–469, 2022. [Online]. Available: https: +//www.sciencedirect.com/science/article/pii/S0019057821006091 +[35] L. Mo, G. Chen, J. Huang, X. Qing, Y. Hu, and X. He, “Graph +theory-based programmable topology derivation of multiport dc– +dc converters with reduced switches,” IEEE Transactions on +Industrial Electronics, vol. 69, no. 6, pp. 5745–5755, 2022. +[36] M. Delgado and G. Profos, “Identifiability of dynamic systems +represented by bond graphs,” Mathematical and Computer Mod- +elling of Dynamical Systems, vol. 5, no. 2, pp. 89–112, 1999. +[37] C. Sueur and G. Dauphin-Tanguy, “Structural controllabil- +ity/observability of linear systems represented by bond graphs,” +Journal of the Franklin Institute, vol. 326, no. 6, pp. 869–883, +1989. +[38] G. Huang, J. Hu, Y. He, J. Liu, M. Ma, Z. Shen, J. Wu, +Y. Xu, H. Zhang, K. Zhong, X. Ning, Y. Ma, H. Yang, +B. Yu, H. Yang, and Y. Wang, “Machine learning for electronic +design automation: A survey,” 2021. [Online]. Available: +https://arxiv.org/abs/2102.03357 +[39] M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and +P. Vandergheynst, “Geometric deep learning: going beyond +euclidean data,” CoRR, vol. abs/1611.08097, 2016. [Online]. +Available: http://arxiv.org/abs/1611.08097 +[40] F. Scarselli, M. Gori, A. C. Tsoi, M. Hagenbuchner, and G. Mon- +fardini, “The graph neural network model,” IEEE Transactions +on Neural Networks, vol. 20, no. 1, pp. 61–80, 2009. +[41] V. Di Massa, G. Monfardini, L. Sarti, F. Scarselli, M. Maggini, +and M. Gori, “A comparison between recursive neural networks +and graph neural networks,” in The 2006 IEEE International Joint +Conference on Neural Network Proceedings, 2006, pp. 778–785. +[42] J. Bruna, W. Zaremba, A. Szlam, and Y. LeCun, “Spectral +networks and locally connected networks on graphs,” 2013. +[Online]. Available: https://arxiv.org/abs/1312.6203 +[43] A. Micheli, “Neural network for graphs: A contextual construc- +tive approach,” IEEE Transactions on Neural Networks, vol. 20, +no. 3, pp. 498–511, 2009. +[44] W. Liao, B. Bak-Jensen, J. R. Pillai, Y. Wang, and Y. Wang, +“A review of graph neural networks and their applications in +power systems,” 2021. [Online]. Available: https://arxiv.org/abs/ +2101.10025 +[45] Y. Ma, Z. He, W. Li, L. Zhang, and B. Yu, Understanding Graphs +in EDA: From Shallow to Deep Learning. New York, NY, +USA: Association for Computing Machinery, 2020, p. 119–126. +[Online]. Available: https://doi.org/10.1145/3372780.3378173 +[46] H. Wang, J. Yang, H.-S. Lee, and S. Han, “Learning to design +circuits,” 2018. [Online]. Available: https://arxiv.org/abs/1812. +02734 +[47] H. Wang, K. Wang, J. Yang, L. Shen, N. Sun, H.-S. Lee, and +S. Han, “Gcn-rl circuit designer: Transferable transistor sizing +with graph neural networks and reinforcement learning,” 2020. +[Online]. Available: https://arxiv.org/abs/2005.00406 +[48] H. He and G. Zhang, “End-to-end learning for distributed circuit +design,” 2018. +[49] G. Zhang, H. He, and D. Katabi, “Circuit-GNN: Graph neural +networks for distributed circuit design,” in Proceedings of +the 36th International Conference on Machine Learning, ser. +Proceedings of Machine Learning Research, K. Chaudhuri and +R. Salakhutdinov, Eds., vol. 97. PMLR, 09–15 Jun 2019, pp. +7364–7373. [Online]. Available: https://proceedings.mlr.press/ +v97/zhang19e.html +[50] K. Hakhamaneshi, M. Nassar, M. Phielipp, P. Abbeel, and +V. Stojanovic´, “Pretraining graph neural networks for few-shot +analog circuit modeling and design,” 2022. [Online]. Available: +https://arxiv.org/abs/2203.15913 +[51] H. Ren, G. F. Kokai, W. J. Turner, and T.-S. Ku, “Paragraph: +Layout parasitics and device parameter prediction using graph + +neural networks,” in Proceedings of the 57th ACM/EDAC/IEEE +Design Automation Conference. IEEE Press, 2020. +[52] X. Gao, C. Deng, M. Liu, Z. Zhang, D. Z. Pan, and Y. Lin, +“Layout symmetry annotation for analog circuits with graph +neural networks,” ser. ASPDAC ’21. New York, NY, USA: +Association for Computing Machinery, 2021, p. 152–157. +[Online]. Available: https://doi.org/10.1145/3394885.3431545 +[53] H. Chen, K. Zhu, M. Liu, X. Tang, N. Sun, and D. Z. +Pan, “Universal symmetry constraint extraction for analog and +mixed-signal circuits with graph neural networks,” in 2021 58th +ACM/IEEE Design Automation Conference (DAC), 2021, pp. +1243–1248. +[54] Y. Ma, Z. He, W. Li, L. Zhang, and B. Yu, Understanding Graphs +in EDA: From Shallow to Deep Learning. New York, NY, +USA: Association for Computing Machinery, 2020, p. 119–126. +[Online]. Available: https://doi.org/10.1145/3372780.3378173 +[55] E. Ustun, C. Deng, D. Pal, Z. Li, and Z. Zhang, “Accurate opera- +tion delay prediction for fpga hls using graph neural networks,” in +2020 IEEE/ACM International Conference On Computer Aided +Design (ICCAD), 2020, pp. 1–9. +[56] Y. Li, Y. Lin, M. Madhusudan, A. Sharma, W. Xu, S. S. Sapat- +nekar, R. Harjani, and J. Hu, “A customized graph neural network +model for guiding analog ic placement,” in 2020 IEEE/ACM +International Conference On Computer Aided Design (ICCAD), +2020, pp. 1–9. +[57] L. Umanand, “Switched junctions in bondgraph for modelling +power electronic systems,” Journal of the Indian Institute of +Science, vol. 80, p. 327, 2013. +[58] A. C. Umarikar and L. Umanand, “Modelling of switching +systems in bond graphs using the concept of switched +power junctions,” Journal of the Franklin Institute, vol. +342, no. 2, pp. 131–147, 2005. [Online]. Available: https: +//www.sciencedirect.com/science/article/pii/S0016003204000833 +[59] J.-E. Stro¨mberg, J. Top, and U. So¨derman, “Variable causality in +bond graphs caused by discrete effects,” 1992. +[60] U. So¨derman and J. erik Stromberg, “The conceptual side of +mode switching,” 1993. +[61] A. Markakis, W. Holderbaum, and B. Potter, “A comparison be- +tween bond graphs switching modelling techniques implemented +on a boost dc-dc converter,” in 2011 IEEE 33rd International +Telecommunications Energy Conference (INTELEC), 2011. +[62] A. Umarikar and L. Umanand, “Modelling of switched mode +power converters using bond graph,” Electric Power Applications, +IEE Proceedings -, vol. 152, pp. 51 – 60, 02 2005. +[63] P. Rodr´ıguez, M. A. Bautista, J. Gonza`lez, and S. Escalera, +“Beyond +one-hot +encoding: +Lower +dimensional +target +embedding,” Image and Vision Computing, vol. 75, pp. 21– +31, +2018. +[Online]. +Available: +https://www.sciencedirect. +com/science/article/pii/S0262885618300623 +[64] “Pyg documentation.” [Online]. Available: https:// pytorch- +geometric.readthedocs.io/ +[65] “Deep graph library.” [Online]. Available: https://www.dgl.ai/ +[66] “Keras documentation.” [Online]. Available: https://keras.io/ +[67] W. Liao, B. Bak-Jensen, J. R. Pillai, Y. Wang, and Y. Wang, +“A review of graph neural networks and their applications in +power systems,” 2021. [Online]. Available: https://arxiv.org/abs/ +2101.10025 +[68] B. Donon, B. Donnot, I. Guyon, and A. Marot, “Graph neural +solver for power systems,” in 2019 International Joint Conference +on Neural Networks (IJCNN), 2019, pp. 1–8. +[69] B. D. A. M. I. G. Balthazar Donon, Re´my Cle´ment and +M. Schoenauer, “Neural networks for power flow: Graph +neural solver,” Electric Power Systems Research, vol. 189, p. +106547, 2020. [Online]. Available: https://www.sciencedirect. +com/science/article/pii/S0378779620303515 +[70] C. Wang, J. An, and G. Mu, “Power system network topology +identification based on knowledge graph and graph neural net- +work,” Frontiers in Energy Research, vol. 8, p. 613331, 2021. +[71] T. N. Kipf and M. Welling, “Semi-supervised classification +with graph convolutional networks,” 2016. [Online]. Available: +https://arxiv.org/abs/1609.02907 +[72] W. L. Hamilton, R. Ying, and J. Leskovec, “Inductive +representation learning on large graphs,” 2017. [Online]. +Available: https://arxiv.org/abs/1706.02216 +[73] Y. Li, D. Tarlow, M. Brockschmidt, and R. Zemel, “Gated +graph sequence neural networks,” 2015. [Online]. Available: +https://arxiv.org/abs/1511.05493 +[74] Y. Shi, Z. Huang, S. Feng, H. Zhong, W. Wang, and Y. Sun, +“Masked label prediction: Unified message passing model +for semi-supervised classification,” 2020. [Online]. Available: +https://arxiv.org/abs/2009.03509 +[75] T. Geng, C. Wu, Y. Zhang, C. Tan, C. Xie, H. You, M. C. +Herbordt, Y. Lin, and A. Li, “I-gcn: A graph convolutional +network accelerator with runtime locality enhancement through +islandization,” 2022. [Online]. Available: https://arxiv.org/abs/ +2203.03606 +[76] C. R. Banbury, C. Zhou, I. Fedorov, R. M. Navarro, +U. Thakker, D. Gope, V. J. Reddi, M. Mattina, and P. N. +Whatmough, “Micronets: Neural network architectures for +deploying tinyml applications on commodity microcontrollers,” +CoRR, vol. abs/2010.11267, 2020. [Online]. Available: https: +//arxiv.org/abs/2010.11267 +[77] D. P. Kingma and J. Ba, “Adam: A method for stochastic +optimization,” arXiv preprint arXiv:1412.6980, 2014. +[78] P. Lemos, N. Jeffrey, M. Cranmer, S. Ho, and P. Battaglia, +“Rediscovering orbital mechanics with machine learning,” 2022. +[Online]. Available: https://arxiv.org/abs/2202.02306 +[79] Z. Chen, J. Xu, C. Alippi, S. X. Ding, Y. Shardt, T. Peng, and +C. Yang, “Graph neural network-based fault diagnosis: a review,” +2021. [Online]. Available: https://arxiv.org/abs/2111.08185 +[80] Z. Chen, L. Zhang, G. Kolhe, H. M. Kamali, S. Rafatirad, +S. M. Pudukotai Dinakarrao, H. Homayoun, C.-T. Lu, and +L. Zhao, “Deep graph learning for circuit deobfuscation,” +Frontiers in Big Data, vol. 4, 2021. [Online]. Available: +https://www.frontiersin.org/articles/10.3389/fdata.2021.608286 + diff --git a/_9E1T4oBgHgl3EQfVAPl/content/tmp_files/load_file.txt b/_9E1T4oBgHgl3EQfVAPl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95f08a54f78d9bfad1012f0f5b4ffc703b167d2d --- /dev/null +++ b/_9E1T4oBgHgl3EQfVAPl/content/tmp_files/load_file.txt @@ -0,0 +1,1698 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf,len=1697 +page_content='Comprehensive Mapping of Continuous/Switching Circuits in CCM and DCM to Machine Learning Domain using Homogeneous Graph Neural Networks Ahmed K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Khamis EE Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=', AASTMT, Alex, Egypt ECE Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=', University at Albany SUNY Albany NY, USA Mohammed Agamy ECE Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=', University at Albany, SUNY Albany NY, USA Abstract: This paper proposes a method of transfer- ring physical continuous and switching/converter circuits working in continuous conduction mode (CCM) and discontinuous conduction mode (DCM) to graph repre- sentation, independent of the connection or the number of circuit components, so that machine learning (ML) algorithms and applications can be easily applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Such methodology is generalized and is applicable to circuits with any number of switches, components, sources and loads, and can be useful in applications such as artificial intelligence (AI) based circuit design automation, layout optimization, circuit synthesis and performance moni- toring and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The proposed circuit representation and feature extraction methodology is applied to seven types of continuous circuits, ranging from second to fourth order and it is also applied to three of the most common converters (Buck, Boost, and Buck-boost) operating in CCM or DCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A classifier ML task can easily differentiate between circuit types as well as their mode of operation, showing classification accuracy of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='37% in continuous circuits and 100% in switching circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Index Terms—Electric circuit, Bond Graph, Graph Neu- ral Networks (GNN), Machine Learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' INTRODUCTION AI algorithms are used to model computationally complex systems or systems/processes with significant parameter uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Modern improvements in com- putation resources enable the incorporation of AI algo- rithms in power converter design and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Complex nonlinear problems such as thermal and electromagnetic designs, modeling of layout parasitics and estimation of component stresses under different operating conditions are some areas where AI algorithms can significantly simplify and optimize the design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [3]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Power A preliminary version of this paper was presented in [1], [2] electronics applications of ML have focused on control, component design and maintenance [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' ML-based sur- rogate/black box models are used for online prediction tasks to reduce computational effort, memory and power used by classical simulation/mathematical-based models [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Design optimization is an additional target as ML models obtain the optimal target without compromising other design constraints or trade-offs of design, which is known in its mathematical formulation as Pareto front [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' ML-based circuit design should be able to reflect circuit component connectivity as well as the effect of varying the values of these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In [9] a graph representation of circuits with a combined feature map for input and output nodes was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, it does not represent details of component types or connec- tivity, rather it is just a numerical input/output transfer characteristic of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Reinforcement Learning (RL) was introduced in [10] to optimize passive com- ponent values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' An updated version of the RL agent was presented in [11], where the RL-based optimization algorithm is used to optimally size transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In this case, based on a given design flow, the RL algorithm updates the node embedding in Graph Neural Network (GNN) representation of the circuit to maximize the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' One-hot encoding is used to represent tran- sistors, in addition to other internal parameters, which are passed as features to a Graph Convolution Network (GCN) to extract node embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Despite the simplicity of this approach, it it is incorrect and does not guarantee a solution in the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In other words, in cir- cuit synthesis/generation problem, there is no guarantee that the circuit synthesis neural network can transform the generated graph to a physically realisable circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Existing methods do not provide a systematic way of circuit feature embedding in GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' These models have several limitations including scalability of circuit size (number of nodes and/or components), mapping con- nectivity and identifying component types within the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This paper proposed a systematic approach for electric circuit representation to enable use of ML design or performance prediction tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This method has the benefits of being scalable and topology agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In this paper the following key contributions are proposed: • A comparative review of different research attempts in mapping circuits to ML domain including cir- cuit representation techniques, feature assignment, intended task and how components and connections are represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Proposing three possible circuit representation tech- niques, listing the advantages and disadvantages while providing mathematical reasons for technique selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • The circuit representation includes different circuit element types and circuit connection types, without indulging the concept with numerical tuning or the empirical hyper-parameters optimization of ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Proposing a unified (applied to all circuit elements) node feature assignment algorithm, irrespective of number of connections present in circuit or circuit order, while combining the feature maps of the nodes to generate the feature map for the whole graph in a GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Proposing a dataset generation algorithm, that is easily applicable to the ML task or application, capturing circuit performance variables of interest in a standardized data format that can be used in ML problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A proof of concept classifier problem applied to variable structure continuous circuits or switching circuits operating in CCM or DCM is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The target ML task covers a wide range of possible tasks or even a combination of tasks including re- gression, classification and clustering tasks, whether it is supervised or unsupervised tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The proposed mapping approach enables a wide range of possible ML tasks or a combination of tasks including regression, classification, clustering, and synthesis of power electronic converter circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' PROBLEM DESCRIPTION Neural networks can construct model from training data after being processed in order to obtain features to characterize the built model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the case of electri- cal circuits, the process does not have an established methodology or criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Problems with interfacing elec- tric circuits to ML tools are highlighted in this section, while different solutions are proposed in next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit Structure Representation Problem The main problem faced when circuits are to be fed to a NN is the fixed size input layer, which has a defined dimension, invalidating the scalability require- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The workaround proposed in [12] pre-processes a matrix consisting of multiple vectors representing circuit components, so that the input to the Convolutional Neural Network (CNN) is of a fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Eventually, this workaround added more computational overhead and increased training time and computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, from a circuit standpoint, it is an incomplete circuit model because it has no explicit representation of the circuit structure or the dynamic behaviour or circuit elements interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In this paper we lay some foundations on how the physical properties of an electric circuit can be mapped to ML space, as follows: 1) Circuit performance is independent of circuit enter- ing order or elements order variation as long as the connection is kept invariant (isomorphic circuits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This makes the circuit representation Permutation Invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2) Circuit connectivity (series or parallel connections) and circuit elements values define the circuit per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3) Circuits may have any numbers of elements and has no upper boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 4) For circuits of similar input/output response (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' dual circuits [13]), circuit type/connection will be the identifying factor in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The realization of the last three definitions necessitates that the ML input layer be independent of the size of the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hence, the representation becomes Scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Dataset Expressiveness Problem Machine learning algorithms gain knowledge by iter- ative training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Datasets contain standardized/normalized data according to the nature of the ML task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Neither a generalized and confirmed methodology to handle circuit datasets nor a feature extraction/definition algo- rithm are defined that independently capture the circuit topology and the effects of component variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' More importantly, a clear measure of dataset expressiveness is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Given the circuits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2, every class has identical component count however, their performance is different and depends on component values, especially at resonance, and the dataset should indicate that differ- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Neural Network Topology Problem The physical circuit topology and the influence of parameter variation on its output variables must be clearly expressed by the selected NN topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' As an example, same circuit performance, can be obtained by using dual components [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In [14] a model of similar purpose employs CNN and takes placement images as its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Arguably, Graph Neural Networks (GNN) are superior in capturing the netlist topology, which is a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, GNN is more efficient in feature encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' For instance, the shape of a transistor can be represented by two real numbers (width and height) in GNN while it requires an array of pixels for CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The spatial features can be easily embraced in GNN by taking the location coordinates as features, which are motivations to take the GNN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' REVIEW OF CIRCUIT REPRESENTATION TECHNIQUES This section offers all possible solutions to presented problems in section II, and highlights the flow of work and derivations made from initial problem statements and better explains available solutions by offering de- tailed comparisons between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' There has been a lot of attempts to better represent circuits in ML domain, which are thoroughly explained in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, the paper will also highlight why solutions offered are insuf- ficient, ungeneralizable and empirical solutions, which either require fixed layout, huge datasets or extensive training and very complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit Representation Methodologies The main problem is to properly encode circuit prob- lem into computer interpretable form, which has been addressed by three modelling techniques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e graph the- ory, Y-Matrix and Bond graph [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A brief is given on every modelling technique, with an expanded illus- tration on the one used in this paper, wile a comparison between the merits and disadvantage of three modelling techniques are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1) Graph Theory Representation Graph theory is a mathematical tool used to model complex systems in a simplified way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the field of power electronics and converters, graph theory has proved to be a powerful tool for representing and an- alyzing the complex network of components and their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='There have been numerous studies in the literature which use graph theory to represent power electronics and converters [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The use of graph theory to represent power electronics and converters has sev- eral advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Graphs provide a concise and intuitive way to represent the components and their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Furthermore, graph algorithms can be used to analyze the system and identify system faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, the use of graph theory to represent power electronics and converters also has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Graphs are lim- ited in their ability to represent complex systems with many components, as the number of nodes and edges increases, the graph becomes cluttered and difficult to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In addition, graph matrices are usually very large and computationally intensive, making it difficult to obtain simulation results in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This can result in inaccurate or unsatisfactory results [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Furthermore, due to the complex relationship between the different components in the power system, the graph model may not be able to accurately represent the real-world system, leading to incorrect results [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Graph theory cannot account for nonlinearity and non-smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Power electronic converters are nonlinear systems and their circuits may contain high-frequency harmonics, which is difficult to capture using graph theory [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Finally, when using graph theory to model a power electronic converter, the system needs to be linearized, which may neglect certain important nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This can lead to incorrect results and further limitations to the accuracy of the model [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2) Y-Matrix Representation The admittance matrix is a powerful tool used to represent power systems and power electronic convert- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This method of representation has been used since its inception in the 1960s, and continues to be an efficient and novel way to model electrical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The admittance matrix is a complex quantity that describes the relationship between the voltage and the current in an electrical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' It consists of a matrix whose ele- ments are admittances of electrical components such as resistors, capacitors, and inductors [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This relationship between the voltage and the current provides a useful representation for solving electrical circuit problems [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The admittance matrix has been used for many appli- cations such as transient analysis and stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In particular, it has been used to study power systems [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In power system analysis, the admittance matrix can represent the components in the power system such as transmission lines, transformers, and loads, which can be analyzed in both the frequency domain and the time domain [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The advantage of the admittance matrix is that it is computationally efficient and provides a concise representation of the wide range of power system components [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='The admittance matrix has also been used for analyzing the stability of power electronic converter systems [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Power electronic converters are devices used to convert AC power to DC power or vice versa, and they generally consist of power switches, capacitors, and inductors [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Using the admittance ma- trix, the stability of the power electronic converter can be accurately analyzed in the frequency and time domains [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This method of representation is relatively old but can provides for accurate and efficient simulations of power electronic converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In a preliminary attempt of this work, different circuits were modelled utilizing Y-bus admittance matrix, where nodes represented buses and admittances serve as node features, while edges represent whether a connection is established between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1 shows a three and four element bus systems and its equivalent Y-bus admittance matrix and the corresponding features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, this representation was proven to be non expressive based on the fact that it is not uniformly scalable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e a three and four element (admittances) systems can both have the same number of nodes, which in this case is two, hence losing a very important feature in graph notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This Graph Representation as Two Nodes with Y-Matrix as node feature Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1 – Early attempt of converting circuit to graph by using Y-Matrix is because the branch elements are lumped together into a single equivalent admittance making it impossible to distinguish between different elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, with this representation, the change in node feature values doesn’t discriminate between whether a new element is added or component value has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3) Bond Graph Representation Bond graphs (BG) were proposed as a graphical language and systematic representation, to overcome limitations of block diagram models [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Using BG, a circuit can be modeled as bonds during all possible series and parallel connection permutations and combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Two key model elements were devised the 0 junction that is used to represent a parallel connection and 1 junction for series connections [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In addition to electric circuits, this approach can be extended to mechanical and chemical models as well [31]–[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The BG representation capturing the dynamics of a system is based on transforming (mapping) system components to their BG model counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The bond graph analogies used to describe physical systems in the form of bonds and paths are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bond graphs in opposition to transfer function which are behavioral models, belong to the class of struc- tural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Controllability and structural observability are applicable to BG, which are structural properties of models [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, it was proven in [36] that BGs are structurally identifiable, which allows a unique set of parameters to associate with given input/output response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In other words, bidirectional transformation governs circuit to graph and graph to circuit transforma- tion and hence, graphs generated from ML algorithms can be translated into a circuit if they match structural identifiability criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' REVIEW OF NEURAL NETWORK TOPOLOGIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Classical Neural Network Topologies Linear regression, random forest (RF) and artificial neural networks (ANN) are classical regression models used as attempts for regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' For classifica- tion tasks, support vector machine (SVM), K-Nearest- Neighbor (KNN) algorithm and RF are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Convolu- tional neural network (CNN) and recurrent neural net- works are extensively used in ML tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' CNN models are composed of convolutional layers and other basic blocks such as non-linear activation functions and down-sample pooling functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' While CNN is suitable for feature extraction on grid structure data like 2-D image, RNN is good at processing sequential data such as text or audio [38] due to their ability to leverage statistical properties of the image as euclidean data such as stationarity and compositionality through local statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' On the contrary, non-Euclidean data has no familiar properties as global parameterization, common system of coordinates, vec- tor space structure, or shift-invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Operations like convolution that are taken for granted in the Euclidean case are even not well defined on non-Euclidean domains [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' From that prospective, it is necessary to use an ML topology that can better represent non-euclidean structures like electric circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Graph Neural Networks GNNs are composed of definite function layers, but unlike other neural networks, the input is a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Acyclic, cyclic, directed, and undirected graphs can be processed by GNN as was stated in the first GNN model in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Scalablity and permutation invariance are unique properties in GNNs allowing input layer to be variable while graph node re-ordering will not affect the NN layer output, which satisfies the requirements needed for physical circuits representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' RNNs and GNNs, capable of directly processing graphs with labeled nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' An image classification task showed that GNNs outperforms RNNs, both in terms of accuracy and error rate [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Convolution operation on graphs is defined by spectral and spatial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In [42], spectral-based GCNs was proposed, which used the spectral graph theory to develop a new variant of graph convolutional operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Graph mutual dependence com- plexity was solved using non-recursive layers presented in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, spatial GCNs have been developed based on the fact that spectral GCNs are difficult to extend to large-scale graphs [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This makes GNNs suitable for circuit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1) Graph definition Graph G is a defined as (V, E) with V the set of vertices/Nodes equals v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=', vN , while set of Edges E ⊑ V × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Let N and M be the number of vertices and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Each graph can be represented by an adjacency matrix A of size N × N : Ai,j = 1 if there 2 Y2 1 Y1 Y4 Y3 G G 1 2 ++-Y2Ya+ Y + Y4Table I – Bond Graph terminologies [34] Terminology Description Strong Bond A single bond that causes effort in the 0 junction and flow in the 1 junction Passive Element A one port element that stores input power as potential energy (C-element), as kinetic energy (I-element) or transforms it into dissipative power (R-element).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Causal BG A BG is called causally completed or causal if the causal stroke known as causality is added on one end of each bond Causal Path A sequence of bonds with/without a transformer in between having causality at the same end of all bonds or a sequence of bonds with a gyrator in between, and all the bonds of one side of the gyrator having same end causality while all the bonds on the other side with causality on opposite end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' That means gyrator switches the direction of efforts/flows on one of its side [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A causal path can be a backward or forward or both depending upon the junction structure, elements and causality Branch A branch is a series of junctions having parent-child relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Two differ-ent sequences of junctions can be connected with a common bond or two-port element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Thus, one of the junction’s sequence acts as parent branch and the other one as child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Causal Loop A causal loop is a closed causal path with bonds (of the child branch) either connected to a similar junction or two different junctions of the parent branch Table II – Comparison between different circuit representation techniques Method Representation Methods Merits Drawbacks Graph Theory Component terminals are nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit Elements are edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Multi-discipline physics based modelling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' More intuitive graph for human reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Converter modelling foundations (duty cycle, CCM & DCM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='.etc) are missing/never been addressed No research on graph identifiability from graph to circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit graph can be defined using three matrices as shown in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bond Graph Elements and connections are nodes with different attributes Solid foundations on circuits/converter modelling in CCM & DCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' BG is a linear transformation and is mathematically identifiable as shown in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Multi-discipline physics based modelling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Generated graph can be defined with one Adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Maintains causality invariance of the system for any operational mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e the state vector resulting from state equation of the system does not change for any operating mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Non-intuitive modelling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Added complexity of causality assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Can yield a bigger graph than graph theory method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Y admittance matrix Circuit buses are nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Connections between buses are edges Well known methodology for circuit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Number of circuit sources can’t be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' System components can be lumped altogether and information about element count is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Used only for power system representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Node count is independent from number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' is an edge from vertex vi to vertex vj , and Ai,j = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Every edge has a set of edge features e V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' REVIEW OF CIRCUIT REPRESENTATION AND DESIGN USING GNN In [45] it was shown that the most intuitive way to rep- resent circuit, netlists or layouts is graph representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' It was also stated that graph neural networks (GNNs) are an opportunity replace shallow methods or mathematical optimization techniques, and Table III shows the state of the art circuit representation trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Many research has utilized GNN in circuit optimizations/classification op- erations and in many applications like transistor sizing, capacitor value optimization and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In [46], [47] , the model leverages reinforcement learning (RL) to learn the optimal policy for best parameter selection by rewarding the model for the best Figure of Merits (FOM) composed of several performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The circuit is embedded into a graph whose vertices are components and edges are wires, while generating a vector for each transistor and passing the graph to the RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Finally, the RL agent processes each vertex in the graph and generates an action vector for each node, then process the graph with an action vector with the purpose of max- imizing the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [48] proposes a model that solves the forward and inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In which, the model maps a given circuit to the corresponding transfer function and vise versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Inversely, the model utilizes gradient descent to optimize the circuit parameters to produce a transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The model leverages the differentiable nature of the neural network and applying gradient descent methods to optimize the input parameters of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, the neural network is trained for a particular circuit topology, and hence cannot be used for general circuit representation, in addition to the lack of switching circuit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, [49] pro- posed a technique for combining the feature maps of the nodes to generate the feature map for the whole graph in a GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' By propagating information from nodes to nodes representing input and output instead of pooling opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The paper represents graphs as a concatenations of the feature maps of the input and output nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In resonator circuits applications, [49] introduced a model that learns to simulate electromagnetic properties of distributed circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit were mapped on system level basis, such that each node refers to a resonator and each edge refers to the interaction between a pair of resonators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=', the electromagnetic coupling) between a pair of resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This representation does not incorporate the resonator internal structure or if the system had different resonators with different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' By propagating information from nodes to nodes, while representing cir- cuits as concatenation of input and output node features instead of pooling operation, regression task is utilized to obtain predictions about circuit performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' On the other hand, feature concatenation is not the correct technique to represent circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Feature concatenation is a numerical representation of circuit inputs and outputs that properly tuned by minimizing the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Attempts has been made to include different circuit topologies and obtain predictions as in [50], where two circuit types were included in the study: the ladder circuits and two stage operational amplifier circuits, with 20k training data instances of resistor ladders with 2 to 10 branches with equal distribution weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The model is based on DeepGEN architecture and was able to make predictions on ladder circuits with higher number of branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, the model’s ability to generalize and applicability to other circuit topologies and types remain questionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, no clue was given on how to distinguish connection type, and its effect on circuit performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, the representation was limited to transistors, without the inclusion of other circuit pa- rameters or elements(Transistor/resistor/voltage sources, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='. etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Also, no guidelines/rules were given on how to model circuit elemtents properties like frequency, phase shift, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='. etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' One major drawback in this representation is the elements with multiple terminals like transistors are represented as four connected nodes, which can cause unnecessary excessive computations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In [51], heteroge- neous GNN were utilized to construct a graph based on a circuit schematic, where each device (transistor, resistor, capacitor, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=') can be mapped into different node and edge type within the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The model target is to predict net capacitance, which was achieved by mapping con- nections as nodes with corresponding node information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' net capacitance), preventing information loss if nets were represented as graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' To complete the struc- ture, circuits were represented as multi-graphs, where graphs have two edges with opposing directions, and are mapped between every net node and the appropri- ate device nodes corresponding to terminal connections within the schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Despite leveraging heterogeneous GNNs to differentiate between circuit elements nodes and netlist nodes, this representation works around the circuit connection type problem (series or parallel) in the netlist nodes by assigning four types of connection signal (Net to transistor gate, transistor gate to net, Net to transistor drain, and transistor drain to net), resulting in an over complicated representation that extensively require more time at training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Physically, connections in series share the same current and connections in parallel share the same voltage, which are not shown in multi- graph heterogeneous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the area of analog circuit layout automation, [52] showed a GNN based model that can identify symmetry constraints in analog circuits That can be extended to other pairwise constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, the graph representation of circuits is simplistic as it treats device instances and device pins as graph nodes, while edges represents connections between pin nodes of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Eventually, this simplistic representation creates a problem of isomorphic graphs, which was mitigated by adding an additional a two-dimensional vector to node feature to distinguish between whether a node is a device or a pin, which eventually increases computational cost at training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Followed by [53] in which circuits was represented as heterogeneous multi-graphs to the purpose of modelling active and passive elements for analog and mixed signal circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In this representation, four types of edges (To transistor (drain), To transistor (source), To transistor (gate), To passive device) are used to represent connections between device/circuit elements, which were represented as nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit representation in previous research can be summarized as: • All methods for circuit to graph representation are arbitrary, without any mathematical/scientific base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' These methods disregards mapping the connection type and hence is substituted by a significant in- crease in the number of hidden layers, number of neurons, training for many epochs, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Other implications of disregarding connection type in previous methods are the limited scope of the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Previous methods cannot be applied to any circuit except what it is intended for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • All methodologies had deficiency in modelling Table III – Review of circuit representation in previous research Node Features Edge Features Circuit Representation Task Network type Circuit components Connections (Series/Parallel) DC operating points, One hot encoding of simulation step, Transistor parameters, Internal capacitances Featureless Every circuit element is represented as node , where node features define the element type and DC operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' No indication was given on connection representation, or its effect on analog circuit performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Learning design policy for selecting optimal circuit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' RNN+RL [46] One hot encoding of element type Circuit order, Passive and active characteristics Featureless GCN+RL [47] [54] Gate logic level, Controllability, Observability Featureless Limited circuit representation in the form of connected nodes according to the physical connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Determine whether an observation point should be added on the output port or not Meta path + GCN Subcircuit coordinates, Center position of the Subcircuit, Angular position of the slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Position of the two subcircuits, Gap length , shift System level representation, where every subcircuit is represented as a node, while edges between two nodes represent distance between two subcircuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Electromagnetic outputs prediction based on resonators relative positions GCN [48] [49] [55] Operation type Bitwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Signal information System level representation, where every node represents a microbench operation, while edges represent signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Operation Delay Prediction for FPGA HLS GraphSAGE [50] One hot encoding of terminal type, Device parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Featureless Edges , but component terminals are represented as nodes No direct indication of connection DC output voltage prediction Deep GEN [51] gate poly length, number of fingers, number of fins, number of copies, length of resistor, Capacitors, number of copies, net N Featureless Nodes No direct indication of connection Net parasitics Predictions based on physical devices parameters GraphSage, Relation GCN and Graph Attention Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [52] One hot encoding (Device/Pin) Path based feature Featureless Nodes represent component terminals and pins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Components can have multiple nodes representing Pins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pin/Components are distinguished by node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Power/GND are represented as I/O nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' No direct indication of connection Binary Classification of layout symmetry GCN [53] Node type, Geometry, layer Featureless Devices and circuit elements No direct indication of connection Binary Classification of layout symmetry Gated Recurrent Unit based GNN [56] Device type, Functional Module, Current mirror, Differential pair, Active load, Device dimension, Device location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Horizontal and vertical distance between pins Pin metal layer, Pin length, Pin type Nodes with different types No direct indication of connection Prediction of IC placement impact on circuit performance GAT + Pooling (PEA) Proposed Element ID, Normalized Component Values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' One for continuous Circuits, Duty Cycle for switching circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Nodes with different types one and zero nodes for every branch/voltage node Different circuit topologies based ML tasks (Classifier, Regression, Clustering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' GCN + Pooling common circuit properties like frequency, phase shift, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Most methodologies mention only elements of in- terest (Transistors and capacitors), but ignores other circuit parameters like inductance, resistance, volt- age source, current source, transformers, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Some methodologies try to simulate the connection type by adding component terminals as nodes and define the circuit as a multi-graph heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Despite the added complexity and extensive computational cost of heterogeneous graphs, This representation suffers a major disadvantage as dif- ferent circuit topologies can have the same graph representations (isomorphic graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This problem is usually addressed by defining another node fea- ture the define whether a node is a pin or a device at the expense of added computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Some representations omits voltage and current ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='V ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Circuit Element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Bondgraph Equivalent Element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Voltage Source (V) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Effort Source (Se) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Current Source (I) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Flow Source (Sf) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Resistance (R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Resistance (R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Inductance (L) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Inertance (I) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='capacitance (C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Compliance (C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Table IV – Circuit to bondgraph equivalent elements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='sources nodes to focus on circuit structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' How- ever, this is incorrect representation since source location can change the circuit behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Some methodologies include one-hot encoding of device position in circuit along with device type, which inherently means the node features vector size per node is linearly proportional to the circuit size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' PROPOSED CONVERTER CIRCUITS MODELING FOR MACHINE LEARNING APPLICATIONS In this section, the proposed formulation of a graph representation of continuous or switching circuits that allow the application of ML algorithms to circuit de- sign and control will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This formulation is completed in several steps: 1) Bond graph modeling of circuit topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2) Generating standardized datasets that capture circuit topology, input and output circuit variables and operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3) Defining a scalable and permutation invariant NN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Graph Creation Using Bond Graph Modeling This section explains how to model electric circuit as a graph for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1) Continuous circuit presentation as Bond Graph An electrical circuit consists of five main compo- nents such as resistors, inductors, capacitors, voltage source, and current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The generalized BG ele- ments and their mathematical relations can describe any continuous circuit and perform analysis of dynamics of electrical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zero-junction is assigned for each distinct voltage node in the circuit where according to Kirchhoff’s voltage law (KVL)—the algebraic sum of all voltage drops around a closed circuit is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Additionally, one-junction is assigned for each element in the circuit, according to Kirchhoff’s current law (KCL)—the algebraic sum of all electrical currents entering and leaving a node is equal to zero), taking into consideration the relative voltage or drops related (a) C Class 2 Se 1 0 R I C (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2 – Converter circuits to Bondgraphs: (a) Two elements circuits, (b) Three elements circuits, (c) Four elements circuits 2) Switching Circuit Representation as Bond Graph A study in [57], [58] showed that switches (unidi- to each element located between two 0-junctions, since 1-junction represents and effort summation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2 shows the bond graph models of seven classes of resonant circuits of increasing order and Table IV shows the equivalent notations used in BGs with their circuit counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' rectional or bidirectional) can be represented in BG by the concept of Switched Power Junctions (SPJ) and activated bonds and hence, BG can be used to model switching circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Other switch modelling techniques including Modulated Transformer (MTF) with Boolean modulation index m and a resistive element R or the 1 0 1 0 Class 1 Class 0 Four Elements Circuits Three Elements Circuits Two Elements Circuits Class 6 Class 5 Class 4 Class 3 Ideal Switch Element method where switch state depends on the junction to which the switch element is connected, an energetic connection is established or broken [59], [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A comparative study in [61] shows that the most convenient method is the SPJ Modelling method as it does not lead to causality conflicts and leads to a unified model, like the Modulated Transformer method, but does not require additional elements (R) to eliminate algebraic loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In this paper, the SPJ method will D C S2 B S1 D D D A C Sf = 0 Sf = 0 D 1s 7 B S2 A be used to represent switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Converter topology and its function are defined by the location of the energy storage/resonance elements (L & C) and the type and order of the switching cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Simplification of Single Pole Double Throw switching cell can be in the form of two Single Pole Single Throw (SPST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Every SPST is modelled as a 1s-junction with two flow decider bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' For the sake of completion, the physical interpretation of current interruption when the SPST switch is OFF is represented when one flow decider bond is modelled as the zero current source (Sf) and the other flow decider bond is connected to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The current source has a zero value, indicating that current falls to zero when Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3 – Switching cell and equivalent BG formulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 4 – Circuit with equivalent BG formulation switch is OFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' D and D¯ are the control signals that control the junction flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This is uniformly analogous to the duty cycle (D) physical concept in converter circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Based on [57], [58], SPST switches combinations can be modelled using (0s and 1s) junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3 shows a switching cell represented as two SPST switches and its equivalent bond graph representation, the flow decider bond and the zero value flow sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Additionally, switched power junctions are a generalisation of the already existing zero and one-junction concepts of the bond graph element set [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Thus, the traditional zero and one-junctions are special cases of the more general switched power zero and switched power one-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' When converters operates in DCM, the inductor current reaches zero before switching cycle is over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This paper utilizes the virtual switch concept to represent converter operation in DCM mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' As the inductor current reaches zero, both switches S1 and S2 are in OFF state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This virtual switch only closes when both switches become OFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' D1, D2, D3 are mutually exclusive control signal to control switches operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The concept of virtual switch presented in [62] is used to express the converter operation in DCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This representation is based on the fact that inductor current reaches zero in DCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The virtual switch shorts the inductor ensuring no current passes through, while connecting certain circuit nodes to maintain voltage balance equations during the DCM time period D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This representation compatible with the predefined physical property namely Scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuits to Graph Representation The second step is to convert the BG formulation to a graph representation containing all gathered and simulated information including circuit types, classes, nodes, edges, node and edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 4 shows a continuous circuit represented as graph following BG formulation, with minor changes in Switching circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Nodes are used to represent circuit element as well as zero and one junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Edges are used to describe circuit connection between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Node and edge features de- scribe operating condition of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In continuous circuits, edge features are set as one describing 100% connection between designated nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The same notation is used for switching circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Node features are used to describe element type as well as the element value placed in circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Some switching circuit properties require special consideration and explained as: a) Duty Cycle Representation The duty cycle is a property in every switching circuit and physically represent the percentage of the connection existence within switching cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Duty cycle is mapped as a feature of the edges the connects to switching nodes (0s & 1s nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' b) Switching Frequency Representation The one/zero switching junctions representing switch- ing cell are connected to zero-valued current source, interrupting the switch current with frequency equal to switching frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In other words, the zero-valued current source works as a control source for every switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Based on the physical properties of the control source, including the switching frequency as a property of the BG control source aligns with the physical properties of VC = DVA + DVB 0s D D D 1s S1 CircuitElementNode Features EdgeFeatures CS Connection NodeFeaturesthe circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' c) Switching Pattern Representation A generalized switching pattern representation is pro- posed, allowing all types of switching patterns and duty cycle variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This adds more flexibility to represent converters that operate differently when subjected to different switching patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e resonant converters oper- ating with different control modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The switching pattern representation is expressed in the control source (flow source in BG representation) node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 6 shows two cases of switching patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the first case, the switching is aligned so that the first switching operation compliments the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The current source node Circuit Element V → I → C → R → L → 0 → 1 → Table VI – Feature matrix assignment Concatenated Feature Matrix features should indicate the same phase shift reference, and by default is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the second case, where switch operations are not aligned either at turn on or turn off, a phase shift φ indicates that delay, and is set the con- trol source of the delayed switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Combining the phase shift information along with duty cycle information, allows complete representation of the switching patterns in switch operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Table V summarizes the switching pattern modes and their node feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Table V – General representation of all possible switching patterns as node features Representation Case 1 *Phase shift is set to φ=0 *Edge Features represents duty cycle *Switches which are controlled dependently are represented with the same phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Case 2 *φ Is the phase shift *Delayed switch include phase shift as node feature C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Dataset Generation Generating a dataset of different circuit topologies, circuit elements and circuit order is shown in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Also, a proposed technique for storing recorded data in a general format for any ML task is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 7 shows a paradigm for such dataset generation step, where a circuit netlist is converted to its equivalent bond graph model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Since BG is a graph notation for modeling circuits, they inherently have all graph characteristics, with all requirements of graph definitions like number of nodes, node types, edge weights and the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Finally, BGs are passed to feature assignment algorithm, where features are assigned to each node in graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1) Feature Assignment Node features are defined based on circuit element type and its behavior in circuit using the proposed algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit simulations are used to obtaining features describing circuit performance such as node voltages and loop currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Simulations run for multiple instances at multiple operating points for all circuits including dif- ferent component values and circuit conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Output values are normalized to common base to avoid sparsity of the feature vector, which is referred in Table VI as ”Normalized Values Vector”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The proposed feature assignment algorithm is expandable and can include many circuit features if it is desired to be included in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Therefore, the normalized values vector can be multiple columns listing not only component’s value, but also different component properties i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e source frequency in continuous circuits or phase shift in switching circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' One main function of feature extraction algorithm is to define the circuit element types, which are defines the concept of Element ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Element ID assigns a binary code based on circuit element type by utilizing one- hot encoding [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The second main function of feature assignment algorithm is to concatenate the assigned one- hot encoded vector with normalized values vector, forming the feature matrix of the whole graph with dimension N × din, where N is number of nodes and din is the dimension of feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2) Dataset Format Extracted features and other graph information like types and number of node, adjacency matrix and edge features are saved in a unique graph dataframe for- mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='This unique dataset format features independent graph dataset of circuits, which allows using this graph representation in any ML library independent of saved graph dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Since there are many graph ML libraries like pytorch-Geometric [64], DGL [65], Keras [66] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='. etc, the final step in the algorithm is to process the dataset to be in a compatible format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pytorch-Geometric GNN library was chosen to build the GNN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Different Circuit Examples Using Proposed Method- ology This section shows some examples from different areas where the proposed methodology is applicable to many ML applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1) Example 1: Power System Power systems (PS) area have a lot of research where ML methodologies has bee applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Recently GNN has been on the spotlight for application in PS, and xx xx xx xx xx xx xx Normalized Value Element ID V I L R C 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 Conveter Circuits Equivalent CCM Bond Graphs Equivalent DCM Bond Graphs R Buck R Boost R Buck Boost Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 5 – Buck, boost and Buck-Boost converters and their equivalent BondGraphs in CCM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 6 – Switching pattern representation as features many publications utilizing GNN in power systems have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A comprehensive overview of GNN appli- cations such as fault scenario application, time series prediction, power flow calculation, and data generation are reviewed in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In [68], [69] the provided network learns to solve load flow problem on random power grids whose size range from 10 to 110 buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A method to identify the topology of a PS network is proposed in [70] based on GNN, avoiding errors in Traditional knowledge graphs in the case of errors or informational conflicts in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' All previously mentioned research empirically transform the PS network into graph without following a circuit-laws-consistent formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 9a shows a PS network example and its graph equivalent with node features, following the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2) Example 2: Two-Stage Amplifier Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 9b shows a two-stage amplifier that was used in [10] as a circuit layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The equivalent graph represen- tation proposed in this work was arbitrarily transformed into a graph by representing every transistor, resistor and capacitor as nodes connected to each other by edges, disregarding the original connection or the physi- cal/electrical consequences of such connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' also shows the proposed graph representation includes component and connection nodes, in addition to node features for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Graph Convolution Network NN have many variants like GCN [71], GraphSage [72], Gated Convolution [73], Transformer convolution [74] and many more, but the most common is GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' GCN was chosen for the following reasons: • Unique ability to extract latent information from graph data compared to other GNN structures as reported in [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Most practical circuit GNN based applications in Table III utilize GCN as their main network model or a part of the model, hence the results from this study can be fairly compared to previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sf Sf C Se 1 0 1 0 Sf 1 Virtual Switch I R Se GND Class 5 Sf 1 Virtual Switch Sf C Se 0 1 0 1 0 I Sf 1 R Se GND Class 4 Sf Sf C 1 Virtual Switch Se 1 0 0 1 Sf 1 R I Se GND Class 3 Sf Sf C Se 1 0 1 0 I R Class 2 I I Sf C Se 1 0 1 0 Sf 1 R Se (GND) Class 1 Sf I I C Se 1 0 1 0 Sf 1 R Se (GND) Class 0 S1 S2 L C L S2 S1 C S1 S2 L C D3 Di D2Di D2DSwitching Circuits Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 7 – From circuit to ML Block diagram (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 8 – Equivalent graph with node and edge features in: a) LCC Continuous circuits, b) Buck converter switching circuit Simple construction and implementation, which can be beneficial if implemented as digital twin on a microcontroller [76] The selection of GCN as the engine for the proposed GNN has allowed better focus on other hyperparame ters and eventually led to better circuit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' GCNs obtain updated features by inspecting neighboring nodes, and aggregating current node information to other neighbours through message passing process then updat ing the node state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Eventually, all the nodes in graph obtain knowledge about self and surrounding neighbor information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 10 shows three layer message passing applied to a single node (node of type 1) of class 1 circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A deeper level of neighbor nodes exploration and better awareness of self node position can be gained by adding an additional GCN layer, at the expense of additional computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Three layer GCN network is utilized in this paper as a mid point between exploration depth and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Node features are repetitively aggregated through the GCN layers via multiple message passing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' At the end of this process, the final node embeddings contain self and all neighbor information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Mathematically, this initial embedding function is rep resented by equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The aggregation layer has mul tiple Graph Convolution Networks (GCN) that performs multiple message passing leaps to collect information about neighbouring nodes and keeps updating the latent dimensional 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' VgndF Circuit Element ID Vall Val2 Element (One hot encoding) Se 1 0 0 0 0 0 xX xX Sf> 0 1 0 0 0 F 0 0 0 0 xX xX 0 0 0 1 0 0 0 xX xX R 0 0 0 0 1 0 0 XX XX 1> 0 0 0 0 1 0 XX XX 10 001000 10 xX XXNxd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' RNxds → R1xd Da R1xd → RIxC RI×C →RIxC2 2 VDD T7 0 VBp Vx Cx 0 1 Tx Rx 1 0 1 0 1 C T4 R 0 1 0 T6 1 1 T3 0 Vin+ T5 T2 0 0 Vin 1 T1 0 VBN Third Message Passing Layer Second Message Passing Layer First Message Passing Layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 10 – Rooted subtree showing message passing applied to node of of type 1 in the circuit of class 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2 with three GCN layers X(l+1) = σ(Dˆ − 1 AˆDˆ − 1 XlΘl) (2) (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 9 – Examples of proposed concept in different applications: a) Power system example b) 65 nm 2 stage amplifier example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [10] where Θl is a weight matrix for the l-th neural network layer and σ is a non-linear activation function like the ReLU, Aˆ= A + I, where I is the identity matrix and Dˆ is the diagonal node degree matrix of Aˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This allows the GCN to scale well, because the number of parameters in the model is not tied to the size of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' GCN Time complexity and Graph Scalability Limit Generally speaking, there are no limitation on the size of the circuit fed to the ML model (theoretically, the circuit order can be infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, the computation time and RAM consumption are the main concerns when feeding circuit graphs to model, which mainly depends on how the model was built, the libraries used to build the model (pytorch or keras or tensorflow .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='etc), the layers depth, operating system used, the model architecture and the output size, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' From a GNN designer prospective, Graph circuit for a GNN input can be represented in two ways: • sparse: As a list of nodes and a list of edge indices • dense: As a list of nodes and an adjacency matrix For any graph G with N vertices of feature vector length F and E edges, the sparse version will operate on the nodes of size N*F and a list of edge indices of size 2*E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The dense representation in contrast will require an adjacency matrix of size N*N, with node degree of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The choice of dense or sparse representation not only affects the memory usage, but also the calculation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Dense and sparse graph tensors require graph convolutions that operate on dense or sparse inputs (or alternatively as seen in some implementations convert between sparse and dense inside the network layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sparse graph tensors would operate on sparse con- volutions that use sparse operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Generally, dense computations would be more expensive but faster than sparse, because sparse graphs would require processing of operations in the shape of a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' For simplicity, we assume the node features at every layer are size- F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' As such, Θl is an F × F matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The time complexity of the convolution operation can be decomposed as: I GCN I 1 GCN Se 0 1 GCN Se GCN C GCN 0 0 GCN R GCN 0 L2 L1 L3 C1 G1 G2 C1 G1 1 0 1 G2 L1 L2 L3 Voc HT R T = C 65nm Two-stage Amplifier 0, 0, 0, 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Il O, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Xa 0, 0, 1, 0, 0, 0, R[0, 0, 0, 1, 0, 0, Ca] [0, 0, 0, 0, 1, 0, Ea [0, 0, 1, 0, 0, 0, La] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 0, 0, 0, 0, VanormalizedFrequency Frequency×C C normalizedFrequency resonanceFrequency normalizedFrequency C L L • Equation (1) : which is a dense matrix multiplica- tion between matrices of size N ×Fl and Fl ×Fl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' We a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='ssumeΣfor all l, Fl = Fl+1 = F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Therefore, this is O NF 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Equation (2): which is a multiplication between ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='trices Σof size N × N and N × F , yielding O N 2F time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hence, the neighbor- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 12 is the second set of experiments, where edge weights were set as the normalized frequency, while nodes that represents capacitive elements were set to have ( 1 ) as edge feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Another experiment is to test whether negative component values would increase the accuracy, or setting the capacitive components as 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' These experiments are reflection from hood aggregation for each node therefore requires C circuit analysis as Xc = −j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' However, the O(dF) work, with a total of O(NdF) = O(EF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' σ(·) : is the activation function which is an element- wise function, so its cost is O(N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Over L layers, this results in computational time results shows that negative capacitive element value and its edge feature as ( 1 ) have negative effect on the accuracy of the classifier, while setting capacitive elements as of inverted value( 1 ) had a com .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' plexity of: Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='LNF 2 + LNdF + LN = O LNF 2 + LNdF Σ significant training accuracy boost to 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' It is imper- = ative to modify node features expression for capacitive O LNF 2+ LEF ) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Optimal Node And Edge Features Exploration To determine the optimal representation of circuit component values, twelve experiments were performed on the continuous circuits of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2 and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 11 - Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The dataset contained 6000 graphs representing the seven circuit types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 70% of the dataset was used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='The data is shuffled before being applied to the model, and there was no mutual data between training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Cross entropy loss function is used in training the model with Adam optimizer [77] with learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Twelve experiments were conducted in order to obtain conclusions and a paradigm of how the node and edge features should represent the circuit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' These experiments were divided into four sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Each set contains three experi- ments and a conclusion based on observations from these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The conditions/modifications applied on the dataset when fed to the classifier are listed on the left of each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The purpose of these experiments is to identify the effect of different component representations, and how would that affect the ML task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Figures also show the classifier problem evolution ranging from three class to seven class classifier problem, along with physical circuit elements representation as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The purpose of the upcoming experiments is to ex- plore the highest impact features on task accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' How- ever, since features are hyper-parameters, some result obtained from edge features may eventually update how the node features are expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the first set of experiments shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 11, edge features are explored and the problem is limited to three classes classifier, edge weights are separately tested as normalized frequency ( circuitF requency ) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' being set as ones, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' being the circuit frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This experiment is concluded with the highest accuracy achieved is when edge weights were set as normalized frequency and as ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' As frequency can be included as edge features, it can be tested if capacitive elements can to be expressed as ( 1 ), which is the purpose of the second experiment set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Eventually, circuit graph dataset was modified to include this change in the the third experiment set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Also, from the first experiment set, edge features set as one had the highest accuracy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The next experiment aims to explore if the concluded node and edge features modifications can enhance the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the third experiment set, the highest accuracy of 100% was achieved in training and testing when edge weights were set to ones and capacitive elements has node feature values of ( 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The first experiment tested whether edge feature can be used as a scaling factor substituted by the node feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The second one tested whether edge weights can be set to one, while the third experiment tested if inductive elements can be set as( 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' From results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 13, it can be concluded that utilizing edge features for scaling deteriorates the classification accuracy as well as representing inductive elements as( 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The optimal edge feature can be defined to be one, without embedding any circuit characteristics or parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In the last set of experiments in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 14, all outcomes and recommendations that was concluded from previous experiments were taken into consideration, while in- creasing the classification problem difficulty to four, five and seven classes classification problem to further verify the optimal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In a four-classes problem, the classifier scored a training accuracy of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='3%, while in five-classes problem the training accuracy score was 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='92%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lastly, the seven-classes problem resulted in training accuracy score of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='37%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The discrepancy of accuracy scores while using the same feature represen- tation is due to the change in dataset number of circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The result is a graph of a circuit with connection nodes and element nodes each has its own features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Nodes are connected by edges having edge features of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' CASE STUDY As a proof of concept, the proposed approach is applied to map two types of topologies: i) continuous circuits and ii) switching circuits, to a ML compatible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Seven resonant circuit topologies of cir- Capacitive elements have node Three Class classification Problem With Frequency As Edge Features Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6898, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6906 (a) Edge Features Set As one Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8606, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8669 (b) Normalized Frequency as edges Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='7574, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='7709 (c) Conclusion When edge weights set as normalized frequency and when they are set as ones are of the highest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 11 – First experiment set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Edge weights are set as: a) Frequency, b) value of one, c)Normalized frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Three Class classification Problem Edge weights represents the normalized frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Capacitive elements have edge weight of 1/norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='722, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='72 (a) Capacitive elements has node feature value of 1/C Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9112, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9175 (b) Capacitive elements has negative node feature value of 1/C Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9107, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9163 (c) Conclusion • Capacitive elements having edge weights of 1/Frequency has negative effect on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Minor decrease in accuracy when capacitors have negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Setting 1/C value has a major accuracy boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 12 – Second experiment set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' a) No change in node features, b) Capacitive element representation is 1 , c) Capacitive element representation −1 C is C Three Class classification Problem feature value of 1/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Edge weight is set to 1000 which is substituted from Node value feature Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6385, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6434 (a) Edge weights are set as one Training Accuracy: 1, Testing Accuracy: 1 (b) Edge weights are set as one Inductive elements have values of 1/L Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='7928, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='7817 (c) Conclusion • Setting Inductive elements values of 1/L deteriorates the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Scale representation as edge weights has a negative effect on the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Edge weights set as ones significantly increase the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 13 – Third experiment set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Edge weights are set as: a) Scaling factor, b) value of one, c) value of one but different inductive element representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' cuit orders ranging from second to fourth order as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2, and three switching circuit topologies in CCM and DCM shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3 are fed to a classifier to show the applicability of the proposed methodology to any ML task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Following the sequence illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 7 and same steps presented in this paper and in [1] and [2], converters are converted to graph form and computer simulations are used to assign normalized node features of the generated graph according to section VI-C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Steady state simulations are run for multiple instances at multiple operating points for all circuits including differ- ent component values and circuit conditions and circuit behavior is recorded and stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The circuit simulation sampling rate is a measure of the accuracy of the circuit simulations in the continuous circuit classifier case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In this case study, a dataset of 6000 graphs with 6000 steady state simulations have been normalized to a common base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This helps to ensure that each feature vector is consistent and not overly sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The normalized values vector is then used to provide a representation of the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 Traing Dataset Testing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8 ccuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6 20.' metadata={'source': 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+page_content='0 + 0 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 Traing Dataset Testing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 + 0 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 Traing Dataset Testing Dataset 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 Traing Dataset Testing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0+ 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 + 0 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 8°0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 Traing Dataset Testing Dataset +00 0 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 TraingDataset Testing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 0 200 400 600 800 1000 1200 Epoch Multi Class classification problem Edges has weight of One Capacitive elements have node feature value of 1/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Four-Class classification problem Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9230, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9246 (a) Five-Class classification problem Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9592, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9611 (b) Seven-Class classification problem Training Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9737, Testing Accuracy: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='9710 (c) Final Conclusion Highest classification accuracy obtained when: • Edge weights are to be set as ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Capacitive elements have node feature value of 1/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 14 – Fourth experiment set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' a) Four-class, b) Five-class, c) Seven-class classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 15 – Circuit classifier structure [1] circuit simulation data that is accurate and reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' To ensure that the sampling rate is accurate, the graphs are divided into a number of subsets based on circuit class, and each subset is simulated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Each of these subsets is tested for accuracy, and any discrepancies are noted and addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' After all the subsets have been tested and corrected, the overall sampling rate of the circuit simulations can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Once the sampling rate has been determined, the normalized values vector is concatunated with element ID to complement the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 15 shows a block diagram of the classifier structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Three GCN layers are used to get information about 3rd level neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The classifier output layer computes a probability score for the class of each topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1) Classifier Problem Formulation Circuit topologies in graph forms (G) are fed to the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Each circuit graph has number of nodes (N) along with their corresponding node features (X) each has dimension (din).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The adjacency matrix (A) defines connections between each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The classifier outputs a probability (Y) of a converter to belong to a certain class (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sub-GCN networks are embedded in each GCN layer, allowing aggregation processes between feature vectors in the neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hyperbolic tangent (“tanh”) is used as the non-linear activation function, while being slower than the Rectified Linear Unit (ReLU) activation function, it helps to avoid the dying ReLU problem due to the very different values of both inputs and outputs [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The global mean readout (GM-Read out) layer returns graph level outputs by aver- aging GCN processed node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A fully Connected (FC) linear layer is a score function for each circuit, while (Softmax) output layer is used to calculate the probability, in range of [0-1], of each circuit belonging to a certain class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The Softmax function formula σ() is stated in equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The classifier uses training datasets and updates weights or GCN layers and linear layers by minimizing the cross entropy loss function, which is shown in equation (11), where: • M - Number of classes log The natural log Y Binary indicator (0 or 1) if class label c is the correct classification for observation O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' p Predicted probability observation O is of class C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A mathematical formulation of the transformations of the designed classifier is stated as: Y = classifier(X, A) (3) Where 3 X GCN layers GCN Global Mean Pooling Linear Transformation Softmax Graphs Representing Circuits Probability for each Class 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 Traing Dataset Testing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0+ 0 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 80 Accur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 Traing Dataset Testing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0+ 0 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 Traning Dataset Testing Dataset +00 200 400 600 800 1000 1200 Epoch000000 RNxdin → RNxd R1xd RIxC RIxc RIxC RNxdn → R1xdX ∈ R(N )×din (4) Y ∈ RC×1 (5) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' of correct predictions belonging to specific class = Total No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' of predictions belonging to that class GCN (k) : RN×din ›→ RN×d, k ∈ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Σ 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='., k − 1 TP (12) (6) GM − Readout : RN×d ›→ R1×d (7) FC : R1×d ›→ R1×C (8) Softmax = R1×C ›→ R1×C (9) Recall = TP + FN No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' of correct predictions belonging to specific class = Total No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' of correct predictions in the dataset (13) 2 × Precision × Recall where F 1score = Precision + Recall (14) ezi σ(zi) = ΣK j=1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' , K (10) ezj Σ M Class two and three have F1 scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='87 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='89, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Since F1 score embeds precision and re- call into one computation, the weighted average of F1 should be used to compare classifier models, not global CrossEntropy = − 2) Results and Analysis c=1 yo,c log(po,c) (11) accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The Recall of class 2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='77, indicating a misclassification occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' On the other hand, the Recall score of class three is 1, indicating all class 3 circuits a) Continuous Circuit Classifier Training and testing accuracy after 1200 epochs are shown in Fig 16, scoring 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='37% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='10 %, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 70% of the dataset containing 6000 graphs rep- resenting the seven circuit classes was used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Cross entropy loss function is used in training the model with Adam optimizer with learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 17 shows the 2-D embedding of the classifier testing dataset output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' It can be clearly seen that graphs falling in the same class cluster together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The confusion matrix shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 18 is used to analyze the classifier behavior and obtain insights about its functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The array gives an insight about overlaps/errors in class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Other classifier as- sessment metrics are listed in Table VII, which shows the precision, recall, F1 and support metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The fol- lowing notations are used to assess binary classifiers performance, but are also extended to multi-classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Positive: The graph is classified as a member of the circuit class the classifier is trying to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • Negative: The instance is classified as not being a member of the class we are trying to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' True or false can be added to Positive or negative to indicate whether the classifier has correctly predicted the class or misclassified it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Generally, precision is a measure of true positive instances, which shows how many of the positive predictions made are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Recall aka sensitivity, is a measure of how many of the positive cases the classifier correctly predicted with respect to the over all the positive cases in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The F1 score is the percentage of correct class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A mathematical formulation of the evaluation metrics are listed in equations (13-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' TP Precision = TP + FP were correctly classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This analysis indicates misclas- sification of 52 class 2 circuit graphs as class 3, resulting in a precision measure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='Additional observations from confusion matrix, classifier metrics and the 2-D vector mapping can be summarized as follows: • Circuits with similar connections are distinctly clas- sified but the clusters appear close in the 2-D vector mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Classes (four and six) are fourth order circuits but are dissimilar in physical connection, hence are mapped in the same vicinity but close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Similarly are classes (Zero and one), follow the same principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' On the other hand, classes (two and three) are second order circuits sharing almost identical circuit connection, hence are mapped very close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The same concept is applied to circuits with dis- similar circuit structures, as they are clustered far from each other in the 2-D map i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e classes 0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' • The similarity between classes two and three in con- nection and number of nodes causes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='63% classifi- cation inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Further tuning of the weights of the linear layer can improve the classifier selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' b) Switching Circuit Classifier The trained classifier scored 100% for training and testing data, when trained for 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 20, a 2-D output representation of 1800 test dataset graphs are plotted and colorized according to their predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuits of the same topology are distinctly identified and clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Further, the operating mode of each of the circuits (CCM or DCM) is also identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The different loci of the 2D plot from every class is a result of convolution operation taking all graph properties representing circuits like component values, type and switches duty cycle and converting it to a lower dimension (2-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' It is also noted that graphs of the same converter topology form groups and cluster in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 16 – Circuit classifier accuracy (a) 3 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 17 – 2-D embeddings of circuit graphs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 18 – Confusion matrix for seven classes circuits 1 0 1 2 3 X (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 20 – a) training and testing data classification accuracy, b) 2D embedding of the three converters in CCM and DCM after classifica- tion 1000 800 600 400 200 Table VII – Continuous circuits classifier assessment metrics Buck CCM Boost CCM Buck Boost CCM Buck DCM Boost DCM 0 Buck-Boost DCM VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' DISCUSSION AND FUTURE WORK Predicted Class Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 19 – Confusion matrix for DC-DC converters close proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This methodology of circuit representation allows incorporating ML techniques in many applications, and can serve the purpose of generating application-specific circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Machine learning and neural network models 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 981 0 0 0 0 0 1013 0 0 0 0 0 0 0 0 0 0 1031 1027 1015 1008 Classes Buck Boost Buck-Boost Buck-DCM Boost-DCM BuckBoost-DCM Buck Converter Boost Converter Buck-Boost Converter 4 2 0 2 4 True Class Buck-Boost DCM Boost DCM Buck DCM Buck-Boost CCM Boost CCM Buck CCM Y Circuit Class Precesion Recall F1 score Support Class 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 310 Class 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 247 Class 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='87 226 Class 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='89 213 Class 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 225 Class 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 289 Class 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 282 Macro avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='97 1792 Weighted avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='97 1792 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='97 1792 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 TraningDataset Testing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 0 200 400 600 800 1000 1200 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2 Traning Dataset Testing D Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 0 25 50 75 100 125 150 175 200 EpochClass0 310 0 0 0 0 0 0 247 0 0 0 0 250 0 174 52 0 0 200 0 0 0 213 0 0 0 150 0 225 100 0 0 0 289 0 50 0 0 0 0 282 - 0 Class0 Class 1 Class2 Class 3 Class 4 Class 5 Class 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='5ClassesTable VIII – DC-DC converters classifier assessment metrics estimation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' CONCLUSION in general are heavily dependent on hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Several aspects are to be included when circuit designer incorporate ML model in circuit design like network depth, number of neuron, activation functions, pooling layers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' These uncertainties in ML models adds more burden when incorporating ML techniques in circuit design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Eventually, a network update is a must at some point of the design process, and eventually designer must fine tweak the ML based design tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The proposed method can be applied to a wide range of applications such as, power electronic converters con- dition monitoring and prognostics, since the developed representation maps the circuit structure and thus voltage stresses at each node and current stresses in each branch can be evaluated and tied to a component/converter reliability function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Another application is network struc- ture and fault detection in large power systems [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Circuit design is another application that fits the pro- posed methodology, where circuit performance parame- ters are set, and the GNN model can generate a circuit topology that meets the input criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Moreover, this study can be further developed to for the purpose of linking finite element modelling software in AI assisted design of magnetic components for the purpose of optimal component values/shape design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Additionally, the proposed methodology has very high potential in circuit obfuscation and reverse engineering when it is required to identify/obscure circuit structure [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' One idea works on the circuit side utilizing the GNN capabil- ity of learning the proper transformation function of the converter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='e can obtain a mathematical transformation of every circuit component and eventually all circuit behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' On the application side, the end goals whether they are gain, current ripples, magnetic design .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='. etc, are transformed into a fictitious statistical domain, and the purpose of the GNN is to generate circuits with similar statistical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This can be beneficial to train AI to generate application specific converters, which eventually will help reduce component size, increase power density, speed and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This methodology is also applicable in power system applications such as network reconstruction and fault detection and load flow In this paper a graph representation of electric circuits is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' This method enables a dynamically scalable interface of different circuit aspects including physical connections, component values and mode of operation, to the machine learning domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Applying the circuit graphs as inputs to a GNN different circuit modeling, design and optimization tasks can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' The effect of bond graph feature selection, scaling and for- mulation was also analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Optimal feature represen- tation results in a more well defined feature matrix and consequently a more accurate circuit and operating mode identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' As a proof of concept case studies of classifiers of continuous and switching circuits were presented where, the proposed algorithms were proven to distinctly identify with high accuracy circuit types based on physical connectivity as well as identifying their mode of operation based on parameter values and control variable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Khamis and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Agamy, “Mapping continuous circuit struc- tures to machine learning space,” in 2022 IEEE 31st International Symposium on Industrial Electronics (ISIE), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 149–155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Khamis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=', “Converter circuits to machine learning: Optimal feature selection,” in 2022 IEEE Energy Conversion Congress and Exposition (ECCE), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 9–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Takamiya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Miyazaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Obara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wada, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sakurai, “Power electronics 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='0: Iot-connected and al- controlled power electronics operating optimally for each user,” in 2017 29th International Symposium on Power Semiconductor Devices and IC’s (ISPSD), 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 29–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [4] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lin, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Blaabjerg, “Artificial-intelligence- based design (ai-d) for circuit parameters of power converters,” IEEE Transactions on Industrial Electronics, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhao and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, “Enabling data-driven condition moni- toring of power electronic systems with artificial intelligence: Concepts, tools, and developments,” IEEE Power Electronics Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 18–27, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Peyghami, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Dragicevic, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Blaabjerg, “Intelligent long- term performance analysis in power electronics systems,” Scien- tific Reports, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1–18, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Blaabjerg, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, “An overview of artificial in- telligence applications for power electronics,” IEEE Transactions on Power Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 4633–4658, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bose, “Neural network applications in power electronics and motor drives—an introduction and perspective,” IEEE Trans- actions on Industrial Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 14–33, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' He, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Katabi, “Circuit-GNN: Graph neural networks for distributed circuit design,” in Proceedings of the 36th International Conference on Machine Learning, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pro- ceedings of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' PMLR, 09–15 Jun 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 7364–7373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lee, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Han, “Learning to design circuits,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' abs/1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='02734, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='02734 [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lee, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Han, “Gcn-rl circuit designer: Transferable transistor sizing with graph neural networks and reinforcement learning,” in Proceedings of the 57th ACM/EDAC/IEEE Design Automation Conference, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' DAC ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' IEEE Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fayyazi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shababi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Nuzzo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Nazarian, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pedram, “Deep learning-based circuit recognition using sparse mapping Circuit Class Precesion Recall F1 score Support Buck-CCM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': 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1013 Buck-Boost-DCM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1031 Macro avg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 6075 Weighted avg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 6075 Accuracy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00 6075 and level-dependent decaying sum circuit representations,” in 2019 Design, Automation Test in Europe Conference Exhibition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 638–641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, “Power converters topological transformation using dual and isomorphic principles,” IEEE Open Journal of Power Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 74–87, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Tang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sun, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pan, “Towards decrypting the art of analog layout: Placement quality prediction via transfer learning,” in 2020 Design, Automation Test in Europe Conference Exhibition (DATE), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 496– 501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Breedveld, “Multibond graph elements in physical systems theory,” Journal of the Franklin Institute, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 319, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1–36, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [16] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Harary, “Graph theory and electric networks,” IRE Transac- tions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 95–109, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [17] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Han, “Generalized method of analysis of simultaneous faults in electric power system,” IEEE Transactions on Power Apparatus and Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' PAS-101, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3933–3942, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kuprat, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liserre, “Graph-theory-based derivation, modeling and control of power converter systems,” IEEE journal of emerging and selected topics in power electron- ics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1–1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [19] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fitzgerald, Graph theory with applications to engineering and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Univ Sci Books, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [20] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Tran and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Canudas de Wit, “Modeling and control of power electronic converters with graph theory and hybrid automata,” IEEE Transactions on Control Systems Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 285–296, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Diaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Mises, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kevork, “Modelling of power electronic converters in graph theory,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 128 285–128 297, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Patel, “Anomalous ringing in circuits and its cancellation,” IET Circuits, Devices & Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 33–39, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kundur, Power system stability and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' McGraw-Hill, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sundar, “Admittance matrix based z-bus formulation for large-scale power systems,” 2018 International Conference on Applied and Theoretical Computing and Communication Tech- nology (iCATccT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1–6, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kettner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Paolone, “On the properties of the compound nodal admittance matrix of polyphase power systems,” IEEE Transactions on Power Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 444– 453, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Saadat, Power system analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' McGraw-Hill Professional, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Nasir, Power electronic converters: modeling, simulation, and control using MATLAB/SIMULINK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' CRC Press, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shuai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hong, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shen, “Stability analysis and location optimization method for multiconverter power systems based on nodal admittance ma- trix,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 529–538, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [29] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gebben, “Bond Graph Bibliography for 1961–1976,” Journal of Dynamic Systems, Measurement, and Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 99, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 143–145, 06 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='1115/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='3427087 [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gawthrop and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bevan, “Bond-graph modeling,” IEEE Control Systems Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 24–45, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Baqqal and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' hammoumi, “A generic approach of mod- elling perturbation of mechanical systems: Concept with bond graph,” in 2018 IEEE International Conference on Technology Management, Operations and Decisions (ICTMOD), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 219–223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sirivadhna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Richards, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Anderson, “The application of bond graphs to electrical machinery and power engineering,” IEEE Power Engineering Review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' PER-3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 35–35, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gawthrop and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Crampin, “Bond graph representation of chemical reaction networks,” IEEE Transactions on NanoBio- science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 449–455, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [34] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Qasim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Rashid, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hasan, “Formalization of bond graph using higher-order-logic theorem proving,” ISA Transac- tions, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 128, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 453–469, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='com/science/article/pii/S0019057821006091 [35] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Mo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Qing, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' He, “Graph theory-based programmable topology derivation of multiport dc– dc converters with reduced switches,” IEEE Transactions on Industrial Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 5745–5755, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Delgado and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Profos, “Identifiability of dynamic systems represented by bond graphs,” Mathematical and Computer Mod- elling of Dynamical Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 89–112, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [37] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sueur and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Dauphin-Tanguy, “Structural controllabil- ity/observability of linear systems represented by bond graphs,” Journal of the Franklin Institute, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 326, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 869–883, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [38] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' He, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ning, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, “Machine learning for electronic design automation: A survey,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='03357 [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bronstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bruna, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' LeCun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Szlam, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Vandergheynst, “Geometric deep learning: going beyond euclidean data,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' abs/1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='08097, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='08097 [40] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Scarselli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gori, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Tsoi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hagenbuchner, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Mon- fardini, “The graph neural network model,” IEEE Transactions on Neural Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 61–80, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [41] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Di Massa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Monfardini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sarti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Scarselli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Maggini, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gori, “A comparison between recursive neural networks and graph neural networks,” in The 2006 IEEE International Joint Conference on Neural Network Proceedings, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 778–785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bruna, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zaremba, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Szlam, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' LeCun, “Spectral networks and locally connected networks on graphs,” 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6203 [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Micheli, “Neural network for graphs: A contextual construc- tive approach,” IEEE Transactions on Neural Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 498–511, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [44] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bak-Jensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pillai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, “A review of graph neural networks and their applications in power systems,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/ 2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='10025 [45] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' He, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yu, Understanding Graphs in EDA: From Shallow to Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery, 2020, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 119–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='1145/3372780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='3378173 [46] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lee, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Han, “Learning to design circuits,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 02734 [47] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lee, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Han, “Gcn-rl circuit designer: Transferable transistor sizing with graph neural networks and reinforcement learning,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='00406 [48] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' He and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, “End-to-end learning for distributed circuit design,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [49] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' He, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Katabi, “Circuit-GNN: Graph neural networks for distributed circuit design,” in Proceedings of the 36th International Conference on Machine Learning, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Proceedings of Machine Learning Research, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Chaudhuri and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Salakhutdinov, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' PMLR, 09–15 Jun 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 7364–7373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='press/ v97/zhang19e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='html [50] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hakhamaneshi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Nassar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Phielipp, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Abbeel, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Stojanovic´, “Pretraining graph neural networks for few-shot analog circuit modeling and design,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='15913 [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ren, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kokai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Turner, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ku, “Paragraph: Layout parasitics and device parameter prediction using graph neural networks,” in Proceedings of the 57th ACM/EDAC/IEEE Design Automation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' IEEE Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [52] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Deng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lin, “Layout symmetry annotation for analog circuits with graph neural networks,” ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' ASPDAC ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery, 2021, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 152–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='1145/3394885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='3431545 [53] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Tang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sun, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pan, “Universal symmetry constraint extraction for analog and mixed-signal circuits with graph neural networks,” in 2021 58th ACM/IEEE Design Automation Conference (DAC), 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1243–1248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [54] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' He, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yu, Understanding Graphs in EDA: From Shallow to Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery, 2020, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 119–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='1145/3372780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='3378173 [55] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ustun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Deng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pal, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, “Accurate opera- tion delay prediction for fpga hls using graph neural networks,” in 2020 IEEE/ACM International Conference On Computer Aided Design (ICCAD), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [56] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Madhusudan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sharma, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sapat- nekar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Harjani, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hu, “A customized graph neural network model for guiding analog ic placement,” in 2020 IEEE/ACM International Conference On Computer Aided Design (ICCAD), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [57] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Umanand, “Switched junctions in bondgraph for modelling power electronic systems,” Journal of the Indian Institute of Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 80, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 327, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [58] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Umarikar and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Umanand, “Modelling of switching systems in bond graphs using the concept of switched power junctions,” Journal of the Franklin Institute, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 342, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 131–147, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='com/science/article/pii/S0016003204000833 [59] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Stro¨mberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Top, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' So¨derman, “Variable causality in bond graphs caused by discrete effects,” 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [60] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' So¨derman and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' erik Stromberg, “The conceptual side of mode switching,” 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [61] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Markakis, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Holderbaum, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Potter, “A comparison be- tween bond graphs switching modelling techniques implemented on a boost dc-dc converter,” in 2011 IEEE 33rd International Telecommunications Energy Conference (INTELEC), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [62] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Umarikar and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Umanand, “Modelling of switched mode power converters using bond graph,” Electric Power Applications, IEE Proceedings -, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 152, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 51 – 60, 02 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [63] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Rodr´ıguez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bautista, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gonza`lez, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Escalera, “Beyond one-hot encoding: Lower dimensional target embedding,” Image and Vision Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 75, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 21– 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' com/science/article/pii/S0262885618300623 [64] “Pyg documentation.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https:// pytorch- geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='io/ [65] “Deep graph library.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='dgl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='ai/ [66] “Keras documentation.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='io/ [67] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Liao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Bak-Jensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pillai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, “A review of graph neural networks and their applications in power systems,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/ 2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='10025 [68] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Donon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Donnot, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Guyon, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Marot, “Graph neural solver for power systems,” in 2019 International Joint Conference on Neural Networks (IJCNN), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [69] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Balthazar Donon, Re´my Cle´ment and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Schoenauer, “Neural networks for power flow: Graph neural solver,” Electric Power Systems Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 189, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 106547, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' com/science/article/pii/S0378779620303515 [70] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' An, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Mu, “Power system network topology identification based on knowledge graph and graph neural net- work,” Frontiers in Energy Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 613331, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [71] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kipf and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Welling, “Semi-supervised classification with graph convolutional networks,” 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='02907 [72] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Hamilton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ying, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Leskovec, “Inductive representation learning on large graphs,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='02216 [73] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Tarlow, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Brockschmidt, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zemel, “Gated graph sequence neural networks,” 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='05493 [74] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Feng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Sun, “Masked label prediction: Unified message passing model for semi-supervised classification,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='03509 [75] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Geng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Tan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' You, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Herbordt, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Li, “I-gcn: A graph convolutional network accelerator with runtime locality enhancement through islandization,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/ 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='03606 [76] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Banbury, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhou, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Fedorov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Navarro, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Thakker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Gope, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Reddi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Mattina, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Whatmough, “Micronets: Neural network architectures for deploying tinyml applications on commodity microcontrollers,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='11267, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='11267 [77] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [78] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lemos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Jeffrey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Cranmer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ho, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Battaglia, “Rediscovering orbital mechanics with machine learning,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='02306 [79] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Alippi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Ding, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Shardt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Peng, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Yang, “Graph neural network-based fault diagnosis: a review,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='08185 [80] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kolhe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Kamali, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Rafatirad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Pudukotai Dinakarrao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Homayoun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Lu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Zhao, “Deep graph learning for circuit deobfuscation,” Frontiers in Big Data, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' 4, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='frontiersin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='org/articles/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='3389/fdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} +page_content='608286' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9E1T4oBgHgl3EQfVAPl/content/2301.03098v1.pdf'} diff --git a/_9FLT4oBgHgl3EQfwi-I/content/tmp_files/2301.12164v1.pdf.txt b/_9FLT4oBgHgl3EQfwi-I/content/tmp_files/2301.12164v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..83e0008e6110d08abd2095d55bc6a8201250e83e --- /dev/null +++ b/_9FLT4oBgHgl3EQfwi-I/content/tmp_files/2301.12164v1.pdf.txt @@ -0,0 +1,1815 @@ +Testing the δ-Kerr metric with black hole X-ray data +Jiahao Tao,1 Shafqat Riaz,1 Biao Zhou,1 Askar B. Abdikamalov,1, 2, 3 Cosimo Bambi,1, ∗ and Daniele Malafarina4 +1Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200438 Shanghai, China +2Ulugh Beg Astronomical Institute, Tashkent 100052, Uzbekistan +3Institute of Fundamental and Applied Research, +National Research University TIIAME, Tashkent 100000, Uzbekistan +4Department of Physics, Nazarbayev University, 010000 Astana, Kazakhstan +The spacetime around astrophysical black holes is thought to be described by the Kerr solution. +However, even within general relativity, there is not yet a proof that the final product of the +complete collapse of an uncharged body can only be a Kerr black hole. We can thus speculate +on the possibility that the spacetime around astrophysical black holes may be described by other +solutions of the Einstein Equations and we can test such a hypothesis with observations. In this +work, we consider the δ-Kerr metric, which is an exact solution of the field equations in vacuum and +can be obtained from a non-linear superposition of the Kerr metric with a static axially symmetric +solution, often referred to as the δ-metric. The parameter δ = 1 + q quantifies the departure of +the source from the Kerr metric and for q = 0 we recover the Kerr solution. From the analysis of +the reflection features in the X-ray spectrum of the Galactic black hole in EXO 1846–031, we find +−0.1 < q < 0.7 (90% CL), which is consistent with the hypothesis that the spacetime around the +compact object in EXO 1846–031 is a Kerr black hole but does not entirely rule out the δ-Kerr +metric. +I. +INTRODUCTION +When a star exhausts all its nuclear fuel, the thermal +pressure of the plasma cannot compensate the star’s own +weight and the body shrinks to find a new equilibrium +configuration. If the collapsing part of the star exceeds +the Oppenheimer-Volkoff limit, which is about 3 M⊙ and +corresponds to the maximum mass for a neutron star, +there is no known mechanism to stop the collapse and +we have the formation of a “gravitationally collapsed ob- +ject” [1, 2]. It is generally assumed that the final outcome +of complete collapse should be a black hole. However, the +exact nature of such a gravitationally collapsed object is +not yet completely understood and therefore theoretical +and observational studies to address this question are ac- +tive lines of research nowadays [3–5]. +In 4-dimensional general relativity, the only vacuum +metric that is stationary, regular on and outside an event +horizon, and asymptotically flat is the Kerr black hole +solution. +This is the celebrated result of a family of +uniqueness theorems, which were pioneered in Refs. [6–8] +and whose final version is still an ongoing research pro- +gram [9]. In the ’60s, Roger Penrose proposed the Cosmic +Censorship Conjecture, according to which all singulari- +ties must be hidden behind an event horizon [10]. If this +is true, within general relativity all gravitationally col- +lapsed objects must be Kerr black holes. However, even +within general relativity, the Cosmic Censorship Conjec- +ture is still unproven and, at the same time, we know +exact solutions of the Einstein Equations that violate the +Cosmic Censorship Conjecture and in which the complete +collapse of a body leads to a spacetime with naked singu- +larities [11]. A viewpoint that is widely accepted today +∗ Corresponding author: bambi@fudan.edu.cn +regarding the appearance of singularities in solutions of +Einstein’s Equations is that they signal a regime where +the theory fails and needs to be replaced by a new the- +ory of gravity. In this sense, the study of singularities in +general relativity may provide hints at the features that +such a new theory must posses and how it may manifest +in astrophysical phenomena [12]. +From astrophysical observations, we know at least +two classes of gravitationally collapsed objects: stellar- +mass compact objects with a mass exceeding the +Oppenheimer-Volkoff limit and supermassive objects in +galactic nuclei. The latter are simply too massive, com- +pact, and old to be clusters of non-luminous bodies like +neutron stars [13]. For both object classes, we have even +a body of observations suggesting that these objects do +not have a normal surface but an event horizon [14, 15]. +The past few years have seen a tremendous progress in +our capability of testing the nature of these compact ob- +jects and today we can use gravitational wave data from +the LIGO-Virgo-KAGRA Collaboration [16–20], X-ray +observations from a number of X-ray missions [21–27], +and the mm images of the supermassive objects in M87∗ +and Sgr A∗ from the Event Horizon Telescope Collabo- +ration [28–33]. +In the present paper, we want to explore the possibil- +ity that the spacetime around these gravitationally col- +lapsed objects is described by the δ-Kerr metric [34, 35], +which is an exact solution of the Einstein Equations +that can be obtained from a non-linear superposition +of the δ-metric (sometimes called Zipoy-Voorhees or γ- +metric)[36–39] and the Kerr metric. +Such a solution, +which can be understood as a stationary extension of +the δ-metric or a deformed extension of the Kerr metric, +has three independent parameters: the mass parameter +M, which is related to the mass of the compact object, +the spin parameter J, which is related to the angular +arXiv:2301.12164v1 [gr-qc] 28 Jan 2023 + +2 +momentum of the source, and a deformation parameter, +q = 1 − δ, which quantifies the departure from the Kerr +solution. For q = 0 and J ̸= 0, the δ-Kerr metric reduces +to the Kerr solution. For J = 0 and q ̸= 0, it reduces +to the δ-metric, while a non-vanishing value of q and J +corresponds to a stationary, axisymmetric, and asymp- +totically flat vacuum solution of the Einstein Equations +with a naked singularity. +The observational properties of the static δ-metric have +been widely studied in the past few years [40–47]. How- +ever, since the δ-metric is static it does not constitute a +good candidate for the gravitational field outside an as- +trophysical source. On the other hand, the δ-Kerr metric +is stationary and continuously linked to the Kerr met- +ric through the value of the deformation parameter and +therefore it is an ideal candidate to test the validity of +the Kerr hypothesis around astrophysical compact ob- +jects. The shadow of the δ-Kerr spacetime was studied +in Ref. [48], while its quasinormal modes were considered +in Ref. [49]. +The δ-metric and the δ-Kerr metric violate the Cosmic +Censorship Conjecture, and for this reason they are nor- +mally not considered as viable solutions for the descrip- +tion of the spacetime around gravitationally collapsed +objects. Therefore in the absence of a proof of the Cos- +mic Censorship Conjecture and/or adopting the idea that +quantum gravity effects may resolve spacetime singulari- +ties and make the Cosmic Censorship Conjecture unnec- +essary [50, 51], it is worth to check whether we can test +and rule out the δ-Kerr metric via astrophysical observa- +tions. To this aim, in this article we construct a reflection +model for the δ-Kerr metric and we analyze a NuSTAR +spectrum of the X-ray binary EXO 1846–031 with strong +reflection features. From the analysis of this observation, +we can constrain the value of the deformation parameter +q of the source and thus test the δ-Kerr spacetime. +The content of the paper is as follows. In Section II, +we briefly review the δ-Kerr metric and, in Section III, +the analysis of the reflection features as a tool for testing +the nature of gravitationally collapsed objects. In Sec- +tion IV, we consider a NuSTAR observation of the X-ray +binary EXO 1846–031 and we describe its data reduction. +In Section V, we present the spectral analysis of the NuS- +TAR observation and from the analysis of the reflection +features we constrain the deformation parameter q of the +δ-Kerr metric. Summary and conclusions are reported in +Section VI. In the present manuscript, we adopt natural +units with GN = c = 1 and the convention of a metric +with signature (− + ++). +II. +δ-KERR METRIC +The δ-Kerr metric was derived in Refs. [34, 35] and can +be obtained as a non-linear superposition of the δ-metric +and the Kerr metric. In Boyer-Lindquist-like coordinates +(t, r, θ, φ), the line element of the δ-Kerr metric is [49] +ds2 = −Fdt2 + 2Fωdtdφ + e2γ +F +B +Adr2+ ++ r2 e2γ +F Bdθ2 + +�r2 +F A sin2 θ − Fω2 +� +dφ2 , +(1) +where +A = 1 − 2M +r ++ a2 +r2 , +B = A + σ2 sin2 θ +r2 +. +(2) +Here M is the mass parameter, which is related to the +gravitational mass of the compact object, J is its spin +parameter, related to the object’s angular momentum, +a = J/M (while the dimensionless spin parameter is a∗ = +a/M), and σ = +√ +M 2 − a2 > 0 is a constant length. F, ω, +and γ are functions of the prolate spheroidal coordinates +x = (r − M) /σ and y = cos θ: +F = A +B , +ω = 2 +� +a − σ C +A +� +, +e2γ = 1 +4 +� +1 + M +σ +�2 +A +(x2 − 1)δ +� x2 − 1 +x2 − y2 +�δ2 +, +(3) +where +A = a+a− + b+b− , +B = a2 ++ + b2 ++ , +C = (x + 1)q � +x +� +1 − y2� +(λ + η) a++ ++y +� +x2 − 1 +� +(1 − λη) b+ +� +; +(4) +a± = (x ± 1)q [x (1 − λη) ± (1 + λη)] , +b± = (x ± 1)q [y (λ + η) ∓ (λ − η)] ; +(5) +λ = α +� +x2 − 1 +�−q (x + y)2q , +η = α +� +x2 − 1 +�−q (x − y)2q ; +(6) +q = δ − 1 , +α = M − σ +a += +a +M + σ . +(7) +Contrary to the Schwarzschild solution, for q ̸= 0 the +δ-Kerr spacetime has a non vanishing mass quadrupole +moment even when a = 0 [48]. +For q = 0, we have +λ = η = α, a± = 2α (r − M ± M) /a, and b± = 2α cos θ. +Therefore, we can get +A = 4α2 +a2 (Σ − 2Mr) , +B = 4α2 +a2 Σ , +C = 4α2 +aσ +� +Σ − Mr +� +1 + cos2 θ +�� +, +(8) + +3 +and +F = 1 − 2Mr +Σ +, +ω = −2Mar sin2 θ +Σ − 2Mr +, +e2γ = +Σ − 2Mr +Σ − 2Mr + M 2 sin2 θ , +(9) +where Σ = r2 + a2 cos2 θ. If we plug these expressions in +the line element in Eq. (1), we recover the familiar Kerr +solution in Boyer-Lindquist coordinates. +Finally note +that for a = 0 and q = 0 we retrieve the Schwarzschild +metric. +III. +X-RAY REFLECTION SPECTROSCOPY +Relativistically blurred reflection features are common +in the X-ray spectra of accreting black holes [52–54]. +These features are produced by illumination of a “cold” +disk by a “hot” corona [55]. The astrophysical system is +shown in Fig. 1. The accretion disk around the black hole +is optically thin and geometrically thick. The gas in the +disk is in local thermal equilibrium and every point on +the surface of the disk emits a blackbody-like spectrum. +The whole disk has a multi-temperature blackbody-like +spectrum because the temperature of the gas increases +approaching the central object. The thermal spectrum +of the accretion disk is normally peaked in the soft X-ray +band (0.1-10 keV) in the case of stellar-mass black holes +in X-ray binary systems and in the UV band (1-100 eV) +in the case of supermassive black holes in active galactic +nuclei. The corona is some hotter plasma (∼ 100 keV) +near the black hole. Thermal photons from the accre- +tion disk can inverse Compton scatter off free electrons +in the corona. The Comptonized photons can illuminate +the disk: Compton scattering and absorption followed by +fluorescent emission generate the reflection spectrum. +In the rest-frame of the gas in the disk, the reflection +spectrum is characterized by narrow fluorescent emission +lines below 10 keV and a Compton hump peaking at 20- +30 keV [57, 58]. The reflection spectrum of the whole +disk detected by a distant observer is blurred because it +is the result of photons coming from all points of the ac- +cretion disk and every point of the disk is characterized +by its own redshift factor, resulting from the combina- +tion of gravitational redshift and Doppler boosting [5]. +X-ray reflection spectroscopy refers to the analysis of the +reflection features in the X-ray spectra of accreting black +holes. In the presence of high-quality data and the cor- +rect astrophysical model, X-ray reflection spectroscopy +can be a powerful technique to probe the strong gravity +regions around black hole candidates [55]. +The idea of using the analysis of reflection features to +test the nature of gravitationally collapsed objects and +the Kerr black hole hypothesis was discussed for the first +time in Ref. [59] and further explored by other authors + + +Black Hole +Accretion Disk +Corona +Thermal Photons +Reflection +Photons +Comptonized +Photons +FIG. 1. Disk-corona system. Figure from Ref. [56] under the +terms of the Creative Commons Attribution 4.0 International +License. +in Refs. [60–64]. In those early works, it was only stud- +ied the shape of the iron Kα line, which is often one of +the most prominent features in the reflection spectrum +and certainly its most informative part about the space- +time metric in the strong gravity region around the com- +pact object. However, none of those models was suitable +to analyze real data. A breakthrough in this field was +the development of the reflection model relxill nk [65– +67], which is an extension of the popular relxill pack- +age [68, 69] for non-Kerr spacetimes. +relxill nk has +been extensively used in the past few years to test the +Kerr black hole hypothesis and specific modified theo- +ries of gravity in which rotating compact objects are not +described by the Kerr solution (e.g. [70–73]). The state- +of-the-art in the field is reviewed in Ref. [27]. +In general, the observed flux of an accretion disk +around a compact object can be calculated as +F(Eo) = +1 +D2 +� +dX dY Io(X, Y ) = += +1 +D2 +� +dX dY Ie(Eo, re, ϑe) , +(10) +where Io and Ie are the specific intensity of the radia- +tion as measured, respectively, by the distant observer +and in the rest-frame of the gas in the disk. X and Y +are the Cartesian coordinates of the image of the disk in +the plane of the distant observer and D is the distance +of the observer from the source. Io = g3Ie follows from +Liouville’s theorem, g = Eo/Ee is the redshift factor, and +Eo and Ee are the photon energies as measured, respec- +tively, by the distant observer and in the rest-frame of +the gas. Here re is the emission radius on the disk and +ϑe is the emission angle, which may differ from the incli- +nation angle of the disk with respect to the line of sight +of the distant observer, i, because of light bending. The +natural way to calculate the observed flux F(Eo) is to +consider a grid on the plane of the distant observer and +follow the trajectories of photons backwards in time from + +4 +every point of the grid to the disk. In this way, we con- +nect every point of the image of the disk on the plane +of the distant observer with its actual emission point on +the disk, we can calculate the redshift factor g, and, if we +know the local spectrum, we can calculate the integral. +In practice, this approach is not doable because the +ray-tracing calculations are too time consuming to be +done during the data analysis process. The current strat- +egy in most reflection models, including also relxill nk, +is to introduce the “transfer function” and rewrite +Eq. (10) as (see, e.g., Refs. [5, 65]) +F(Eo) = +1 +D2 +� rout +rin +dre +� 1 +0 +dg∗ +πreg2 +� +g∗(1 − g∗) +×f(g∗, re, i) Ie(Eo, re, ϑe) , (11) +where rin and rout are, respectively, the inner and the +outer edges of the disk, f is the transfer function [74] +f(g∗, re, i) = g +� +g∗(1 − g∗) +πre +J(X, Y ; g∗, re) , +(12) +g∗ is the relative redshift at the emission radius re for an +observer with viewing angle i +g∗ = +g − gmin +gmax − gmin +, +(13) +and gmin = gmin(re, i) and gmax = gmax(re, i) are, re- +spectively, the minimum and the maximum values of the +redshift factor g for photons emitted at the emission ra- +dius re and detected by an observer with viewing angle i. +Finally J(X, Y ; g∗, re) is the Jacobian between the Carte- +sian coordinates of the image of the disk in the plane of +the distant observer and the two variables re and g∗ used +to map the emission points on the accretion disk. +The transfer function and the non-relativistic reflection +spectrum can be calculated before the data analysis pro- +cess on a computer cluster and tabulated in FITS files for +a grid of their input parameters. During the data analy- +sis process, the model calls the FITS files and can quickly +calculate the integral to obtain the observed spectrum. +If we want to construct a model for a different spacetime +metric, we just need to replace the old FITS file of the +transfer function with a new one, which is calculated for +the new metric of interest. This is what we have done +to implement the δ-Kerr metric in relxill nk: we have +considered a grid of spin parameters a∗, viewing angles +i, and deformation parameters q and for every point of +the grid we have calculated the transfer function with a +ray-tracing code for 100 values of re and 40 values of g∗. +If, instead of the full non-relativistic reflection spec- +trum, we consider only a narrow iron line at 6.4 keV in Ie, +the calculation of F(Eo) produces relativistically broad- +ened iron lines. +While any precise measurement from +the analysis of the reflection features requires to consider +the full reflection spectrum and not only an iron line, a +single iron line can show better the impact of the param- +eter q on the shape of the spectrum. Fig. 2 shows some +relativistically broadened iron lines for two values of the +inclination angle (i = 30◦ and 60◦) and two values of +the spin parameter (a∗ = 0.7 and 0.998). For every line, +the emissivity profile is supposed to be a power law with +emissivity index p = 8, the inner edge of the disk is set at +the radius of the innermost stable circular orbit (ISCO), +and the outer edge of the disk is set at 400 gravitational +radii. In every panel, we show the iron line for q = 0 +(Kerr spacetime), ±0.5, and ±1. Notice that the value +q = −1, corresponding to δ = 0 is the limiting case of +an extremely flattened source, which in the static case +corresponds to the Curzon solution [75]. +IV. +OBSERVATION AND DATA REDUCTION +EXO 1846–031 is a low mass X-ray binary [76]. It was +discovered by the European X-ray Observatory Satellite +(EXOSAT) on April 3, 1985 [77]. +A second outburst +was detected by CGRO/BATSE in 1994 [78]. After be- +ing in quiescence for about 25 years, the source had a +new outburst in 2019, which was first detected by MAXI +on July 23 [79]. This third outburst was then observed +by other instruments; e.g., Swift/XRT [80], VLA [81], +and MeerKAT [82]. The Nuclear Spectroscopic Telescope +Array mission (NuSTAR) [83] observed EXO 1846–031 +on August 3, 2019 (observation ID 90501334002) with a +22.2 ks exposure time. In what follows, we will consider +this NuSTAR observation, which was first analyzed in +Ref. [84]. +For the data reduction, we follow Ref. [84]. NuSTAR +has two detectors, which are called Focal Plane Mod- +ules (FPM) A and B. We download the raw data from +the HEASARC website and use the HEASOFT module +nupipeline to convert the raw data into cleaned event +files with NuSTARDAS and the CALDB 20220301 cali- +bration database, so that we can get the source and back- +ground information. For the source, we select a 180 arc- +seconds radius circular region at the center of the source +for both FPMA and FPMB. For the background, we take +a region of the same size of the source as far as possible +from the source but on the same detector, so that the +influence of the source’s photons can be ignored. +Af- +terwards, we use the HEASOFT module nuproducts to +generate the source and background spectra, the response +matrix file, and the ancillary file. Last, we use grppha to +group the spectra to have at least 30 counts per bin. Since +the new CALDB corrects the calibration in the 3-7 keV +energy range, we do not need the table nuMLIv1.mod used +in Ref. [84]. +V. +SPECTRAL ANALYSIS +For the spectral analysis, we use XSPEC v12.12.1 [85]. +First, we fit the data with an absorbed power law to +see the reflection features in the spectrum. In XSPEC +language, the model reads +const × tbabs × (diskbb + cutoffpl) . + +5 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Flux [arbitrary units] +i = 30◦ +Kerr : q = 0 +q = 0.5 +q = −0.5 +q = 1 +q = −1 +a∗ = 0.7 +i = 60◦ +0 +2 +4 +6 +8 +10 +E [keV ] +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +Flux [arbitrary units] +0 +2 +4 +6 +8 +10 +E [keV ] +a∗ = 0.998 +FIG. 2. Iron line profiles in δ-Kerr spacetimes. The inclination angle of the disk with respect to the line of sight of the distant +observer is i = 30◦ (left panels) and 60◦ (right panels). The dimensionless spin parameter is a∗ = 0.7 (top panels) and 0.998 +(bottom panels). These profiles are calculated assuming that the emissivity profile is described by a power law with emissivity +index p = 8, the inner edge of the disk is at the ISCO radius, and the outer edge is at 400 rg, where rg is the gravitational +radius. +const is used to have a cross-calibration constant be- +tween the detectors FPMA and FPMB: the constant is +frozen to 1 for FPMA and is free for FPMB. tbabs de- +scribes the Galactic absorption [86]: the hydrogen col- +umn density, NH, is the only parameter of the model and +is left free in the fit. diskbb describes the thermal spec- +trum of the accretion disk [87]: the temperature at the +inner edge of the disk, Tin, and the normalization of the +component are left free in the fit. cutoffpl describes +the continuum from the corona: the photon index, Γ, +the high energy cutoff, Ecut, and the normalization of +this component are left free in the fit. The ratio between +the data and the best-fit model is shown in Fig. 3 and we +clearly see unresolved strong reflection features: a broad- +ened iron Kα line peaking around 7 keV and a Compton +hump peaking at 20-30 keV. Such a strong blurred re- +flection features suggest that this NuSTAR spectrum is +suitable to test the nature of the gravitationally collapsed +object in EXO 1846–031 with relxill nk. +To fit the reflection features, we add relxill nk to the +total model. We employ the flavor relxillion nk, which +describes the relativistically blurred reflection spectrum +of an accretion disk with a non-trivial ionization gradi- +ent [88, 89]. In XSPEC language, the total model now +reads +const × tbabs × (diskbb + relxillion nk) . +relxill nk has several parameters. The spacetime met- +ric is described by the spin a∗ and the deformation pa- +rameter q of the δ-Kerr metric and both parameters are +left free in the fit. The inner edge of the accretion disk +is set at the ISCO and therefore it is not a free param- +eter but directly depends on the values a∗ and q. The + +6 +5 +10 +20 +50 +Energy (keV) +0.8 + +0.9 + +1.0 + +1.1 + +1.2 +Ratio +FPMA +FPMB +FIG. 3. Data to best-fit model ratio for an absorbed power +law. +We clearly see a broadened iron line peaking around +7 keV and a Compton hump peaking at 20-30 keV. +outer edge of the disk is fixed to 900 rg, where rg is the +gravitational radius and 900 rg is the maximum value al- +lowed by the model. The emissivity profile of the accre- +tion disk can potentially be described by a twice broken +power law and there are thus five parameters: the emis- +sivity indices of the inner, central, and outer regions (p1, +p2, and p3, respectively) and the breaking radii between +the inner and the central parts, Rbr1, and between the +central and outer parts, Rbr21. To model the emissivity +profile with a broken power law (instead of a twice broken +power law), we simply set p2 = p3 and Rbr1 = Rbr2 (i.e., +the central region collapses and we have only the inner +and outer regions). The viewing angle, i, the iron abun- +dance, AFe, the ionization at the inner edge of the disk, +ξin, and the ionization index, αξ, are all free parameters +in the fit. The model includes the continuum from the +corona and the reflection fraction, Rf, regulates the rel- +ative strength between the reflection component and the +continuum. The photon index, Γ, and the high-energy +cutoff, Ecut, of the continuum illuminating the disk are +free in the fit. +From previous analyses [88, 89], we know that this +spectrum requires a non-vanishing ionization gradient +and for this reason we use the flavor relxillion nk. +If we fit the data with a model with a disk with con- +stant ionization, we need to add a Gaussian to the total +model [84]. We fit the data with four models (Models 1- +4), which are listed in Tab. I. +In our first fit (Model 1), we fit the data assuming +that the emissivity profile of the disk is described by a +1 This means that the emission of the disk scales as r−p1 in the +inner part (r < Rbr1), as r−p2 in the central part (Rbr1 < r < +Rbr2), and as r−p3 in the outer part (r > Rbr2). +broken power law (so p2 = p3 and Rbr1 = Rbr2). The +best-fit values are reported in Tab. II. The best-fit model +and the data to best-fit model ratio are shown in the +top-left panel in Fig. 4. As we can see from Tab. II, we +find a very high emissivity index for the inner region of +the accretion disk and an almost vanishing emissivity in- +dex for the outer part. Such an emissivity profile may +be generated by a corona covering a large portion of the +accretion disk [90–93] and the data may prefer a twice +broken power law. As Model 2, we thus fit the spectrum +with a twice broken power law. The best-fit values are +reported in the third column of Tab. II and the best-fit +model and the data to best-fit model ratio are shown in +the top-right panel in Fig. 4. We do not see any improve- +ment in the fit and Rbr2 is stuck at the outer edge of the +accretion disk. Unfortunately, for the outer edge of the +disk we have already chosen the maximum value allowed +by the model and we cannot try to fit the data with a +larger disk. As Model 3, we reconsider an emissivity pro- +file described by a broken power law, but this time we +freeze the emissivity index of the outer region of the disk +to 3, which is the value normally expected for the outer +emissivity index when the corona is compact. The results +are still shown in Tab. II and Fig. 4, but the fit is clearly +worse with ∆χ2 = +65 with respect to Model 1. Last, we +consider the possibility of the presence of a distant cold +reflector and we add xillver [94] to the total model. As +shown in Tab. II, these data clearly do not require any +distant reflector. +VI. +DISCUSSION AND CONCLUSIONS +This observation of EXO 1846–031 has been already +analyzed and extensively discussed in the literature [84, +88, 89, 95], so here we will focus on the possibility to use +such observation to constrain the value of the deforma- +tion parameter q of the δ-Kerr metric. +Model 1 is the simplest model and fits the data well. +Our constraint on q is (90% confidence level, only statis- +tical uncertainty) +−0.1 < q < 0.7 , +(14) +and therefore our analysis is consistent with the hy- +pothesis that the gravitationally collapsed object in the +X-ray binary EXO 1846–031 is a Kerr black hole (for +which q = 0). However, our analysis does not exclude +q = O(0.1) and therefore natural values of the defor- +mation parameter are allowed. Also notice that positive +value of q implies that a non rotating source is oblate. +Therefore the above bounds on q are consistent with a +spinning oblate compact object, which is more physically +realistic than a prolate (i.e. q < 0) one. +From Model 1, we find that the emissivity profile is +very steep around the central object and almost flat at +larger radii. While this is not the emissivity profile ex- +pected from a compact corona, it is common in Galac- +tic black holes; see, for instance, the discussion in [93] + +7 +Model +XSPEC Model +Emissivity Profile +1 +tbabs×(diskbb+relxillion nk) +p1, p2, Rbr1 +2 +tbabs×(diskbb+relxillion nk) +p1, p2, p3 = 3, Rbr1, Rbr2 +3 +tbabs×(diskbb+relxillion nk) +p1, p2 = 3, Rbr1 +4 +tbabs×(diskbb+relxill nk+xillver) +p1, p2, Rbr1 +TABLE I. Summary of the models used in the spectral analysis of this work. +Model +1 +2 +3 +4 +tbabs +NH/1022 cm−2 +4.3+0.3 +−0.4 +4.2+0.5 +−0.4 +5.8+0.4 +−0.4 +4.2+0.3 +−0.4 +diskbb +Tin [keV] +0.31+0.10 +−0.08 +0.31+0.09 +−0.09 +0.497+0.013 +−0.012 +0.31+0.09 +−0.09 +Norm/105 +1.4+10 +−1.2 +1.1+8 +−0.9 +0.084+0.018 +−0.009 +1.2+8 +−1.1 +relxillion nk +p1 +10.0−2.4 +10.0−2.3 +10.0−0.4 +10.0−2.3 +p2 +0.5+0.7 +0.4+0.6 +3∗ +0.5+0.7 +p3 +– +3∗ +– +– +Rbr1 [rg] +5.2+2.1 +−1.5 +5.5+2.9 +−1.5 +2.86+0.16 +−0.10 +5.1+2.8 +−2.3 +Rbr2 [rg] +– +900−490 +– +– +a∗ +0.998−0.004 +0.998−0.004 +0.998−0.007 +0.998−0.004 +i [deg] +78.2+0.6 +−1.2 +78.0+0.9 +−1.2 +69.3+1.6 +−0.9 +78.2+0.7 +−1.1 +Γ +2.04+0.05 +−0.09 +2.01+0.05 +−0.06 +1.84+0.02 +−0.03 +2.02+0.07 +−0.07 +log ξin [erg · cm · s−1] +3.00+0.08 +−0.14 +3.05+0.07 +−0.19 +3.61+0.11 +−0.07 +3.03+0.09 +−0.17 +AFe +1.5+0.4 +−0.5 +1.5+0.5 +−0.5 +2.7+0.7 +−0.3 +1.5+0.5 +−0.5 +Ecut [keV] +110+18 +−25 +103+22 +−6 +80+5 +−7 +106+19 +−12 +Rf +0.237+0.019 +−0.015 +0.222+0.014 +−0.014 +0.25+0.06 +−0.04 +0.25+0.12 +−0.02 +αξ +0.19+0.06 +−0.05 +0.19+0.04 +−0.03 +0.00+0.07 +0.19+0.06 +−0.04 +q +0.57+0.11 +−0.7 +0.57+0.10 +−0.5 +1.89−0.21 +0.57+0.10 +−0.5 +Norm/10−2 +2.58+0.18 +−0.34 +2.48+0.19 +−0.21 +1.72+0.16 +−0.08 +2.3+0.4 +−0.5 +xillver +Norm/10−3 +– +– +– +2+5 +constant +FPMA +1∗ +1∗ +1∗ +1∗ +FPMB +1.0152+0.0014 +−0.0014 +1.0152+0.0014 +−0.0014 +1.0152+0.0014 +−0.0014 +1.0152+0.0014 +−0.0014 +χ2/ν +2659.62/2599 +2659.59/2598 +2724.98/2600 +2659.41/2598 += 1.02332 += 1.02371 += 1.04807 += 1.02364 +TABLE II. Best-fit table of Models 1-4. The reported uncertainties correspond to the 90% confidence level for one relevant +parameter (∆χ2 = 2.71). ∗ means the value of the parameter is frozen during the fit. When there is no lower/upper uncertainty, +the boundary of the range in which the parameter is allowed to vary is within the 90% confidence limit. + +8 +5 +10 +20 +50 +Energy (keV) +0.01 +0.1 +1 +keV²(phs/cm²/s/keV) +FPMA +FPMB +5 +10 +20 +50 +Energy (keV) +0.95 +1.0 +1.05 +Ratio +Model 1 +5 +10 +20 +50 +Energy (keV) +0.01 +0.1 +1 +keV²(phs/cm²/s/keV) +FPMA +FPMB +5 +10 +20 +50 +Energy (keV) +0.95 +1.0 +1.05 +Ratio +Model 2 +5 +10 +20 +50 +Energy (keV) +0.01 +0.1 +1 +keV²(phs/cm²/s/keV) +FPMA +FPMB +5 +10 +20 +50 +Energy (keV) +0.95 +1.0 +1.05 +Ratio +Model 3 +5 +10 +20 +50 +Energy (keV) +0.01 +0.1 +1 +keV²(phs/cm²/s/keV) +FPMA +FPMB +5 +10 +20 +50 +Energy (keV) +0.95 +1.0 +1.05 +Ratio +Model 4 +FIG. 4. Best-fit model and data to best-fit model ratio for Models 1-4. + +9 +and references therein. If we try to fit the data with a +twice broken power law (Model 2) or by adding a non- +relativistic reflection component (Model 4), we do not see +any significant difference: the value of the second break- +ing radius would be large and the normalization of the +non-relativistic reflection component would be very low. +The estimate of the model parameters are thus consis- +tent with the measurements inferred with Model 1. If we +model the emissivity profile with a broken power law and +we impose that the outer emissivity index is 3, the esti- +mate of some model parameters would be different (and +we find that the spacetime significantly deviates from +the Kerr solution!), but the fit is definitively worse and +Model 3 can be ruled out (∆χ2 = +65 with respect to +Model 1). +We are not aware of other tests of the δ-Kerr metric +published in the literature and observational constraints +on the deformation parameter q, though models for the +shadow and quasinormal modes have been studied in +Refs. +[48, 49]. +Since the δ-Kerr spacetime is an ex- +act vacuum solutions of the field equations in general +relativity which relates to the Kerr black hole through +the variation of one continuous parameter with a clear +physical interpretation, we would argue that experimen- +tal tests to constrain the allowed values of q from obser- +vations are important towards a possible resolution of the +Kerr hypothesis. We could certainly constrain q from the +available gravitational wave data from the LIGO-Virgo- +KAGRA Collaboration following the approach employed +in Refs. [19, 20] for testing other non-Kerr metrics. The +deformation parameter q may also be constrained from +the available mm black hole images from the Event Hori- +zon Telescope Collaboration (see, e.g., Ref. [33]). While +these three techniques (X-ray, gravitational waves, and +black hole imaging) are sensitive to different relativistic +effects, in general, X-ray tests are those that can pro- +vide the most stringent constraints on possible deviations +from the Kerr solution, while gravitational wave con- +straints are normally a bit weaker and black hole imaging +constraints are more than an order of magnitude weaker; +see, for example, Ref. [27]. This is the typical situation +with the current data. However, gravitational wave con- +straints are expected to improve quickly in the coming +years. +Concerning X-ray tests, the constraint reported in the +present work is likely close to the best that we can do +today. Somewhat more stringent constraints may be ob- +tained from sources in which we can test the Kerr metric +from the simultaneous analysis of the reflection features +and the thermal spectrum, as done in Refs. [23, 25, 26]. +This is not possible for the NuSTAR spectrum analyzed +here because the thermal component is too weak and +we do not have independent measurements of the mass +and distance of the compact object. More stringent con- +straints on the deformation parameter q require higher +quality data, which will be available from the next gen- +eration of X-ray missions, starting from eXTP [96], which +is currently scheduled to be launched in 2027. +Acknowledgments – This work was supported by +the National Natural Science Foundation of China +(NSFC), +Grant +No. +12250610185, +11973019, +and +12261131497, the Natural Science Foundation of Shang- +hai, Grant No. 22ZR1403400, the Shanghai Municipal +Education Commission, Grant No. +2019-01-07-00-07- +E00035, and Fudan University, Grant No. JIH1512604. +DM acknowledges support from Nazarbayev University +Faculty Development Competitive Research Grant No. +11022021FD2926. +[1] R. Ruffini and S. Bonazzola, Phys. Rev. 187, 1767-1783 +(1969) doi:10.1103/PhysRev.187.1767 +[2] J. +M. +Lattimer, +Ann. +Rev. +Nucl. +Part. +Sci. +62, +485-515 (2012) doi:10.1146/annurev-nucl-102711-095018 +[arXiv:1305.3510 [nucl-th]]. +[3] C. +Bambi, +Rev. +Mod. +Phys. +89, +025001 +(2017) +doi:10.1103/RevModPhys.89.025001 +[arXiv:1509.03884 +[gr-qc]]. +[4] K. +Yagi +and +L. +C. +Stein, +Class. +Quant. +Grav. +33, 054001 (2016) doi:10.1088/0264-9381/33/5/054001 +[arXiv:1602.02413 [gr-qc]]. +[5] C. Bambi, Black Holes: A Laboratory for Testing Strong +Gravity (Springer Singapore, 2017), ISBN 978-981-10- +4524-0 doi:10.1007/978-981-10-4524-0 +[6] W. +Israel, +Phys. +Rev. +164, +1776-1779 +(1967) +doi:10.1103/PhysRev.164.1776 +[7] B. +Carter, +Phys. +Rev. +Lett. +26, +331-333 +(1971) +doi:10.1103/PhysRevLett.26.331 +[8] D. C. Robinson, Phys. Rev. Lett. 34, 905-906 (1975) +doi:10.1103/PhysRevLett.34.905 +[9] P. T. Chrusciel, +J. Lopes Costa and M. Heusler, +Living Rev. Rel. 15, 7 (2012) doi:10.12942/lrr-2012-7 +[arXiv:1205.6112 [gr-qc]]. +[10] R. +Penrose, +Riv. +Nuovo +Cim. +1, +252-276 +(1969) +doi:10.1023/A:1016578408204 +[11] P. S. Joshi and D. Malafarina, Int. J. Mod. Phys. +D 20, 2641-2729 (2011) doi:10.1142/S0218271811020792 +[arXiv:1201.3660 [gr-qc]]. +[12] D. +Malafarina, +Universe +3, +48 +(2017) +doi:10.3390/universe3020048 +[arXiv:1703.04138 +[gr- +qc]]. +[13] E. Maoz, Astrophys. J. Lett. 494, L181-L184 (1998) +doi:10.1086/311194 [arXiv:astro-ph/9710309 [astro-ph]]. +[14] R. +Narayan +and +J. +E. +McClintock, +New +Astron. +Rev. 51, 733-751 (2008) doi:10.1016/j.newar.2008.03.002 +[arXiv:0803.0322 [astro-ph]]. +[15] A. +E. +Broderick, +A. +Loeb +and +R. +Narayan, +As- +trophys. J. 701, 1357-1366 (2009) doi:10.1088/0004- +637X/701/2/1357 [arXiv:0903.1105 [astro-ph.HE]]. +[16] B. +P. +Abbott +et +al. +[LIGO +Scientific +and +Virgo], +Phys. +Rev. +Lett. +116, +221101 +(2016) +[erra- +tum: +Phys. +Rev. +Lett. +121, +129902 +(2018)] +doi:10.1103/PhysRevLett.116.221101 [arXiv:1602.03841 +[gr-qc]]. + +10 +[17] N. +Yunes, +K. +Yagi +and F. +Pretorius, +Phys. +Rev. +D 94, 084002 (2016) doi:10.1103/PhysRevD.94.084002 +[arXiv:1603.08955 [gr-qc]]. +[18] B. +P. +Abbott +et +al. +[LIGO +Scientific +and +Virgo], +Phys. +Rev. +D +100, +104036 +(2019) +doi:10.1103/PhysRevD.100.104036 +[arXiv:1903.04467 +[gr-qc]]. +[19] A. Cardenas-Avendano, S. Nampalliwar and N. Yunes, +Class. Quant. Grav. 37, 135008 (2020) doi:10.1088/1361- +6382/ab8f64 [arXiv:1912.08062 [gr-qc]]. +[20] S. +Shashank +and +C. +Bambi, +Phys. +Rev. +D +105, +104004 +(2022) +doi:10.1103/PhysRevD.105.104004 +[arXiv:2112.05388 [gr-qc]]. +[21] Z. Cao, S. Nampalliwar, C. Bambi, T. Dauser and +J. A. Garcia, Phys. Rev. Lett. 120, 051101 (2018) +doi:10.1103/PhysRevLett.120.051101 [arXiv:1709.00219 +[gr-qc]]. +[22] A. +Tripathi, +S. +Nampalliwar, +A. +B. +Abdikamalov, +D. Ayzenberg, +C. Bambi, +T. Dauser, +J. A. Gar- +cia +and +A. +Marinucci, +Astrophys. +J. +875, +56 +(2019) doi:10.3847/1538-4357/ab0e7e [arXiv:1811.08148 +[gr-qc]]. +[23] A. +Tripathi, +A. +B. +Abdikamalov, +D. +Ayzenberg, +C. +Bambi, +V. +Grinberg +and +M. +Zhou, +Astro- +phys. J. 907, 31 (2021) doi:10.3847/1538-4357/abccbd +[arXiv:2010.13474 [astro-ph.HE]]. +[24] A. Tripathi, Y. Zhang, A. B. Abdikamalov, D. Ayzen- +berg, C. Bambi, J. Jiang, H. Liu and M. Zhou, Astro- +phys. J. 913, 79 (2021) doi:10.3847/1538-4357/abf6cd +[arXiv:2012.10669 [astro-ph.HE]]. +[25] Z. Zhang, H. Liu, A. B. Abdikamalov, D. Ayzenberg, +C. Bambi and M. Zhou, Astrophys. J. 924, 72 (2022) +doi:10.3847/1538-4357/ac350e [arXiv:2106.03086 [astro- +ph.HE]]. +[26] A. +Tripathi, +A. +B. +Abdikamalov, +D. +Ayzenberg, +C. Bambi, V. Grinberg, H. Liu and M. Zhou, JCAP +01, +019 +(2022) +doi:10.1088/1475-7516/2022/01/019 +[arXiv:2106.10982 [astro-ph.HE]]. +[27] C. Bambi, [arXiv:2210.05322 [gr-qc]]. +[28] C. +Bambi, +K. +Freese, +S. +Vagnozzi +and +L. +Visinelli, +Phys. +Rev. +D +100, +044057 +(2019) +doi:10.1103/PhysRevD.100.044057 +[arXiv:1904.12983 +[gr-qc]]. +[29] S. +W. +Wei +and +Y. +X. +Liu, +Eur. +Phys. +J. +Plus +136, 436 (2021) doi:10.1140/epjp/s13360-021-01398-9 +[arXiv:2003.07769 [gr-qc]]. +[30] R. Kumar and S. G. Ghosh, JCAP 07, 053 (2020) +doi:10.1088/1475-7516/2020/07/053 +[arXiv:2003.08927 +[gr-qc]]. +[31] D. +Psaltis +et +al. +[Event +Horizon +Tele- +scope], +Phys. +Rev. +Lett. +125, +141104 +(2020) +doi:10.1103/PhysRevLett.125.141104 [arXiv:2010.01055 +[gr-qc]]. +[32] K. Akiyama et al. [Event Horizon Telescope], Astrophys. +J. Lett. 930, L17 (2022) doi:10.3847/2041-8213/ac6756 +[33] S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, +A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and +P. S. Joshi, et al. [arXiv:2205.07787 [gr-qc]]. +[34] H. Quevedo and B. Mashhoon, Phys. Lett. A 109, 13-18 +(1985) doi:10.1016/0375-9601(85)90381-0. +[35] H. Quevedo and B. Mashhoon, Phys. Rev. D 43, 3902 +(1991) doi:10.1103/PhysRevD.43.3902. +[36] D. M. Zipoy, Journ. Math. Phys. 22, 1137 (1970) +https://doi.org/10.1063/1.1705005. +[37] B. +H. +Voorhees, +Phys. +Rev. +D +2, +2119 +(1970) +doi:10.1103/PhysRevD.2.2119. +[38] D. Papadopoulos, B. Stewart and L. Witten, Phys. Rev. +D 24, 320 (1981) doi:10.1103/PhysRevD.24.320. +[39] L. Herrera, F. M. Paiva and N. Santos, Int. J. Mod. Phys. +D 9, 649 (2000) doi:10.1142/S021827180000061X, +[40] L. Herrera, +Int. J. Mod. Phys. D 17, +557 (2008) +doi:10.1142/S0218271808012255 +[arXiv:0711.4723 +[gr- +qc]]. +[41] L. Herrera, G. Magli and D. Malafarina, Gen. Rel. +Grav. 37, 1371 (2005) doi:10.1007/s10714-005-0120-1 +[arXiv:gr-qc/0407037]. +[42] A.N. +Chowdhury, +M. +Patil, +D. +Malafarina +and +P.S. +Joshi, +Phys. +Rev. +D +85, +104031 +(2012) +doi:10.1103/PhysRevD.85.104031 +[arXiv:1112.2522 +[gr-qc]]. +[43] A.B. Abdikamalov, A.A. Abdujabbarov, D. Ayzenberg, +D. Malafarina, +C. Bambi and B. Ahmedov, +Phys. +Rev. D 100, 024014 doi:10.1103/PhysRevD.100.024014 +[arXiv:1904.06207 [gr-qc]]. +[44] K. Boshkayev, E. Gasperin, A.C. Guti´errez-Pi˜neres, +H. +Quevedo +and +S. +Toktarbay, +Phys. +Rev. +D +93, +024024 +(2016) +doi:10.1103/PhysRevD.93.024024 +[arXiv:1509.03827 [gr-qc]]. +[45] C.A. Benavides-Gallego, A.A. Abdujabbarov, D. Mala- +farina, +B. +Ahmedov +and +C. +Bambi, +Phys. +Rev. +D 99, 044012 (2019) doi:10.1103/PhysRevD.99.044012 +[arXiv:1812.04846 [gr-qc]]. +[46] B. Toshmatov, D. Malafarina and N. Dadhich, Phys. Rev. +D 100, 044001 (2019) doi:10.1103/PhysRevD.100.044001 +[arXiv;1905.01088 [gr-qc]]. +[47] B. +Toshmatov +and +D. +Malafarina, +Phys. +Rev. +D +100, 104052 (2019) doi:10.1103/PhysRevD.100.104052 +[arXiv:1910.11565 [gr-qc]]. +[48] S. Li, T. Mirzaev, A.A. Abdujabbarov, D. Malafa- +rina, +B. Ahmedov and W.-B. Han. Phys. Rev. D +106, 084041 (2022) doi:10.1103/PhysRevD.106.084041 +[arXiv:2207.10933 [gr-qc]]. +[49] A. Allahyari, H. Firouzjahi and B. Mashhoon, Class. +Quant. +Grav. +37, +055006 +(2020) +doi:10.1088/1361- +6382/ab6860 [arXiv:1908.10813 [gr-qc]]. +[50] E. +G. +Gimon +and +P. +Horava, +Phys. +Lett. +B +672, 299-302 (2009) doi:10.1016/j.physletb.2009.01.026 +[arXiv:0706.2873 [hep-th]]. +[51] C. Bambi and K. Freese, Phys. Rev. D 79, 043002 (2009) +doi:10.1103/PhysRevD.79.043002 +[arXiv:0812.1328 +[astro-ph]]. +[52] Y. +Tanaka, +K. +Nandra, +A. +C. +Fabian, +H. +Inoue, +C. +Otani, +T. +Dotani, +K. +Hayashida, +K. +Iwasawa, +T. Kii and H. Kunieda, et al. Nature 375, 659 (1995) +doi:10.1038/375659a0 +[53] K. Nandra, P. M. O’Neill, I. M. George and J. N. Reeves, +Mon. +Not. +Roy. +Astron. +Soc. +382, +194 +(2007) +doi:10.1111/j.1365-2966.2007.12331.x +[arXiv:0708.1305 +[astro-ph]]. +[54] J. M. Miller, C. S. Reynolds, A. C. Fabian, G. Miniutti +and L. C. Gallo, Astrophys. J. 697, 900-912 (2009) +doi:10.1088/0004-637X/697/1/900 +[arXiv:0902.2840 +[astro-ph.HE]]. +[55] C. Bambi, L. W. Brenneman, T. Dauser, J. A. Gar- +cia, +V. +Grinberg, +A. +Ingram, +J. +Jiang, +H. +Liu, +A. M. Lohfink and A. Marinucci, et al. Space Sci. +Rev. 217, +65 (2021) doi:10.1007/s11214-021-00841-8 +[arXiv:2011.04792 [astro-ph.HE]]. + +11 +[56] C. +Bambi, +Arab. +J. +Math. +11, +81-90 +(2022) +doi:10.1007/s40065-021-00336-y +[arXiv:2106.04084 +[gr-qc]]. +[57] R. R. Ross and A. C. Fabian, Mon. Not. Roy. As- +tron. +Soc. +358, +211-216 +(2005) +doi:10.1111/j.1365- +2966.2005.08797.x [arXiv:astro-ph/0501116 [astro-ph]]. +[58] J. Garcia and T. Kallman, Astrophys. J. 718, 695 (2010) +doi:10.1088/0004-637X/718/2/695 +[arXiv:1006.0485 +[astro-ph.HE]]. +[59] Y. Lu and D. F. Torres, Int. J. Mod. Phys. D 12, 63- +78 (2003) doi:10.1142/S0218271803002718 [arXiv:astro- +ph/0205418 [astro-ph]]. +[60] J. Schee and Z. Stuchlik, Gen. Rel. Grav. 41, 1795-1818 +(2009) doi:10.1007/s10714-008-0753-y [arXiv:0812.3017 +[astro-ph]]. +[61] T. +Johannsen +and +D. +Psaltis, +Astrophys. +J. +773, +57 +(2013) +doi:10.1088/0004-637X/773/1/57 +[arXiv:1202.6069 [astro-ph.HE]]. +[62] C. +Bambi, +Phys. +Rev. +D +87, +023007 +(2013) +doi:10.1103/PhysRevD.87.023007 +[arXiv:1211.2513 +[gr-qc]]. +[63] C. +Bambi, +Phys. +Rev. +D +87, +084039 +(2013) +doi:10.1103/PhysRevD.87.084039 +[arXiv:1303.0624 +[gr-qc]]. +[64] C. +Bambi +and +D. +Malafarina, +Phys. +Rev. +D +88, +064022 +(2013) +doi:10.1103/PhysRevD.88.064022 +[arXiv:1307.2106 [gr-qc]]. +[65] C. Bambi, A. Cardenas-Avendano, T. Dauser, J. A. Gar- +cia +and +S. +Nampalliwar, +Astrophys. +J. +842, +76 +(2017) doi:10.3847/1538-4357/aa74c0 [arXiv:1607.00596 +[gr-qc]]. +[66] A. B. Abdikamalov, D. Ayzenberg, C. Bambi, T. Dauser, +J. A. Garcia and S. Nampalliwar, Astrophys. J. 878, 91 +(2019) doi:10.3847/1538-4357/ab1f89 [arXiv:1902.09665 +[gr-qc]]. +[67] A. B. Abdikamalov, D. Ayzenberg, C. Bambi, T. Dauser, +J. +A. +Garcia, +S. +Nampalliwar, +A. +Tripathi +and +M. Zhou, Astrophys. J. 899, 80 (2020) doi:10.3847/1538- +4357/aba625 [arXiv:2003.09663 [astro-ph.HE]]. +[68] T. Dauser, J. Garcia, J. Wilms, M. Bock, L. W. Bren- +neman, M. Falanga, K. Fukumura and C. S. Reynolds, +Mon. +Not. +Roy. +Astron. +Soc. +430, +1694 +(2013) +doi:10.1093/mnras/sts710 +[arXiv:1301.4922 +[astro- +ph.HE]]. +[69] J. Garc´ıa, T. Dauser, A. Lohfink, T. R. Kallman, +J. Steiner, J. E. McClintock, L. Brenneman, J. Wilms, +W. Eikmann and C. S. Reynolds, et al. Astrophys. +J. +782, +76 +(2014) +doi:10.1088/0004-637X/782/2/76 +[arXiv:1312.3231 [astro-ph.HE]]. +[70] M. Zhou, Z. Cao, A. Abdikamalov, D. Ayzenberg, +C. Bambi, L. Modesto and S. Nampalliwar, Phys. Rev. +D 98, 024007 (2018) doi:10.1103/PhysRevD.98.024007 +[arXiv:1803.07849 [gr-qc]]. +[71] J. Zhu, A. B. Abdikamalov, D. Ayzenberg, M. Azreg- +Ainou, C. Bambi, M. Jamil, S. Nampalliwar, A. Tri- +pathi and M. Zhou, Eur. Phys. J. C 80, 622 (2020) +doi:10.1140/epjc/s10052-020-8198-x +[arXiv:2005.00184 +[gr-qc]]. +[72] B. Zhou, A. B. Abdikamalov, D. Ayzenberg, C. Bambi, +S. Nampalliwar and A. Tripathi, JCAP 01, 047 (2021) +doi:10.1088/1475-7516/2021/01/047 +[arXiv:2005.12958 +[astro-ph.HE]]. +[73] S. +Riaz, +S. +Shashank, +R. +Roy, +A. +B. +Abdika- +malov, +D. +Ayzenberg, +C. +Bambi, +Z. +Zhang +and +M. Zhou, +JCAP 10, +040 (2022) doi:10.1088/1475- +7516/2022/10/040 [arXiv:2206.03729 [gr-qc]]. +[74] C. T. Cunningham, Astrophys. J. 202, 788-802 (1975) +doi:10.1086/154033 +[75] H. E. J. Curzon, Proc. London Math Soc. 23, 477 (1924) +doi:10.1112/plms/s2-23.1.477. +[76] A. N. Parmar, L. Angelini, P. Roche and N. E. White, +A&A 279, 179 (1993). +[77] A. N. Parmar and N. E. White, IAU Circ. 4051, 1 (1985). +[78] S. N. Zhang, B. A. Harmon, C. A. Wilson and et al, IAU +Circ. 6096, 1 (1994). +[79] H. Negoro, M. Nakajima, S. Sugita and et al, The As- +tronomer’s Telegram 12968, 1 (2019). +[80] I. Mereminskiy, R. Krivonos, P. Medvedev and et al, The +Astronomer’s Telegram 12969, 1 (2019). +[81] J. Miller-Jones, T. Russell, G. Sivakoff and A. Tetarenko, +The Astronomer’s Telegram 12977, 1 (2019). +[82] D. Williams, R. Fender, P. Woudt and J. Miller-Jones, +The Astronomer’s Telegram 12992, 1 (2019). +[83] F. +A. +Harrison +et +al. +[NuSTAR], +Astrophys. +J. +770, +103 +(2013) +doi:10.1088/0004-637X/770/2/103 +[arXiv:1301.7307 [astro-ph.IM]]. +[84] P. A. Draghis, J. M. Miller, E. M. Cackett, E. S. Kam- +moun, M. T. Reynolds, J. A. Tomsick and A. Zoghbi, As- +trophys. J. 900, 78 (2020) doi:10.3847/1538-4357/aba2ec +[arXiv:2007.04324 [astro-ph.HE]]. +[85] K. A. Arnaud, ASP Conf. Ser. 101 (1996). +[86] J. Wilms, A. Allen and R. McCray, Astrophys. J. +542, 914-924 (2000) doi:10.1086/317016 [arXiv:astro- +ph/0008425 [astro-ph]]. +[87] K. Makishima, Y. Maejima, K. Mitsuda, H. V. Bradt, +R. A. Remillard, I. R. Tuohy, R. Hoshi and M. Nakagawa, +Astrophys. J. 308, 635-643 (1986) doi:10.1086/164534 +[88] A. B. Abdikamalov, D. Ayzenberg, C. Bambi, H. Liu +and Y. Zhang, +Phys. Rev. D 103, +103023 (2021) +doi:10.1103/PhysRevD.103.103023 +[arXiv:2101.10100 +[astro-ph.HE]]. +[89] A. B. Abdikamalov, D. Ayzenberg, C. Bambi, H. Liu +and +A. +Tripathi, +Astrophys. +J. +923, +175 +(2021) +doi:10.3847/1538-4357/ac3237 [arXiv:2108.00375 [astro- +ph.HE]]. +[90] G. Miniutti, A. C. Fabian, R. Goyder and A. N. Lasenby, +Mon. Not. Roy. Astron. Soc. 344, +L22-L26 (2003) +doi:10.1046/j.1365-8711.2003.06988.x +[arXiv:astro- +ph/0307163 [astro-ph]]. +[91] D. R. Wilkins and L. C. Gallo, Mon. Not. Roy. As- +tron. Soc. 449, 129-146 (2015) doi:10.1093/mnras/stv162 +[arXiv:1501.05302 [astro-ph.HE]]. +[92] S. Riaz, A. B. Abdikamalov, D. Ayzenberg, C. Bambi, +H. Wang and Z. Yu, Astrophys. J. 925, 51 (2022) +doi:10.3847/1538-4357/ac3827 [arXiv:2012.07469 [astro- +ph.HE]]. +[93] Q. +Liu, +H. +Liu, +C. +Bambi +and +L. +Ji, +Mon. +Not. +Roy. +Astron. +Soc. +512, +2082-2092 +(2022) +doi:10.1093/mnras/stac616 +[arXiv:2111.00719 +[astro- +ph.HE]]. +[94] J. Garcia, T. Dauser, C. S. Reynolds, T. R. Kallman, +J. E. McClintock, J. Wilms and W. Eikmann, Astrophys. +J. 768, 146 (2013) doi:10.1088/0004-637X/768/2/146 +[arXiv:1303.2112 [astro-ph.HE]]. +[95] Z. Yu, +Q. Jiang, +A. B. Abdikamalov, +D. Ayzen- +berg, +C. +Bambi, +H. +Liu, +S. +Nampalliwar +and +A. +Tripathi, +Phys. +Rev. +D +104, +084035 +(2021) +doi:10.1103/PhysRevD.104.084035 +[arXiv:2106.11658 + +12 +[astro-ph.HE]]. +[96] S. N. Zhang et al. [eXTP], Proc. SPIE Int. Soc. +Opt. Eng. 9905, 99051Q (2016) doi:10.1117/12.2232034 +[arXiv:1607.08823 [astro-ph.IM]]. + diff --git a/_9FLT4oBgHgl3EQfwi-I/content/tmp_files/load_file.txt b/_9FLT4oBgHgl3EQfwi-I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34d75bb5298fd68a28d6b65b8d48a1183c2deb7a --- /dev/null +++ b/_9FLT4oBgHgl3EQfwi-I/content/tmp_files/load_file.txt @@ -0,0 +1,1523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf,len=1522 +page_content='Testing the δ-Kerr metric with black hole X-ray data Jiahao Tao,1 Shafqat Riaz,1 Biao Zhou,1 Askar B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 3 Cosimo Bambi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' ∗ and Daniele Malafarina4 1Center for Field Theory and Particle Physics and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 200438 Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' China 2Ulugh Beg Astronomical Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tashkent 100052,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Uzbekistan 3Institute of Fundamental and Applied Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' National Research University TIIAME,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tashkent 100000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Uzbekistan 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nazarbayev University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 010000 Astana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kazakhstan The spacetime around astrophysical black holes is thought to be described by the Kerr solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' However, even within general relativity, there is not yet a proof that the final product of the complete collapse of an uncharged body can only be a Kerr black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We can thus speculate on the possibility that the spacetime around astrophysical black holes may be described by other solutions of the Einstein Equations and we can test such a hypothesis with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In this work, we consider the δ-Kerr metric, which is an exact solution of the field equations in vacuum and can be obtained from a non-linear superposition of the Kerr metric with a static axially symmetric solution, often referred to as the δ-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The parameter δ = 1 + q quantifies the departure of the source from the Kerr metric and for q = 0 we recover the Kerr solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' From the analysis of the reflection features in the X-ray spectrum of the Galactic black hole in EXO 1846–031, we find −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 < q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='7 (90% CL), which is consistent with the hypothesis that the spacetime around the compact object in EXO 1846–031 is a Kerr black hole but does not entirely rule out the δ-Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' INTRODUCTION When a star exhausts all its nuclear fuel, the thermal pressure of the plasma cannot compensate the star’s own weight and the body shrinks to find a new equilibrium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If the collapsing part of the star exceeds the Oppenheimer-Volkoff limit, which is about 3 M⊙ and corresponds to the maximum mass for a neutron star, there is no known mechanism to stop the collapse and we have the formation of a “gravitationally collapsed ob- ject” [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' It is generally assumed that the final outcome of complete collapse should be a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' However, the exact nature of such a gravitationally collapsed object is not yet completely understood and therefore theoretical and observational studies to address this question are ac- tive lines of research nowadays [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In 4-dimensional general relativity, the only vacuum metric that is stationary, regular on and outside an event horizon, and asymptotically flat is the Kerr black hole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' This is the celebrated result of a family of uniqueness theorems, which were pioneered in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [6–8] and whose final version is still an ongoing research pro- gram [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In the ’60s, Roger Penrose proposed the Cosmic Censorship Conjecture, according to which all singulari- ties must be hidden behind an event horizon [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If this is true, within general relativity all gravitationally col- lapsed objects must be Kerr black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' However, even within general relativity, the Cosmic Censorship Conjec- ture is still unproven and, at the same time, we know exact solutions of the Einstein Equations that violate the Cosmic Censorship Conjecture and in which the complete collapse of a body leads to a spacetime with naked singu- larities [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A viewpoint that is widely accepted today ∗ Corresponding author: bambi@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='cn regarding the appearance of singularities in solutions of Einstein’s Equations is that they signal a regime where the theory fails and needs to be replaced by a new the- ory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In this sense, the study of singularities in general relativity may provide hints at the features that such a new theory must posses and how it may manifest in astrophysical phenomena [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' From astrophysical observations, we know at least two classes of gravitationally collapsed objects: stellar- mass compact objects with a mass exceeding the Oppenheimer-Volkoff limit and supermassive objects in galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The latter are simply too massive, com- pact, and old to be clusters of non-luminous bodies like neutron stars [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For both object classes, we have even a body of observations suggesting that these objects do not have a normal surface but an event horizon [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The past few years have seen a tremendous progress in our capability of testing the nature of these compact ob- jects and today we can use gravitational wave data from the LIGO-Virgo-KAGRA Collaboration [16–20], X-ray observations from a number of X-ray missions [21–27], and the mm images of the supermassive objects in M87∗ and Sgr A∗ from the Event Horizon Telescope Collabo- ration [28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In the present paper, we want to explore the possibil- ity that the spacetime around these gravitationally col- lapsed objects is described by the δ-Kerr metric [34, 35], which is an exact solution of the Einstein Equations that can be obtained from a non-linear superposition of the δ-metric (sometimes called Zipoy-Voorhees or γ- metric)[36–39] and the Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Such a solution, which can be understood as a stationary extension of the δ-metric or a deformed extension of the Kerr metric, has three independent parameters: the mass parameter M, which is related to the mass of the compact object, the spin parameter J, which is related to the angular arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='12164v1 [gr-qc] 28 Jan 2023 2 momentum of the source, and a deformation parameter, q = 1 − δ, which quantifies the departure from the Kerr solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For q = 0 and J ̸= 0, the δ-Kerr metric reduces to the Kerr solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For J = 0 and q ̸= 0, it reduces to the δ-metric, while a non-vanishing value of q and J corresponds to a stationary, axisymmetric, and asymp- totically flat vacuum solution of the Einstein Equations with a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The observational properties of the static δ-metric have been widely studied in the past few years [40–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' How- ever, since the δ-metric is static it does not constitute a good candidate for the gravitational field outside an as- trophysical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' On the other hand, the δ-Kerr metric is stationary and continuously linked to the Kerr met- ric through the value of the deformation parameter and therefore it is an ideal candidate to test the validity of the Kerr hypothesis around astrophysical compact ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The shadow of the δ-Kerr spacetime was studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [48], while its quasinormal modes were considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The δ-metric and the δ-Kerr metric violate the Cosmic Censorship Conjecture, and for this reason they are nor- mally not considered as viable solutions for the descrip- tion of the spacetime around gravitationally collapsed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Therefore in the absence of a proof of the Cos- mic Censorship Conjecture and/or adopting the idea that quantum gravity effects may resolve spacetime singulari- ties and make the Cosmic Censorship Conjecture unnec- essary [50, 51], it is worth to check whether we can test and rule out the δ-Kerr metric via astrophysical observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' To this aim, in this article we construct a reflection model for the δ-Kerr metric and we analyze a NuSTAR spectrum of the X-ray binary EXO 1846–031 with strong reflection features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' From the analysis of this observation, we can constrain the value of the deformation parameter q of the source and thus test the δ-Kerr spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The content of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In Section II, we briefly review the δ-Kerr metric and, in Section III, the analysis of the reflection features as a tool for testing the nature of gravitationally collapsed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In Sec- tion IV, we consider a NuSTAR observation of the X-ray binary EXO 1846–031 and we describe its data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In Section V, we present the spectral analysis of the NuS- TAR observation and from the analysis of the reflection features we constrain the deformation parameter q of the δ-Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Summary and conclusions are reported in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In the present manuscript, we adopt natural units with GN = c = 1 and the convention of a metric with signature (− + ++).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' δ-KERR METRIC The δ-Kerr metric was derived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [34, 35] and can be obtained as a non-linear superposition of the δ-metric and the Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In Boyer-Lindquist-like coordinates (t, r, θ, φ), the line element of the δ-Kerr metric is [49] ds2 = −Fdt2 + 2Fωdtdφ + e2γ F B Adr2+ + r2 e2γ F Bdθ2 + �r2 F A sin2 θ − Fω2 � dφ2 , (1) where A = 1 − 2M r + a2 r2 , B = A + σ2 sin2 θ r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' (2) Here M is the mass parameter, which is related to the gravitational mass of the compact object, J is its spin parameter, related to the object’s angular momentum, a = J/M (while the dimensionless spin parameter is a∗ = a/M), and σ = √ M 2 − a2 > 0 is a constant length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' F, ω, and γ are functions of the prolate spheroidal coordinates x = (r − M) /σ and y = cos θ: F = A B , ω = 2 � a − σ C A � , e2γ = 1 4 � 1 + M σ �2 A (x2 − 1)δ � x2 − 1 x2 − y2 �δ2 , (3) where A = a+a− + b+b− , B = a2 + + b2 + , C = (x + 1)q � x � 1 − y2� (λ + η) a++ +y � x2 − 1 � (1 − λη) b+ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' (4) a± = (x ± 1)q [x (1 − λη) ± (1 + λη)] , b± = (x ± 1)q [y (λ + η) ∓ (λ − η)] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' (5) λ = α � x2 − 1 �−q (x + y)2q , η = α � x2 − 1 �−q (x − y)2q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' (6) q = δ − 1 , α = M − σ a = a M + σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' (7) Contrary to the Schwarzschild solution, for q ̸= 0 the δ-Kerr spacetime has a non vanishing mass quadrupole moment even when a = 0 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For q = 0, we have λ = η = α, a± = 2α (r − M ± M) /a, and b± = 2α cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Therefore, we can get A = 4α2 a2 (Σ − 2Mr) , B = 4α2 a2 Σ , C = 4α2 aσ � Σ − Mr � 1 + cos2 θ �� , (8) 3 and F = 1 − 2Mr Σ , ω = −2Mar sin2 θ Σ − 2Mr , e2γ = Σ − 2Mr Σ − 2Mr + M 2 sin2 θ , (9) where Σ = r2 + a2 cos2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If we plug these expressions in the line element in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' (1), we recover the familiar Kerr solution in Boyer-Lindquist coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Finally note that for a = 0 and q = 0 we retrieve the Schwarzschild metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' X-RAY REFLECTION SPECTROSCOPY Relativistically blurred reflection features are common in the X-ray spectra of accreting black holes [52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' These features are produced by illumination of a “cold” disk by a “hot” corona [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The astrophysical system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The accretion disk around the black hole is optically thin and geometrically thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The gas in the disk is in local thermal equilibrium and every point on the surface of the disk emits a blackbody-like spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The whole disk has a multi-temperature blackbody-like spectrum because the temperature of the gas increases approaching the central object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The thermal spectrum of the accretion disk is normally peaked in the soft X-ray band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1-10 keV) in the case of stellar-mass black holes in X-ray binary systems and in the UV band (1-100 eV) in the case of supermassive black holes in active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The corona is some hotter plasma (∼ 100 keV) near the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Thermal photons from the accre- tion disk can inverse Compton scatter off free electrons in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The Comptonized photons can illuminate the disk: Compton scattering and absorption followed by fluorescent emission generate the reflection spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In the rest-frame of the gas in the disk, the reflection spectrum is characterized by narrow fluorescent emission lines below 10 keV and a Compton hump peaking at 20- 30 keV [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The reflection spectrum of the whole disk detected by a distant observer is blurred because it is the result of photons coming from all points of the ac- cretion disk and every point of the disk is characterized by its own redshift factor, resulting from the combina- tion of gravitational redshift and Doppler boosting [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' X-ray reflection spectroscopy refers to the analysis of the reflection features in the X-ray spectra of accreting black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In the presence of high-quality data and the cor- rect astrophysical model, X-ray reflection spectroscopy can be a powerful technique to probe the strong gravity regions around black hole candidates [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The idea of using the analysis of reflection features to test the nature of gravitationally collapsed objects and the Kerr black hole hypothesis was discussed for the first time in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [59] and further explored by other authors Black Hole Accretion Disk Corona Thermal Photons Reflection Photons Comptonized Photons FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Disk-corona system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Figure from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [56] under the terms of the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [60–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In those early works, it was only stud- ied the shape of the iron Kα line, which is often one of the most prominent features in the reflection spectrum and certainly its most informative part about the space- time metric in the strong gravity region around the com- pact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' However, none of those models was suitable to analyze real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A breakthrough in this field was the development of the reflection model relxill nk [65– 67], which is an extension of the popular relxill pack- age [68, 69] for non-Kerr spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' relxill nk has been extensively used in the past few years to test the Kerr black hole hypothesis and specific modified theo- ries of gravity in which rotating compact objects are not described by the Kerr solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [70–73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The state- of-the-art in the field is reviewed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In general, the observed flux of an accretion disk around a compact object can be calculated as F(Eo) = 1 D2 � dX dY Io(X, Y ) = = 1 D2 � dX dY Ie(Eo, re, ϑe) , (10) where Io and Ie are the specific intensity of the radia- tion as measured, respectively, by the distant observer and in the rest-frame of the gas in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' X and Y are the Cartesian coordinates of the image of the disk in the plane of the distant observer and D is the distance of the observer from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Io = g3Ie follows from Liouville’s theorem, g = Eo/Ee is the redshift factor, and Eo and Ee are the photon energies as measured, respec- tively, by the distant observer and in the rest-frame of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Here re is the emission radius on the disk and ϑe is the emission angle, which may differ from the incli- nation angle of the disk with respect to the line of sight of the distant observer, i, because of light bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The natural way to calculate the observed flux F(Eo) is to consider a grid on the plane of the distant observer and follow the trajectories of photons backwards in time from 4 every point of the grid to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In this way, we con- nect every point of the image of the disk on the plane of the distant observer with its actual emission point on the disk, we can calculate the redshift factor g, and, if we know the local spectrum, we can calculate the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In practice, this approach is not doable because the ray-tracing calculations are too time consuming to be done during the data analysis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The current strat- egy in most reflection models, including also relxill nk, is to introduce the “transfer function” and rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' (10) as (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [5, 65]) F(Eo) = 1 D2 � rout rin dre � 1 0 dg∗ πreg2 � g∗(1 − g∗) ×f(g∗, re, i) Ie(Eo, re, ϑe) , (11) where rin and rout are, respectively, the inner and the outer edges of the disk, f is the transfer function [74] f(g∗, re, i) = g � g∗(1 − g∗) πre J(X, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' g∗, re) , (12) g∗ is the relative redshift at the emission radius re for an observer with viewing angle i g∗ = g − gmin gmax − gmin , (13) and gmin = gmin(re, i) and gmax = gmax(re, i) are, re- spectively, the minimum and the maximum values of the redshift factor g for photons emitted at the emission ra- dius re and detected by an observer with viewing angle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Finally J(X, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' g∗, re) is the Jacobian between the Carte- sian coordinates of the image of the disk in the plane of the distant observer and the two variables re and g∗ used to map the emission points on the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The transfer function and the non-relativistic reflection spectrum can be calculated before the data analysis pro- cess on a computer cluster and tabulated in FITS files for a grid of their input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' During the data analy- sis process, the model calls the FITS files and can quickly calculate the integral to obtain the observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If we want to construct a model for a different spacetime metric, we just need to replace the old FITS file of the transfer function with a new one, which is calculated for the new metric of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' This is what we have done to implement the δ-Kerr metric in relxill nk: we have considered a grid of spin parameters a∗, viewing angles i, and deformation parameters q and for every point of the grid we have calculated the transfer function with a ray-tracing code for 100 values of re and 40 values of g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If, instead of the full non-relativistic reflection spec- trum, we consider only a narrow iron line at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='4 keV in Ie, the calculation of F(Eo) produces relativistically broad- ened iron lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' While any precise measurement from the analysis of the reflection features requires to consider the full reflection spectrum and not only an iron line, a single iron line can show better the impact of the param- eter q on the shape of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 2 shows some relativistically broadened iron lines for two values of the inclination angle (i = 30◦ and 60◦) and two values of the spin parameter (a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='7 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For every line, the emissivity profile is supposed to be a power law with emissivity index p = 8, the inner edge of the disk is set at the radius of the innermost stable circular orbit (ISCO), and the outer edge of the disk is set at 400 gravitational radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In every panel, we show the iron line for q = 0 (Kerr spacetime), ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='5, and ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Notice that the value q = −1, corresponding to δ = 0 is the limiting case of an extremely flattened source, which in the static case corresponds to the Curzon solution [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' OBSERVATION AND DATA REDUCTION EXO 1846–031 is a low mass X-ray binary [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' It was discovered by the European X-ray Observatory Satellite (EXOSAT) on April 3, 1985 [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A second outburst was detected by CGRO/BATSE in 1994 [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' After be- ing in quiescence for about 25 years, the source had a new outburst in 2019, which was first detected by MAXI on July 23 [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' This third outburst was then observed by other instruments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=', Swift/XRT [80], VLA [81], and MeerKAT [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The Nuclear Spectroscopic Telescope Array mission (NuSTAR) [83] observed EXO 1846–031 on August 3, 2019 (observation ID 90501334002) with a 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2 ks exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In what follows, we will consider this NuSTAR observation, which was first analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For the data reduction, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' NuSTAR has two detectors, which are called Focal Plane Mod- ules (FPM) A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We download the raw data from the HEASARC website and use the HEASOFT module nupipeline to convert the raw data into cleaned event files with NuSTARDAS and the CALDB 20220301 cali- bration database, so that we can get the source and back- ground information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For the source, we select a 180 arc- seconds radius circular region at the center of the source for both FPMA and FPMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' For the background, we take a region of the same size of the source as far as possible from the source but on the same detector, so that the influence of the source’s photons can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Af- terwards, we use the HEASOFT module nuproducts to generate the source and background spectra, the response matrix file, and the ancillary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Last, we use grppha to group the spectra to have at least 30 counts per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Since the new CALDB corrects the calibration in the 3-7 keV energy range, we do not need the table nuMLIv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='mod used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' SPECTRAL ANALYSIS For the spectral analysis, we use XSPEC v12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' First, we fit the data with an absorbed power law to see the reflection features in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In XSPEC language, the model reads const × tbabs × (diskbb + cutoffpl) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='06 Flux [arbitrary units] i = 30◦ Kerr : q = 0 q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='5 q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='5 q = 1 q = −1 a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='7 i = 60◦ 0 2 4 6 8 10 E [keV ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0150 Flux [arbitrary units] 0 2 4 6 8 10 E [keV ] a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='998 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Iron line profiles in δ-Kerr spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The inclination angle of the disk with respect to the line of sight of the distant observer is i = 30◦ (left panels) and 60◦ (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The dimensionless spin parameter is a∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='7 (top panels) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='998 (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' These profiles are calculated assuming that the emissivity profile is described by a power law with emissivity index p = 8, the inner edge of the disk is at the ISCO radius, and the outer edge is at 400 rg, where rg is the gravitational radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' const is used to have a cross-calibration constant be- tween the detectors FPMA and FPMB: the constant is frozen to 1 for FPMA and is free for FPMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' tbabs de- scribes the Galactic absorption [86]: the hydrogen col- umn density, NH, is the only parameter of the model and is left free in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' diskbb describes the thermal spec- trum of the accretion disk [87]: the temperature at the inner edge of the disk, Tin, and the normalization of the component are left free in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' cutoffpl describes the continuum from the corona: the photon index, Γ, the high energy cutoff, Ecut, and the normalization of this component are left free in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The ratio between the data and the best-fit model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 3 and we clearly see unresolved strong reflection features: a broad- ened iron Kα line peaking around 7 keV and a Compton hump peaking at 20-30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Such a strong blurred re- flection features suggest that this NuSTAR spectrum is suitable to test the nature of the gravitationally collapsed object in EXO 1846–031 with relxill nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' To fit the reflection features, we add relxill nk to the total model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We employ the flavor relxillion nk, which describes the relativistically blurred reflection spectrum of an accretion disk with a non-trivial ionization gradi- ent [88, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In XSPEC language, the total model now reads const × tbabs × (diskbb + relxillion nk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' relxill nk has several parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The spacetime met- ric is described by the spin a∗ and the deformation pa- rameter q of the δ-Kerr metric and both parameters are left free in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The inner edge of the accretion disk is set at the ISCO and therefore it is not a free param- eter but directly depends on the values a∗ and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The 6 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2 Ratio FPMA FPMB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Data to best-fit model ratio for an absorbed power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We clearly see a broadened iron line peaking around 7 keV and a Compton hump peaking at 20-30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' outer edge of the disk is fixed to 900 rg, where rg is the gravitational radius and 900 rg is the maximum value al- lowed by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The emissivity profile of the accre- tion disk can potentially be described by a twice broken power law and there are thus five parameters: the emis- sivity indices of the inner, central, and outer regions (p1, p2, and p3, respectively) and the breaking radii between the inner and the central parts, Rbr1, and between the central and outer parts, Rbr21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' To model the emissivity profile with a broken power law (instead of a twice broken power law), we simply set p2 = p3 and Rbr1 = Rbr2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=', the central region collapses and we have only the inner and outer regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The viewing angle, i, the iron abun- dance, AFe, the ionization at the inner edge of the disk, ξin, and the ionization index, αξ, are all free parameters in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The model includes the continuum from the corona and the reflection fraction, Rf, regulates the rel- ative strength between the reflection component and the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The photon index, Γ, and the high-energy cutoff, Ecut, of the continuum illuminating the disk are free in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' From previous analyses [88, 89], we know that this spectrum requires a non-vanishing ionization gradient and for this reason we use the flavor relxillion nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If we fit the data with a model with a disk with con- stant ionization, we need to add a Gaussian to the total model [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We fit the data with four models (Models 1- 4), which are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' In our first fit (Model 1), we fit the data assuming that the emissivity profile of the disk is described by a 1 This means that the emission of the disk scales as r−p1 in the inner part (r < Rbr1), as r−p2 in the central part (Rbr1 < r < Rbr2), and as r−p3 in the outer part (r > Rbr2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' broken power law (so p2 = p3 and Rbr1 = Rbr2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The best-fit values are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The best-fit model and the data to best-fit model ratio are shown in the top-left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' As we can see from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' II, we find a very high emissivity index for the inner region of the accretion disk and an almost vanishing emissivity in- dex for the outer part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Such an emissivity profile may be generated by a corona covering a large portion of the accretion disk [90–93] and the data may prefer a twice broken power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' As Model 2, we thus fit the spectrum with a twice broken power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The best-fit values are reported in the third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' II and the best-fit model and the data to best-fit model ratio are shown in the top-right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We do not see any improve- ment in the fit and Rbr2 is stuck at the outer edge of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Unfortunately, for the outer edge of the disk we have already chosen the maximum value allowed by the model and we cannot try to fit the data with a larger disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' As Model 3, we reconsider an emissivity pro- file described by a broken power law, but this time we freeze the emissivity index of the outer region of the disk to 3, which is the value normally expected for the outer emissivity index when the corona is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The results are still shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' II and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 4, but the fit is clearly worse with ∆χ2 = +65 with respect to Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Last, we consider the possibility of the presence of a distant cold reflector and we add xillver [94] to the total model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' II, these data clearly do not require any distant reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS This observation of EXO 1846–031 has been already analyzed and extensively discussed in the literature [84, 88, 89, 95], so here we will focus on the possibility to use such observation to constrain the value of the deforma- tion parameter q of the δ-Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Model 1 is the simplest model and fits the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Our constraint on q is (90% confidence level, only statis- tical uncertainty) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 < q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='7 , (14) and therefore our analysis is consistent with the hy- pothesis that the gravitationally collapsed object in the X-ray binary EXO 1846–031 is a Kerr black hole (for which q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' However, our analysis does not exclude q = O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1) and therefore natural values of the defor- mation parameter are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Also notice that positive value of q implies that a non rotating source is oblate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Therefore the above bounds on q are consistent with a spinning oblate compact object, which is more physically realistic than a prolate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' q < 0) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' From Model 1, we find that the emissivity profile is very steep around the central object and almost flat at larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' While this is not the emissivity profile ex- pected from a compact corona, it is common in Galac- tic black holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' see, for instance, the discussion in [93] 7 Model XSPEC Model Emissivity Profile 1 tbabs×(diskbb+relxillion nk) p1, p2, Rbr1 2 tbabs×(diskbb+relxillion nk) p1, p2, p3 = 3, Rbr1, Rbr2 3 tbabs×(diskbb+relxillion nk) p1, p2 = 3, Rbr1 4 tbabs×(diskbb+relxill nk+xillver) p1, p2, Rbr1 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Summary of the models used in the spectral analysis of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Model 1 2 3 4 tbabs NH/1022 cm−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='4 4.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='89−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='57+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='5 Norm/10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='58+0.' metadata={'source': 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+page_content='0014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0152+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0152+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0152+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0014 χ2/ν 2659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='62/2599 2659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='59/2598 2724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='98/2600 2659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='41/2598 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='02332 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='02371 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04807 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='02364 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Best-fit table of Models 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The reported uncertainties correspond to the 90% confidence level for one relevant parameter (∆χ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' ∗ means the value of the parameter is frozen during the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' When there is no lower/upper uncertainty, the boundary of the range in which the parameter is allowed to vary is within the 90% confidence limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 8 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 1 keV²(phs/cm²/s/keV) FPMA FPMB 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05 Ratio Model 1 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 1 keV²(phs/cm²/s/keV) FPMA FPMB 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05 Ratio Model 2 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 1 keV²(phs/cm²/s/keV) FPMA FPMB 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05 Ratio Model 3 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1 1 keV²(phs/cm²/s/keV) FPMA FPMB 5 10 20 50 Energy (keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05 Ratio Model 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Best-fit model and data to best-fit model ratio for Models 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 9 and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If we try to fit the data with a twice broken power law (Model 2) or by adding a non- relativistic reflection component (Model 4), we do not see any significant difference: the value of the second break- ing radius would be large and the normalization of the non-relativistic reflection component would be very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The estimate of the model parameters are thus consis- tent with the measurements inferred with Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' If we model the emissivity profile with a broken power law and we impose that the outer emissivity index is 3, the esti- mate of some model parameters would be different (and we find that the spacetime significantly deviates from the Kerr solution!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' ), but the fit is definitively worse and Model 3 can be ruled out (∆χ2 = +65 with respect to Model 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We are not aware of other tests of the δ-Kerr metric published in the literature and observational constraints on the deformation parameter q, though models for the shadow and quasinormal modes have been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Since the δ-Kerr spacetime is an ex- act vacuum solutions of the field equations in general relativity which relates to the Kerr black hole through the variation of one continuous parameter with a clear physical interpretation, we would argue that experimen- tal tests to constrain the allowed values of q from obser- vations are important towards a possible resolution of the Kerr hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' We could certainly constrain q from the available gravitational wave data from the LIGO-Virgo- KAGRA Collaboration following the approach employed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [19, 20] for testing other non-Kerr metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' The deformation parameter q may also be constrained from the available mm black hole images from the Event Hori- zon Telescope Collaboration (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' While these three techniques (X-ray, gravitational waves, and black hole imaging) are sensitive to different relativistic effects, in general, X-ray tests are those that can pro- vide the most stringent constraints on possible deviations from the Kerr solution, while gravitational wave con- straints are normally a bit weaker and black hole imaging constraints are more than an order of magnitude weaker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' see, for example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' This is the typical situation with the current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' However, gravitational wave con- straints are expected to improve quickly in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Concerning X-ray tests, the constraint reported in the present work is likely close to the best that we can do today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Somewhat more stringent constraints may be ob- tained from sources in which we can test the Kerr metric from the simultaneous analysis of the reflection features and the thermal spectrum, as done in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [23, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' This is not possible for the NuSTAR spectrum analyzed here because the thermal component is too weak and we do not have independent measurements of the mass and distance of the compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' More stringent con- straints on the deformation parameter q require higher quality data, which will be available from the next gen- eration of X-ray missions, starting from eXTP [96], which is currently scheduled to be launched in 2027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Acknowledgments – This work was supported by the National Natural Science Foundation of China (NSFC), Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 12250610185, 11973019, and 12261131497, the Natural Science Foundation of Shang- hai, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 22ZR1403400, the Shanghai Municipal Education Commission, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 2019-01-07-00-07- E00035, and Fudan University, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' JIH1512604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' DM acknowledges support from Nazarbayev University Faculty Development Competitive Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 11022021FD2926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ruffini and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bonazzola, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 187, 1767-1783 (1969) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1767 [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lattimer, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 62, 485-515 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1146/annurev-nucl-102711-095018 [arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3510 [nucl-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 89, 025001 (2017) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='025001 [arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='03884 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Yagi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Stein, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 33, 054001 (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0264-9381/33/5/054001 [arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='02413 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Black Holes: A Laboratory for Testing Strong Gravity (Springer Singapore, 2017), ISBN 978-981-10- 4524-0 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1007/978-981-10-4524-0 [6] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Israel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 164, 1776-1779 (1967) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1776 [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Carter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 26, 331-333 (1971) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='331 [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Robinson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 34, 905-906 (1975) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='905 [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Chrusciel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lopes Costa and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Heusler, Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 15, 7 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='12942/lrr-2012-7 [arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='6112 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Penrose, Riv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 1, 252-276 (1969) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1023/A:1016578408204 [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Joshi and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 20, 2641-2729 (2011) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1142/S0218271811020792 [arXiv:1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3660 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina, Universe 3, 48 (2017) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3390/universe3020048 [arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04138 [gr- qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [13] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Maoz, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 494, L181-L184 (1998) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1086/311194 [arXiv:astro-ph/9710309 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Narayan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' McClintock, New Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 51, 733-751 (2008) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='newar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='002 [arXiv:0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0322 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Broderick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Loeb and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Narayan, As- trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 701, 1357-1366 (2009) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0004- 637X/701/2/1357 [arXiv:0903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1105 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [LIGO Scientific and Virgo], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 116, 221101 (2016) [erra- tum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 121, 129902 (2018)] doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='221101 [arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='03841 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 10 [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Yunes, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Yagi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Pretorius, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 94, 084002 (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='084002 [arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='08955 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [LIGO Scientific and Virgo], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 100, 104036 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='104036 [arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04467 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Cardenas-Avendano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Yunes, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 37, 135008 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/1361- 6382/ab8f64 [arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='08062 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Shashank and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 105, 104004 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='104004 [arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05388 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [21] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Cao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Garcia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 120, 051101 (2018) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='051101 [arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='00219 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Gar- cia and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Marinucci, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 875, 56 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/ab0e7e [arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='08148 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grinberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, Astro- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 907, 31 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/abccbd [arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='13474 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzen- berg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, Astro- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 913, 79 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/abf6cd [arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='10669 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [25] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 924, 72 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/ac350e [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='03086 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grinberg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, JCAP 01, 019 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/1475-7516/2022/01/019 [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='10982 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, [arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05322 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Freese, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Vagnozzi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Visinelli, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 100, 044057 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='044057 [arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='12983 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wei and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Plus 136, 436 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1140/epjp/s13360-021-01398-9 [arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='07769 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kumar and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ghosh, JCAP 07, 053 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/1475-7516/2020/07/053 [arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='08927 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Psaltis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [Event Horizon Tele- scope], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 125, 141104 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='141104 [arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01055 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [Event Horizon Telescope], Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 930, L17 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/2041-8213/ac6756 [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Vagnozzi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tsai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Visinelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Afrin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Allahyari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambhaniya, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ghosh and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Joshi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='07787 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Quevedo and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mashhoon, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A 109, 13-18 (1985) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1016/0375-9601(85)90381-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [35] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Quevedo and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mashhoon, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 43, 3902 (1991) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [36] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zipoy, Journ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 22, 1137 (1970) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1705005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [37] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Voorhees, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 2, 2119 (1970) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [38] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Papadopoulos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Stewart and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Witten, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 24, 320 (1981) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [39] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Herrera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Paiva and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Santos, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 9, 649 (2000) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1142/S021827180000061X, [40] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Herrera, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 17, 557 (2008) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1142/S0218271808012255 [arXiv:0711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='4723 [gr- qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [41] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Herrera, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Magli and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 37, 1371 (2005) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1007/s10714-005-0120-1 [arXiv:gr-qc/0407037].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Chowdhury, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Patil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Joshi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 85, 104031 (2012) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='104031 [arXiv:1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2522 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdujabbarov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ahmedov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 100, 024014 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='024014 [arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='06207 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [44] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Boshkayev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Gasperin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Guti´errez-Pi˜neres, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Quevedo and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Toktarbay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 93, 024024 (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='024024 [arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='03827 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [45] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Benavides-Gallego, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdujabbarov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mala- farina, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ahmedov and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 99, 044012 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='044012 [arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04846 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [46] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Toshmatov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dadhich, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 100, 044001 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='044001 [arXiv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01088 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [47] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Toshmatov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 100, 104052 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='104052 [arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='11565 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mirzaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdujabbarov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafa- rina, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ahmedov and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 106, 084041 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='084041 [arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='10933 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Allahyari, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Firouzjahi and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mashhoon, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 37, 055006 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/1361- 6382/ab6860 [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='10813 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [50] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Gimon and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Horava, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B 672, 299-302 (2009) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='026 [arXiv:0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2873 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [51] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Freese, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 79, 043002 (2009) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='043002 [arXiv:0812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1328 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [52] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tanaka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nandra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fabian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Inoue, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Otani, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dotani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Hayashida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Iwasawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kii and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kunieda, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nature 375, 659 (1995) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1038/375659a0 [53] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nandra, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' O’Neill, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' George and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Reeves, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 382, 194 (2007) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='12331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='x [arXiv:0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1305 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Miller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Reynolds, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fabian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Miniutti and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Gallo, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 697, 900-912 (2009) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0004-637X/697/1/900 [arXiv:0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2840 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [55] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Brenneman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Gar- cia, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grinberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ingram, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lohfink and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Marinucci, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 217, 65 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1007/s11214-021-00841-8 [arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04792 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 11 [56] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Arab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 11, 81-90 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1007/s40065-021-00336-y [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04084 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [57] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ross and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fabian, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 358, 211-216 (2005) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1365- 2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='08797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='x [arXiv:astro-ph/0501116 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [58] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Garcia and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kallman, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 718, 695 (2010) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0004-637X/718/2/695 [arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0485 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [59] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lu and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Torres, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 12, 63- 78 (2003) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1142/S0218271803002718 [arXiv:astro- ph/0205418 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [60] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Schee and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Stuchlik, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 41, 1795-1818 (2009) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1007/s10714-008-0753-y [arXiv:0812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3017 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [61] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Johannsen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Psaltis, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 773, 57 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0004-637X/773/1/57 [arXiv:1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='6069 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [62] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 87, 023007 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='023007 [arXiv:1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2513 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [63] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 87, 084039 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='084039 [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='0624 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [64] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Malafarina, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 88, 064022 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='064022 [arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2106 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [65] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Cardenas-Avendano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Gar- cia and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 842, 76 (2017) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/aa74c0 [arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='00596 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [66] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Garcia and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 878, 91 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/ab1f89 [arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='09665 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [67] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Garcia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 899, 80 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538- 4357/aba625 [arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='09663 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [68] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Garcia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wilms, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bock, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bren- neman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Falanga, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fukumura and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Reynolds, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 430, 1694 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1093/mnras/sts710 [arXiv:1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='4922 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [69] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Garc´ıa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lohfink, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kallman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Steiner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' McClintock, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Brenneman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wilms, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Eikmann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Reynolds, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 782, 76 (2014) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0004-637X/782/2/76 [arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3231 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [70] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Cao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Modesto and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 98, 024007 (2018) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='024007 [arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='07849 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [71] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Azreg- Ainou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Jamil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tri- pathi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C 80, 622 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1140/epjc/s10052-020-8198-x [arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='00184 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [72] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi, JCAP 01, 047 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/1475-7516/2021/01/047 [arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='12958 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [73] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Riaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Shashank, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdika- malov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhang and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhou, JCAP 10, 040 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/1475- 7516/2022/10/040 [arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='03729 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [74] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Cunningham, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 202, 788-802 (1975) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1086/154033 [75] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Curzon, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' London Math Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 23, 477 (1924) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1112/plms/s2-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [76] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Parmar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Angelini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roche and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' White, A&A 279, 179 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [77] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Parmar and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' White, IAU Circ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 4051, 1 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [78] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Harmon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wilson and et al, IAU Circ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 6096, 1 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [79] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Negoro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nakajima, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Sugita and et al, The As- tronomer’s Telegram 12968, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [80] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mereminskiy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Krivonos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Medvedev and et al, The Astronomer’s Telegram 12969, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [81] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Miller-Jones, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Russell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Sivakoff and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tetarenko, The Astronomer’s Telegram 12977, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [82] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Williams, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fender, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Woudt and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Miller-Jones, The Astronomer’s Telegram 12992, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [83] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [NuSTAR], Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 770, 103 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0004-637X/770/2/103 [arXiv:1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='7307 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='IM]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [84] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Draghis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Miller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Cackett, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kam- moun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Reynolds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tomsick and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zoghbi, As- trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 900, 78 (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/aba2ec [arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='04324 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [85] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Arnaud, ASP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 101 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [86] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wilms, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Allen and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' McCray, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 542, 914-924 (2000) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1086/317016 [arXiv:astro- ph/0008425 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [87] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Makishima, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Maejima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Mitsuda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bradt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Remillard, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tuohy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Hoshi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nakagawa, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 308, 635-643 (1986) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1086/164534 [88] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, H.' metadata={'source': 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[arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='10100 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [89] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 923, 175 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/ac3237 [arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='00375 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [90] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Miniutti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Fabian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Goyder and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Lasenby, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 344, L22-L26 (2003) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1046/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1365-8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='06988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='x [arXiv:astro- ph/0307163 [astro-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [91] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wilkins and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Gallo, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 449, 129-146 (2015) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1093/mnras/stv162 [arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='05302 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [92] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Riaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Yu, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 925, 51 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='3847/1538-4357/ac3827 [arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='07469 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [93] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ji, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 512, 2082-2092 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1093/mnras/stac616 [arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='00719 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [94] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Garcia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Dauser, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Reynolds, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Kallman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' McClintock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Wilms and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Eikmann, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 768, 146 (2013) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1088/0004-637X/768/2/146 [arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2112 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [95] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Yu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Jiang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Abdikamalov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Ayzen- berg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Bambi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Nampalliwar and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Tripathi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' D 104, 084035 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='084035 [arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='11658 12 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [96] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' [eXTP], Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' SPIE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content=' 9905, 99051Q (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='2232034 [arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='08823 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} +page_content='IM]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FLT4oBgHgl3EQfwi-I/content/2301.12164v1.pdf'} diff --git a/_NE1T4oBgHgl3EQfogR8/content/2301.03321v1.pdf b/_NE1T4oBgHgl3EQfogR8/content/2301.03321v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e099dbbeff91e1e9cfc9080eb0f97b12f46cc380 --- /dev/null +++ b/_NE1T4oBgHgl3EQfogR8/content/2301.03321v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/atFIT4oBgHgl3EQflys0/content/tmp_files/2301.11306v1.pdf.txt b/atFIT4oBgHgl3EQflys0/content/tmp_files/2301.11306v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..06ba4f376f9ea0d5f0efdac4c253bc0e998f7eb8 --- /dev/null +++ b/atFIT4oBgHgl3EQflys0/content/tmp_files/2301.11306v1.pdf.txt @@ -0,0 +1,1499 @@ +arXiv:2301.11306v1 [math.CO] 26 Jan 2023 +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +Abstract. We establish that if d ≥ 2k + 6 and q is odd and sufficiently large with respect to α ∈ (0, 1), +then every set A ⊆ Fd +q of size |A| ≥ αqd will contain an isometric copy of every spherical (k + 2)-point +configuration that spans k dimensions. +1. Introduction +Geometric Ramsey theory has its origins in series of papers by Erd˝os et al. [6, 7, 8], where they studied +geometric configurations which cannot be destroyed by partitioning Euclidean space into finitely many classes. +The fundamental problem is to classify those finite sets X which are Ramsey, in the sense that for every +number of colors r ∈ N there is a dimension d = d(r, X) for which every r-coloring of Rd contains a +monochromatic, congruent copy of X. +The simplest example of a Ramsey set is a regular k-simplex; that is, k + 1 equidistant points. Indeed, for +any dimension d ≥ kr, any r-coloring of a regular d-simplex contains k + 1 points of the same color, forming +a monochromatic regular k-simplex. On the other hand, a simple construction using the geometry of the +Euclidean metric shows that any set of three collinear points is not Ramsey. In fact, Erd˝os et al. [6] showed +that every Ramsey set must be spherical; that is, contained in some sphere. This has led to the conjecture +by Graham [13] that a finite set X is Ramsey if and only if it is spherical. +This conjecture is far from settled. Examples of sets known to be Ramsey include vertices of “bricks” +(k-dimensional rectangles) [6], non-degenerate simplices [9], trapezoids [18], regular polygons and regular +polyhedra [17]. +Common to many of these results was the exploitation of additional symmetries of the +configuration. It was observed by Leader, Russell and Walters [20] that all known examples of Ramsey sets +are subtransitive in the sense that they can be embedded in a higher dimensional set on which the rotation +group acts transitively. They introduced a rival conjecture that a finite set X is Ramsey if and only if it is +subtransitive, and further showed [19] that almost all 4-point subsets of a circle are not subtransitive. This +was later extended by Eberhard [5] to show that almost all (k + 2)-point sets on the (k − 1)-sphere are not +subtransitive. It remains an open question whether or not such configurations are Ramsey. +The aim of this article is to show that an analogue of Graham’s conjecture holds in finite field geometries +for 4-point spherical configurations spanning two dimensions, and more generally for spherical (k + 2)-point +configurations spanning k dimensions. We in fact prove a stronger density version; namely that if d ≥ 2k + 6 +and q is taken to be odd and sufficiently large with respect to α ∈ (0, 1), then every set A ⊆ Fd +q of size +|A| ≥ αqd contains an isometric copy of every such configuration X. To be clear, here we say that two sets +X and X′ are isometric if there is a bijection φ : X → X′ such that |φ(x) − φ(x′)|2 = |x − x′|2 for all x ∈ X +and x′ ∈ X′, where |x|2 = x · x is the usual dot product of the vector x ∈ Fd +q with itself. +Our approach takes the point of view modern arithmetic combinatorics which has been very successful +in the study of linear patterns in subsets of Z of positive density [10, 11]. In fact, one of the main purpose +of this article is to extend these techniques to the setting of geometric Ramsey theory, where one counts +configurations determined by both linear as well as certain non-linear relations, i.e. isometries. +The setting of vector spaces over finite fields provides a useful model to study many problems in arithmetic +combinatorics; see especially the surveys [14, 25, 28]. In the context of geometric Ramsey theory over finite +fields, notable results have been obtained by a number of authors [3, 15, 16, 24]. However, those results +2010 Mathematics Subject Classification. 11B30. +The first author was partially supported by NSF-DMS grant 1702411 and Simons Foundation Collaboration Grant for +Mathematicians 245792. The second author was partially supported by Grants NSF-DMS 1600840 and ERC-AdG 321104. +1 + +2 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +concern patterns consisting of points in general position, with no linear relations between them, and hence +are fundamentally different. +Analogous results for simplices in Euclidean spaces and the integer lattice have been given in [1, 22], and +it reasonable to expect that our approach here may be successfully adapted to these settings. In the context +of geometric (density) Ramsey theory in Rd, some results using this approach were recently obtained in +[4, 21]. We hope to address further adaptation in the near future. +1.1. Outline of paper. The main results of the paper are stated is Section 2 below, and some preliminaries +and reductions are presented in Sections 3 and 4. +A key observation of the paper, see Proposition 6 in Section 5.1, is that the count of isometric copies of a +fixed configuration X along bounded functions is controlled by a certain uniformity norm. This norm, which +we introduce in Section 5.1, measures the uniformity or randomness of a function along geometric rectangles, +and it should be compared with the so-called U 2-uniformity norm of Gowers [10] which measures uniformity +along combinatorial rectangles. If a set A is sufficiently uniform with respect to this norm, then it quickly +follows from Proposition 6 that A contains many, in fact the statistically correct number of, isometric copies +of X. The proof of Proposition 6 is presented in Section 6. +In order to handle arbitrary sets A we prove an inverse theorem, see Theorem 10 in Section 7.1, to establish +that functions with large uniformity norm correlate with structured sets. Given such an inverse theorem +there are then various standard iterative procedures that one may hope to adapt to this setting to complete +the argument. We follow an energy increment route which to leads a so-called arithmetic regularity lemma, +namely Proposition 7, that allows us to decompose the indicator function of A as 1A = f1 + f2 + f3, where +f1 is highly structured, f2 has small L2 norm and f3 has small uniformity norm. The proof of Proposition 7 +is presented in Section 7. +In Sections 5.2 and 5.3 we demonstrate how Proposition 7 leads to a proof of our main results. This +consists of counting the isometric copies of X along the main term f1 and showing the contribution of the +functions f2 and f3 are negligible. The setback of this approach is that it leads to very weak bounds, in fact +the dependence of q on α is tower-exponential. It seems quite possible that one could instead proceed via a +density increment argument and obtain better, exponential type bounds, but we do not pursue this here. +We conclude the paper with an appendix in which we discuss the necessity of the spherical condition in +the statement of our main result. +2. Main results +We will always work with a finite field Fq of odd characteristic. For vectors v, w ∈ Fd +q, we define their dot +product v · w := �d +j=1 vjwj as usual and we will work with the isotropic measurements of length |v|2 := v · v +and distance |v − w|2. For any u ∈ Fd +q and λ ∈ Fq, we define the sphere Sλ(u) = {x ∈ Fd +q : |x − u|2 = λ}, +and we will simply write Sλ when u = 0. For k ∈ N, we will say that X ⊆ Fd +q spans k dimensions when +dim(Span(X − X)) = k, and we will call X spherical provided exists a sphere Sλ(u) ⊆ Fd +q with X ⊆ Sλ(u). +Note the (k + 1)-point spherical configurations spanning k dimensions are exactly the k-simplices, which +were shown in [24] to appear as isometric copies in sufficiently dense subsets of Fd +q provided d > k. +In Euclidean spaces it is easy to see that if the finite sets X and X′ are isometric, then X′ = z + U(X) for +some vector z and orthogonal transformation U ⊆ O(d), i.e. X′ can be obtained from X by a rigid motion. +The same may not hold in finite field geometries due to the presence of self-orthogonal vectors x for which +|x|2 = 0. However, it follows from Witt’s extension theorem [27] that if the subspace V := Span(X − X) +is non-degenerate in the sense that V ∩ V ⊥ = {0}, then isometric copies of X are indeed obtained by +rigid motions. We use this fact in the appendix to construct dense subsets avoiding isometric copies of +non-spherical sets, establishing the necessity of restricting our attention to spherical configurations X. +The main result of this paper is the following. +Theorem 1. Let d, k ∈ N with d ≥ 2k + 6, α ∈ (0, 1), and q ≥ q(α, k). If A ⊆ Fd +q with |A| ≥ αqd, then +A contains at least c(α, k)q(k+1)d−k(k+1)/2 isometric copies of every (k + 2)-point spherical configuration +spanning k dimensions. + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +3 +Here we write c(α, k) to stand in for some positive constant depending only on α and k, and q ≥ q(α, k) +to indicate q is taken sufficiently large with respect to α and k. We will use similar notation to indicate the +dependency of constants c, C > 0 that may change between occurrences. It is helpful to think of α and k as +fixed with q allowed to tend toward infinity, and implicit constants in our big-O notation may depend on k. +Note that if A would be a random subset of Fd +q of density α then it would contain αk+2q(k+1)d−k(k+1)/2 +isometric copies of a (k + 2)-point spherical configuration X up to an error. +This is because there are +k(k + 1)/2 quadratic relations, given by the length of the edges, between the points of the configuration X +and each vertex is contained in A with probability α. +It immediately follows from Theorem 1 that for any fixed number of colors r, if we take q sufficiently +large with respect to r, then any r-coloring of F10 +q +contains monochromatic, isometric copies of all 4-point +spherical sets spanning 2 dimensions; this establishes a finite field version of Graham’s conjecture that all +cyclic quadrilaterals are Ramsey. In fact, we prove a stronger statement relative to spheres; see [12, 23] for +some so-called “sphere Ramsey” results in the Euclidean setting. +Theorem 2. Let d, k ∈ N with d ≥ 2k + 6, α ∈ (0, 1), and q ≥ q(α, k). If λ ∈ F∗ +q and A ⊆ Sλ with +|A| ≥ αqd−1, then A contains at least c(α, k)q(k+1)d−(k+1)(k+2)/2 isometric copies of every (k + 2)-point +spherical configuration spanning k dimensions. +A straightforward counting argument reveals that Theorem 2 quickly implies Theorem 1, this argument +is presented in Section 4 below. +We remark here that the relationship between d and k in both theorems could be improved if one were +only interested in “high rank” configurations; see the comments following Lemma 3. Our methods are further +able to prove a version of Theorem 2 when λ = 0 provided d > 2k + 6, but we do not pursue this since it +does not impact our proof of Theorem 1. +3. Preliminaries +Here we record notation and ingredients that we will require for the proof of Theorem 2. Given any +function f : Fd +q → C and B ⊆ Fd +q, we write +Ex∈Bf(x) := +1 +|B| +� +x∈B +f(x) +for the average of f over B, and we will understand the average Exf(x) is taken over Fd +q. We will condense +multiple averages Ey1Ey2 · · · Eyk as Ey1,...,yk, and to indicate linear independence we will use the notation +E∗ +y1,...,yk := +1 +qkd +� +y1,...,yk∈Fd +q +linearly independent +Letting χ denote the canonical additive character of Fq, we define the Fourier transform �f : Fd +q → C by +�f(ξ) := Exf(x)χ(−ξ · x) +and we recall the Fourier inversion formula f(x) = � +ξ∈Fdq �f(ξ)χ(ξ · x). Given two functions f, g : Fd +q → C, +we recall Plancherel’s identity +Exf(x)g(x) = +� +ξ∈Fdq +�f(ξ)�g(ξ) +and, defining the convolution f ∗ g(x) = Eyf(y)g(x − y), we also recall � +f ∗ g(ξ) = �f(ξ)�g(ξ). +We will write σλ = q1Sλ for a normalized indicator function of Sλ, where we have the asymptotic +(1) +�σλ(ξ) = +� +1 + O(q−1/2) +if ξ = 0 +O(q−1/2) +otherwise +valid for d ≥ 2 and λ ∈ F∗ +q; see, for instance, [16, Lemma 2.2] exploiting Weil’s bounds on Kloosterman +sums. + +4 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +To vectors y1, . . . , yj−1 ∈ Fd +q and constants c1, . . . , cj ∈ Fq, we will associate the spherical measure +(2) +σc1,...,cj +y1,...,yj−1(yj) = +� +qj +if yi · yj = ci for all 1 ≤ i ≤ j +0 +otherwise +. +This is essentially an L1-normalized indicator function for the intersection of the sphere Scj with j − 1 +hyperplanes, so one should expect Fourier decay in appropriate directions. To import the corresponding +Fourier asymptotics, we set +δy1,...,yj−1(ξ) := +� +1 +if ξ ∈ Span(y1, . . . , yj−1) +0 +otherwise, +and record the simplest case of [24, Lemma 7]. +Lemma 3. Let c1, . . . , cj ∈ Fq with cj ̸= 0 and y1, . . . , yj−1 ∈ Fd +q linearly independent. If d ≥ 2j, then +|�σc1,...,cj +y1,...,yj−1(ξ)| = δy1,...,yj−1(ξ) + O(q−1/2). +It is worth mentioning if one is willing to impose technical conditions on the defining vectors and scalars, +then results within [24] include stronger asymptotics valid in the range d > j. These would allow us to +improve the required relationship of d ≥ 2k + 6 in Theorem 2 for configurations avoiding self-orthogonal +subspaces, but we opt instead for a uniform result valid for all (k+2)-point spherical configurations spanning +k dimensions. For clarity of presentation, we will often suppress the scalars c1, . . . , cj in the notation (2) +since we will always restrict ourselves to cj ̸= 0 and Lemma 3 does not depend on the other scalars. +We will also require the notion of Bohr sets, which provide a substitute for fine subgroup structure. We +define the Bohr set of spectrum Γ ⊆ Fd +q and radius ρ ∈ (0, 1] by +(3) +B(Γ, ρ) := {x ∈ Fd +q : |χ(ξ · x) − 1| ≤ ρ for all ξ ∈ Γ}. +Setting β = |B(Γ, ρ)|/qd, we record the standard bound +(4) +β ≥ +� ρ +2π +�|Γ| +which can be found in [26]. Setting B = B(Γ, ρ), we will use the L1 normalized indicator function µB = +β−11B and its repeated convolution νB = µB ∗µB ∗µB ∗µB. The structure provided by repeated convolution +will frequently be useful. For example, provided d ≥ 2, we can apply Fourier inversion, (1) and Plancherel +to see +(5) +µB ∗ µB ∗ σ(x) = +� +ξ∈Fd +q +�µB(ξ)2�σ(ξ)χ(ξ · x) = 1 + O +� +q−1/2 � +ξ∈Fd +q +|�µB(ξ)|2� += 1 + O(β−1q−1/2), +from which it immediately follows that +(6) +νB ∗ σ(x) = 1 + O(β−1q−1/2). +In other words, provided q sufficiently large with respect to β, repeated convolutions of µB with σ can be +considered essentially constant. +We call the Bohr set B(Γ, ρ) regular if for every ǫ > 0, we have both +|B(Γ, (1 + ǫ)ρ)| ≤ (1 + 100ǫ|Γ|)|B(Γ, ρ)| +|B(Γ, (1 − ǫ)ρ)| ≥ (1 − 100ǫ|Γ|)|B(Γ, ρ)|; +This definition, due to Bourgain [2], ensures that all Bohr sets are only a small dilation away from being +regular (see [26, Lemma 4.25]), and regular Bohr sets are essentially closed under addition by elements of +their interior. Given a regular Bohr set B = B(Γ, ρ), we will write B′ ≺ǫ B in the case that B′ = B(Γ′, ρ′) +is another regular Bohr set with Γ′ ⊇ Γ and ρ′ ≤ ǫρ/(200|Γ|). The following standard lemma provides the +main consequence of regularity for our purposes; we include the proof for completeness. +Lemma 4. Let ǫ ∈ (0, 1), B = B(Γ, ρ) regular with B′ ≺ǫ B, and f : Fd +q → C with |f| ≤ 1. For any y ∈ B′, +|Ex∈Bf(x) − Ex∈Bf(x + y)| ≤ ǫ. + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +5 +Proof. We have +|Ex∈Bf(x) − Ex∈Bf(x + y)| = +1 +|B| +��� +� +x∈Fdq +f(x)(1B(x) − 1B(x − y)) +��� ≤ |B△(y + B)| +|B| +Since B′ ≺ǫ B and y ∈ B′ we have the relationship +B△(y + B) ⊆ B(Γ, ρ + ρ′) \ B(Γ, ρ − ρ′) +and our claim follows from the bound |B(Γ, ρ + ρ′) \ B(Γ, ρ − ρ′)| ≤ ǫ|B| resulting from regularity. +□ +4. Reduction to Dense Spherical Sets +In this section we present the straightforward counting argument that estblishes Theorem 1 as a conse- +quence of Theorem 2. The main observation is that the collection of spheres of a fixed radius λ ∈ F∗ +q provides +a uniform cover of Fd +q. Hence, any subset A ⊆ Fd +q with density α has density nearly α on a large number, +in fact a positive proportion, of these spheres. Theorem 2 then implies that within each of these spheres, A +contains many of the sought after configurations. By counting the contribution of each of these spheres, it is +easy to see that A contains a positive proportion of the count of all such configurations within Fd +q, as each +fixed configuration is contained in approximately the same number of spheres. +We first record the characterization of spherical configurations that will be most useful as we proceed. +This follows in a straightforward way from Lemma 16 in the Appendix. +Lemma 5. Let X ⊆ Fd +q be any (k + 2)-point spherical configuration spanning k dimensions. If any (k + 1)- +point subset of X that spans k dimensions is contained in a sphere Sλ(u), then X ⊆ Sλ(u) as well. +Let us now see how Theorem 2 implies Theorem 1. +Proof of Theorem 1. Fixing λ ∈ F∗ +q, we first establish that for many centers u ∈ Fd +q, |A ∩ Sλ(u)| is large. +For d ≥ 2 and x ∈ Fd +q, (1) implies Euσλ(u − x) = 1 + O(q−1/2), in which case we can ensure q is sufficiently +large for +(α/2)q2d−1 ≤ +� +u∈Fdq +|A ∩ Sλ(u)| += +� +u∈Fd +q +|A∩Sλ(u)|≥(α/4)qd−1 +|A ∩ Sλ(u)| + +� +u∈Fd +q +|A∩Sλ(u)|<(α/4)qd−1 +|A ∩ Sλ(u)| +≤ |Sλ||{u ∈ Fd +q : |A ∩ Sλ(u)| ≥ (α/4)qd−1}| + (α/4)q2d. +Again using (1) and ensuring q sufficiently large, we can conclude +|{u ∈ Fd +q : |(A − u) ∩ Sλ| ≥ (α/4)qd−1}| ≥ (α/8)qd. +Fixing now some (k + 2)-point spherical configuration X spanning k dimensions, we see that for each of +these good centers u, Theorem 2 guarantees that A ∩ Sλ(u) contains at least c(α, k)q(k+1)d−(k+2 +2 ) isometric +copies of X. We need to account for how many spheres we count each isometric copy within. By translating +X if necessary, we can parametrize it as +X = {0, v1, . . . , vk, a1v1 + · · · + akvk} +for linearly independent v1, . . . , vk ∈ Fd +q and a1, . . . , ak ∈ Fq. Then for a translation x ∈ Fd +q and linearly +independent vectors y1, . . . , yk ∈ Fd +q with yi · yj = vi · vj for all 1 ≤ i ≤ j ≤ k, we wish to count how many +spheres Sλ(u) contain the configuration +X′ = {x, x + y1, . . . , x + yk, x + a1y1 + · · · + akyk}. +Since y1, . . . , yk are linearly independent, Lemma 3 applies to the spherical measure +σy1,...,yk(u) := σλ(u) +k +� +j=1 +σλ(yj − u) + +6 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +so that, for d ≥ 2k + 2, +Eu∈Fdqσy1,...,yk(x − u) = 1 + O(q−1/2). +Lemma 5 ensures that {x, x + y1, . . . , x + yk} ⊆ Sλ(u) implies X′ ⊆ Sλ(u) as well, so we have shown +|{u ∈ Fd +q : X′ ⊆ Sλ(u)}| = (1 + O(q−1/2))qd−k−1. +Hence, within each of the (α/8)qd good spheres Sλ(u), A ∩ Sλ(u) contains c(α, k)q(k+1)d−( +k+2 +2 ) isometric +copies of X, and each of these copies is contained in roughly qd−k−1 spheres. In total this yields that A +contains c(α, k)q(k+1)d−(k+1 +2 ) isometric copies of X as claimed. +□ +5. Proof of Theorem 2 +In this section we reduce the task of proving Theorem 2 to that of establishing Propositions 6 and 7 below. +5.1. Counting Configurations, a Uniformity Norm, and two key Propositions. For the remainder, +we fix λ ∈ F∗ +q and aim to establish Theorem 2; there is no harm in assuming λ = 1. For brevity, we set +S = Sλ and σ = σλ. We will be considering (k + 2)-point spherical configurations spanning k dimensions +typically parameterized as +X := {0, v1, . . . , vk, a1v1 + · · · + akvk} ⊆ Fd +q, +with v1, . . . , vk ∈ Fd +q linearly independent and coefficients a1, . . . , ak ∈ Fq. We will consider other sets of +k + 2 points of the form +X′ = {x0, x0 + x1, . . . , x0 + xk, x0 + a1x1 + · · · + akxk} ⊆ Fd +q +and check whether these are indeed isometric copies of X contained within S by checking whether x1, . . . , xk +are linearly independent and further satisfy the conditions +|x0|2 = |x0 + x1|2 = · · · = |x0 + xk|2 = λ and xi · xj = vi · vj for each 1 ≤ i ≤ j ≤ k; +when all of this is true, we write X′ ≃ X. We remark that this notation only explicitly insists that k + 1 +points of X′ lie on S. However, since we are working with the same coefficients a1, . . . , ak, Lemma 16 ensures +X′ is spherical and Lemma 5 ensures X′ ⊂ S as well. That is, if X′ ≃ X, then it is also the case that X′ ⊆ S. +To count copies of X parameterized as X′ above, we define the weight +SX(x0, . . . , xk) := +� +q(k+1)(k+2)/2 +if X′ ≃ X +0 +otherwise +and a normalized counting operator on functions f0, f1, . . . , fk+1 : Fd +q → C by +(7) NX(f0, . . . , fk+1) := Ex0f0(x0)E∗ +x1,...,xk +� +k +� +j=1 +fj(x0 + xj) +� +fk+1(x0 + a1x1 + · · · + akxk)SX(x0, . . . , xk). +Note that as long as we restrict our attention to linearly independent x1, . . . , xk, we can write out +SX(x0, . . . , xk) = σ(x0) +k +� +j=1 +σx0,...,xj−1(xj) +for appropriate spherical measures with implicit scalars determined by our sphere S and the dot products +between the defining vectors v1, . . . , vk of X. Moreover, the contribution of linearly dependent x1, . . . , xk is +negligible, in the sense that we trivially have +1 +qd +� +xj∈Span(x1,...,xj−1) +σx1,...,xj−1(xj) ≤ q−1 +whenever d ≥ 2j. Together with Lemma 3, this allows us to freely add in linearly dependent x1, . . . , xk to +our count (7) at the cost of an acceptable O(q−1) error, provided each fj is bounded as will typically be the +case. + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +7 +The starting point for our argument is to show that the counting operator (7) is controlled by what we +call the U 2 +⊥(S) norm, defined for f : Fd +q → C by +∥f∥U2 +⊥(S) := +� +Ex,h,h′fσ(x)fσ(x + h)fσ(x + h′)fσ(x + h + h′) +�1/4 +, +which is the usual Gowers U 2(Fd +q) norm of the function fσ. While the U 2(Fd +q) norm averages over combina- +torial rectangles, the U 2 +⊥(S) norm averages instead over geometric rectangles contained within our sphere S. +Note that if +|x|2 = |x + h|2 = |x + h′|2 = |x + h + h′|2 = λ, +then it is also the case that h · h′ = 0. +We will show that the operator (7) is controlled by the U 2 +⊥(S) norm in the following sense. +Proposition 6. Let f0, . . . , fk+1 : Fd +q → C with |fj| ≤ 1. If d ≥ 2k + 6, then +|NX(f0, . . . , fk+1)| ≤ +min +0≤j≤k+1∥fj∥U2 +⊥(S) + O(q−1/8) +Results of this type are often called generalized von-Neumann inequalities in the arithmetic combinatorics +literature. The proof of Proposition 6 is presented in Section 6 below. +To see the utility of such a result, consider a set A ⊆ S with |A| ≥ αqd−1 and fA = 1A − α. If A +is sufficiently uniform, in the sense that ∥fA∥U2 +⊥(S) is sufficiently small with respect to α and k, then the +decomposition 1A = α + fA along with Proposition 6 provides +NX(1A, . . . , 1A) ≳ αk+2 +provided q is taken sufficiently large with respect to α and k. Of course, not all sets must be uniform in this +sense, and we will require a more sophisticated decomposition. Defining the L2(S) norm by +∥f∥L2(S) := +� +Ex|f(x)|2σ(x) +�1/2 +, +we will use the following decomposition. +Proposition 7. Let η ∈ (0, 1), f : Fd +q → [−1, 1] and ϕ : (0, 1]2 → (0, 1) increasing in both coordinates. If +d ≥ 2 and q ≥ q(η, f), then there exists B = B(Γ, ρ) regular with |Γ| ≤ C(η, f) and ρ ≥ c(η, f), and there +exist functions f1, f2, f3 : Fd +q → [−2, 2] with +f = f1 + f2 + f3 +f1 = νB ∗ (fσ) +∥f2∥U2 +⊥(S) ≤ ϕ(|Γ|−1, ρ) +∥f3∥L2(S) ≤ η. +Results of this type are often called arithmetic regularity lemmas in the arithmetic combinatorics literature. +In light of (6), it should at least be clear that ensuring f1 is bounded amounts to ensuring q is sufficiently +large with respect to the other parameters. The proof of Proposition 7 is presented in Section 7 below. +We close this section by demonstrating, as promised, how Propositions 6 and 7 can be applied to give a +proof of Theorem 2. We initially specialize, in Section 5.2 below, to the case of spherical quadrilaterals, that +is when k = 2. The proof of the general case follows along similar lines and is presented in Section 5.3. +5.2. Proof that Propositions 6 and 7 imply the k = 2 case of Theorem 2. It clearly suffices to +establish the following +Theorem 8. Let α ∈ (0, 1) and A ⊆ S with |A| ≥ αqd−1. If d ≥ 10, q ≥ q(α), and X ⊆ Fd +q is a spherical +4-point configuration spanning 2 dimensions, then +NX(1A, 1A, 1A, 1A) ≥ c(α). +In other words, A contains c(α)q3d−6 isometric copies of X. + +8 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +Proof. To set up, let ǫ > 0 be a parameter to be determined only in terms of α and let ϕ : (0, 1]2 → (0, 1) be +a function increasing in both coordinates to be determined only in terms of ǫ. We apply the decomposition +theorem to obtain a regular Bohr set B = B(Γ, ρ) with |Γ| ≤ C(ǫ, ϕ) and ρ ≥ c(ǫ, ϕ) and functions f1, f2, f3 : +Fd +q → [−2, 2] such that +1A = f1 + f2 + f3 +f1 = νB ∗ (1Aσ) +∥f2∥U2 +⊥(S) ≤ ϕ(|Γ|−1, ρ) +∥f3∥L2(S) ≤ ǫ. +We take ρ′ ∈ [ǫρ/(400|Γ|), ǫρ/(200|Γ|)] so that B1 := B(Γ, ρ′) ≺ǫ B. Throughout the argument we will +continue to take q(α) large enough so that, with parameters other than q fixed, we can absorb error terms +that tend to zero as q → ∞ into a single O(ǫ) error. We now fix a spherical 4-point configuration spanning +2 dimensions X, parameterize it as +X = {0, v, w, av + bw} +for v, w linearly independent, and search for isometric copies of the form +{x, x + y, x + z, x + ay + bz} ⊆ S +where |y|2 = |v|2, |z|2 = |w|2, and y · z = v · w. To detect these copies, we define the spherical measures +σx(y) := σ−|v|2/2,|v|2 +x +(y), and +σx,y(z) := σ−|w|2/2,v·w,|w|2 +x,y +(z) +in which case we can parametrize our counting operator as +NX(g0, g1, g2, g3) = Exg0(x)σ(x)E∗ +y,zg1(x + y)g2(x + z)g3(x + ay + bz)σ(x)σx(y)σx,y(z). +Setting +B2 := B(Γ ∪ {a · ξ : ξ ∈ Γ} ∪ {b · ξ : ξ ∈ Γ}, ρ′/4), +we note that restricting y, z ∈ B2 + B2 ensures y, z, ay + bz ∈ B1. This is useful to us, since for our main +term function f1, for any x ∈ Fd +q and any x′ ∈ B1, Lemma 4 provides +(8) +f1(x + x′) = f1(x) + O(ǫ). +We will in fact work with the further restricted count +N B +X (g0, g1, g2, g3) := Exg0σ(x)E∗ +y,zg1(x + y)g2(x + z)g3(x + ay + bz)σx(y)µB2 ∗ µB2(y)σx,y(z)µB2 ∗ µB2(z). +Setting β := |B2|/qd, the size estimate (4) and the dependency of our parameters guarantees +β−1 ≤ +�C|Γ| +ǫρ +�3|Γ| +≤ C(α). +We should establish that this restricted count is well normalized. That is, we will show +(9) +N B +X (1, 1, 1, 1) = 1 + O(ǫ), +where here 1 stands for the constant 1 function. Applying Parseval and extracting the ξ = 0 term, +EzµB2 ∗ µB2(z)σx,y(z) = 1 + +� +ξ∈Fdq\{0} +�µB2(ξ)2�σx,y(ξ) + O(ǫ), +so we can apply Lemma 3 for +N B +X (1, 1, 1, 1) = Ex,yσ(x)σx(y)µB2 ∗ µB2(y) + O + +β−1 +� +ξ∈Fdq\{0} +|�µB2(ξ)|2Ex,yσ(x)σx(y)δx,y(ξ) + + + O(ǫ). +Without trying to be too careful, notice that uniformly in ξ ̸= 0, +Ex,yσ(x)σx(y)δx,y(ξ) ≤ q3Ex,yδx,y(ξ) = O(q−1), +so we can apply Plancherel and ensure q is sufficiently large to conclude +N B +X (1, 1, 1, 1) = Ex,yσ(x)σx(y)µB2 ∗ µB2(y) + O(ǫ). + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +9 +Arguing similarly to eliminate the average in y, we conclude (9). This restricted count will be useful for us +since we trivially have the lower bound +(10) +NX(1A, 1A, 1A, 1A) ≥ β2N B +X (1A, 1A, 1A, 1A), +and we will spend the rest of the proof establishing the lower bound +N B +X (1A, 1A, 1A, 1A) ≥ α4 + O(ǫ). +From the decomposition 1A = f1 + f2 + f3, +N B +X (1A, 1A, 1A, 1A) = +� +1≤i0,i1,i2,i3≤3 +N B +X (fi0, fi1, fi2, fi3), +and we handle separately the three cases (i) when each of i0, i1, i2, i3 equals 1, (ii) when one of i0, i1, i2, i3 +equals 2, and (iii) when one of i0, i1, i2, i3 equals 3. Case (i) will yield our main term of α4, and we will +argue that cases (ii) and (iii) contribute a negligible O(ǫ) error. +For the first case, applying (8) and (9) yields +N B +X (f1, f1, f1, f1) = Exf1(x)4σ(x)Ey,zµB2 ∗ µB2(y)σx(y)µB2 ∗ µB2(z)σx,y(z) + O(ǫ). +Arguing as we did to establish (9), +N B +X (f1, f1, f1, f1) = Exf1(x)4σ(x) + O(ǫ). +Then from H¨older’s inequality, +N B +X (f1, f1, f1, f1) ≥ (Exf1(x)σ(x))4 + O(ǫ), +and writing +Exf1(x)σ(x) = ExνB ∗ 1A(x)σ(x) += Ex1A(x)νB ∗ σ(x), +we can apply (6) to conclude +N B +X (f1, f1, f1, f1) ≥ α4 + O(ǫ). +For the second case, with one of i0, i1, i2, i3 = 2, we apply Fourier inversion to write +µB2 ∗ µB2(y)µB2 ∗ µB2(z) = +� +ξ1,ξ2∈Fdq +�µB2(ξ1)2�µB2(ξ2)2χ(−ξ1 · y)χ(−ξ2 · z). +Setting χξ(x) = χ(ξ · x), this allows us to express +N B +X (fi0, fi1, fi2, fi3) = +� +ξ1,ξ2∈Fdq +�µB2(ξ1)2�µB2(ξ2)2TX,S(fi0χξ1+ξ2, fi1χ−ξ1, fi2χ−ξ2, fi3). +As ∥fχξ∥U2 +⊥(S) = ∥f∥U2 +⊥(S), we can apply the triangle inequality, Proposition 6 and Plancherel to conclude +|N B +X (fi0, fi1, fi2, fi3)| ≤ β−2∥f2∥U2 +⊥(S) + O(ǫ) +≤ +�C|Γ| +ǫρ +�6|Γ| +ϕ(|Γ|−1, ρ) + O(ǫ). +We see that by taking +ϕ(|Γ|−1, ρ) = ǫ +�cǫρ +|Γ| +�6|Γ| +, +we can ensure case (ii) contributes at most O(ǫ). +For the third case, we will assume i0 = 3, but each case is similar by reindexing in x0. Applying the +triangle inequality and using that fi1, fi2, fi3 are each at most 2 in absolute value, +N B +X (f3, fi1, fi2, fi3) ≤ 8Ex|f3(x)|σ(x)E∗ +y,zσx(y)µB2 ∗ µB2(y)σx,y(z)µB2 ∗ µB2(z). +Then arguing as we did to establish (9) we have +N B +X (f3, fi1, fi2, fi3) ≤ 8Ex|f3(x)|σ(x) + O(ǫ). + +10 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +Applying Cauchy-Schwarz and the fact that ∥f3∥L2(S) ≤ ǫ, we conclude case (iii) again contributes at most +an O(ǫ) error term. In total we have shown +N B +X (1A, 1A, 1A, 1A) ≥ α4 + O(ǫ) +which we can insist is at least α4/2 by taking ǫ sufficiently small with respect to α. Then since β ≥ c(α), we +are done by recalling (10). +□ +5.3. Proof that Propositions 6 and 7 imply Theorem 2 in its full generality. It clearly suffices to +establish the following +Theorem 9. Let α ∈ (0, 1) and A ⊆ S with |A| ≥ αqd−1. If d ≥ 2k + 6, q ≥ q(α, k), and X ⊆ Fd +q is a +spherical (k + 2)-point configuration spanning k dimensions, then +NX(1A, . . . , 1A) ≥ c(α, k). +In other words, A contains c(α, k)q(k+1)d−(k+1)(k+2)/2 isometric copies of X. +Proof. The proof is essentially the same as for 4-point configurations spanning 2 dimensions, although the +notation becomes more cumbersome. As before, to set up, let ǫ > 0 be a parameter to be determined only +in terms of α and k and let ϕ : (0, 1]2 → (0, 1) be a function increasing in both coordinates to be determined +only in terms of ǫ. We apply the decomposition theorem to obtain a regular Bohr set B = B(Γ, ρ) with +|Γ| ≤ C(ǫ, ϕ) and ρ ≥ c(ǫ, ϕ) and functions f1, f2, f3 : Fd +q → [−2, 2] such that +1A = f1 + f2 + f3 +f1 = νB ∗ (1Aσ) +∥f2∥U2 +⊥(S) ≤ ϕ(|Γ|−1, ρ) +∥f3∥L2(S) ≤ ǫ. +We take ρ′ ∈ [ǫρ/(400|Γ|), ǫρ/(200|Γ|)] so that B1 := B(Γ, ρ′) ≺ǫ B. Throughout the argument we will +continue to take q(α, k) large enough so that, with parameters other than q fixed, we can absorb error terms +that tend to zero as q → ∞ into a single O(ǫ) error. We now fix a spherical (k + 2)-point configuration +spanning k dimensions X parameterized as +X = {0, v1, . . . , vk+1} +where vk+1 = a1v1 + · · · + akvk and search for isometric copies of the form +{x0, x0 + x1, . . . , x0 + xk+1} ⊂ S +where xk+1 = a1x1 + · · · + akxk and xi · xj = vi · vj for 1 ≤ i, j ≤ k. To detect these copies, we define the +spherical measures +σx0,...,xj−1(xj) := (σ)−|vj|2/2,v1·vj,...,vj−1·vj,|vj|2 +x0,...,xj−1 +(xj) +in which case we can parametrize our count as +NX(g0, . . . , gk+1) = Ex0g0(x0)σ(x0)E∗ +x1,...,xkg(x0 + xk+1) +k +� +j=1 +g(x0 + xj)σx0,...,xj−1(xj). +Setting +B2 := B + +Γ ∪ +k� +j=1 +{aj · ξ : ξ ∈ Γ} , ρ′ +2k + + , +ensures that if x1, . . . , xk ∈ B2 + B2, then x1, . . . , xk+1 ∈ B1. Again we will use that for x ∈ Fd +q and x′ ∈ B1, +(11) +f1(x + x′) = f1(x) + O(ǫ). +Our restricted count is given by +N B +X (g0, . . . , gk+1) = Ex0g0(x0)σ(x0)E∗ +x1,...,xkg(x0 + xk+1) +k +� +j=1 +g(x0 + xj)µB2 ∗ µB2(xj)σ(j+1)(xj). + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +11 +As before, setting β := |B2|/qd, we have β−1 ≤ C(α, k) and a well-normalized restricted count +(12) +N B +X (1, . . . , 1) = 1 + O(ǫ). +Applying the straightforward lower bound +(13) +NX(1A, . . . , 1A) ≥ βkN B +X (1A, . . . , 1A), +we will spend the rest of the proof establishing +N B +X (1A, . . . , 1A) ≥ αk+2 + O(ǫ). +From the decomposition 1A = f1 + f2 + f3, +N B +X (1A, . . . , 1A) = +� +1≤i0,...,ik+1≤3 +N B +X (fi0, . . . , fik+1), +and we handle separately the three cases (i) when each of ij equals 1, (ii) when one of ij equals 2, and (iii) +when one of ij equals 3. Case (i) will yield our main term of αk+2, and we will argue that cases (ii) and (iii) +contribute a negligible O(ǫ) error. +For the first case, applying (11) and (12) yields +N B +X (f1, . . . , f1) = Ex0f1(x0)k+2σ(x)Ex1,...,xk +k +� +j=1 +µB2 ∗ µB2(xj)σ(j+1)(xj) + O(ǫ). +Arguing as we did to establish (9), +N B +X (f1, f1, f1, f1) = Exf1(x)k+2σ(x) + O(ǫ). +Then from H¨older’s inequality, +N B +X (f1, f1, f1, f1) ≥ (Exf1(x)σ(x))k+2 + O(ǫ) += (Ex1A(x)νB ∗ σ(x))k+2 + O(ǫ) += αk+2 + O(ǫ). +For the second case, we apply Fourier inversion to write +k +� +j=1 +µB2 ∗ µB2(xj) = +� +ξ1,...,ξk∈Fdq +k +� +j=1 +�µB2(ξj)2χ(−ξj · xj) +in order to express +N B +X (fi0, . . . , fik+1) = +� +ξ1,...,ξk∈Fd +q +k +� +j=1 +�µB2(ξj)2NX(fi0χξ1+···+ξk, fi1χ−ξ1, . . . fikχ−ξk, fik+1). +As before, ∥fχξ∥U2 +⊥(S) = ∥f∥U2 +⊥(S), so we apply the triangle inequality, Proposition 6 and Plancherel to +conclude +|N B +X (fi0, . . . , fik+1)| ≤ β−k∥f2∥U2 +⊥(S) + O(ǫ) +≤ +�C|Γ| +ǫρ +�C|Γ| +ϕ(|Γ|−1, ρ) + O(ǫ). +We see that by taking +ϕ(|Γ|−1, ρ) = ǫ +�cǫρ +|Γ| +�C|Γ| +, +we can ensure case (ii) contributes at most O(ǫ). +For the third case, we assume i0 = 3 and apply the triangle inequality for +N B +X (f3, fi1, . . . , fik+1) ≤ 2k+1Ex0|f3(x0)|σ(x0)Ex1,...,xk +k +� +j=1 +µB2 ∗ µB2(xj)σ(j+1)(xj) + O(ǫ). + +12 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +Applying Cauchy-Schwarz and the fact that ∥f3∥L2(S) ≤ ǫ, we conclude case (iii) again contributes at most +an O(ǫ) error term. In total we have shown +N B +X (1A, . . . , 1A) ≥ αk+2 + O(ǫ) +which we can insist is at least αk+2/2 by taking ǫ sufficiently small with respect to α and k. Then since +β ≥ c(α, k), we are done by recalling (13). +□ +It remains to prove both Proposition 6 and Proposition 7. +6. Proof of Proposition 6 +We establish Proposition 6 through two careful applications of Cauchy-Schwarz. In a sense, each ap- +plication of Cauchy-Schwarz replaces our configuration with a more regular configuration, and we are left +considering averages over geometric rectangles rather than more complicated spherical configurations. +We fix a spherical (k + 2)-point configuration X spanning k dimensions parameterized by +X = {v0, v1, . . . , vk+1} +with v1 − v0, . . . , vk − v0 linearly independent and +vk+1 = +k +� +j=0 +ajvj +for coefficients aj ∈ Fq. It is enough by symmetry to show +|NX(f0, . . . , fk+1)| ≤ ∥fk∥U2 +⊥(S) + O(q−1/8). +For x0, . . . , xk ∈ Fd +q, we will understand that +xk+1 = +k +� +j=0 +aixi, +and we will detect if {x0, . . . , xk+1} ≃ X with the spherical measures σ(x0) and, for 1 ≤ j ≤ k, +σ(j)(xj) := σc0,j,...,cj−1,j,λ +x0,...,xj−1 +(xj) +where we define ci,j = |vi − vj|2/2 − λ, since if |xi|2 = |xj|2 = λ, checking whether |xi − xj|2 = |vi − vj|2 +amounts to checking whether xi · xj = |vi − vj|2/2 − λ. Then we can express our counting operator +NX(f0, . . . , fk+1) = Ex0,...,xkσ(x0) +k+1 +� +j=0 +fj(xj) +k +� +j=1 +σ(j)(xj) + O(q−1), +where we have included the negligible amount of linearly dependent vectors. Rearranging and applying the +triangle inequality, +|NX(f0, . . . , fk+1)| ≤ Ex0,...,xk−1σ(x0) +k−1 +� +j=1 +σ(j)(xj) +���Exkfk(xk)fk+1(xk+1)σ(k)(xk) +��� + O(q−1). +Introducing the differencing notation +∆hf(x) := f(x)f(x + h), +we square both sides and apply Cauchy-Schwarz for +|NX(f0, . . . , fk+1)|2 ≤ Ex0,...,xk−1σ(x0) +k−1 +� +j=1 +σ(j)(xj)Exk,h∆hfk(xk)∆hfk+1(xk+1)∆hσ(k)(xk) + O(q−1). +We introduce new spherical measures with an additional condition involving h defined by +σh(x0) := σ0 +h(x0) +σ(j) +h (xj) := σ0,c0,j,...,cj−1,j,λ +h,x0,...,xj−1 +(xj). + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +13 +which allows us to rewrite +σ(x0) +k−1 +� +j=1 +σ(j)(xj)∆hσ(k)(xk) = σh(x0) +k−1 +� +j=1 +σ(j) +h (xj)σ(k)(xk)σ(xk + h). +Our bound from above can then be rearranged as +|NX(f0, . . . , fk+1)|2 ≤ Eh,x0,...,xk−1σh(x0) +k−1 +� +j=1 +σ(j) +h (xj)Exk∆hfk(xk)∆hfk+1(xk+1)σ(k)(xk)σ(xk+h)+O(q−1). +We claim that in the average above, the dependence of xk+1 on xk is superficial. That is, while +xk+1 = +k +� +j=0 +ajxj, +it must be the case that at least two coefficients aj are nonzero since we are working with a (k + 2)-point +spherical configuration. In particular, aj ̸= 0 for some 0 ≤ j < k, and for this distinguished j, we can reindex +in xj in the average above to replace xk+1 with +x′ +k+1 := +k−1 +� +j=0 +a′ +jxj, +allowing us to rearrange our bound above as +|NX(f0, . . . , fk+1)|2 ≤ Eh,x0,...,xk−1∆hfk+1(x′ +k+1)σh(x0) +k−1 +� +j=1 +σ(j) +h (xj)Exk∆hfk(xk)σ(k)(xk)σ(xk+h)+O(q−1), +where we may have needed to adjust the implicit scalars in our spherical measures. This allows us to proceed +as before by applying the triangle inequality for +|NX(f0, . . . , fk+1)|2 ≤ Eh,x0,...,xk−1σh(x0) +k−1 +� +j=1 +σ(j) +h (xj) +���Exk∆hfk(xk)σ(k)(xk)σ(xk + h) +��� + O(q−1), +and Cauchy-Schwarz once more for +|NX(f0, . . . , fk+1)|4 ≤ Eh,x0,...,xk−1σh(x0) +k−1 +� +j=1 +σ(j) +h (xj)Exk,h′∆h′∆hfk(xk)∆h′σ(k)(xk)∆h′σ(xk +h)+O(q−1). +We can again reorganize our spherical measures, rewriting +σh(x0) +k−1 +� +j=1 +σ(j) +h (xj)∆h′σ(k)(xk)∆h′σ(xk + h) = ∆h′∆hσ(xk)σxk,h,h′(x0) +k−1 +� +j=1 +σxk,h,h′,x0,...,xj−1(xj) +for appropriate implicit scalars. Using this to rearrange our bound above, we have +|NX(f0, . . . , fk+1)|4 ≤ Exk,h,h′∆h′∆hfkσ(xk)Ex0,...,xk−1σxk,h,h′(x0) +k−1 +� +j=1 +σxk,h,h′,x0,...,xj−1(xj) + O(q−1). +Applying Lemma 3, we have, uniformly in xk, h, h′ when d ≥ 2k + 6, +Ex0,...,xk−1σxk,h,h′(x0) +k−1 +� +j=1 +σxk,h,h′,x0,...,xj−1(xj) = 1 + O(q−1/2), +establishing +|NX(f0, . . . , fk+1)|4 ≤ Exk,h,h′∆h′∆hfkσ(xk) + O(q−1/2) +from which we see |NX(f0, . . . , fk+1)| ≤ ∥fk∥U2 +⊥(S) + O(q−1/8) as required. +□ +7. An Inverse Theorem and Proof of Proposition 7 +This section is dedicated to establishing Proposition 7. We begin in Section 7.1 by establishing Theorem 10, +an inverse theorem which reveals that functions with large U 2 +⊥(S) norm must exhibit Fourier bias. + +14 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +7.1. An Inverse Theorem. One way to see that functions with large U 2 +⊥(S) norm exhibit Fourier bias is +to relate the U 2 +⊥(S) norm to the usual U 2(Fd +q) norm. In order to do so, define for v, w ∈ Fd +q the normalized +indicator function +ℓv=w = +� +qd +if v = w +0 +otherwise +Then one can express +∥f∥4 +U2 +⊥(S) = Ex,y,z,wfσ(x)fσ(y)fσ(z)fσ(w)ℓx+w=y+z +Expanding ℓx+w=y+z via orthogonality, +ℓx+w=y+z = +� +ξ∈Fdq +χ(ξ(x + w − y − z)), +providing the identity +∥f∥4 +U2 +⊥(S) = +� +ξ∈Fdq +|� +fσ(ξ)|4. +The right hand side is precisely ∥fσ∥U2(Fdq). It is tempting to here use Plancherel to bound ∥f∥4 +U2 +⊥(S) above +by supξ∈Fd +q |� +fσ(ξ)|2∥fσ∥2 +L2, but this is not generally helpful since the L2 term may grow with q. By being a +bit more careful, we establish the following inverse theorem. +Theorem 10. Let f : Fd +q → C with |f| ≤ 1. If d ≥ 8, then +∥f∥U2 +⊥(S) ≤ sup +ξ∈Fdq +|� +fσ(ξ)|1/4 + O(q−1/32). +Proof. We apply absolute values and the triangle inequality for +∥f∥4 +U2 +⊥(S) ≤ Exσ(x)|Ey,z,wfσ(y)fσ(z)fσ(w)ℓx=y+z−w| +Applying Cauchy-Schwarz, +∥f∥8 +U2 +⊥(S) ≤ (Exσ(x)) +� +Exσ(x)|Ey,z,wfσ(y)fσ(z)fσ(w)ℓx=y+z−w|2� +. +For any fixed x and y, we have +Ez,wσ(z)σ(w)ℓx=y+z−w = Ezσx+y(z) +for some measure σx+y. Then since d ≥ 4, we can apply (1) and Lemma 3 for +∥f∥8 +U2 +⊥(S) ≤ Exσ(x)Ey1,z1,w1 +y2,z2,w2fσ(y1)fσ(y2)fσ(z1)fσ(z2)fσ(w1)fσ(w2)ℓx=y1+z1−w1ℓx=y2+z2−w2 + O(q−1/2) +Moving the average in x inside and rearranging a bit, this simplifies to +∥f∥8 +U2 +⊥(S) ≤ Ey1,z1,w1 +y2,z2,w2fσ(y1)fσ(y2)fσ(z1)fσ(z2)fσ(w1)fσ(w2)σ(y1 +z1−w1)ℓw2−w1=y2−y1+z2−z1 +O(q−1/2) +Setting f1 = f and f2 = f, rearranging a bit more and applying the triangle inequality, +∥f∥8 +U2 +⊥(S) ≤ Ew1,w2σ(w1)σ(w2)|Ey1,y2,z1,z2σ(y1+z1−w1)ℓw2−w1=y2−y1+z2−z1 +� +j=1,2 +fjσ(yj)fjσ(zj)|+O(q−1/2) +Setting f3 = f and f4 = f and again requiring d ≥ 4, we apply Cauchy-Schwarz as before and rearrange for +(14) +∥f∥16 +U2 +⊥(S) ≤ Ey1,...,y4 +z1,...,z4 +4 +� +j=1 +fjσ(yj)fjσ(zj)ℓy2−y1+z2−z1=y4−y3+z4−z3Wy1,...,y4 +z1,...,z4 + O(q−1/2) +where +Wy1,...,y4 +z1,...,z4 = Ewσ(w)σ(y2 − y1 + z2 − z1 + w)σ(y1 + z1 − w)σ(y3 + z3 − w) +Since d ≥ 4, we can restrict the sum in (14) to only consider the terms when the vectors y2 − y1 + z2 − +z1, y1 + z1, y3 + z3 are linearly independent at the cost of an error that can be absorbed in our current error +of O(q−1/2). For these vectors, the function +σ′(w) = σ(w)σ(y2 − y1 + z2 − z1 + w)σ(y1 + z1 − w)σ(y3 + z3 − w) + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +15 +is a measure for which Lemma 3 applies, allowing us to conclude +Wy1,...,y4 +z1,...,z4 = Ewσ′(w) = 1 + O(q−1/2), +valid for d ≥ 8. We have arrived at the estimate +∥f∥16 +U2 +⊥(S) ≤ Ey1,...,y4 +z1,...,z4 +4 +� +j=1 +fjσ(yj)fjσ(zj)ℓy2−y1+z2−z1=y4−y3+z4−z3 + O(q−1/2) +Expanding ℓy2−y1+z2−z1=y4−y3+z4−z3 via orthogonality, we have +ℓy2−y1+z2−z1=y4−y3+z4−z3 = +� +ξ∈Fdq +χ(ξ · (y4 − y3 − y2 + y1 + z4 − z3 − z2 + z1)), +from which we have the identity +Ey1,...,y4 +z1,...,z4 +4 +� +j=1 +fjσ(yj)fjσ(zj)ℓy2−y1+z2−z1=y4−y3+z4−z3 = +� +ξ∈Fdq +|� +fσ(ξ)|8 +It follows that we can conclude +∥f∥16 +U2 +⊥(S) ≤ +� +ξ∈Fd +q +|� +fσ(ξ)|8 + O(q−1/2) +≤ sup +ξ∈Fdq +|� +fσ(ξ)|4 � +ξ∈Fdq +|� +fσ(ξ)|4 + O(q−1/2) +≤ sup +ξ∈Fd +q +|� +fσ(ξ)|4 + O(q−1/2), +where we have used both |� +fσ(ξ)| ≤ 1 + O(q−1/2) and � +ξ∈Fdq |� +fσ(ξ)|4 = ∥f∥4 +U2 +⊥(S) ≤ 1 + O(q−1/2). +□ +7.2. Proof of Proposition 7. In order to use the Fourier bias from Theorem 10 to prove Proposition 7, we +will construct a sequence of Bohr sets, refining at each stage, until we arrive at B′ ≺ǫ B for which νB′ ∗ (fσ) +and νB ∗ (fσ) are close in an L2 sense. This will require a number of technical lemmas, the first of which +indicates that these two convolutions are somewhat orthogonal. +Lemma 11. Let ǫ ∈ (0, 1), B = B(Γ, ρ) a regular Bohr set with B′ ≺ǫ B, and f : Fd +q → C with |f| ≤ 1. If +d ≥ 2 and q ≥ q(|Γ|, ρ, ǫ), then +∥νB′ ∗ (fσ)∥2 +2 − ∥νB ∗ (fσ)∥2 +2 ≥ ∥νB′ ∗ (fσ) − νB ∗ (fσ)∥2 +2 + O(ǫ). +Proof. By expanding the square, ∥νB′ ∗ (fσ) − νB ∗ (fσ)∥2 +2 is equal to +∥νB′ ∗ (fσ)∥2 +2 − 2ExνB′ ∗ (fσ)(x)νB ∗ (fσ)(x) + ∥νB ∗ (fσ)∥2 +2, +so it suffices to show that +∥νB ∗ (fσ)∥2 +2 ≤ ExνB′ ∗ (fσ)(x)νB ∗ (fσ)(x) + O(ǫ) +Using the relationship B′ ≺ǫ B, we can apply Lemma 4 to see +|�µB(ξ)| = |Ex∈Bχ(x · ξ)| +≤ |Ex∈BEy∈B′χ((x + y) · ξ)| + ǫ += |Ex∈Bχ(x · ξ)Ey∈B′χ(y · ξ)| + ǫ +≤ |�µB′(ξ)| + ǫ. +in which case we apply Plancherel and this observation for +∥νB ∗ (fσ)∥2 +2 = +� +ξ∈Fdq +|�µB(ξ)|8|� +fσ(ξ)|2 ≤ +� +ξ∈Fdq +|�µB(ξ)|4 (|�µB′(ξ)| + ǫ)4 |� +fσ(ξ)|2 + +16 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +Writing β = |B|/qd, we can apply Plancherel and (5) for +� +ξ∈Fdq +|� +µB(ξ)|4|� +fσ(ξ)|2 = ∥µB ∗ µB ∗ fσ(x)∥2 +2 ≤ 1 + O(β−1q−1/2). +Using the bound (4), we see that we can take q sufficiently large and apply Plancherel again to conclude +∥νB ∗ fσ∥2 +2 ≤ +� +ξ∈Fd +q +|�µB(ξ)|4|�µB′(ξ)|4|� +fσ(ξ)|2 + O(ǫ) += ExνB′ ∗ (fσ)(x)νB ∗ (fσ)(x) + O(ǫ). +□ +With Lemma 11 in hand, we are ready to translate Theorem 10 into an energy increment, showing that +Fourier bias leads to a Bohr set refinement with increased L2 energy. +Proposition 12. Let η ∈ (0, 1), ǫ ∈ (0, cη8) for c > 0 sufficiently small, B = B(Γ, ρ) a regular Bohr set, +and f : Fd +q → C with |f| ≤ 1. If d ≥ 2, q ≥ q(|Γ|, ρ, η, ǫ), and +∥f − νB ∗ (fσ)∥U2 +⊥(S) ≥ η, +then there exists B′ = B(Γ′, ρ′) with B′ ≺ǫ B, |Γ′| ≤ |Γ| + 1, ρ′ ≥ c(|Γ|, ǫ)ρ and +∥νB′ ∗ (fσ)∥2 +L2(S) ≥ ∥νB ∗ (fσ)∥2 +L2(S) + cη8. +Proof. From the inverse theorem, ∥f − νB ∗ (fσ)∥U2 +⊥(S) ≥ η implies the existence of some γ ∈ Fd +q with +|� +fσ(γ) − [(νB ∗ (fσ))σ]∧(γ)| ≥ η4 + O(q−1/32). +Expanding [(νB ∗ (fσ))σ]∧(γ), we can apply (1) for +[(νB ∗ (fσ))σ]∧(γ) = +� +ξ∈Fd +q +�σ(ξ)�µB(γ − ξ)4 � +fσ(γ − ξ) += �µB(γ)4 � +fσ(γ) + O +� +q−1/2 � +γ +|�µB(ξ)|4|� +fσ(ξ)| +� +Applying the uniform bounds |�µB(ξ)| ≤ 1 and |� +fσ(ξ)| ≤ 1 + O(q−1/2) together with Plancherel, we have +(15) +[(νB ∗ (fσ))σ]∧(γ) = �µB(γ)4 � +fσ(γ) + O(β−1q−1/2). +in which case we can ensure q is sufficiently large and refine our inverse statement to +(16) +|� +fσ(γ)||1 − �µB(γ)4| ≥ η4/2. +Set Γ′ = Γ ∪ {γ}. It follows from [26, Lemma 4.25] that for any ǫ ∈ (0, 1) we can choose a radius ρ′ ∈ +[ǫρ/(400|Γ|), ǫρ/(200|Γ|)] such that B′ := B(Γ′, ρ′) is regular, in which case B′ ≺ǫ B. From Lemma 11, +∥νB′ ∗ (fσ)∥2 +2 − ∥νB ∗ (fσ)∥2 +2 ≥ ∥νB′ ∗ (fσ) − νB ∗ (fσ)∥2 +2 + O(ǫ), +so we would like to provide a lower bound for +∥νB′ ∗ fσ − νB ∗ fσ∥2 +2 = +� +ξ∈Fdq +|�µB′(ξ)4 − �µB(ξ)4|2|� +fσ(ξ)|2 ≥ |�µB′(γ)4 − �µB(γ)4|2|� +fσ(γ)|2 +As γ ∈ Γ′, |�µB′(γ) − 1| ≤ ρ′ ≤ ǫ. Combining this with |� +fσ(γ)| ≤ 1 + O(q−1/2) and (16), +|�µB′(γ)4 − �µB(γ)4|2|� +fσ(γ)|2 ≥ |1 − �µB(γ)4|2|� +fσ(γ)|2 + O(ǫ) ≥ η8/4 + O(ǫ). +Provided ǫ ≤ cη8 for a sufficiently small absolute constant c > 0, we have managed to show +∥νB′ ∗ (fσ)∥2 +2 − ∥νB ∗ (fσ)∥2 +2 ≥ cη8. +For the same conclusion with L2(S) norms, we can apply (15) for +∥νB ∗ (fσ)∥2 +L2(S) = +� +ξ∈Fd +q +[νB ∗ (fσ)σ]∧(ξ)�µB(ξ)4 � +fσ(ξ) += ∥νB ∗ (fσ)∥2 +2 + O(β−2q−1/2). +and ensure q is sufficiently large. +□ + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +17 +Next we iterate Proposition 12 in order to find a Bohr set refinement for which νB′ ∗ (fσ) approximates +f well in a U 2 +⊥(S) sense. +Proposition 13. Let η ∈ (0, 1), ǫ ∈ (0, cη8) for c > 0 sufficiently small, B = B(Γ, ρ) a regular Bohr set, +and f : Fd +q → C with |f| ≤ 1. If d ≥ 2 and q ≥ q(|Γ|, ρ, η, ǫ), then there exists B′ = B(Γ′, ρ′) with B′ ≺ǫ B, +|Γ′| ≤ C(|Γ|, η), ρ′ ≥ c(|Γ|, η, ǫ)ρ, and +∥f − νB′ ∗ (fσ)∥U2 +⊥(S) ≤ η. +Proof. We will construct a sequence of Bohr sets +Bj+1 ≺ǫ Bj ≺ǫ · · · ≺ǫ B1 ≺ǫ B0 +where we write Bj = B(Γj, ρj) and start with B0 = B. Our goal is find j ≥ 1 for which +(17) +∥f − νBj ∗ (fσ)∥U2 +⊥(S) ≤ η. +We set Γ1 = Γ and choose ρ1 ∈ [ǫρ/(400|Γ|), ǫρ/(200|Γ|)] so that B1 ≺ǫ B0. Starting with j = 1 and iterating, +we are done if (17) is satisfied. If (17) is false for j ≥ 1, then by applying Proposition 12 with q sufficiently +large, we obtain Bj+1 for which Bj+1 ≺ǫ Bj, |Γj+1| ≤ |Γ| + j, ρj+1 ≥ c(|Γ|, η, j)ρ, and +∥νBj+1 ∗ (fσ)∥2 +L2(S) ≥ ∥νBj ∗ (fσ)∥2 +L2(S) + cη8. +From (6), we can ensure that that for every j ≥ 0, we have +∥νBj+1 ∗ (fσ)∥2 +L2(S) ≤ 2, +in which case it is clear that we can iterate this process at most O(η−8) times before we arrive at some j for +which (17) is satisfied. +□ +Finally, we iterate Proposition 13 to find a Bohr set B for which νB ∗(fσ) approximates f arbitrarily well +in the U 2 +⊥(S) norm, at the cost of an L2-error term. This is necessary since shrinking η in Proposition 13 +leads to a loss of control of the size of the approximating Bohr set. +Letting ǫ > 0, we will construct a sequence of Bohr sets +BJ ≺ǫ · · · ≺ǫ B1 ≺ǫ B0 +where we write Bj = B(Γj, ρj). We start with Γ0 = {0}, ρ0 = 1, and η0 = ϕ(1, 1), in which case B0 = Fd +q. +For 1 ≤ j ≤ J, we set ηj = ϕ(|Γj−1|−1, ρj−1), ensure ǫ ∈ (0, cη8 +j ), and apply Proposition 13 to obtain Bj +with Bj ≺ǫ Bj−1, |Γj| ≤ C(|Γj−1|, ηj−1) and ρj ≥ c(|Γj−1|, ηj−1) such that +∥f − νBj ∗ (fσ)∥U2 +⊥(S) ≤ ηj +for sufficiently large p. We claim there exists some j for which +(18) +∥νBj+1 ∗ (fσ) − νBj ∗ (fσ)∥2 +L2(S) ≤ η2. +Using Lemma 11, we can insist ǫ is sufficiently small with respect to η to guarantee +∥νBj+1 ∗ (fσ) − νBj ∗ (fσ)∥2 +L2(S) ≤ ∥νBj+1 ∗ (fσ)∥2 +L2(S) − ∥νBj ∗ (fσ)∥2 +L2(S) + η2/2. +If ∥νBj+1 ∗ (fσ)∥2 +L2(S) ≤ ∥νBj ∗ (fσ)∥2 +L2(S), then (18) would follow. Suppose instead that +∥νBj+1 ∗ (fσ)∥2 +L2(S) > ∥νBj ∗ (fσ)∥2 +L2(S) +for all j. From (6), ∥νBj ∗ (fσ)∥2 +L2(S) ≤ 2 for sufficiently large q, in which case the ∥νBj ∗ (fσ)∥2 +L2(S) form a +bounded increasing sequence. It follows that we can take J = O(η−2) and conclude (18) for some now fixed +1 ≤ j ≤ J. We are then finished by setting B = Bj and +f1 = νBj ∗ (fσ) +f2 = f − νBj+1 ∗ (fσ) +f3 = νBj+1 ∗ (fσ) − νBj ∗ (fσ). +This completes the proof of Proposition 7. +□ + +18 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +Appendix: Necessity of the Spherical Condition +By an isometry of Fd +q we mean a linear map U : Fd +q → Fd +q so that |U(x)|2 = |x|2 for all x. It is easy +to see that U(x) · U(y) = x · y for all x and y, hence U T U = UU T = I. We call a set X = {x0, . . . , xk} +non-degenerate if V ∩ V ⊥ = {0} for the subspace V = Span(X − X). +Lemma 14. Suppose X is a non-degenerate k + 1-point configuration and Y = {y0, . . . , yk} is isometric to +X. Then there exists a vector z and an isometry U of Fd +q so that Y = z + U(X). +Proof. By performing a translation we can assume x0 = y0 = 0. Assume that the vertices are labeled so +that |yi − yj|2 = |xi − xj|2 for all 0 ≤ i ≤ j ≤ k. Let V and W be the subspace spanned by the sets X +and Y respectively. Since X and hence Y are non-degenerate, we have that the map φ : xi → yi, 1 ≤ i ≤ k +extends to an isometry Φ : V → W. By Witt’s extension theorem [27] there is an orthogonal transformation +U : Fd +q → Fd +q extending the map Φ. Clearly U(X) = Y . +□ +Let us reformulate the property that a set X is non-degenerate. Assume X = {0, x1, . . . xk} and X′ = +{x1, . . . , xk} is a linearly independent set such that Span(X′) = Span(X). Let M be the symmetric l × l +matrix with entries mij = xi · xj for 1 ≤ i, j ≤ l. +Lemma 15. The set X is non-degenerate if and only if the associated inner product matrix M has maximal +rank. +Proof. Clearly X is non-degenerate if and only if X′ is linearly independent. Let V = Span(X′) and assume +that 0 ̸= v ∈ V such that v⊥V . Then v = �l +i=1 bixi and v · xj = �l +i=1 mjibi = Mbj = 0 for all 1 ≤ j ≤ l. +Thus Mb = 0 for a non-zero vector b = (bj) and M has rank less than l. Conversely if rank(M) < l then +Mb = 0 for some b = (bj) ̸= 0 and the vector v = � +i bixi is orthogonal to V = Span(X′) = Span(X). +□ +Note that the inner product matrix drops rank if some non-trivial algebraic relations between the distances +|xi − xj|2 of the points xi, xj of X, hence generic configurations are non-degenerate. Next, we show that +the spherical condition is necessary over finite fields as well, at least for non-degenerate sets. We follow the +proof in [6] with some minor modifications. First we give the following characterization of spherical sets. +Lemma 16. Let X = {x0, x1, . . . , xk} ⊆ Fd +q. Then X is spherical if and only if the following holds. For +every c0, c1, . . . , ck ∈ Fq if +(I) +k +� +i=0 +ci = 0, +and +k +� +i=0 +cixi = 0, +then also +(II) +k +� +i=0 +ci|xi|2 = 0. +Proof. Suppose X is spherical, that is |xi − z|2 = r for 0 ≤ i ≤ k for some z ∈ Fd +q and r ∈ Fq. Then +|xi|2 = r − |z|2 + 2xi · z for all i, hence +k +� +i=0 +ci|xi|2 = +k +� +i=0 +ci(|z|2 + r) + 2 +k +� +i=0 +cixi · z = 0. +Conversely, assume X is not spherical. We show that there exists c0, . . . , ck satisfying (I) but not (II). One +may assume that X is minimal that is X′ is spherical for X′ ⊆ X, X′ ̸= X. Since a simplex is spherical +there is a non-trivial linear combination �k +i=1 ai(xi −x0) = 0 and by reindexing the vertices one may assume +ak ̸= 0. Taking X′ = {x0, . . . , xk−1} one has |xi − z|2 = r for 1 ≤ i < k but |xk − z|2 − r = b ̸= 0. Then +|xi|2 − |x0|2 = 2(xi − x0) · z for 1 ≤ i < k and |xk|2 − |x0|2 = 2(xk − x0) · z + b. Thus +k +� +i=1 +ai(|xi|2 − |x0|2) = 2 +k +� +i=1 +ak(xi − x0) · z + akb = akb ̸= 0. +Taking c0 = − �k +i=1 ai and ci = ai for 1 ≤ i ≤ k the claim follows. +□ + +SPHERICAL CONFIGURATIONS OVER FINITE FIELDS +19 +Lemma 17. Let X ⊆ Fd +q be a non-spherical set of k elements. Then there exists a set A ⊆ Fd +q of size +|A| ≥ ckqd which does not contain any set Y of the form Y = z + U(X), where ck > 0 is a constant +depending only on k. +Proof. Let t ∈ F∗ +q such that |χ(t) − 1| ≥ 1/2, χ being a non-trivial character. Such a t exists as � +t χ(t) = 0. +If X is not spherical then, by scaling, there exists c0, . . . , ck so that (I) holds and � +i=0 ci|xi|2 = t. +If +Y = z + U(X) then �k +i=0 ciyi = 0 and moreover +k +� +i=0 +ci|yi|2 = +k +� +i=0 +ci(|z|2 + 2z · U(xi) + |xi|2) = t. +Let B = B(Γ, ρ) be the Bohr set with Γ = {c0, . . . , ck} and ρ = +1 +4(k+1). By the estimate (4), |B| ≥ ckq, +for some constant ck > 0 depending only on k. Define A = {x ∈ Fd +q : +|x|2 ∈ B}. Since the number of +solutions to |x|2 = b is at least 1 +2qd−1 uniformly for b ∈ Fq for q ≥ q0, we have that |A| ≥ ck +2 qd. We show +that Y = z + U(x) for some z ∈ Fd +q and an orthogonal transformation U then Y ⊊ A. Indeed if Y ⊆ A then +|χ(ci|yi|2) − 1| ≤ +1 +4(k+1) for 0 ≤ i ≤ k + 1. It follows +|χ( +k +� +i=0 +ci|yi|2) − 1| = | +k +� +i=0 +χ(ci|yi|2) − 1| ≤ +k +� +i=0 +|χ(ci|yi|2) − 1| ≤ 1/4. +This implies |χ(t) − 1| ≤ 1/4 contradicting our choice of t. +□ +References +[1] J. Bourgain, A Szemer´edi type theorem for sets of positive density in Rk, Israel J. Math. 54 (1986), no. 3, 307–316. +MR 853455 +[2] J. Bourgain, On triples in arithmetic progression, Geom. Funct. Anal. 9 (1999), no. 5, 968–984. MR 1726234 +[3] J. Chapman, M. B. Erdo˘gan, D. Hart, A. Iosevich, and D. Koh, Pinned distance sets, k-simplices, Wolff’s exponent in +finite fields and sum-product estimates, Math. Z. 271 (2012), no. 1-2, 63–93. MR 2917133 +[4] B. Cook, ´A. Magyar, M. Pramanik, A Roth type theorem for dense subsets of Rd, Bull. London Math. Soc. 49 (2017), no. +4, 676–689. MR 3725488 +[5] S. Eberhard, Almost all sets of d + 2 points on the (d − 1)-sphere are not subtransitive, Mathematika 59 (2013), no. 2, +267–268. MR 3081771 +[6] P. Erd˝os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey theorems. I, +J. Combinatorial Theory Ser. A 14 (1973), 341–363. MR 0316277 +[7] P. Erd˝os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey theorems. +II, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erd˝os on his 60th birthday), Vol. I, North-Holland, +Amsterdam, 1975, pp. 529–557. Colloq. Math. Soc. J´anos Bolyai, Vol. 10. MR 0382047 +[8] P. Erd˝os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey theorems. +III, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erd˝os on his 60th birthday), Vol. I, North-Holland, +Amsterdam, 1975, pp. 559–583. Colloq. Math. Soc. J´anos Bolyai, Vol. 10. MR 0382048 +[9] P. Frankl and V. R¨odl, A partition property of simplices in Euclidean space, J. Amer. Math. Soc. 3 (1990), no. 1, 1–7. +MR 1020148 +[10] W. T. Gowers, A new proof of Szemer´edi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), +no. 3, 529–551. MR 1631259 +[11] W. T. Gowers, A new proof of Szemer´edi’s theorem, Geom. Funct. Anal. 11 (2001), no. 3, 465–588. MR 1844079 +[12] R. L. Graham, Euclidean Ramsey theorems on the n-sphere, J. Graph Theory 7 (1983), no. 1, 105–114. MR 693027 +[13] R. L. Graham, Recent trends in Euclidean Ramsey theory, Discrete Math. 136 (1994), no. 1-3, 119–127, Trends in discrete +mathematics. MR 1313284 +[14] B. Green, Finite field models in additive combinatorics, Surveys in combinatorics 2005, London Math. Soc. Lecture Note +Ser., vol. 327, Cambridge Univ. Press, Cambridge, 2005, pp. 1–27. MR 2187732 +[15] D. Hart and A. Iosevich, Ubiquity of simplices in subsets of vector spaces over finite fields, Anal. Math. 34 (2008), no. 1, +29–38. MR 2379694 +[16] A. Iosevich and M. Rudnev, Erd˝os distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007), +no. 12, 6127–6142 (electronic). MR 2336319 +[17] I. Kˇr´ıˇz, Permutation groups in Euclidean Ramsey theory, Proc. Amer. Math. Soc. 112 (1991), no. 3, 899–907. MR 1065087 +[18] I. Kˇr´ıˇz, All trapezoids are Ramsey, Discrete Math. 108 (1992), no. 1-3, 59–62, Topological, algebraical and combinatorial +structures. Frol´ık’s memorial volume. MR 1189829 +[19] I. Leader, P. A. Russell, and M. Walters, Transitive sets and cyclic quadrilaterals, J. Comb. 2 (2011), no. 3, 457–462. +MR 2913203 + +20 +NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL +[20] I. Leader, P. A. Russell, and M. Walters, Transitive sets in Euclidean Ramsey theory, J. Combin. Theory Ser. A 119 +(2012), no. 2, 382–396. MR 2860600 +[21] N. Lyall and ´A. Magyar, Product of simplices and sets of positive upper density in Rd, Math. Proc. Cambridge Philos. +Soc. (2017), 1-27. doi:10.1017/S0305004117000184 +[22] ´A. Magyar, k-point configurations in sets of positive density of Zn, Duke Math. J. 146 (2009), no. 1, 1–34. MR 2475398 +[23] J. Matouˇsek and V. R¨odl, On Ramsey sets in spheres, J. Combin. Theory Ser. A 70 (1995), no. 1, 30–44. MR 1323999 +[24] H. Parshall, Simplices over finite fields, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2323–2334. MR 3626492 +[25] I. E. Shparlinski, Additive combinatorics over finite fields: new results and applications, Finite fields and their applications, +Radon Ser. Comput. Appl. Math., vol. 11, De Gruyter, Berlin, 2013, pp. 233–271. MR 3157428 +[26] T. Tao and V. Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University +Press, Cambridge, 2006. MR 2289012 +[27] E. Witt, Theorie der quadratischen Formen in beliebigen K¨orpern, J. Reine Angew. Math. 176 (1937), 31–44. MR 1581519 +[28] J. Wolf, Finite field models in arithmetic combinatorics—ten years on, Finite Fields Appl. 32 (2015), 233–274. MR 3293412 +Department of Mathematics, The University of Georgia, Athens, GA 30602, USA +Email address: lyall@math.uga.edu +Department of Mathematics, The University of Georgia, Athens, GA 30602, USA +Email address: magyar@math.uga.edu +Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA +Email address: parshall.6@osu.edu + diff --git a/atFIT4oBgHgl3EQflys0/content/tmp_files/load_file.txt b/atFIT4oBgHgl3EQflys0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4bd2a75d8e6828534e8a3b421b0f49c7e3645505 --- /dev/null +++ b/atFIT4oBgHgl3EQflys0/content/tmp_files/load_file.txt @@ -0,0 +1,996 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf,len=995 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='11306v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='CO] 26 Jan 2023 SPHERICAL CONFIGURATIONS OVER FINITE FIELDS NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We establish that if d ≥ 2k + 6 and q is odd and sufficiently large with respect to α ∈ (0, 1), then every set A ⊆ Fd q of size |A| ≥ αqd will contain an isometric copy of every spherical (k + 2)-point configuration that spans k dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Introduction Geometric Ramsey theory has its origins in series of papers by Erd˝os et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' [6, 7, 8], where they studied geometric configurations which cannot be destroyed by partitioning Euclidean space into finitely many classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The fundamental problem is to classify those finite sets X which are Ramsey, in the sense that for every number of colors r ∈ N there is a dimension d = d(r, X) for which every r-coloring of Rd contains a monochromatic, congruent copy of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The simplest example of a Ramsey set is a regular k-simplex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' that is, k + 1 equidistant points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Indeed, for any dimension d ≥ kr, any r-coloring of a regular d-simplex contains k + 1 points of the same color, forming a monochromatic regular k-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' On the other hand, a simple construction using the geometry of the Euclidean metric shows that any set of three collinear points is not Ramsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In fact, Erd˝os et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' [6] showed that every Ramsey set must be spherical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' that is, contained in some sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This has led to the conjecture by Graham [13] that a finite set X is Ramsey if and only if it is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This conjecture is far from settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Examples of sets known to be Ramsey include vertices of “bricks” (k-dimensional rectangles) [6], non-degenerate simplices [9], trapezoids [18], regular polygons and regular polyhedra [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Common to many of these results was the exploitation of additional symmetries of the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It was observed by Leader, Russell and Walters [20] that all known examples of Ramsey sets are subtransitive in the sense that they can be embedded in a higher dimensional set on which the rotation group acts transitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' They introduced a rival conjecture that a finite set X is Ramsey if and only if it is subtransitive, and further showed [19] that almost all 4-point subsets of a circle are not subtransitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This was later extended by Eberhard [5] to show that almost all (k + 2)-point sets on the (k − 1)-sphere are not subtransitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It remains an open question whether or not such configurations are Ramsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The aim of this article is to show that an analogue of Graham’s conjecture holds in finite field geometries for 4-point spherical configurations spanning two dimensions, and more generally for spherical (k + 2)-point configurations spanning k dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We in fact prove a stronger density version;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' namely that if d ≥ 2k + 6 and q is taken to be odd and sufficiently large with respect to α ∈ (0, 1), then every set A ⊆ Fd q of size |A| ≥ αqd contains an isometric copy of every such configuration X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' To be clear, here we say that two sets X and X′ are isometric if there is a bijection φ : X → X′ such that |φ(x) − φ(x′)|2 = |x − x′|2 for all x ∈ X and x′ ∈ X′, where |x|2 = x · x is the usual dot product of the vector x ∈ Fd q with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Our approach takes the point of view modern arithmetic combinatorics which has been very successful in the study of linear patterns in subsets of Z of positive density [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In fact, one of the main purpose of this article is to extend these techniques to the setting of geometric Ramsey theory, where one counts configurations determined by both linear as well as certain non-linear relations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The setting of vector spaces over finite fields provides a useful model to study many problems in arithmetic combinatorics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' see especially the surveys [14, 25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In the context of geometric Ramsey theory over finite fields, notable results have been obtained by a number of authors [3, 15, 16, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' However, those results 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 11B30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The first author was partially supported by NSF-DMS grant 1702411 and Simons Foundation Collaboration Grant for Mathematicians 245792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The second author was partially supported by Grants NSF-DMS 1600840 and ERC-AdG 321104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1 2 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL concern patterns consisting of points in general position, with no linear relations between them, and hence are fundamentally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Analogous results for simplices in Euclidean spaces and the integer lattice have been given in [1, 22], and it reasonable to expect that our approach here may be successfully adapted to these settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In the context of geometric (density) Ramsey theory in Rd, some results using this approach were recently obtained in [4, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We hope to address further adaptation in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Outline of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The main results of the paper are stated is Section 2 below, and some preliminaries and reductions are presented in Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' A key observation of the paper, see Proposition 6 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1, is that the count of isometric copies of a fixed configuration X along bounded functions is controlled by a certain uniformity norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This norm, which we introduce in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1, measures the uniformity or randomness of a function along geometric rectangles, and it should be compared with the so-called U 2-uniformity norm of Gowers [10] which measures uniformity along combinatorial rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If a set A is sufficiently uniform with respect to this norm, then it quickly follows from Proposition 6 that A contains many, in fact the statistically correct number of, isometric copies of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The proof of Proposition 6 is presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In order to handle arbitrary sets A we prove an inverse theorem, see Theorem 10 in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1, to establish that functions with large uniformity norm correlate with structured sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Given such an inverse theorem there are then various standard iterative procedures that one may hope to adapt to this setting to complete the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We follow an energy increment route which to leads a so-called arithmetic regularity lemma, namely Proposition 7, that allows us to decompose the indicator function of A as 1A = f1 + f2 + f3, where f1 is highly structured, f2 has small L2 norm and f3 has small uniformity norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The proof of Proposition 7 is presented in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='3 we demonstrate how Proposition 7 leads to a proof of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This consists of counting the isometric copies of X along the main term f1 and showing the contribution of the functions f2 and f3 are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The setback of this approach is that it leads to very weak bounds, in fact the dependence of q on α is tower-exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It seems quite possible that one could instead proceed via a density increment argument and obtain better, exponential type bounds, but we do not pursue this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We conclude the paper with an appendix in which we discuss the necessity of the spherical condition in the statement of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Main results We will always work with a finite field Fq of odd characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For vectors v, w ∈ Fd q, we define their dot product v · w := �d j=1 vjwj as usual and we will work with the isotropic measurements of length |v|2 := v · v and distance |v − w|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For any u ∈ Fd q and λ ∈ Fq, we define the sphere Sλ(u) = {x ∈ Fd q : |x − u|2 = λ}, and we will simply write Sλ when u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For k ∈ N, we will say that X ⊆ Fd q spans k dimensions when dim(Span(X − X)) = k, and we will call X spherical provided exists a sphere Sλ(u) ⊆ Fd q with X ⊆ Sλ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Note the (k + 1)-point spherical configurations spanning k dimensions are exactly the k-simplices, which were shown in [24] to appear as isometric copies in sufficiently dense subsets of Fd q provided d > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In Euclidean spaces it is easy to see that if the finite sets X and X′ are isometric, then X′ = z + U(X) for some vector z and orthogonal transformation U ⊆ O(d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' X′ can be obtained from X by a rigid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The same may not hold in finite field geometries due to the presence of self-orthogonal vectors x for which |x|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' However, it follows from Witt’s extension theorem [27] that if the subspace V := Span(X − X) is non-degenerate in the sense that V ∩ V ⊥ = {0}, then isometric copies of X are indeed obtained by rigid motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We use this fact in the appendix to construct dense subsets avoiding isometric copies of non-spherical sets, establishing the necessity of restricting our attention to spherical configurations X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The main result of this paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let d, k ∈ N with d ≥ 2k + 6, α ∈ (0, 1), and q ≥ q(α, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If A ⊆ Fd q with |A| ≥ αqd, then A contains at least c(α, k)q(k+1)d−k(k+1)/2 isometric copies of every (k + 2)-point spherical configuration spanning k dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 3 Here we write c(α, k) to stand in for some positive constant depending only on α and k, and q ≥ q(α, k) to indicate q is taken sufficiently large with respect to α and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will use similar notation to indicate the dependency of constants c, C > 0 that may change between occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It is helpful to think of α and k as fixed with q allowed to tend toward infinity, and implicit constants in our big-O notation may depend on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Note that if A would be a random subset of Fd q of density α then it would contain αk+2q(k+1)d−k(k+1)/2 isometric copies of a (k + 2)-point spherical configuration X up to an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This is because there are k(k + 1)/2 quadratic relations, given by the length of the edges, between the points of the configuration X and each vertex is contained in A with probability α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It immediately follows from Theorem 1 that for any fixed number of colors r, if we take q sufficiently large with respect to r, then any r-coloring of F10 q contains monochromatic, isometric copies of all 4-point spherical sets spanning 2 dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' this establishes a finite field version of Graham’s conjecture that all cyclic quadrilaterals are Ramsey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In fact, we prove a stronger statement relative to spheres;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' see [12, 23] for some so-called “sphere Ramsey” results in the Euclidean setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let d, k ∈ N with d ≥ 2k + 6, α ∈ (0, 1), and q ≥ q(α, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If λ ∈ F∗ q and A ⊆ Sλ with |A| ≥ αqd−1, then A contains at least c(α, k)q(k+1)d−(k+1)(k+2)/2 isometric copies of every (k + 2)-point spherical configuration spanning k dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' A straightforward counting argument reveals that Theorem 2 quickly implies Theorem 1, this argument is presented in Section 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We remark here that the relationship between d and k in both theorems could be improved if one were only interested in “high rank” configurations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' see the comments following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Our methods are further able to prove a version of Theorem 2 when λ = 0 provided d > 2k + 6, but we do not pursue this since it does not impact our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Preliminaries Here we record notation and ingredients that we will require for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Given any function f : Fd q → C and B ⊆ Fd q, we write Ex∈Bf(x) := 1 |B| � x∈B f(x) for the average of f over B, and we will understand the average Exf(x) is taken over Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will condense multiple averages Ey1Ey2 · · · Eyk as Ey1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yk, and to indicate linear independence we will use the notation E∗ y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yk := 1 qkd � y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yk∈Fd q linearly independent Letting χ denote the canonical additive character of Fq, we define the Fourier transform �f : Fd q → C by �f(ξ) := Exf(x)χ(−ξ · x) and we recall the Fourier inversion formula f(x) = � ξ∈Fdq �f(ξ)χ(ξ · x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Given two functions f, g : Fd q → C, we recall Plancherel’s identity Exf(x)g(x) = � ξ∈Fdq �f(ξ)�g(ξ) and, defining the convolution f ∗ g(x) = Eyf(y)g(x − y), we also recall � f ∗ g(ξ) = �f(ξ)�g(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will write σλ = q1Sλ for a normalized indicator function of Sλ, where we have the asymptotic (1) �σλ(ξ) = � 1 + O(q−1/2) if ξ = 0 O(q−1/2) otherwise valid for d ≥ 2 and λ ∈ F∗ q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' see, for instance, [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='2] exploiting Weil’s bounds on Kloosterman sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 4 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL To vectors y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , yj−1 ∈ Fd q and constants c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , cj ∈ Fq, we will associate the spherical measure (2) σc1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',cj y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yj−1(yj) = � qj if yi · yj = ci for all 1 ≤ i ≤ j 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This is essentially an L1-normalized indicator function for the intersection of the sphere Scj with j − 1 hyperplanes, so one should expect Fourier decay in appropriate directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' To import the corresponding Fourier asymptotics, we set δy1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yj−1(ξ) := � 1 if ξ ∈ Span(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , yj−1) 0 otherwise, and record the simplest case of [24, Lemma 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , cj ∈ Fq with cj ̸= 0 and y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , yj−1 ∈ Fd q linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 2j, then |�σc1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',cj y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yj−1(ξ)| = δy1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yj−1(ξ) + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It is worth mentioning if one is willing to impose technical conditions on the defining vectors and scalars, then results within [24] include stronger asymptotics valid in the range d > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' These would allow us to improve the required relationship of d ≥ 2k + 6 in Theorem 2 for configurations avoiding self-orthogonal subspaces, but we opt instead for a uniform result valid for all (k+2)-point spherical configurations spanning k dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For clarity of presentation, we will often suppress the scalars c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , cj in the notation (2) since we will always restrict ourselves to cj ̸= 0 and Lemma 3 does not depend on the other scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will also require the notion of Bohr sets, which provide a substitute for fine subgroup structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We define the Bohr set of spectrum Γ ⊆ Fd q and radius ρ ∈ (0, 1] by (3) B(Γ, ρ) := {x ∈ Fd q : |χ(ξ · x) − 1| ≤ ρ for all ξ ∈ Γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Setting β = |B(Γ, ρ)|/qd, we record the standard bound (4) β ≥ � ρ 2π �|Γ| which can be found in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Setting B = B(Γ, ρ), we will use the L1 normalized indicator function µB = β−11B and its repeated convolution νB = µB ∗µB ∗µB ∗µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The structure provided by repeated convolution will frequently be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For example, provided d ≥ 2, we can apply Fourier inversion, (1) and Plancherel to see (5) µB ∗ µB ∗ σ(x) = � ξ∈Fd q �µB(ξ)2�σ(ξ)χ(ξ · x) = 1 + O � q−1/2 � ξ∈Fd q |�µB(ξ)|2� = 1 + O(β−1q−1/2), from which it immediately follows that (6) νB ∗ σ(x) = 1 + O(β−1q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In other words, provided q sufficiently large with respect to β, repeated convolutions of µB with σ can be considered essentially constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We call the Bohr set B(Γ, ρ) regular if for every ǫ > 0, we have both |B(Γ, (1 + ǫ)ρ)| ≤ (1 + 100ǫ|Γ|)|B(Γ, ρ)| |B(Γ, (1 − ǫ)ρ)| ≥ (1 − 100ǫ|Γ|)|B(Γ, ρ)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This definition, due to Bourgain [2], ensures that all Bohr sets are only a small dilation away from being regular (see [26, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='25]), and regular Bohr sets are essentially closed under addition by elements of their interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Given a regular Bohr set B = B(Γ, ρ), we will write B′ ≺ǫ B in the case that B′ = B(Γ′, ρ′) is another regular Bohr set with Γ′ ⊇ Γ and ρ′ ≤ ǫρ/(200|Γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The following standard lemma provides the main consequence of regularity for our purposes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' we include the proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let ǫ ∈ (0, 1), B = B(Γ, ρ) regular with B′ ≺ǫ B, and f : Fd q → C with |f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For any y ∈ B′, |Ex∈Bf(x) − Ex∈Bf(x + y)| ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We have |Ex∈Bf(x) − Ex∈Bf(x + y)| = 1 |B| ��� � x∈Fdq f(x)(1B(x) − 1B(x − y)) ��� ≤ |B△(y + B)| |B| Since B′ ≺ǫ B and y ∈ B′ we have the relationship B△(y + B) ⊆ B(Γ, ρ + ρ′) \\ B(Γ, ρ − ρ′) and our claim follows from the bound |B(Γ, ρ + ρ′) \\ B(Γ, ρ − ρ′)| ≤ ǫ|B| resulting from regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Reduction to Dense Spherical Sets In this section we present the straightforward counting argument that estblishes Theorem 1 as a conse- quence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The main observation is that the collection of spheres of a fixed radius λ ∈ F∗ q provides a uniform cover of Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Hence, any subset A ⊆ Fd q with density α has density nearly α on a large number, in fact a positive proportion, of these spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Theorem 2 then implies that within each of these spheres, A contains many of the sought after configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' By counting the contribution of each of these spheres, it is easy to see that A contains a positive proportion of the count of all such configurations within Fd q, as each fixed configuration is contained in approximately the same number of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We first record the characterization of spherical configurations that will be most useful as we proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This follows in a straightforward way from Lemma 16 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let X ⊆ Fd q be any (k + 2)-point spherical configuration spanning k dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If any (k + 1)- point subset of X that spans k dimensions is contained in a sphere Sλ(u), then X ⊆ Sλ(u) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let us now see how Theorem 2 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Fixing λ ∈ F∗ q, we first establish that for many centers u ∈ Fd q, |A ∩ Sλ(u)| is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For d ≥ 2 and x ∈ Fd q, (1) implies Euσλ(u − x) = 1 + O(q−1/2), in which case we can ensure q is sufficiently large for (α/2)q2d−1 ≤ � u∈Fdq |A ∩ Sλ(u)| = � u∈Fd q |A∩Sλ(u)|≥(α/4)qd−1 |A ∩ Sλ(u)| + � u∈Fd q |A∩Sλ(u)|<(α/4)qd−1 |A ∩ Sλ(u)| ≤ |Sλ||{u ∈ Fd q : |A ∩ Sλ(u)| ≥ (α/4)qd−1}| + (α/4)q2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Again using (1) and ensuring q sufficiently large, we can conclude |{u ∈ Fd q : |(A − u) ∩ Sλ| ≥ (α/4)qd−1}| ≥ (α/8)qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Fixing now some (k + 2)-point spherical configuration X spanning k dimensions, we see that for each of these good centers u, Theorem 2 guarantees that A ∩ Sλ(u) contains at least c(α, k)q(k+1)d−(k+2 2 ) isometric copies of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We need to account for how many spheres we count each isometric copy within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' By translating X if necessary, we can parametrize it as X = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk, a1v1 + · · · + akvk} for linearly independent v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk ∈ Fd q and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , ak ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then for a translation x ∈ Fd q and linearly independent vectors y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , yk ∈ Fd q with yi · yj = vi · vj for all 1 ≤ i ≤ j ≤ k, we wish to count how many spheres Sλ(u) contain the configuration X′ = {x, x + y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , x + yk, x + a1y1 + · · · + akyk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Since y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , yk are linearly independent, Lemma 3 applies to the spherical measure σy1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yk(u) := σλ(u) k � j=1 σλ(yj − u) 6 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL so that, for d ≥ 2k + 2, Eu∈Fdqσy1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',yk(x − u) = 1 + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 5 ensures that {x, x + y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , x + yk} ⊆ Sλ(u) implies X′ ⊆ Sλ(u) as well, so we have shown |{u ∈ Fd q : X′ ⊆ Sλ(u)}| = (1 + O(q−1/2))qd−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Hence, within each of the (α/8)qd good spheres Sλ(u), A ∩ Sλ(u) contains c(α, k)q(k+1)d−( k+2 2 ) isometric copies of X, and each of these copies is contained in roughly qd−k−1 spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In total this yields that A contains c(α, k)q(k+1)d−(k+1 2 ) isometric copies of X as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof of Theorem 2 In this section we reduce the task of proving Theorem 2 to that of establishing Propositions 6 and 7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Counting Configurations, a Uniformity Norm, and two key Propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the remainder, we fix λ ∈ F∗ q and aim to establish Theorem 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' there is no harm in assuming λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For brevity, we set S = Sλ and σ = σλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will be considering (k + 2)-point spherical configurations spanning k dimensions typically parameterized as X := {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk, a1v1 + · · · + akvk} ⊆ Fd q, with v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk ∈ Fd q linearly independent and coefficients a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , ak ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will consider other sets of k + 2 points of the form X′ = {x0, x0 + x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , x0 + xk, x0 + a1x1 + · · · + akxk} ⊆ Fd q and check whether these are indeed isometric copies of X contained within S by checking whether x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk are linearly independent and further satisfy the conditions |x0|2 = |x0 + x1|2 = · · · = |x0 + xk|2 = λ and xi · xj = vi · vj for each 1 ≤ i ≤ j ≤ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' when all of this is true, we write X′ ≃ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We remark that this notation only explicitly insists that k + 1 points of X′ lie on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' However, since we are working with the same coefficients a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , ak, Lemma 16 ensures X′ is spherical and Lemma 5 ensures X′ ⊂ S as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' That is, if X′ ≃ X, then it is also the case that X′ ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' To count copies of X parameterized as X′ above, we define the weight SX(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk) := � q(k+1)(k+2)/2 if X′ ≃ X 0 otherwise and a normalized counting operator on functions f0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1 : Fd q → C by (7) NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1) := Ex0f0(x0)E∗ x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk � k � j=1 fj(x0 + xj) � fk+1(x0 + a1x1 + · · · + akxk)SX(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Note that as long as we restrict our attention to linearly independent x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk, we can write out SX(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk) = σ(x0) k � j=1 σx0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1(xj) for appropriate spherical measures with implicit scalars determined by our sphere S and the dot products between the defining vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Moreover, the contribution of linearly dependent x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk is negligible, in the sense that we trivially have 1 qd � xj∈Span(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1) σx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1(xj) ≤ q−1 whenever d ≥ 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Together with Lemma 3, this allows us to freely add in linearly dependent x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk to our count (7) at the cost of an acceptable O(q−1) error, provided each fj is bounded as will typically be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 7 The starting point for our argument is to show that the counting operator (7) is controlled by what we call the U 2 ⊥(S) norm, defined for f : Fd q → C by ∥f∥U2 ⊥(S) := � Ex,h,h′fσ(x)fσ(x + h)fσ(x + h′)fσ(x + h + h′) �1/4 , which is the usual Gowers U 2(Fd q) norm of the function fσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' While the U 2(Fd q) norm averages over combina- torial rectangles, the U 2 ⊥(S) norm averages instead over geometric rectangles contained within our sphere S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Note that if |x|2 = |x + h|2 = |x + h′|2 = |x + h + h′|2 = λ, then it is also the case that h · h′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will show that the operator (7) is controlled by the U 2 ⊥(S) norm in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1 : Fd q → C with |fj| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 2k + 6, then |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)| ≤ min 0≤j≤k+1∥fj∥U2 ⊥(S) + O(q−1/8) Results of this type are often called generalized von-Neumann inequalities in the arithmetic combinatorics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The proof of Proposition 6 is presented in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' To see the utility of such a result, consider a set A ⊆ S with |A| ≥ αqd−1 and fA = 1A − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If A is sufficiently uniform, in the sense that ∥fA∥U2 ⊥(S) is sufficiently small with respect to α and k, then the decomposition 1A = α + fA along with Proposition 6 provides NX(1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1A) ≳ αk+2 provided q is taken sufficiently large with respect to α and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Of course, not all sets must be uniform in this sense, and we will require a more sophisticated decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Defining the L2(S) norm by ∥f∥L2(S) := � Ex|f(x)|2σ(x) �1/2 , we will use the following decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let η ∈ (0, 1), f : Fd q → [−1, 1] and ϕ : (0, 1]2 → (0, 1) increasing in both coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 2 and q ≥ q(η, f), then there exists B = B(Γ, ρ) regular with |Γ| ≤ C(η, f) and ρ ≥ c(η, f), and there exist functions f1, f2, f3 : Fd q → [−2, 2] with f = f1 + f2 + f3 f1 = νB ∗ (fσ) ∥f2∥U2 ⊥(S) ≤ ϕ(|Γ|−1, ρ) ∥f3∥L2(S) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Results of this type are often called arithmetic regularity lemmas in the arithmetic combinatorics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In light of (6), it should at least be clear that ensuring f1 is bounded amounts to ensuring q is sufficiently large with respect to the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The proof of Proposition 7 is presented in Section 7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We close this section by demonstrating, as promised, how Propositions 6 and 7 can be applied to give a proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We initially specialize, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='2 below, to the case of spherical quadrilaterals, that is when k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The proof of the general case follows along similar lines and is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof that Propositions 6 and 7 imply the k = 2 case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It clearly suffices to establish the following Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let α ∈ (0, 1) and A ⊆ S with |A| ≥ αqd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 10, q ≥ q(α), and X ⊆ Fd q is a spherical 4-point configuration spanning 2 dimensions, then NX(1A, 1A, 1A, 1A) ≥ c(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In other words, A contains c(α)q3d−6 isometric copies of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 8 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' To set up, let ǫ > 0 be a parameter to be determined only in terms of α and let ϕ : (0, 1]2 → (0, 1) be a function increasing in both coordinates to be determined only in terms of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We apply the decomposition theorem to obtain a regular Bohr set B = B(Γ, ρ) with |Γ| ≤ C(ǫ, ϕ) and ρ ≥ c(ǫ, ϕ) and functions f1, f2, f3 : Fd q → [−2, 2] such that 1A = f1 + f2 + f3 f1 = νB ∗ (1Aσ) ∥f2∥U2 ⊥(S) ≤ ϕ(|Γ|−1, ρ) ∥f3∥L2(S) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We take ρ′ ∈ [ǫρ/(400|Γ|), ǫρ/(200|Γ|)] so that B1 := B(Γ, ρ′) ≺ǫ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Throughout the argument we will continue to take q(α) large enough so that, with parameters other than q fixed, we can absorb error terms that tend to zero as q → ∞ into a single O(ǫ) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We now fix a spherical 4-point configuration spanning 2 dimensions X, parameterize it as X = {0, v, w, av + bw} for v, w linearly independent, and search for isometric copies of the form {x, x + y, x + z, x + ay + bz} ⊆ S where |y|2 = |v|2, |z|2 = |w|2, and y · z = v · w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' To detect these copies, we define the spherical measures σx(y) := σ−|v|2/2,|v|2 x (y), and σx,y(z) := σ−|w|2/2,v·w,|w|2 x,y (z) in which case we can parametrize our counting operator as NX(g0, g1, g2, g3) = Exg0(x)σ(x)E∗ y,zg1(x + y)g2(x + z)g3(x + ay + bz)σ(x)σx(y)σx,y(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Setting B2 := B(Γ ∪ {a · ξ : ξ ∈ Γ} ∪ {b · ξ : ξ ∈ Γ}, ρ′/4), we note that restricting y, z ∈ B2 + B2 ensures y, z, ay + bz ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This is useful to us, since for our main term function f1, for any x ∈ Fd q and any x′ ∈ B1, Lemma 4 provides (8) f1(x + x′) = f1(x) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will in fact work with the further restricted count N B X (g0, g1, g2, g3) := Exg0σ(x)E∗ y,zg1(x + y)g2(x + z)g3(x + ay + bz)σx(y)µB2 ∗ µB2(y)σx,y(z)µB2 ∗ µB2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Setting β := |B2|/qd, the size estimate (4) and the dependency of our parameters guarantees β−1 ≤ �C|Γ| ǫρ �3|Γ| ≤ C(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We should establish that this restricted count is well normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' That is, we will show (9) N B X (1, 1, 1, 1) = 1 + O(ǫ), where here 1 stands for the constant 1 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Applying Parseval and extracting the ξ = 0 term, EzµB2 ∗ µB2(z)σx,y(z) = 1 + � ξ∈Fdq\\{0} �µB2(ξ)2�σx,y(ξ) + O(ǫ), so we can apply Lemma 3 for N B X (1, 1, 1, 1) = Ex,yσ(x)σx(y)µB2 ∗ µB2(y) + O \uf8eb \uf8edβ−1 � ξ∈Fdq\\{0} |�µB2(ξ)|2Ex,yσ(x)σx(y)δx,y(ξ) \uf8f6 \uf8f8 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Without trying to be too careful, notice that uniformly in ξ ̸= 0, Ex,yσ(x)σx(y)δx,y(ξ) ≤ q3Ex,yδx,y(ξ) = O(q−1), so we can apply Plancherel and ensure q is sufficiently large to conclude N B X (1, 1, 1, 1) = Ex,yσ(x)σx(y)µB2 ∗ µB2(y) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 9 Arguing similarly to eliminate the average in y, we conclude (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This restricted count will be useful for us since we trivially have the lower bound (10) NX(1A, 1A, 1A, 1A) ≥ β2N B X (1A, 1A, 1A, 1A), and we will spend the rest of the proof establishing the lower bound N B X (1A, 1A, 1A, 1A) ≥ α4 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' From the decomposition 1A = f1 + f2 + f3, N B X (1A, 1A, 1A, 1A) = � 1≤i0,i1,i2,i3≤3 N B X (fi0, fi1, fi2, fi3), and we handle separately the three cases (i) when each of i0, i1, i2, i3 equals 1, (ii) when one of i0, i1, i2, i3 equals 2, and (iii) when one of i0, i1, i2, i3 equals 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Case (i) will yield our main term of α4, and we will argue that cases (ii) and (iii) contribute a negligible O(ǫ) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the first case, applying (8) and (9) yields N B X (f1, f1, f1, f1) = Exf1(x)4σ(x)Ey,zµB2 ∗ µB2(y)σx(y)µB2 ∗ µB2(z)σx,y(z) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Arguing as we did to establish (9), N B X (f1, f1, f1, f1) = Exf1(x)4σ(x) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then from H¨older’s inequality, N B X (f1, f1, f1, f1) ≥ (Exf1(x)σ(x))4 + O(ǫ), and writing Exf1(x)σ(x) = ExνB ∗ 1A(x)σ(x) = Ex1A(x)νB ∗ σ(x), we can apply (6) to conclude N B X (f1, f1, f1, f1) ≥ α4 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the second case, with one of i0, i1, i2, i3 = 2, we apply Fourier inversion to write µB2 ∗ µB2(y)µB2 ∗ µB2(z) = � ξ1,ξ2∈Fdq �µB2(ξ1)2�µB2(ξ2)2χ(−ξ1 · y)χ(−ξ2 · z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Setting χξ(x) = χ(ξ · x), this allows us to express N B X (fi0, fi1, fi2, fi3) = � ξ1,ξ2∈Fdq �µB2(ξ1)2�µB2(ξ2)2TX,S(fi0χξ1+ξ2, fi1χ−ξ1, fi2χ−ξ2, fi3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' As ∥fχξ∥U2 ⊥(S) = ∥f∥U2 ⊥(S), we can apply the triangle inequality, Proposition 6 and Plancherel to conclude |N B X (fi0, fi1, fi2, fi3)| ≤ β−2∥f2∥U2 ⊥(S) + O(ǫ) ≤ �C|Γ| ǫρ �6|Γ| ϕ(|Γ|−1, ρ) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We see that by taking ϕ(|Γ|−1, ρ) = ǫ �cǫρ |Γ| �6|Γ| , we can ensure case (ii) contributes at most O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the third case, we will assume i0 = 3, but each case is similar by reindexing in x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Applying the triangle inequality and using that fi1, fi2, fi3 are each at most 2 in absolute value, N B X (f3, fi1, fi2, fi3) ≤ 8Ex|f3(x)|σ(x)E∗ y,zσx(y)µB2 ∗ µB2(y)σx,y(z)µB2 ∗ µB2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then arguing as we did to establish (9) we have N B X (f3, fi1, fi2, fi3) ≤ 8Ex|f3(x)|σ(x) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 10 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL Applying Cauchy-Schwarz and the fact that ∥f3∥L2(S) ≤ ǫ, we conclude case (iii) again contributes at most an O(ǫ) error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In total we have shown N B X (1A, 1A, 1A, 1A) ≥ α4 + O(ǫ) which we can insist is at least α4/2 by taking ǫ sufficiently small with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then since β ≥ c(α), we are done by recalling (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof that Propositions 6 and 7 imply Theorem 2 in its full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It clearly suffices to establish the following Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let α ∈ (0, 1) and A ⊆ S with |A| ≥ αqd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 2k + 6, q ≥ q(α, k), and X ⊆ Fd q is a spherical (k + 2)-point configuration spanning k dimensions, then NX(1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1A) ≥ c(α, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In other words, A contains c(α, k)q(k+1)d−(k+1)(k+2)/2 isometric copies of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The proof is essentially the same as for 4-point configurations spanning 2 dimensions, although the notation becomes more cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' As before, to set up, let ǫ > 0 be a parameter to be determined only in terms of α and k and let ϕ : (0, 1]2 → (0, 1) be a function increasing in both coordinates to be determined only in terms of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We apply the decomposition theorem to obtain a regular Bohr set B = B(Γ, ρ) with |Γ| ≤ C(ǫ, ϕ) and ρ ≥ c(ǫ, ϕ) and functions f1, f2, f3 : Fd q → [−2, 2] such that 1A = f1 + f2 + f3 f1 = νB ∗ (1Aσ) ∥f2∥U2 ⊥(S) ≤ ϕ(|Γ|−1, ρ) ∥f3∥L2(S) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We take ρ′ ∈ [ǫρ/(400|Γ|), ǫρ/(200|Γ|)] so that B1 := B(Γ, ρ′) ≺ǫ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Throughout the argument we will continue to take q(α, k) large enough so that, with parameters other than q fixed, we can absorb error terms that tend to zero as q → ∞ into a single O(ǫ) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We now fix a spherical (k + 2)-point configuration spanning k dimensions X parameterized as X = {0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk+1} where vk+1 = a1v1 + · · · + akvk and search for isometric copies of the form {x0, x0 + x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , x0 + xk+1} ⊂ S where xk+1 = a1x1 + · · · + akxk and xi · xj = vi · vj for 1 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' To detect these copies, we define the spherical measures σx0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1(xj) := (σ)−|vj|2/2,v1·vj,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',vj−1·vj,|vj|2 x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1 (xj) in which case we can parametrize our count as NX(g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , gk+1) = Ex0g0(x0)σ(x0)E∗ x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xkg(x0 + xk+1) k � j=1 g(x0 + xj)σx0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Setting B2 := B \uf8eb \uf8edΓ ∪ k� j=1 {aj · ξ : ξ ∈ Γ} , ρ′ 2k \uf8f6 \uf8f8 , ensures that if x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk ∈ B2 + B2, then x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk+1 ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Again we will use that for x ∈ Fd q and x′ ∈ B1, (11) f1(x + x′) = f1(x) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Our restricted count is given by N B X (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , gk+1) = Ex0g0(x0)σ(x0)E∗ x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xkg(x0 + xk+1) k � j=1 g(x0 + xj)µB2 ∗ µB2(xj)σ(j+1)(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 11 As before, setting β := |B2|/qd, we have β−1 ≤ C(α, k) and a well-normalized restricted count (12) N B X (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1) = 1 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Applying the straightforward lower bound (13) NX(1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1A) ≥ βkN B X (1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1A), we will spend the rest of the proof establishing N B X (1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1A) ≥ αk+2 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' From the decomposition 1A = f1 + f2 + f3, N B X (1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1A) = � 1≤i0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',ik+1≤3 N B X (fi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fik+1), and we handle separately the three cases (i) when each of ij equals 1, (ii) when one of ij equals 2, and (iii) when one of ij equals 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Case (i) will yield our main term of αk+2, and we will argue that cases (ii) and (iii) contribute a negligible O(ǫ) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the first case, applying (11) and (12) yields N B X (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , f1) = Ex0f1(x0)k+2σ(x)Ex1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk k � j=1 µB2 ∗ µB2(xj)σ(j+1)(xj) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Arguing as we did to establish (9), N B X (f1, f1, f1, f1) = Exf1(x)k+2σ(x) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then from H¨older’s inequality, N B X (f1, f1, f1, f1) ≥ (Exf1(x)σ(x))k+2 + O(ǫ) = (Ex1A(x)νB ∗ σ(x))k+2 + O(ǫ) = αk+2 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the second case, we apply Fourier inversion to write k � j=1 µB2 ∗ µB2(xj) = � ξ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',ξk∈Fdq k � j=1 �µB2(ξj)2χ(−ξj · xj) in order to express N B X (fi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fik+1) = � ξ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',ξk∈Fd q k � j=1 �µB2(ξj)2NX(fi0χξ1+···+ξk, fi1χ−ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' fikχ−ξk, fik+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' As before, ∥fχξ∥U2 ⊥(S) = ∥f∥U2 ⊥(S), so we apply the triangle inequality, Proposition 6 and Plancherel to conclude |N B X (fi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fik+1)| ≤ β−k∥f2∥U2 ⊥(S) + O(ǫ) ≤ �C|Γ| ǫρ �C|Γ| ϕ(|Γ|−1, ρ) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We see that by taking ϕ(|Γ|−1, ρ) = ǫ �cǫρ |Γ| �C|Γ| , we can ensure case (ii) contributes at most O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the third case, we assume i0 = 3 and apply the triangle inequality for N B X (f3, fi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fik+1) ≤ 2k+1Ex0|f3(x0)|σ(x0)Ex1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk k � j=1 µB2 ∗ µB2(xj)σ(j+1)(xj) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 12 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL Applying Cauchy-Schwarz and the fact that ∥f3∥L2(S) ≤ ǫ, we conclude case (iii) again contributes at most an O(ǫ) error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In total we have shown N B X (1A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , 1A) ≥ αk+2 + O(ǫ) which we can insist is at least αk+2/2 by taking ǫ sufficiently small with respect to α and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then since β ≥ c(α, k), we are done by recalling (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ It remains to prove both Proposition 6 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof of Proposition 6 We establish Proposition 6 through two careful applications of Cauchy-Schwarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In a sense, each ap- plication of Cauchy-Schwarz replaces our configuration with a more regular configuration, and we are left considering averages over geometric rectangles rather than more complicated spherical configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We fix a spherical (k + 2)-point configuration X spanning k dimensions parameterized by X = {v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk+1} with v1 − v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , vk − v0 linearly independent and vk+1 = k � j=0 ajvj for coefficients aj ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It is enough by symmetry to show |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)| ≤ ∥fk∥U2 ⊥(S) + O(q−1/8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk ∈ Fd q, we will understand that xk+1 = k � j=0 aixi, and we will detect if {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk+1} ≃ X with the spherical measures σ(x0) and, for 1 ≤ j ≤ k, σ(j)(xj) := σc0,j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',cj−1,j,λ x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1 (xj) where we define ci,j = |vi − vj|2/2 − λ, since if |xi|2 = |xj|2 = λ, checking whether |xi − xj|2 = |vi − vj|2 amounts to checking whether xi · xj = |vi − vj|2/2 − λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then we can express our counting operator NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1) = Ex0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xkσ(x0) k+1 � j=0 fj(xj) k � j=1 σ(j)(xj) + O(q−1), where we have included the negligible amount of linearly dependent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Rearranging and applying the triangle inequality, |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)| ≤ Ex0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1σ(x0) k−1 � j=1 σ(j)(xj) ���Exkfk(xk)fk+1(xk+1)σ(k)(xk) ��� + O(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Introducing the differencing notation ∆hf(x) := f(x)f(x + h), we square both sides and apply Cauchy-Schwarz for |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)|2 ≤ Ex0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1σ(x0) k−1 � j=1 σ(j)(xj)Exk,h∆hfk(xk)∆hfk+1(xk+1)∆hσ(k)(xk) + O(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We introduce new spherical measures with an additional condition involving h defined by σh(x0) := σ0 h(x0) σ(j) h (xj) := σ0,c0,j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',cj−1,j,λ h,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1 (xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 13 which allows us to rewrite σ(x0) k−1 � j=1 σ(j)(xj)∆hσ(k)(xk) = σh(x0) k−1 � j=1 σ(j) h (xj)σ(k)(xk)σ(xk + h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Our bound from above can then be rearranged as |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)|2 ≤ Eh,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1σh(x0) k−1 � j=1 σ(j) h (xj)Exk∆hfk(xk)∆hfk+1(xk+1)σ(k)(xk)σ(xk+h)+O(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We claim that in the average above, the dependence of xk+1 on xk is superficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' That is, while xk+1 = k � j=0 ajxj, it must be the case that at least two coefficients aj are nonzero since we are working with a (k + 2)-point spherical configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In particular, aj ̸= 0 for some 0 ≤ j < k, and for this distinguished j, we can reindex in xj in the average above to replace xk+1 with x′ k+1 := k−1 � j=0 a′ jxj, allowing us to rearrange our bound above as |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)|2 ≤ Eh,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1∆hfk+1(x′ k+1)σh(x0) k−1 � j=1 σ(j) h (xj)Exk∆hfk(xk)σ(k)(xk)σ(xk+h)+O(q−1), where we may have needed to adjust the implicit scalars in our spherical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This allows us to proceed as before by applying the triangle inequality for |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)|2 ≤ Eh,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1σh(x0) k−1 � j=1 σ(j) h (xj) ���Exk∆hfk(xk)σ(k)(xk)σ(xk + h) ��� + O(q−1), and Cauchy-Schwarz once more for |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)|4 ≤ Eh,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1σh(x0) k−1 � j=1 σ(j) h (xj)Exk,h′∆h′∆hfk(xk)∆h′σ(k)(xk)∆h′σ(xk +h)+O(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We can again reorganize our spherical measures, rewriting σh(x0) k−1 � j=1 σ(j) h (xj)∆h′σ(k)(xk)∆h′σ(xk + h) = ∆h′∆hσ(xk)σxk,h,h′(x0) k−1 � j=1 σxk,h,h′,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1(xj) for appropriate implicit scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Using this to rearrange our bound above, we have |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)|4 ≤ Exk,h,h′∆h′∆hfkσ(xk)Ex0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1σxk,h,h′(x0) k−1 � j=1 σxk,h,h′,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1(xj) + O(q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Applying Lemma 3, we have, uniformly in xk, h, h′ when d ≥ 2k + 6, Ex0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xk−1σxk,h,h′(x0) k−1 � j=1 σxk,h,h′,x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',xj−1(xj) = 1 + O(q−1/2), establishing |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)|4 ≤ Exk,h,h′∆h′∆hfkσ(xk) + O(q−1/2) from which we see |NX(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , fk+1)| ≤ ∥fk∥U2 ⊥(S) + O(q−1/8) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' An Inverse Theorem and Proof of Proposition 7 This section is dedicated to establishing Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We begin in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1 by establishing Theorem 10, an inverse theorem which reveals that functions with large U 2 ⊥(S) norm must exhibit Fourier bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 14 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' An Inverse Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' One way to see that functions with large U 2 ⊥(S) norm exhibit Fourier bias is to relate the U 2 ⊥(S) norm to the usual U 2(Fd q) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In order to do so, define for v, w ∈ Fd q the normalized indicator function ℓv=w = � qd if v = w 0 otherwise Then one can express ∥f∥4 U2 ⊥(S) = Ex,y,z,wfσ(x)fσ(y)fσ(z)fσ(w)ℓx+w=y+z Expanding ℓx+w=y+z via orthogonality, ℓx+w=y+z = � ξ∈Fdq χ(ξ(x + w − y − z)), providing the identity ∥f∥4 U2 ⊥(S) = � ξ∈Fdq |� fσ(ξ)|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The right hand side is precisely ∥fσ∥U2(Fdq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It is tempting to here use Plancherel to bound ∥f∥4 U2 ⊥(S) above by supξ∈Fd q |� fσ(ξ)|2∥fσ∥2 L2, but this is not generally helpful since the L2 term may grow with q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' By being a bit more careful, we establish the following inverse theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let f : Fd q → C with |f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 8, then ∥f∥U2 ⊥(S) ≤ sup ξ∈Fdq |� fσ(ξ)|1/4 + O(q−1/32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We apply absolute values and the triangle inequality for ∥f∥4 U2 ⊥(S) ≤ Exσ(x)|Ey,z,wfσ(y)fσ(z)fσ(w)ℓx=y+z−w| Applying Cauchy-Schwarz, ∥f∥8 U2 ⊥(S) ≤ (Exσ(x)) � Exσ(x)|Ey,z,wfσ(y)fσ(z)fσ(w)ℓx=y+z−w|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For any fixed x and y, we have Ez,wσ(z)σ(w)ℓx=y+z−w = Ezσx+y(z) for some measure σx+y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then since d ≥ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' we can apply (1) and Lemma 3 for ∥f∥8 U2 ⊥(S) ≤ Exσ(x)Ey1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='w1 y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='w2fσ(y1)fσ(y2)fσ(z1)fσ(z2)fσ(w1)fσ(w2)ℓx=y1+z1−w1ℓx=y2+z2−w2 + O(q−1/2) Moving the average in x inside and rearranging a bit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' this simplifies to ∥f∥8 U2 ⊥(S) ≤ Ey1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='w1 y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='w2fσ(y1)fσ(y2)fσ(z1)fσ(z2)fσ(w1)fσ(w2)σ(y1 +z1−w1)ℓw2−w1=y2−y1+z2−z1 +O(q−1/2) Setting f1 = f and f2 = f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' rearranging a bit more and applying the triangle inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' ∥f∥8 U2 ⊥(S) ≤ Ew1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='w2σ(w1)σ(w2)|Ey1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='z2σ(y1+z1−w1)ℓw2−w1=y2−y1+z2−z1 � j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='2 fjσ(yj)fjσ(zj)|+O(q−1/2) Setting f3 = f and f4 = f and again requiring d ≥ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' we apply Cauchy-Schwarz as before and rearrange for (14) ∥f∥16 U2 ⊥(S) ≤ Ey1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',y4 z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',z4 4 � j=1 fjσ(yj)fjσ(zj)ℓy2−y1+z2−z1=y4−y3+z4−z3Wy1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',y4 z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',z4 + O(q−1/2) where Wy1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',y4 z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',z4 = Ewσ(w)σ(y2 − y1 + z2 − z1 + w)σ(y1 + z1 − w)σ(y3 + z3 − w) Since d ≥ 4, we can restrict the sum in (14) to only consider the terms when the vectors y2 − y1 + z2 − z1, y1 + z1, y3 + z3 are linearly independent at the cost of an error that can be absorbed in our current error of O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For these vectors, the function σ′(w) = σ(w)σ(y2 − y1 + z2 − z1 + w)σ(y1 + z1 − w)σ(y3 + z3 − w) SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 15 is a measure for which Lemma 3 applies, allowing us to conclude Wy1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',y4 z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',z4 = Ewσ′(w) = 1 + O(q−1/2), valid for d ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We have arrived at the estimate ∥f∥16 U2 ⊥(S) ≤ Ey1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',y4 z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',z4 4 � j=1 fjσ(yj)fjσ(zj)ℓy2−y1+z2−z1=y4−y3+z4−z3 + O(q−1/2) Expanding ℓy2−y1+z2−z1=y4−y3+z4−z3 via orthogonality, we have ℓy2−y1+z2−z1=y4−y3+z4−z3 = � ξ∈Fdq χ(ξ · (y4 − y3 − y2 + y1 + z4 − z3 − z2 + z1)), from which we have the identity Ey1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',y4 z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=',z4 4 � j=1 fjσ(yj)fjσ(zj)ℓy2−y1+z2−z1=y4−y3+z4−z3 = � ξ∈Fdq |� fσ(ξ)|8 It follows that we can conclude ∥f∥16 U2 ⊥(S) ≤ � ξ∈Fd q |� fσ(ξ)|8 + O(q−1/2) ≤ sup ξ∈Fdq |� fσ(ξ)|4 � ξ∈Fdq |� fσ(ξ)|4 + O(q−1/2) ≤ sup ξ∈Fd q |� fσ(ξ)|4 + O(q−1/2), where we have used both |� fσ(ξ)| ≤ 1 + O(q−1/2) and � ξ∈Fdq |� fσ(ξ)|4 = ∥f∥4 U2 ⊥(S) ≤ 1 + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' In order to use the Fourier bias from Theorem 10 to prove Proposition 7, we will construct a sequence of Bohr sets, refining at each stage, until we arrive at B′ ≺ǫ B for which νB′ ∗ (fσ) and νB ∗ (fσ) are close in an L2 sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This will require a number of technical lemmas, the first of which indicates that these two convolutions are somewhat orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let ǫ ∈ (0, 1), B = B(Γ, ρ) a regular Bohr set with B′ ≺ǫ B, and f : Fd q → C with |f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 2 and q ≥ q(|Γ|, ρ, ǫ), then ∥νB′ ∗ (fσ)∥2 2 − ∥νB ∗ (fσ)∥2 2 ≥ ∥νB′ ∗ (fσ) − νB ∗ (fσ)∥2 2 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' By expanding the square, ∥νB′ ∗ (fσ) − νB ∗ (fσ)∥2 2 is equal to ∥νB′ ∗ (fσ)∥2 2 − 2ExνB′ ∗ (fσ)(x)νB ∗ (fσ)(x) + ∥νB ∗ (fσ)∥2 2, so it suffices to show that ∥νB ∗ (fσ)∥2 2 ≤ ExνB′ ∗ (fσ)(x)νB ∗ (fσ)(x) + O(ǫ) Using the relationship B′ ≺ǫ B, we can apply Lemma 4 to see |�µB(ξ)| = |Ex∈Bχ(x · ξ)| ≤ |Ex∈BEy∈B′χ((x + y) · ξ)| + ǫ = |Ex∈Bχ(x · ξ)Ey∈B′χ(y · ξ)| + ǫ ≤ |�µB′(ξ)| + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' in which case we apply Plancherel and this observation for ∥νB ∗ (fσ)∥2 2 = � ξ∈Fdq |�µB(ξ)|8|� fσ(ξ)|2 ≤ � ξ∈Fdq |�µB(ξ)|4 (|�µB′(ξ)| + ǫ)4 |� fσ(ξ)|2 16 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL Writing β = |B|/qd, we can apply Plancherel and (5) for � ξ∈Fdq |� µB(ξ)|4|� fσ(ξ)|2 = ∥µB ∗ µB ∗ fσ(x)∥2 2 ≤ 1 + O(β−1q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Using the bound (4), we see that we can take q sufficiently large and apply Plancherel again to conclude ∥νB ∗ fσ∥2 2 ≤ � ξ∈Fd q |�µB(ξ)|4|�µB′(ξ)|4|� fσ(ξ)|2 + O(ǫ) = ExνB′ ∗ (fσ)(x)νB ∗ (fσ)(x) + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ With Lemma 11 in hand, we are ready to translate Theorem 10 into an energy increment, showing that Fourier bias leads to a Bohr set refinement with increased L2 energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let η ∈ (0, 1), ǫ ∈ (0, cη8) for c > 0 sufficiently small, B = B(Γ, ρ) a regular Bohr set, and f : Fd q → C with |f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 2, q ≥ q(|Γ|, ρ, η, ǫ), and ∥f − νB ∗ (fσ)∥U2 ⊥(S) ≥ η, then there exists B′ = B(Γ′, ρ′) with B′ ≺ǫ B, |Γ′| ≤ |Γ| + 1, ρ′ ≥ c(|Γ|, ǫ)ρ and ∥νB′ ∗ (fσ)∥2 L2(S) ≥ ∥νB ∗ (fσ)∥2 L2(S) + cη8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' From the inverse theorem, ∥f − νB ∗ (fσ)∥U2 ⊥(S) ≥ η implies the existence of some γ ∈ Fd q with |� fσ(γ) − [(νB ∗ (fσ))σ]∧(γ)| ≥ η4 + O(q−1/32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Expanding [(νB ∗ (fσ))σ]∧(γ), we can apply (1) for [(νB ∗ (fσ))σ]∧(γ) = � ξ∈Fd q �σ(ξ)�µB(γ − ξ)4 � fσ(γ − ξ) = �µB(γ)4 � fσ(γ) + O � q−1/2 � γ |�µB(ξ)|4|� fσ(ξ)| � Applying the uniform bounds |�µB(ξ)| ≤ 1 and |� fσ(ξ)| ≤ 1 + O(q−1/2) together with Plancherel, we have (15) [(νB ∗ (fσ))σ]∧(γ) = �µB(γ)4 � fσ(γ) + O(β−1q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' in which case we can ensure q is sufficiently large and refine our inverse statement to (16) |� fσ(γ)||1 − �µB(γ)4| ≥ η4/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Set Γ′ = Γ ∪ {γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It follows from [26, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='25] that for any ǫ ∈ (0, 1) we can choose a radius ρ′ ∈ [ǫρ/(400|Γ|), ǫρ/(200|Γ|)] such that B′ := B(Γ′, ρ′) is regular, in which case B′ ≺ǫ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' From Lemma 11, ∥νB′ ∗ (fσ)∥2 2 − ∥νB ∗ (fσ)∥2 2 ≥ ∥νB′ ∗ (fσ) − νB ∗ (fσ)∥2 2 + O(ǫ), so we would like to provide a lower bound for ∥νB′ ∗ fσ − νB ∗ fσ∥2 2 = � ξ∈Fdq |�µB′(ξ)4 − �µB(ξ)4|2|� fσ(ξ)|2 ≥ |�µB′(γ)4 − �µB(γ)4|2|� fσ(γ)|2 As γ ∈ Γ′, |�µB′(γ) − 1| ≤ ρ′ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Combining this with |� fσ(γ)| ≤ 1 + O(q−1/2) and (16), |�µB′(γ)4 − �µB(γ)4|2|� fσ(γ)|2 ≥ |1 − �µB(γ)4|2|� fσ(γ)|2 + O(ǫ) ≥ η8/4 + O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Provided ǫ ≤ cη8 for a sufficiently small absolute constant c > 0, we have managed to show ∥νB′ ∗ (fσ)∥2 2 − ∥νB ∗ (fσ)∥2 2 ≥ cη8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For the same conclusion with L2(S) norms, we can apply (15) for ∥νB ∗ (fσ)∥2 L2(S) = � ξ∈Fd q [νB ∗ (fσ)σ]∧(ξ)�µB(ξ)4 � fσ(ξ) = ∥νB ∗ (fσ)∥2 2 + O(β−2q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' and ensure q is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 17 Next we iterate Proposition 12 in order to find a Bohr set refinement for which νB′ ∗ (fσ) approximates f well in a U 2 ⊥(S) sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let η ∈ (0, 1), ǫ ∈ (0, cη8) for c > 0 sufficiently small, B = B(Γ, ρ) a regular Bohr set, and f : Fd q → C with |f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If d ≥ 2 and q ≥ q(|Γ|, ρ, η, ǫ), then there exists B′ = B(Γ′, ρ′) with B′ ≺ǫ B, |Γ′| ≤ C(|Γ|, η), ρ′ ≥ c(|Γ|, η, ǫ)ρ, and ∥f − νB′ ∗ (fσ)∥U2 ⊥(S) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We will construct a sequence of Bohr sets Bj+1 ≺ǫ Bj ≺ǫ · · · ≺ǫ B1 ≺ǫ B0 where we write Bj = B(Γj, ρj) and start with B0 = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Our goal is find j ≥ 1 for which (17) ∥f − νBj ∗ (fσ)∥U2 ⊥(S) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We set Γ1 = Γ and choose ρ1 ∈ [ǫρ/(400|Γ|), ǫρ/(200|Γ|)] so that B1 ≺ǫ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Starting with j = 1 and iterating, we are done if (17) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If (17) is false for j ≥ 1, then by applying Proposition 12 with q sufficiently large, we obtain Bj+1 for which Bj+1 ≺ǫ Bj, |Γj+1| ≤ |Γ| + j, ρj+1 ≥ c(|Γ|, η, j)ρ, and ∥νBj+1 ∗ (fσ)∥2 L2(S) ≥ ∥νBj ∗ (fσ)∥2 L2(S) + cη8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' From (6), we can ensure that that for every j ≥ 0, we have ∥νBj+1 ∗ (fσ)∥2 L2(S) ≤ 2, in which case it is clear that we can iterate this process at most O(η−8) times before we arrive at some j for which (17) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ Finally, we iterate Proposition 13 to find a Bohr set B for which νB ∗(fσ) approximates f arbitrarily well in the U 2 ⊥(S) norm, at the cost of an L2-error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This is necessary since shrinking η in Proposition 13 leads to a loss of control of the size of the approximating Bohr set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Letting ǫ > 0, we will construct a sequence of Bohr sets BJ ≺ǫ · · · ≺ǫ B1 ≺ǫ B0 where we write Bj = B(Γj, ρj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We start with Γ0 = {0}, ρ0 = 1, and η0 = ϕ(1, 1), in which case B0 = Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For 1 ≤ j ≤ J, we set ηj = ϕ(|Γj−1|−1, ρj−1), ensure ǫ ∈ (0, cη8 j ), and apply Proposition 13 to obtain Bj with Bj ≺ǫ Bj−1, |Γj| ≤ C(|Γj−1|, ηj−1) and ρj ≥ c(|Γj−1|, ηj−1) such that ∥f − νBj ∗ (fσ)∥U2 ⊥(S) ≤ ηj for sufficiently large p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We claim there exists some j for which (18) ∥νBj+1 ∗ (fσ) − νBj ∗ (fσ)∥2 L2(S) ≤ η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Using Lemma 11, we can insist ǫ is sufficiently small with respect to η to guarantee ∥νBj+1 ∗ (fσ) − νBj ∗ (fσ)∥2 L2(S) ≤ ∥νBj+1 ∗ (fσ)∥2 L2(S) − ∥νBj ∗ (fσ)∥2 L2(S) + η2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If ∥νBj+1 ∗ (fσ)∥2 L2(S) ≤ ∥νBj ∗ (fσ)∥2 L2(S), then (18) would follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Suppose instead that ∥νBj+1 ∗ (fσ)∥2 L2(S) > ∥νBj ∗ (fσ)∥2 L2(S) for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' From (6), ∥νBj ∗ (fσ)∥2 L2(S) ≤ 2 for sufficiently large q, in which case the ∥νBj ∗ (fσ)∥2 L2(S) form a bounded increasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It follows that we can take J = O(η−2) and conclude (18) for some now fixed 1 ≤ j ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We are then finished by setting B = Bj and f1 = νBj ∗ (fσ) f2 = f − νBj+1 ∗ (fσ) f3 = νBj+1 ∗ (fσ) − νBj ∗ (fσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This completes the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ 18 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL Appendix: Necessity of the Spherical Condition By an isometry of Fd q we mean a linear map U : Fd q → Fd q so that |U(x)|2 = |x|2 for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It is easy to see that U(x) · U(y) = x · y for all x and y, hence U T U = UU T = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We call a set X = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk} non-degenerate if V ∩ V ⊥ = {0} for the subspace V = Span(X − X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Suppose X is a non-degenerate k + 1-point configuration and Y = {y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , yk} is isometric to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then there exists a vector z and an isometry U of Fd q so that Y = z + U(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' By performing a translation we can assume x0 = y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Assume that the vertices are labeled so that |yi − yj|2 = |xi − xj|2 for all 0 ≤ i ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let V and W be the subspace spanned by the sets X and Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Since X and hence Y are non-degenerate, we have that the map φ : xi → yi, 1 ≤ i ≤ k extends to an isometry Φ : V → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' By Witt’s extension theorem [27] there is an orthogonal transformation U : Fd q → Fd q extending the map Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Clearly U(X) = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ Let us reformulate the property that a set X is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Assume X = {0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' xk} and X′ = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk} is a linearly independent set such that Span(X′) = Span(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let M be the symmetric l × l matrix with entries mij = xi · xj for 1 ≤ i, j ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' The set X is non-degenerate if and only if the associated inner product matrix M has maximal rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Clearly X is non-degenerate if and only if X′ is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let V = Span(X′) and assume that 0 ̸= v ∈ V such that v⊥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then v = �l i=1 bixi and v · xj = �l i=1 mjibi = Mbj = 0 for all 1 ≤ j ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Thus Mb = 0 for a non-zero vector b = (bj) and M has rank less than l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Conversely if rank(M) < l then Mb = 0 for some b = (bj) ̸= 0 and the vector v = � i bixi is orthogonal to V = Span(X′) = Span(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ Note that the inner product matrix drops rank if some non-trivial algebraic relations between the distances |xi − xj|2 of the points xi, xj of X, hence generic configurations are non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Next, we show that the spherical condition is necessary over finite fields as well, at least for non-degenerate sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We follow the proof in [6] with some minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' First we give the following characterization of spherical sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let X = {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk} ⊆ Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then X is spherical if and only if the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' For every c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , ck ∈ Fq if (I) k � i=0 ci = 0, and k � i=0 cixi = 0, then also (II) k � i=0 ci|xi|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Suppose X is spherical, that is |xi − z|2 = r for 0 ≤ i ≤ k for some z ∈ Fd q and r ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then |xi|2 = r − |z|2 + 2xi · z for all i, hence k � i=0 ci|xi|2 = k � i=0 ci(|z|2 + r) + 2 k � i=0 cixi · z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Conversely, assume X is not spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We show that there exists c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , ck satisfying (I) but not (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' One may assume that X is minimal that is X′ is spherical for X′ ⊆ X, X′ ̸= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Since a simplex is spherical there is a non-trivial linear combination �k i=1 ai(xi −x0) = 0 and by reindexing the vertices one may assume ak ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Taking X′ = {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , xk−1} one has |xi − z|2 = r for 1 ≤ i < k but |xk − z|2 − r = b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then |xi|2 − |x0|2 = 2(xi − x0) · z for 1 ≤ i < k and |xk|2 − |x0|2 = 2(xk − x0) · z + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Thus k � i=1 ai(|xi|2 − |x0|2) = 2 k � i=1 ak(xi − x0) · z + akb = akb ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Taking c0 = − �k i=1 ai and ci = ai for 1 ≤ i ≤ k the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ SPHERICAL CONFIGURATIONS OVER FINITE FIELDS 19 Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let X ⊆ Fd q be a non-spherical set of k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Then there exists a set A ⊆ Fd q of size |A| ≥ ckqd which does not contain any set Y of the form Y = z + U(X), where ck > 0 is a constant depending only on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let t ∈ F∗ q such that |χ(t) − 1| ≥ 1/2, χ being a non-trivial character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Such a t exists as � t χ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If X is not spherical then, by scaling, there exists c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , ck so that (I) holds and � i=0 ci|xi|2 = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' If Y = z + U(X) then �k i=0 ciyi = 0 and moreover k � i=0 ci|yi|2 = k � i=0 ci(|z|2 + 2z · U(xi) + |xi|2) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Let B = B(Γ, ρ) be the Bohr set with Γ = {c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' , ck} and ρ = 1 4(k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' By the estimate (4), |B| ≥ ckq, for some constant ck > 0 depending only on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Define A = {x ∈ Fd q : |x|2 ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Since the number of solutions to |x|2 = b is at least 1 2qd−1 uniformly for b ∈ Fq for q ≥ q0, we have that |A| ≥ ck 2 qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' We show that Y = z + U(x) for some z ∈ Fd q and an orthogonal transformation U then Y ⊊ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Indeed if Y ⊆ A then |χ(ci|yi|2) − 1| ≤ 1 4(k+1) for 0 ≤ i ≤ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' It follows |χ( k � i=0 ci|yi|2) − 1| = | k � i=0 χ(ci|yi|2) − 1| ≤ k � i=0 |χ(ci|yi|2) − 1| ≤ 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' This implies |χ(t) − 1| ≤ 1/4 contradicting our choice of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' □ References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Bourgain, A Szemer´edi type theorem for sets of positive density in Rk, Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 54 (1986), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 3, 307–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 853455 [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Bourgain, On triples in arithmetic progression, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 9 (1999), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 5, 968–984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1726234 [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Chapman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Erdo˘gan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Hart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Iosevich, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Koh, Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 271 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1-2, 63–93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2917133 [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Cook, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Magyar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Pramanik, A Roth type theorem for dense subsets of Rd, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 49 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 4, 676–689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 3725488 [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Eberhard, Almost all sets of d + 2 points on the (d − 1)-sphere are not subtransitive, Mathematika 59 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 2, 267–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 3081771 [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Erd˝os, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Graham, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Montgomery, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Rothschild, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Spencer, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Straus, Euclidean Ramsey theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' I, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Combinatorial Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' A 14 (1973), 341–363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 0316277 [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Erd˝os, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Graham, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Montgomery, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Rothschild, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Spencer, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Straus, Euclidean Ramsey theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' II, Infinite and finite sets (Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=', Keszthely, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' dedicated to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Erd˝os on his 60th birthday), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' I, North-Holland, Amsterdam, 1975, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 529–557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' J´anos Bolyai, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 0382047 [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Erd˝os, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Graham, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Montgomery, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Rothschild, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Spencer, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Straus, Euclidean Ramsey theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' III, Infinite and finite sets (Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=', Keszthely, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' dedicated to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Erd˝os on his 60th birthday), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' I, North-Holland, Amsterdam, 1975, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 559–583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' J´anos Bolyai, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 0382048 [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Frankl and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' R¨odl, A partition property of simplices in Euclidean space, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 3 (1990), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1, 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1020148 [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Gowers, A new proof of Szemer´edi’s theorem for arithmetic progressions of length four, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 8 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 3, 529–551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1631259 [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Gowers, A new proof of Szemer´edi’s theorem, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 11 (2001), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 3, 465–588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1844079 [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Graham, Euclidean Ramsey theorems on the n-sphere, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Graph Theory 7 (1983), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1, 105–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 693027 [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Graham, Recent trends in Euclidean Ramsey theory, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 136 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1-3, 119–127, Trends in discrete mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1313284 [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Green, Finite field models in additive combinatorics, Surveys in combinatorics 2005, London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lecture Note Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 327, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Press, Cambridge, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2187732 [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Hart and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Iosevich, Ubiquity of simplices in subsets of vector spaces over finite fields, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 34 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1, 29–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2379694 [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Iosevich and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Rudnev, Erd˝os distance problem in vector spaces over finite fields, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 359 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 12, 6127–6142 (electronic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2336319 [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Kˇr´ıˇz, Permutation groups in Euclidean Ramsey theory, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 112 (1991), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 3, 899–907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1065087 [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Kˇr´ıˇz, All trapezoids are Ramsey, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 108 (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1-3, 59–62, Topological, algebraical and combinatorial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Frol´ık’s memorial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1189829 [19] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Leader, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Russell, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Walters, Transitive sets and cyclic quadrilaterals, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 2 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 3, 457–462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2913203 20 NEIL LYALL, ´AKOS MAGYAR, HANS PARSHALL [20] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Leader, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Russell, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Walters, Transitive sets in Euclidean Ramsey theory, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' A 119 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 2, 382–396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2860600 [21] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Lyall and ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Magyar, Product of simplices and sets of positive upper density in Rd, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' (2017), 1-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='1017/S0305004117000184 [22] ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Magyar, k-point configurations in sets of positive density of Zn, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 146 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1, 1–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2475398 [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Matouˇsek and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' R¨odl, On Ramsey sets in spheres, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' A 70 (1995), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 1, 30–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1323999 [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Parshall, Simplices over finite fields, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 145 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 6, 2323–2334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 3626492 [25] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Shparlinski, Additive combinatorics over finite fields: new results and applications, Finite fields and their applications, Radon Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 11, De Gruyter, Berlin, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 233–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 3157428 [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Tao and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 105, Cambridge University Press, Cambridge, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 2289012 [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Witt, Theorie der quadratischen Formen in beliebigen K¨orpern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 176 (1937), 31–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 1581519 [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' Wolf, Finite field models in arithmetic combinatorics—ten years on, Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' 32 (2015), 233–274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content=' MR 3293412 Department of Mathematics, The University of Georgia, Athens, GA 30602, USA Email address: lyall@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='edu Department of Mathematics, The University of Georgia, Athens, GA 30602, USA Email address: magyar@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='edu Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA Email address: parshall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} +page_content='6@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFIT4oBgHgl3EQflys0/content/2301.11306v1.pdf'} 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+Abstract—Local feature matching between images remains +a challenging task, especially in the presence of significant +appearance variations, e.g., extreme viewpoint changes. In this +work, we propose DeepMatcher, a deep Transformer-based +network built upon our investigation of local feature matching +in detector-free methods. The key insight is that local feature +matcher with deep layers can capture more human-intuitive and +simpler-to-match features. Based on this, we propose a Slimming +Transformer (SlimFormer) dedicated for DeepMatcher, which +leverages vector-based attention to model relevance among all +keypoints and achieves long-range context aggregation in an +efficient and effective manner. A relative position encoding is +applied to each SlimFormer so as to explicitly disclose relative +distance information, further improving the representation of +keypoints. A layer-scale strategy is also employed in each Slim- +Former to enable the network to assimilate message exchange +from the residual block adaptively, thus allowing it to simulate the +human behaviour that humans can acquire different matching +cues each time they scan an image pair. To facilitate a better +adaption of the SlimFormer, we introduce a Feature Transition +Module (FTM) to ensure a smooth transition in feature scopes +with different receptive fields. By interleaving the self- and cross- +SlimFormer multiple times, DeepMatcher can easily establish +pixel-wise dense matches at coarse level. Finally, we perceive +the match refinement as a combination of classification and +regression problems and design Fine Matches Module to predict +confidence and offset concurrently, thereby generating robust and +accurate matches. Experimentally, we show that DeepMatcher +significantly outperforms the state-of-the-art methods on several +benchmarks, demonstrating the superior matching capability of +DeepMatcher. The code is available at https://github.com/XT- +1997/DeepMatcher. +Index Terms—Local feature matching, Pose Estimation, Trans- +former. +I. INTRODUCTION +L +OCAL feature matching [1]–[4] is the prerequisite for +a variety of geometric computer vision applications, in- +cluding Simultaneous Localization and Mapping (SLAM) [5], +[6] and Structure-from-Motion (SFM) [7], [8]. As a broadly +acknowledged matching pipeline, detector-based matching [3], +[9]–[18] is typically accomplished by (i) detecting and de- +scribing a set of sparse keypoints such as SIFT [9], ORB +[10], and learning-based equivalents [13], [19], (2) instituting +point-to-point correspondences via nearest neighbour search +This work was in part by National Natural Science Foundation of China +under Grant 62176072 and 62073101. (Corresponding author: Ruifeng Li and +Ke Wang. †: These authors contribute equally.) +Tao Xie, Kun Dai, Ke Wang, Ruifeng Li, Lijun Zhao are with State Key +Laboratory of Robotics and System, Harbin Institute of Technology, Harbin +150006, China (email: xietao1997@hit.edu.cn; 20s108237@stu.hit.edu.cn; +wangke@hit.edu.cn; lrf100@hit.edu.cn; zhaolj@hit.edu.cn). +Tao Xie is also with SenseTime Group Inc., Beijing 100080, China (email: +xietao@sensetime.com). +LoFTR +DeepMatcher +DeepMatcher P:93.9% +Matches:423 +DeepMatcher P:97.7% +Matches:215 +DeepMatcher P:93.6% +Matches:188 +DeepMatcher +LoFTR P:84.1% +Matches:679 +LoFTR P:89.2% +Matches:167 +LoFTR P:73.3% +Matches:15 +LoFTR +LoFTR +DeepMatcher +DeepMatcher-L +624 +GFLOPs +328 +303 +255 +196 +352 +Matches +Precision +20.9% +69.9% +80.8% +DeepMatcher-L P:80.8% +Matches:624 +ΔR:1.81°, Δt:2.26° +LoFTR P:20.9% +Matches:196 +ΔR:6.02°, Δt:13.13° +DeepMatcher P:69.9% +Matches:352 +ΔR:4.57°, Δt:6.03° +Fig. 1. The comparison between LoFTR and DeepMatcher under large +viewpoint changes. DeepMatcher families considerably outperform LoFTR +with more dense and precise matches while using less GFLOPs. +or more advanced matching algorithms [20]. The use of a +feature detector narrows the matching search space, revealing +the general efficacy of such detector-based matching process. +Nonetheless, when dealing with image pairs with severe +viewpoint variations, such pipeline struggles to build reliable +correspondences since the detectors are essentially incapable +of extracting repeated keypoints in this case. +Parallel to the detector-based matching, another stream +of research seeks to establish correspondences directly from +original images by extracting visual descriptors on dense +grids across an image, thus assuring that substantial repeat- +ing keypoints could well be captured [2], [21]–[28]. Earlier +detector-free matching works [21]–[25] generally depend on +iterative convolution based on correlation or cost volume to +identify probable neighbourhood consensus. Transformer [29] +has recently attracted considerable interest in computer vision +due to its excellent model capability and superior potentials +for capturing long-range relationships. On the basis of this +insight, various studies base their modelling of long-range +relationships on Transformer backbone [2], [26]–[28]. As a +representative work, LoFTR [2] updates features by repeatedly +interleaving the self- and cross-attention layers, and replace +vanilla Transformer with linear Transformer [30] to achieve +manageable computation cost. These detector-free methdos +can generate repeatable keypoints in indistinct regions with +poor textures or motion blur, thus yielding impressive results. +After witnessing the success of detector-free methods, an +intriguing issue arises: could we build a deeper yet compact +local feature matcher to further improve performance while +reducing computing costs? Intuitively, when individuals match +images, they scan the images back and forth, and the more +times they scan them, the easier it is for them to remember +the easier-to-match features, which indicates that a deep lo- +cal feature matcher could display superior matching ability. +arXiv:2301.02993v1 [cs.CV] 8 Jan 2023 + +L0FTR, P: 84:1% +Matches: 679DeepMatcherP:9 +Matches: +ate +ESEARN +fonicL0FTR, P: 89.2% +Matches: 167DeepMatcher, P:97.7% +Matches: 215DeepMatcher, P:93.6% +Matches:188L0FTR,P:73.3% +Matches:15ate +ESEARK +ronicJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +2 +However, as demonstrated in LoFTR, doubling the number +of Transformer layers has little effect on the results, which +is contrary to expectations. In this work, we argue that the +following obstacles hinder us from developing a deep local +feature matcher for detector-free methods: +(i) Typical detector-free methods begin with a convolution +neural network (CNN) as the basic feature extractor, followed +by Transformer layers to capture long-range relevance so +that generating credible correspondences. In terms of context +ranges, there is apparently a gap between the global receptive +field of Transformer and the local neighborhood of CNN, +which is detrimental to subsequent stages involving deep +feature interaction. +(ii) The translation invariance of CNN causes ambiguity +in scenes with recurring geometry patterns or symmetrical +structures. Current detector-free methods utilize absolute po- +sition encodings before Transformer layers to tackle this +issue, while the position information would disappear as the +Transformer layers grow deeper. Moreover, humans naturally +associate items across observations by referring to not only +their absolute position but also their relative position. +(iii) Intuitively, the depth of the network is more prominent +than width in the field of feature matching. However, as the +linear Transformer layer in LoFTR goes deeper, the model +fails to learn effective context aggregation from deeper layers +since the linear Transformer uses a context-agnostic manner +to approximate self-attention, which cannot efficaciously sim- +ulate relevance among all keypoints. +To this end, we propose DeepMatcher, a deep local feature +matching network that can produce more human-intuitive and +simpler-to-match features for accurate correspondence with +less computational complexity, as shown in Fig. 1. Firstly, +we utilize a CNN network to generate pixel tokens with +enriched features. Secondly, an Feature Transition Module +(FTM) is introduced to ensure a smooth transition from the +locally aggregated features extracted by CNN to features with +a global receptive field extracted by Transformer. Then, we +propose a Slimming Transformer (SlimFormer) to build +deep network that strengthens long-range global context mod- +elling intra-/inter-images. Technically, SlimFormer leverages +vector-based attention that efficiently handles pixel tokens +with linear complexity for robust long-range global context +aggregation. Besides, a relative position encoding is applied to +each SlimFormer to clearly express relative distance informa- +tion, boosting the network’s capacity to convey information, +particularly in deeper layers. Moreover, SlimFormer utilizes +a layer-scale strategy that enables the network to assimilate +message exchange from the residual block adaptively, thus +allowing it to simulate the human behavior that human can +receive different matching information each time they scan +an image pair. By interleaving the self- and cross-SlimFormer +multiple times, DeepMatcher learns the discriminative features +to construct dense matches at the coarse level by Coarse +Matches Module (CMM). Ultimately, we view the match +refinement as a combination of classification and regression +problems and devise Fine Matches Module (FMM) to pre- +dict confidence and offset concurrently, obtaining robust and +accurate matches. +To summarize, the main contributions of this work are as +follows: +• We propose DeepMatcher, a deep Transformer-based +network for local feature matching, achieving state-of- +the-art results on various benchmarks. +• We propose a Feature Transition Module (FTM) to ensure +a smooth transition from the locally aggregated features +extracted by CNN to features with a global receptive field +extracted by SlimFormer. +• We propose a Slimming Transformer (SlimFormer) that +integrates long-range global context aggregation, relative +position encoding, and layer-scale strategy to enable +DeepMatcher to be extended into dozen layers. +• We propose Fine Matches Module (FMM) that views the +match refinement as a combination of classification and +regression problems to optimze coarse matches, deriving +robust and accurate matches. +II. RELATED WORK +A. Detector-based Methods +The conventional pipeline of detector-based matching sys- +tems detects two sets of keypoints, describes them with +high-dimensional vectors, and then implements a matching +algorithm to generate matches between the two sets of key- +points [31], [32]. +Regarding feature detection and description, there are nu- +merous handcrafted methods that seek to strike a balance +between accuracy and efficiency [9], [10], [33]. However, +the handcrafted descriptors are fragile when coping with +image pairs with extreme appearance variations. With the +development of deep learning [34]–[37], numerous approaches +leverage elaborate convolution neural network (CNN) to ex- +tract robust feature representations, hence achieving superior +performance. SuperPoint [11] builds a large dataset of pseudo- +ground truth interest point locations in real images, supervised +by the interest point detector itself, as opposed to a large- +scale human annotation. D2-Net [12] makes the collected +keypoints more stable by delaying the detection to a later +stage. Subsequently, the aforementioned methods utilize the +nearest neighbor search, followed by a robust estimator, such +as RANSAC or its variants [38]–[42], to find matches between +the retrieved keypoints. +Recent researches [3], [16]–[18] has interpreted local feature +matching as a graph matching problem involving two sets of +features. These methods utilize keypoints as nodes to construct +graph neural network (GNN), employ the self- and cross- +attention layers in Transformer to exchange global visual +and geometric messages across nodes, and then generate +the matches in accordance with soft assignment matrixes. +Typically, SuperGlue [3] utilizes self- and cross-attention in +Transformer to integrate global context information, followed +by the Sinkhorn algorithm to generate matches according to +the soft assignment matrix. Nonetheless, the matrix multipli- +cation in vanilla Transformer results in quadratic complexity +with respect to the number of keypoints, making SuperGlue +costly to deal with substantial keypoints. To tackle this prob- +lem, many approaches attempt to ameliorate the structure of + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +3 +SuperGlue. SGMNet [16] exploits the sparsity of graph neural +network to lower the computation complexity. ClusterGNN +[18] employs a progressive clustering module adaptively to +divide keypoints into different subgraphs to reduce compu- +tation. However, limited to inherent essence, detector-based +approaches are incapable of extracting repeated keypoints +when handling image pairs with large appearance variations. +B. Detector-free Methods +Detector-free methods exclude the feature detector and gen- +erate dense matches directly from the original images. Earlier +detector-free matching researches [21]–[25] generally utilize +convolutional neural network (CNN) based on correlation +or cost volume to identify probable neighbourhood consen- +sus. DRC-Net [23] generates a 4D correlation tensor from +the coarse-resolution features, which is refined by a learn- +able neighborhood consensus module to generate matches. +Patch2Pix [24] proposes a weakly supervised approach to +learn matches that are consistent with the epipolar geometry of +image pairs. DFM [25] uses pre-trained VGG architecture as a +feature extractor and captures matches without any additional +training strategy. Although elevating the matching accuracy, +these methods extract ambiguous feature representations and +fail to discriminate incorrect matches owing to the limited +receptive field of CNN. +To handle this issue, LoFTR [2], the pioneering detector-free +GNN method, utlizes Transformer to realize global context +information exchange and extracts matches in a coarse-to-fine +manner. Matchformer [26] proposes a human-intuitive extract- +and-match scheme that interleaves self- and cross-attention in +each stage of the hierarchical encoder. Such a match-aware +encoder releases the overloaded decoder and makes the model +highly efficient. QuadTree [43] proposes a novel Transformer +structure that builds token pyramids and computes attention +in a coarse-to-fine manner. Then, the QuadTree Transformer +is integrated into LoFTR and achieves superior matching +performance. TopicFM [27] applies a topic-modeling strategy +to encode high-level contexts in images, which improves the +robustness of matching by focusing on the same semantic +areas between the images. ASpanFormer [28] proposes a +Transformer-based detector-free architecture, in which the flow +maps are regressed in each cross-attention phase to perform +local attention. These detector-free methods are capable of +generating repeatable keypoints in indistinct regions with poor +textures or motion blur, thus resulting in amazing results. +However, the architecture of existing detector-free methods +is designed as shallow-broad, and building a deeper and more +compact local feature matcher to further improve performance +while reducing computational costs has not been investigated. +C. Efficient Transformer +In the vanilla Transformer, the memory cost is quadratic +to the length of sequences due to the matrix multiplication, +which has become a bottleneck for Transformer when dealing +with long sequences. Recently, several approaches have been +proposed to improve the efficiency of Transformer [30], [44], +[45]. Linear Transformer [30] expresses self-attention as a +LoFTR +CNN +Transformer Layer +PE +correlation-based +refinement block +DeepMatcher +CNN +… +Transformer Layer +PE +PE +PE +PE +PE +PE +network-based +refinement block +CNN +Transformer Layer +PE +correlation-based +refinement block +LoFTR P:20.9% +Matches:196 +ΔR:6.02°, Δt:13.13° +LoFTR +DeepMatcher +… +Transformer Layer +PE +PE +PE +PE +PE +network-based +refinement block +DeepMatcher-L P:80.8% +Matches:624 +ΔR:1.81°, Δt:2.26° +CNN +Fig. 2. The comparison between LoFTR and DeepMatcher. DeepMatcher +designs a deep-narrow Transformer layers to capture more human-intuitive and +simpler-to-match features. Besides, the position encoding (PE) is integrated +to each Transformer layer to convey position information in deep layers. +Moreover, a network-based refinement block is proposed to extract more +precise matches. +linear dot product of kernel feature maps and makes use +of the associativity property of matrix products to reduce +the computational complexity. BigBird [44] combines local +attention and global attention at certain positions and utilizes +random attention on several randomly selected token pairs. +FastFormer [45] uses additive attention mechanism to model +global contexts, achieving effective context modeling with +linear complexity. +III. METHODOLOGY +A. Overall +Intuitively, when humans match images, they scan the +images back and forth, and the more times they scan them, +the simpler it is for them to recall the easier-to-match fea- +tures, which suggests that a deep local feature matcher can +exhibit higher matching abilities. Thus, as shown in Fig. 2, +we consider depth of the network is more prominent than +width and present a deep Transformer-based network, namely +DeepMatcher. As shown in Fig. 3, given the image pair IA and +IB, our network produces reliable and accurate matches across +images in an end-to-end manner. The matching process starts +with a CNN-based encoder to extract the fine-level features +¯FA, ¯FB and coarse-level features ˆFA, ˆFB. Before feeding these +features to Slimming Transformer (SlimFormer), we utilize the +Feature Transition Module (FTM) to guarantee a smooth tran- +sition to SlimFormer. Then, we utilize SlimFormer to achieve +long-range global context aggregation intra-/inter-images in +an efficient and effective way. A relative position encoding +is applied on each SlimFormer to explicitly model relative +distance information, hence enhancing the DeepMatcher’s +ability to convey information, particularly in deeper layers. +A layer-scale strategy is also leveraged in each SlimFormer to +enables the network to assimilate message exchange from the +feed-forward modules adaptively, thus allowing it to simulate +the human behavior that human can receive different matching +information each time they scan an image pair. After interleav- +ing SlimFormer by L times, the enhanced features LF seq +A +and +LF seq +B +are utilized to establish coarse matches Hc, which are +further optimized to fine matches Hf using Correspondence +Refine Module (CRM). In the following part, we introduce the +details and underlying insights of each individual block. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +4 +(a) Local Feature Extractor +(b) Feature-transition Module +1×1Conv +1×1DWConv +1×1Conv +3×3DWConv +1×1Conv +5×5DWConv +1×1Conv +7×7DWConv +Fine Matches +(d) Coarse Matches Module +Score Matrix +Softmax +Softmax +Soft Assignment Matrix +Coarse Matches +1×1Conv +Img2 +Seq +Img2 +Seq +SlimFormer×L2 +Seq2 +Img +Seq2 +Img +Concat +1×1Conv +Max +Pooling +1×1Conv +1×1Conv +1×1Conv +1×1Conv +Sigmoid +Confidence +Offset +(e) Fine Matches Module +Img +2Seq +Img +2Seq +… +c. Locality Feed-forward Network +(c) Slimming Transformer +Concat +MLP +GELU +MLP +×L +Vector-based Attention +(c) Slimming Transformer +… +… +Feed-forward Network +Layer Norm +… +MLP +Softmax +MLP +MLP +Softmax +MLP +MLP +MLP +Value vector +Position Encoding +Fig. 3. The network architecture of DeepMatcher. DeepMatcher takes an image pair(IA, IB) as input and generates transitional features from Local Feature +Extractor and Feature Transition Module. Then, DeepMatcher interleaves Slimming Transformer by L times to perform long-range context aggregation. Coarse +Matches Module is utilized to establish coarse matches, which are optimize to fine matches by Fine Matches Module. +B. Local Feature Extractor +As the first part of DeepMatcher, we use a standard con- +volutional neural network (CNN) with FPN [46] to extract +the coarse-level features +ˆ +FA, +ˆ +FB ∈ R ˆ +C×H/8×W/8, and the +fine-level features ¯FA, ¯FB ∈ R ¯ +C×H/2×W/2 for the image +pair IA and IB, where H and W are the height and width +of the original images, ˆC, ¯C denote feature dimension. For +convenience, we denote N = H/8 × W/8 as the number +of pixel tokens. Since each pixel in ˆFA, ˆFB represents an +8 × 8 grid in the original images IA, IB, we view the central +position of all grids as the pixel coordinates PA, PB ∈ RN×2 +of keypoints. +C. Feature Transition Module (FTM) +In the subsequent steps, we construct graph neural network +(GNN) and propose SlimFormer that leverages self-/cross- +attention in Transformer to aggregate global context infor- +mation intra-/inter-image. Nevertheless, there is apparently a +gap between the feature extractor and SlimFormer in terms +of context ranges, which is deleterious to subsequent steps in- +volving deep feature interaction. Besides, representing features +at multiple scales is so critical for discriminating objects or +regions of varying sizes that it can ensure prominent features at +various scales can be preserved for deep features aggregation. +Thus, we propose a Feature Transition Module (FTM) inserted +between the local feature extractor and SlimFormer to adjust +the receptive fields of the extracted features, ensuring effective +deep feature interaction in SlimFormer. Specifically, instead of +directly using (1 × 1, 3 × 3, 5 × 5, 7 × 7) convolution, FTM +adopts (1 × 1, 3 × 3, 5 × 5, 7 × 7) depth-wise convolution [47] +followed by 1 × 1 point-wise convolution to reduce model +parameters and computation, obtaining multi-scale feature rep- +resentations. Then, we concatenate the features along channel +dimension, hence enlarging the receptive fields of +ˆ +FA, ˆ +FB. +Finally, we derive the updated F ftm +A +, F ftm +B +∈ R ˆ +C×H/8×W/8, +which can be formulated as: +FTM(F) = [C1/4 +1 +(DW1(F))||C1/4 +1 +(DW3(F))|| +C1/4 +1 +(DW5(F))||C1/4 +1 +(DW7(F))], +F ftm +A += FTM( ˆFA), +F ftm +B += FTM( ˆFB), +(1) +where C1/4 +1 +means using 1×1 convolution to squeeze the chan- +nel dimension to ˆC/4; DW1, DW3, DW5, DW7 mean depth- +wise convolution with kernel size of 1, 3, 5, 7, respectively; +[·||·] means concatenation along the channel dimension. +D. Slimming Transformer (SlimFormer) +We flatten the updated enhanced features F ftm +A +, F ftm +B +to +be the input sequence for deep feature aggregation, obtaining +F seq +A , F seq +B +∈ RN× ˆ +C. Following [3], we view keypoints with +features F seq +A , F seq +B +in image pairs as nodes to construct +GNN, in which the global context aggregation intra-/inter- +image is performed. Intuitively, more observations between +images can result in more precise matches, indicating that +deep feature interaction is essential for local features matching +task. Nevertheless, the ablation study of LoFTR demonstrates +that the matching performance has not been significantly +improved with more Transformer layers. We attribute this +phenomenon to the following reasons: (i) LoFTR only uti- +lizes absolute position encoding before Transformer layers, +where the position information would disappear when the +Transformer layers grow deeper. Moreover, humans primarily +associate objects by referring to their relative positions. (ii) +The linear Transformer utilized in LoFTR uses a context- +agnostic manner to approximate self-attention, which cannot + +UJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +5 +fully model relevance among all keypoints, especially in deep +layers. To handle this dilemma, we propose SlimFormer that +leverages relative position information and global context +information to boost the capability of DeepMatcher to convey +abundant information, hence extracting discriminative feature +representations LF seq +A ,L F seq +B +∈ R ˆ +C×H/8×W/8. +Vector-based Attention (VAtt) Layer. Instead of using +a context-agnostic manner to approximate self-attention, we +convert query vector to global query contexts and leverage +element-wise product to model relevance among all keypoints. +Technically, during each feature enhancement process, we uti- +lize self-/cross-attention to aggregate long-range context infor- +mation intra-/inter-images. For self-attention, the input features +U and R are same (either (F seq +A , F seq +A ) or (F seq +B , F seq +B )). For +cross-attention, the input features U and R are different (either +(F seq +A , F seq +B ) or (F seq +B , F seq +A )). Firstly, SlimFormer transforms +the input features U and R into the query, key, and value +vectors Q, K, V ∈ RN× ˆ +C. +Q = UWQ, +K = RWK, +V = RWV , +(2) +where WQ, WK, WV +∈ R ˆ +C× ˆ +C denote learnable weights +for feature transformation. Then, we perform relative position +encoding on query vector Q and key K. +˜Q = DPE(Q), +˜K = DPE(K), +(3) +where DPE(·) means relative position encoding operation, +described below. +Next, modeling the context information of the input fea- +tures based on the interactions among ˜Q, ˜K, and V is a +critical problem for Transformer-like architectures. In the +vanilla Transformer, dot-product attention mechanism leads to +quadratic complexity, making it unrealistic to establish deep +Transformer layers. A potential method to reduce the com- +putational complexity is to summarize the attention matrices +before modeling their interactions. Inspired by [45], we intro- +duce vector-based attention that effectively models long-range +interactions among pixel tokens to alleviate this bottleneck. +Instead of computing a quadratic attention map QKT that +encodes all possible interactions between candidate matches, +we form a compact representation of query-key interactions via +vector-based attention that computes the correlation between +global query vector and each key vector. Specifically, we firstly +leverage MLP to calculate the weight ˜Qimp ∈ R1×N of each +query vector: +˜Qimp = Softmax(MLP( ˜Q)), +(4) +where Softmax(·) means softmax operation. +The global query vector ˘Q ∈ R1× ˆ +C is set to be a linear +combination of ˜Q: +˘Q = ˜Qimp ⊗ ˜Q +(5) +where ⊗ means matrix multiplication. +Then, we utilize the element-wise multiplication between +the global query vector ˘Q and each key vector to model their +interaction, obtaining context-aware key vector ˜KQ ∈ RN× ˆ +C: +˜KQ = ˘Q ⊙ ˜K, +(6) +where ⊙ denotes element-wise multiplication. +We utilize a similar vector-based attention to extract global +context-aware key vector ˘KQ and model the interaction be- +tween ˘KQ and V : +˜KQimp = Softmax(MLP( ˜KQ)) +˘KQ = ˜KQimp ⊗ ˜KQ +Λ = ˘KQ ⊙ V +(7) +Subsequently, we employ a MLP and short-cut structure to +derive the global message M ∈ RN× ˆ +C. +M = MLP(Λ) + ˜Q +(8) +For convenience, we define the process of vector-based +attention layer as: +M = VAtt(U, R) +(9) +Feed-forward Network (FFN). Inspired by conventional +Transformers, we employ a feed-forward network applied +to M to extract discriminative features for effectively deep +features aggregation. The feed-forward network consists of +two fully-connected layers and a GELU activation function. +The hidden dimension between the two fully-connected layers +is extended by a scale rate γ to learn abundant feature +representation. This process can be formulated as: +FFN(U, M) = MLP1/γ(GELU(MLPγ/2([U||M]))), +(10) +where MLP1/γ, MLPγ/2 mean expand the channel dimen- +sion by 1/γ, γ/2 times with a MLP, respectively; [·||·] means +concatenation along channel dimension; GELU(·) means +GELU activation function. Ultimately, we obtain enhanced +message ˘ +M ∈ RN× ˆ +C. +Layer Scale Strategy. Intuitively, people obtain different +message after observing images each time, which inspires us +to propose a layer-scale strategy. Specifically, in accordance +with ResNet [48], we utilize a shortcut structure to realize +efficient training. Then, we design a learnable scaling factor +ξ to adaptively balance original features U and enhanced +message ˘ +M, which is formulated as. +˘U = U + ξ ˘ +M +(11) +By incorporating ξ into SlimFormer, SlimFormer can easily +simulate the human behaviour that humans acquire different +matching cues each time they scan an image pair. +Relative Position Encoding (RPE). The local feature ex- +tractor learns strict translation invariant features, which could +cause ambiguity in scenes that have repetitive geometry texture +or symmetric structures. Previous works [2], [3], [16], [26] +attach a distinctive absolute positional embedding to each +keypoint, thus alleviating such ambiguity. However, compared +with absolute position, relative position is more conducive for +humans to establish connections between objects. Therefore, +we argue that incorporating the explicit relative position de- +pendency during each deep feature aggregation is essential for +distinguishing identical features. However, relative position is +not applicable to transformers with linear complexity as they +do not explicitly calculate the quadratic complexity attention + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +6 +matrix. To this end, we employ rotary positional embedding +(RoPE) [49] that leverages absolute position encoding to +achieve relative position encoding without manipulating the +attention matrix. Given a pixel token Ti and its features +Fi ∈ R ˆ +C, the rotary position encoding function is defined +by: +Pos(Ti, Fi) = Θ(Ti)Fi = +� +� +� +� +� +� +M1 +0 +. . . +0 +0 +M2 +... +... +... +0 +0 +. . . +0 +M ˆ +C/2 +� +� +� +� +� +� +Fi, +(12) +where Θ(Ti) ∈ R ˆ +C× ˆ +C is a block diagonal matrix. Each block +with size of 2 × 2 is defined by: +Mk = +�cos iθk +−sin iθk +sin iθk +cos iθk +� +, +θk = +1 +100002(k−1)/ ˆ +C (13) +where θk encodes the index of the feature channel. +Compared to sinusoidal encoding [2], [3], [26], rotary +positional embedding has two advantages: (i) Θ(·) is an +orthogonal function, the encoding only changes the feature’s +direction but not the feature’s length, which could stabilize +the learning process. (ii) The dot product of two encoded +features < Pos(Ti, Fi), Pos(Tj, Fj) > in self-attention of +vanilla Transformer can be derived to: +[Θ(Ti)Fi]T Θ(Tj)Fj = (Fi)T Θ(Tj − Ti)Fj +(14) +which means the relative 2D distance information can be +explicitly revealed by the dot product. +Since RoPE injects position information by rotation, which +maintains the norm of hidden representations unchanged, such +positional encoding can be directly applied to linear com- +plexity transformers as demonstrated in [49]. In SlimFormer, +we implement this by employing rotary positional embedding +into Q, K to incorporate relative position information, as +illustrated in Fig. 3 or Eq. (3). For more details about RoPE, +we encourage readers to refer to original papers. +Self-/Cross-SlimFormer. In summary, the SlimFormer is +formatted as: +Slim(U, R) = U + ξFFN(U, VAtt(U, R)) +(15) +We perform L times of SlimFormer for feature enhance- +ment. During the l-th feature enhancement, we use self-/cross- +attention mechanism to integrate intra-/inter-image informa- +tion, which can be formulated as: +l−1F seq +A += Slim(l−1F seq +A , l−1F seq +A ), +l−1F seq +B += Slim(l−1F seq +B , l−1F seq +B ), +lF seq +A += Slim(l−1F seq +A , l−1F seq +B ), +lF seq +B += Slim(l−1F seq +B , lF seq +A ) +(16) +Ultimately, we incorporate relative position information and +global context message into enhanced features LF seq +A ,L F seq +B . +: in Coarse Matches +c +B +P +: in Fine Matches +f +B +P +: Ground Truth +Fig. 4. Visualization of refinement result. The keypoints (green) in the fine +matches approximate the ground-truth (blue). +E. Coarse Matches Module (CMM) +Given +LF seq +A +and +LF seq +B , we utilize inner product of +LF seq +A , LF seq +B +to calculate the score matrix S ∈ RN×N. +S(i, j) = ⟨LF seq +A , LF seq +B ⟩, +(17) +where ⟨·, ·⟩ means the inner product. Subsequently, we apply +softmax operator on both dimensions (denoted as dual-softmax +operation) to convert the S to soft assignment matrix G ∈ +RN×N: +G = Softmax(S)col · Softmax(S)row, +(18) +where Softmax(·)col, +Softmax(·)row mean performing +softmax on each column and row of S, respectively. +Then, for the i-th keypoint in IA and the j-th keypoint in IB, +we regard them as a pair of predicted coarse matches if they +satisfy the following two conditions: (i) The soft assignment +score is higher than a predefined threshold λ: G(i, j) > λ. (ii) +They satisfy the mutual nearest neighbor (MNN) criteria, i.e., +G(i, j) is the maximum value in the corresponding row and +column. Ultimately, we derive the index D of anchor points +in coarse matches: +D = {(i, j)|(i, j) ∈ MNN(M), M(i, j) > λ} +(19) +Given the index D and the keypoints coordinates PA, PB, +the coarse matches Hc = {(P c +A, P c +B)} are formulated as: +Hc = {(PA(i), PB(j)) | ∀(i, j) ∈ D} +(20) +F. Fine Matches Module (FMM) +After establishing coarse matches, a coarse-to-fine module +is applied to refine these matches to the original picture +resolution. However, the coarse-to-fine module in LoFTR only +predicts the offset of the coarse matches without appraising +whether the predicted matches are reliable. To tackle this issue, +we view the match refinement as a combination of classifica- +tion and regression problems and design Fine Matches Module +to predict confidence and offset concurrently. +As shown in Fig. 3, for each coarse match, we locate its +position at fine-level feature maps and crop two sets of local +image patches with the size of w ×w, obtaining local features +¯F w +A , ¯F w +B ∈ RK× ¯ +C×w×w, where K is the number of coarse +matches. Then, we flatten ¯F w +A , ¯F w +B to be sequences, imple- +ment SlimFormer to perform L2 times of global information +passing, and rearrange the sequences into 2D feature maps, + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +7 +obtaining L2 ¯F w +A ,L2 ¯F w +B . The feature maps are concatenated +along channel dimension and fed into a network, which is +comprised of two convolution layers, a max pooling layer, +and four convolution layers. The network predicts the offset +∆ ∈ RK×2 of the P c +B and the confidence c ∈ RK×1 of the +predicted coarse matches: +¯Fmid = C1(C1(Pmax(C1(C1([L2 ¯F w +A ||L2 ¯F w +B ]))))), +c = Sig(C1( ¯Fmid)), +∆ = C1( ¯Fmid), +(21) +where Pmax means global max pooling operation; C1(·) +means 1 × 1 convolution; [·||·] denotes concatenation along +the channel dimension; Sig(·) means sigmoid function. +Ultimately, we obtain the fine matches Hf = {(P f +A, P f +B)}: +Hf = {(P c +A(i), P c +B(i) + ∆(i)) | i ∈ {1, 2, 3, ..., K}} +(22) +G. Loss +DeepMatcher generates final dense matches according to +soft assignment matrix G and offset ∆. Therefore, the total +loss Lall of DeepMatcher comprises of matching loss Lm, +regression loss Lr, and classification loss Lc. +Lall = Lm + βLr + φLc, +(23) +where β and φ are weighting coefficient. +Matching Loss. Following [2], we calculate the index Egt +of the ground truth matches, which are utilized in conjunction +with soft assignment matrix G to calculate matching loss Lm +defined as focal loss [50]. +Lm = −[ +1 +|Egt| +� +(i,j)∈Egt +α(1 − G(i, j))ηlog G(i, j)+ +1 +N − |Egt| +� +(i,j)/∈Egt +(1 − α)G(i, j)ηlog (1 − G(i, j))], +(24) +where α is a weighting factor; η is a focusing parameter; |Egt| +means the number of ground truth matches. +Regression Loss. For predicted matches {(P f +A, P f +B)}, we +project P f +A in the first image to second image, deriving P gt +B . +Then, the ground truth offset ∆gt is formulated as: +∆gt = P gt +B − P f +B +(25) +According to predicted offset ∆ and ground truth offset ∆gt, +we define the regression loss Lr as: +Lr = 1 +K +K +� +i=1 +∥∆gt(i) − ∆(i)∥2 +2, +(26) +where K is the number of predicted matches. Notably, we +ignore the predicted matches with ∆gt larger than predefined +threshold ψ. +Classification Loss. For the predicted matches with ground +truth offset less than ψ, we regard them as positive and +define the classification label as 1, while other matches are +viewed as negative. Ultimately, we obtain the ground truth +confidence cgt, while are utilized to calculate classification +loss Lc together with predicted confidence c. +Lc = − 1 +K +K +� +i=1 +� +cgt(i)log c(i) + (1 − cgt(i))log (1 − c(i)) +� +(27) +IV. EXPERIMENTS +A. Implementation Details +Architecture details. We adopt a slightly modified ResNet- +18 with FPN for local feature extraction. We use a width of +96 for the stem layer, followed by widths of [96, 128, 192] +for the next three stages. We construct the FPN with levels P1 +through P3 and take P3 features as the coarse-level features, +P1 features as the fine-level features. Thus, the dimensions of +fine-level and coarse-level feature maps are C = 96, ˆC = 192, +respectively. The scale rate γ in feed-forward network is set +to 4. Following SuperGlue, we set the confidence threshold +λ = 0.2 to obtain coarse matches. Besides, we choose w = 5 +to crop local windows in fine-level feature maps for matches +refinement. To reconcile the coarse matching loss, regression +loss, and classification loss, we set both weighting coefficients +β and φ to 0.2. For matching loss, we set the weighting +factor α = 0.25 and the focusing parameter η = 2. When +making classification labels, we set ψ to 8. In this work, we +elaborately design two versions of DeepMatcher that interleave +SlimFormer by L = 6, 10 times for feature enhancement, +resulting in DeepMatcher and DeepMatcher-L. +Training scheme for Scannet [51]. We train DeepMatcher +on Scannet [51] dataset with 32 Tesla V100 GPUs for indoor +local feature matching. In accordance with LoFTR, we sample +200 image pairs per scene at each epoch and balance scene +variants over iterations. We employ the AdamW solver for +optimization with a weight decay of 0.1. The initial learning +rate is set to 6 ×10−4 and will decrease by 0.5 every 3 epochs. +We use gradient clipping that is set to 0.5 to avoid exploding +gradients. +Training scheme for MegaDepth [52]. We train Deep- +Matcher on MegaDepth [52] datasets with 32 Tesla V100 +GPUs for outdoor local feature matching. Following LoFTR, +we randomly sample 100 pairs from each sub-scene during +each epoch of training. We train DeepMatcher for 30 epochs +in total. We also employ the AdamW solver for optimization +with a weight decay of 0.1. The initial learning rate is set to 8 +× 10−4, with a linear learning rate warm-up in 3 epochs from +0.1 to the initial learning rate. We decay the learning rate by +0.5 every 4 epochs starting from the 4-th epoch. +B. Indoor Pose Estimation +Typically, indoor pose estimation task is hampered by +motion blur and significant viewpoint shifts. There are com- +monly extensive regions of low textures in indoor scenes. To +evaluate the performance of DeepMatcher in such situations, +we conducted indoor pose estimation experiments. +Dataset. We use ScanNet [51] dataset to validate the +effectiveness of DeepMatcher on indoor pose estimation task. +ScanNet consists of 1513 RGB-D sequences with RGB images + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +8 +TABLE I +INDOOR POSE ESTIMATION EVALUATION ON SCANNET DATASET. THE +AUC@(5◦, 10◦, 20◦) IS REPORTED. +Local +features +Matcher +Pose estimation AUC +@5◦ +@10◦ +@20◦ +Detector-based Methods +D2-Net [12] +NN +5.25 +14.53 +27.96 +ContextDesc [53] +ratio test [9] +6.64 +15.01 +25.75 +SuperPoint [11] +NN +9.43 +21.53 +36.40 +NN + OANet [54] +11.76 +26.90 +43.85 +SuperGlue [3] +16.16 +33.81 +51.84 +SGMNet [16] +15.40 +32.06 +48.32 +DenseGAP [17] +17.01 +36.07 +55.66 +HTMatch [55] +15.11 +31.42 +48.23 +Detector-free Methods +—— +LoFTR [2] +22.06 +40.80 +57.62 +QuadTree [43] +24.90 +44.70 +61.80 +MatchFormer [26] +24.31 +43.90 +61.41 +ASpanFormer [28] +25.60 +46.00 +63.30 +DeepMatcher +25.38 +44.38 +60.35 +DeepMatcher-L +27.32 +46.25 +62.49 +and corresponding ground-truth poses in indoor environments. +Following [3], we select 230M image pairs with overlap values +ranging from 0.4 to 0.8 as the training set and 1500 image +pairs as the test set. All images are resized to 640 × 480. +Evaluation Protocol. In accordance with [2], [3], we report +the area under the cumulative curve (AUC) of pose errors at +the thresholds (5◦, 10◦, 20◦), where pose errors are defined as +the maximum of translational and rotational errors between +ground-truth poses and predicted poses by DeepMatcher. +Specifically, given the predicted dense matches, we utilize +OPENCV to calculate the essential matrix E and relative pose +˜T of image pairs. Then, the pose errors ∆T are defined as +the maximum of translational and rotational errors between +ground-truth relative pose T = [R|t] and estimated relative +pose ˜T = [ ˜R|˜t]: +∆T =max(∆t, ∆R), +∆t = arccos( +˜t · t +||˜t||2 · ||t||2 +), ∆R = arccos(tr( ˜RT R) − 1 +2 +), +(28) +where ∆t and ∆R denote the translational error and rotational +error, respectively; R, t is the ground-truth rotation matrix and +translation vector; ˜R, ˜t mean the predicted rotation matrix and +translation vector; tr(·) means the trace of a matrix. +Given the pose errors of all image pairs, we plot the cumu- +lative error distribution curve, whose area at three thresholds +(5◦, 10◦, 20◦) are computed as AUC@(5◦, 10◦, 20◦). +Results. +As +illustrated +in +Table +I, +we +observe +that +the +detector-free +methods +achieve +superior +performance +than +detector-based +methods +since +the +detector +strug- +gles to extract repeatable keypoints when handling image +pairs with significant viewpoint change. Wherein, Deep- +Matcher and DeepMatcher-L outperform all cutting-edge +detector-based and detector-free methods by a great mar- +gin. More specifically, DeepMatcher-L surpasses detector- +based method DenseGAP by (10.31%, 10.18%, 6.83%) in +TABLE II +OUTDOOR POSE ESTIMATION EVALUATION ON MEGADEPTH DATASET. +THE AUC@(5◦, 10◦, 20◦) IS REPORTED. +Local +features +Matcher +Pose estimation AUC +@5◦ +@10◦ +@20◦ +Detector-based Methods +SuperPoint [11] +SuperGlue [3] +42.18 +61.16 +75.96 +DenseGAP [17] +41.17 +56.87 +70.22 +ClusterGNN [18] +44.19 +58.54 +70.33 +Detector-free Methods +—— +DRC-Net [23] +27.01 +42.96 +58.31 +LoFTR [2] +52.80 +69.19 +81.18 +QuadTree [43] +54.60 +70.50 +82.20 +TopicFM [27] +54.10 +70.10 +81.60 +MatchFormer [26] +52.91 +69.74 +82.00 +ASpanFormer [28] +55.30 +71.50 +83.10 +DeepMatcher +55.71 +72.25 +83.49 +DeepMatcher-L +56.98 +73.11 +84.15 +terms of AUC@(5◦, 10◦, 20◦), demonstrating the superior- +ity of detector-free structure. Compared with the pioneer- +ing method LoFTR, DeepMatcher-L realizes superior per- +formance with the improvement of (5.26%, 5.45%, 4.87%). +Furthermore, DeepMatcher-L outperforms the state-of-the- +art detector-free method ASpanFormer by (1.72%, 0.25%) +in terms of AUC@(5◦, 10◦), proving the deep Transformer +architecture is essential to extract more human-intuitive and +easier-to-match features. Additionally, DeepMatcher-L only +consumes 77.65% GFLOPs and achieves 26.95% inference +speed boost compared with ASpanFormer, as demonstrated in +Table IV. +C. Outdoor Pose Estimation +Outdoor pose estimation remains a challenging task owing +to the intricate 3D geometry, extreme illumination and view- +point changes. To demonstrate the efficacy of DeepMatcher +in overcoming these obstacles, an outdoor pose estimation +experiment is conducted. +Dataset. We utilize MegaDepth [52] to conduct the outdoor +pose estimation experiment. MegaDepth contains 1M internet +images from 196 different scenes. These images come from +photo-tourism and contain challenging conditions, including +large viewpoint and illumination variations. Following [2], +[14], we select 100 image pairs each scene for training and +1500 image pairs for testing. Images are resized such that their +longer dimensions are equal to 840. +Evaluation Protocol. We use the same evaluation metrics +AUC@(5◦, 10◦, 20◦) as the indoor pose estimation task. +Results. As shown in Table II, we can observe that +DeepMatcher families surpass other methods in all evalua- +tion metrics. Specifically, DeepMatcher-L noticeably outper- +forms the cutting-edge detector-based method ClusterGNN +by (12.79%, 14.57%, 13.82%) in AUC@(5◦, 10◦, 20◦) since +the detector struggles to extract repeatable keypoints in im- +age pairs with extreme viewpoint change. Besides, com- +pared with the baseline approach LoFTR, DeepMatcher- +L achieves superior performance with the improvement of + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +9 +1 2 3 4 5 6 7 8 9 10 +threshold(px) +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +MMA +Overall +DeepMatcher-L +DeepMatcher +MatchFormer +LoFTR +SuperGlue +D2-Net +R2D2 +SparseNCNet +1 2 3 4 5 6 7 8 9 10 +threshold(px) +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Illumination +1 2 3 4 5 6 7 8 9 10 +threshold(px) +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Viewpoint +Fig. 5. Image matching evaluation on HPatches dataset. +(4.18%, 3.92%, 2.97%). Moreover, DeepMatcher-L also sur- +passes the state-of-the-art detector-free method ASpanFormer +by (1.68%, 1.61%, 1.05%), further validating the rationality of +the deep Transformer structures. +D. Image Matching +As a fundamental visual task, image matching plays an +important role in several applications. Therefore, we conduct +an image matching experiment to validate the performance of +DeepMatcher. +Dataset. We conduct homography estimation experiments +on the HPatches dataset [56]. Following [12], we select 108 +sequences from HPatches. Each sequence consists of a ground- +truth homography matrix and 6 images of progressively larger +illumination (52 sequences with illumination changes) or view- +point changes (56 sequences with viewpoint changes). +Evaluation Protocol. We adopt the generally employed +mean matching accuracy (MMA) as metric, i.e., the average +proportion of correct correspondences per image pair [12]. +Specifically, the keypoints from the i-th query image are +projected into the reference image by using the provided +homography matrix Hi. Then, the matches with reprojection +errors that are lower than a predefined threshold t are deemed +correct. Finally, we compute the average percentage of correct +matches across all image pairs and define MMA as: +MMA(t) = +1 +HP +HP +� +i=1 +( +�N f +j=1 1(t − ||Hi(P f +A,i,j) − P f +B,i,j||2) +N f +), +(29) +where HP means the number of image pairs in HPatches; N f +means the number of predicted matches; 1(·) is a binary indi- +cator function whose output is 1 for non-negative value and 0 +otherwise; t is the threshold of reprojection error, varying from +1 to 10 pixels; Hi(·) means warping the keypoints in the i-th +query image to reference image by ground-truth homography +matrix; (P f +A,i,j, P f +B,i,j) means the pixel coordinates of the j-th +match in the i-th image pair. +Results. As illustrated in Fig. 5, we can observe that Deep- +Matcher families achieve superior performance than detector- +free methods (i.e. MatchFormer, LoFTR, SparseNCNet). Un- +der varying illumination conditions, DeepMatcher yields in- +ferior performance at low thresholds, while achieving out- +standing performance when the threshold is larger than 5. +Moreover, when handling image pairs with viewpoint changes, +DeepMatcher exhibits extremely superior robotness compared +LoFTR +DeepMatcher +DeepMatcher-L P:93.9% +Matches:423 +DeepMatcher-L P:97.7% +Matches:215 +DeepMatcher-L P:93.6% +Matches:188 +DeepMatcher +LoFTR P:84.1% +Matches:679 +LoFTR P:89.2% +Matches:167 +LoFTR P:73.3% +Matches:15 +LoFTR +Fig. 6. +Visualization of the predicted matches. The mismatches, whose +reprojection errors are larger than 5px, are colored red. +TABLE III +HOMOGRAPHY ESTIMATION EVALUATION ON HPATCHES DATASET. +Local +features +Matcher +Overall +Illumination +Viewpoint +CCM (ε<1/3/5 pixels) +Detector-based Methods +D2-Net [12] +NN +0.38/0.71/0.82 0.66/0.95/0.98 0.12/0.49/0.67 +R2D2 [13] +NN +0.47/0.77/0.82 0.63/0.93/0.98 0.32/0.64/0.70 +ASLFeat [57] +NN +0.48/0.81/0.88 0.62/0.94/0.98 0.34/0.69/0.78 +SuperPoint [11] +NN +0.46/0.78/0.85 0.57/0.92/0.97 0.35/0.65/0.74 +SuperGlue [3] +0.51/0.82/0.89 0.60/0.92/0.98 0.42/0.71/0.81 +SGMNet [16] +0.52/0.85/0.91 0.59/0.94/0.98 0.46/0.74/0.84 +ClusterGNN [18] +0.52/0.84/0.90 0.61/0.93/0.98 0.44/0.74/0.81 +Detector-free Methods +—— +SparseNCNet [22] 0.36/0.65/0.76 0.62/0.92/0.97 0.13/0.40/0.58 +Patch2Pix [24] +0.50/0.79/0.87 0.71/0.95/0.98 0.30/0.64/0.76 +LoFTR [2] +0.55/0.81/0.86 0.74/0.95/0.98 0.38/0.69/0.76 +MatchFormer [26] 0.55/0.81/0.87 0.75/0.95/0.98 0.37/0.68/0.78 +DeepMatcher +0.50/0.81/0.90 0.62/0.93/0.98 0.38/0.70/0.81 +DeepMatcher-L +0.51/0.83/0.91 0.64/0.94/0.98 0.39/0.72/0.84 +with other detector-free methods. As shown in Fig. 6, we +select three pairs of image pairs from HPatches dataset and +visualize the matches predicted by LoFTR and DeepMatcher- +L to further validate the robustness of DeepMatcher-L to +viewpoint variations. +E. Homography Estimation +Since the distribution and number of matches are essential +to estimate reliable geometry relationship between image +pairs, we conduct a homography estimation experiment to +comprehensively evaluate the performance of DeepMatcher. +Dataset. We assess DeepMatcher on HPatches dataset, +which is widely used for homography estimation task. +Evaluation Protocol. Following the corner correctness met- +ric (CCM) utilized in [24], we report the percentage of image +pairs with average corner errors ε smaller than 1/3/5 pixels. +Specifically, based on the predicted dense matches, we use +OPENCV to calculate the homography matrix ˜Hi for the i-th +image pair. Subsequently, four corners in the query image are +projected into the reference image by using the ground-truth +homography matrix Hi and the predicted homography matrix + +L0FTR, P: 84:1% +Matches: 679DeepMatcherP:9 +Matches: +ate +ESEARN +fonicL0FTR, P: 89.2% +Matches: 167DeepMatcher, P:97.7% +Matches: 215DeepMatcher, P:93.6% +Matches:188L0FTR,P:73.3% +Matches:15ate +ESEARK +ronicJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +10 +LoFTR +DeepMatcher +SuperPoint + SuperGlue +Indoor +Outdoor +SGMNet +LoFTR +DeepMatcher +SuperPoint + SuperGlue +Indoor +Outdoor +SGMNet +LoFTR +DeepMatcher-L +SuperPoint + SuperGlue +Indoor +Outdoor +SGMNet +DeepMatcher-L P:100.0% +Matches:1268 +ΔR:1.02°, Δt:1.83° +SuperGlue P:82.9% +Matches:117 +ΔR:5.87°, Δt:6.13° +LoFTR P:90.4% +Matches:1025 +ΔR:3.16°, Δt:2.99° +SGMNet P:85.8% +Matches:358 +ΔR:5.21°, Δt:7.47° +DeepMatcher-L P:97.7% +Matches:576 +ΔR:1.51°, Δt:3.05° +LoFTR P:91.1% +Matches:492 +ΔR:2.44°, Δt:7.42° +SGMNet P:30.7% +Matches:127 +ΔR:18.23°, Δt:35.46° +SuperGlue P:51.8% +Matches:56 +ΔR:17.38°, Δt:77.89° +DeepMatcher-L P:96.6% +Matches:293 +ΔR:1.34°, Δt:1.74° +LoFTR P:59.9% +Matches:167 +ΔR:3.28°, Δt:4.01° +SuperGlue P:51.2% +Matches:168 +ΔR:5.18°, Δt:6.15° +SGMNet P:0.0% +Matches:92 +ΔR:inf °, Δt:inf ° +DeepMatcher-L P:86.2% +Matches:203 +ΔR:0.98°, Δt:2.55° +LoFTR P:79.2% +Matches:125 +ΔR:1.98°, Δt:5.24° +SGMNet P:59.5% +Matches:37 +ΔR:2.92°, Δt:7.69° +SuperGlue P:74.0% +Matches:154 +ΔR:2.41°, Δt:6.49° +Fig. 7. Visualization of the predicted matches. The matches are colored by their reprojection errors (green indicates correct matches, and red indicates +mismatches). We set the error threshold to 10 and 15 pixels for indoor and outdoor scenes. +L=1 +L=2 +L=3 +L=4 +L=5 +L=10 +L=9 +L=8 +L=7 +L=6 +L=1 +L=2 +L=3 +L=4 +L=5 +L=10 +L=9 +L=8 +L=7 +L=6 +DeepMatcher P:27.7% +Matches:358 +ΔR:8.19°, Δt:11.93° +DeepMatcher P:0.0% +Matches:3532 +ΔR:inf °, Δt:inf ° +DeepMatcher P:0.9% +Matches:747 +ΔR:inf °, Δt:inf ° +DeepMatcher P:2.7% +Matches:580 +ΔR:inf °, Δt:inf ° +DeepMatcher P:8.7% +Matches:447 +ΔR:inf °, Δt:inf ° +DeepMatcher P:20.8% +Matches:356 +ΔR:14.10°, Δt:30.03° +DeepMatcher P:45.9% +Matches:401 +ΔR:4.62°, Δt:10.60° +DeepMatcher P:55.1% +Matches:403 +ΔR:3.44°, Δt:9.13° +DeepMatcher P:73.6% +Matches:648 +ΔR:3.15°, Δt:7.74° +DeepMatcher P:74.1% +Matches:690 +ΔR:1.08°, Δt:3.07° +L=1 +L=2 +L=3 +L=4 +L=5 +L=6 +L=7 +L=8 +L=9 +L=10 +DeepMatcher-L P:54.9% +Matches:636 +ΔR:8.21°, Δt:10.70° +DeepMatcher-L P:0.8% +Matches:3124 +ΔR:inf °, Δt:inf ° +DeepMatcher-L P:12.2% +Matches:810 +ΔR:30.05°, Δt:78.64° +DeepMatcher-L P:25.8% +Matches:718 +ΔR:28.32°, Δt:76.97° +DeepMatcher-L P:36.2% +Matches:640 +ΔR:18.65 °, Δt:47.37 ° +DeepMatcher-L P:44.9% +Matches:657 +ΔR:14.92°, Δt:12.50° +DeepMatcher-L P:62.6% +Matches:755 +ΔR:3.01°, Δt:5.80° +DeepMatcher-L P:68.6% +Matches:892 +ΔR:1.45°, Δt:3.01° +DeepMatcher-L P:75.9% +Matches:1140 +ΔR:1.06°, Δt:2.03° +DeepMatcher-L P:94.1% +Matches:909 +ΔR:0.35°, Δt:0.23° +L=1 +L=2 +L=3 +L=4 +L=5 +L=6 +L=7 +L=8 +L=9 +L=10 +Fig. 8. The predicted matches of DeepMatcher-L after each SlimFormer. The threshold λ of the soft assignment matrix is set to 0. +˜Hi, respectively. Ultimately, we calculate average reprojection +error as corner error εi, thereby obtaining the CCM: +εi = +� +p∈Pco ||Hi(p) − ˜Hi(p)||2 +4 +, +CCM(t) = +�HP +i=1 1(t − εi) +HP +, +(30) +where Pco = {(0, 0), (Wo − 1, 0), (0, Ho − 1), (Wo − 1, Ho − +1)} means the four corners coordinates of the query image; +Hi(·) and ˜Hi(·) mean warping the corners in the i-th query +image to reference image by ground-truth homography matrix +and predicted homography matrix, respectively; t ∈ {1, 3, 5} +means the predefined threshold. +Results. As shown in Table III, DeepMatcher achieves +the best performance among the detector-free methods under +extreme viewpoint changes. Specifically, DeepMatcher out- +performs LoFTR and MatchFormer with the improvement of +(1%, 5%) and (2%, 3%) when thresholds are set to 3, 5 pixels, +repsectively. Furthermore, DeepMatcher-L surpasses LoFTR +and MatchFormer with the improvement of (1%, 3%, 8%) and +(2%, 4%, 6%). Besides, the detector-based methods are more +robust to viewpoint variations, while the detector-free methods +realize superior performance when handling image pairs with +extreme illumination changes. In comparison, DeepMatcher +strikes a decent balance when handling image pairs with +various viewpoint and illumination changes. +F. Understanding DeepMatcher +Qualitative Results Visualization. To further exhibit the +capability of DeepMatcher to handle image pairs with extreme +appearance settings, e.g., sparse texture, motion blur, large +viewpoint and illumination changes, we visualize the matches +predicted by SGMNet, SuperGlue, LoFTR, and DeepMatcher- +L. As shown in Fig. 7, we can observe that DeepMatcher-L +achieves dense and accurate matching performance. +Visual Descriptors Enhancement Efficacy Analysis. To +validate the effectiveness of performing L times of Slim- +Former for feature enhancment, we visualize the matching +results of DeepMatcher after each SlimFormer. As illustrated +in Fig. 8, we can observe that the matching precision is +promoted consistently, demonstrating interleaving SlimFormer +can effectively integrate intra-/inter-image information, hence +extracting easier-to-match features. +Efficiency Analysis. To validate the efficiency of Deep- +Matcher, we compare several cutting-edge detector-free meth- +ods in terms of parameters, flops, and inference speed to +determine their computational cost and storage consumption. +We resize the input images to 640 × 480 and conduct all +experiments on a single NVIDIA TITAN RTX GPU. When +counting runtime, we run the test code 500 times and re- +port the average time to eliminate occasionality. Notably, +we only compare DeepMatcher families with other detector- +free methods. As shown in Table IV, we can observe that +DeepMatcher families realize competitive inference speed with + +LoFTR, P: 59.9% +Matchesi167DeepMatcher, P: 96.6% +Matches:293SuperGlue, P: 50.3% +Matches.161SuperGlue, P: 82.9% +Matches:117SGMNet, P: 47.8% +Matches:90SuperGlue, P: 71.9% +Matches:320FTR. P: 87.8% +Matches:98DeepMatcher,P:91.1 +Matches:381SGMNet, P: 85.8% +Matches:358 +△R:5.21°At:1 47DeepMatcherP:100.0% +Matches:1268 +△R:1.02°△t:1.83°L0FTR. P: 90.4% +Matches:1025 +△R:3.16°△t:2.99°SuperGlue, P: 82.9% +Matches: 117 +△R:5.87°Zt:6.13L0FTR, P: 79.2% +Matches125SGMNet. P:307% +Matches:127 +△R:18.23°. At:35.46DeepMatcher,P:977% +Matches:576 +△R:1.51°△t:3.05°LoFTR, P: 91.1% +Matches:49 +AR:2.44°.△t74SuperGlue,P:51.8% +Matches:56 +AR:17.38°.At:7789DeepMatcher, P: 96.6% +Matches:293 +AR:1349At:1.74LoFTR, P: 59.9% +Matches:-167 +AR:3289AL:401SuperGlue, P: 51.2% +Matches.168 +AR.5.18°.AL6SGMNetP: 0.0% +Matches:92 +△R:info△t:infsDeepMatcher, P: 86.2% +Matches203 +△R:0.98Zt:2.55L0FTR, P: 79.2% +Matches125 +AR:198At:5DeepMatcher, P: 86.2% +Matches203SuperGlue, P: 74.0% +Matches154 +△R:2.41:At6.498SGMNet,P:59.5% +Matches3 +AR:2-92AL7698SGMNetP: 0.0% +Matches:9SuperGlue, P: 74.8% +Matches15SGMNet, P: 59.5% +Matches03DeepMatcherP:100.0% +Matches:1268L0ETR. P: 90.4% +Matches:1025SGMNet. P: 91.5% +Matches:259DeepMatcher.P: 0.0% +Matches:3532DeepMatcher:P:0.5% +Matches.747DeepMatcher, P: 74.1% +Matches690 +△R.1.08°,△t3.07DeepMatcher.P: 0.0% +Mathes3532 +ARinfAt:infoDeepMatcher:P:0.9% +Matches.747 +AR.infAt.infoDeepMatcherP:2.8% +Matches580 +ARinfoAtinfsDeepMatcherP:8.7% +Matches.447 +ARinfoAtimtsDeepMatcherP:20.8% +Matches.356 +AR:410:t12.039DeepMatcher.P:27.7% +Matches358 +AR:4 19°_At:11.939DeepMatcher, P:45.9% +Matches: +401 +AR.3.62 +2:t10.60DeepMatcherP:55.1% +Matches403 +△R.3.44°△t:DeepMatcher, P:73.6% +Matches.648 +△R.3.15°,△t:7.74°DeepMatcherP:1.4% +Matches:580DeepMatcherP:5.6% +Matches.447DeepMatcherP:149% +Matches356DeepMatcher.P:20.9% +Matches.358DeepMatcher P:39.2% +Matches401DeepMatcher, P: 49.4% +Matches +403DeepMatcher, P:71.9% +Matches.648DeepMatcher, P: 74.1% +Matches690JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +11 +TABLE IV +EFFICIENCY ANALYSIS. SEVERAL APPROACHES ARE COMPARED IN +TERMS OF PARAMETERS (MB), GFLOPS, AND RUNTIME (S). WE ALSO +RECORD THE COMPUTATIONAL COMPLEXITY (TC) OF THE ATTENTION +LAYER. N DENOTES THE PIXLE TOKEN NUMBER, C DENOTES THE +FEATURE DIMENSION, k DENOTES SELECTED TOKEN NUMBER, r DENOTES +DOWN-SCALE RATIO, w DENOTES THE SAMPLE NUMBER. +Methods +Params GFLOPs Runtime +TC +LoFTR [2] +11.06 +328.67 +0.079 +O(NC2) +QuadTree [43] +13.21 +382.01 +0.152 +O(kNC) +MatchFormer [26] +22.37 +396.95 +0.357 +O(N 2C/r) +AspanFormer [28] +15.05 +391.38 +0.141 +O((N/r)2C + NwC) +DeepMatcher +10.96 +255.99 +0.089 +O(NC) +DeepMatcher-L +15.51 +303.90 +0.103 +O(NC) +Original Image +Fig. 9. +Visualization of weights in SlimFormer structure. SlimFormer +emphasises keypoints at object boundaries to incorporate global context. +less GFLOPs since SlimFormer leverages element-wise prod- +uct to model relevance among all keypoints. Compared with +the baseline LoFTR, DeepMatcher and DeepMatcher-L only +consume (77.89%, 92.46%) GFLOPs. Moreover, compared +with cutting-edge detector-free methods QuadTree, Match- +Former and AspanFormer, DeepMatcher-L exhibits more effi- +cient matching performance with (20.45%, 23.44%, 22.35%) +less GFLOPs and (32.24%, 71.15%, 26.95%) inference speed +boost. +Furthermore, we also record the dominant computational +complexity of the attention layers in various methods. As +shown in Table IV, DeepMatcher achieves the minimum +computational complexity of the attention layer. Specifically, +compared with the baseline LoFTR, DeepMatcher reduces +the dominant computational complexity from O(NC2) to +O(NC). +Weight Analysis. To explore which keypoints SlimFormer +pays attention to when extracting global vectors, we visualize +the weight ˜Qimp and ˜KQimp in Eq. (4) and Eq. (7), respec- +tively. As shown in Fig. 9, we can observe that SlimFormer +primarily pays attention to the prominent keypoints at object +boundaries that involve tremendous visual and geometry infor- +mation. Consequently, SlimFormer exhibits puissant capability +to aggregate global context information effectively. +2 +4 +6 +8 +10 +12 +14 +Transformer Layers Number +18 +20 +22 +24 +26 +28 +AUC@5° Values +DeepMatcher +LoFTR +2 +4 +6 +8 +10 +12 +14 +Transformer Layers Number +38 +40 +42 +44 +46 +AUC@10° Values +DeepMatcher +LoFTR +2 +4 +6 +8 +10 +12 +14 +Transformer Layers Number +54 +56 +58 +60 +62 +AUC@20° Values +DeepMatcher +LoFTR +2 +4 +6 +8 +10 +12 +14 +Transformer Layers Number +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +25.0 +Paeameters (MB) +DeepMatcher +LoFTR +Fig. 10. +The AUC values and parameters of DeepMatcher and LoFTR +with the Transformer layers increasing. +Deep Transformer Architecture Analysis. To verify the +opinion that deep Transformer architecture is essential to +extract more human-intuitive and easier-to-match features, we +record the indoor pose estimation precision with the num- +ber of Transformer layers increasing. As shown in Fig. 10, +since SlimFormer leverages vector-based attention for robust +long-range global context aggregation and utilizes layer-scale +strategy and relative position encoding to enhance the rep- +resentation of keypoints, the AUC values of DeepMatcher +are promoted consistently with the Transformer layers going +deep, while the accuracy of LoFTR is constantly fluctuating. +Besides, the matching performance of DeepMatcher is sig- +nificantly superior to LoFTR. Moreover, the parameters of +DeepMatcher and LoFTR increase linearly with the number +of Transformer layers, while DeepMatcher occupies fewer +parameters. +G. Ablation Study +Effect of the Proposed Modules. To thoroughly validate +the rationality of each module, we conduct indoor pose esti- +mation experiments using different variants of DeepMatcher. +As illustrated in Table V, we can observe that all components +contribute to the outstanding performance of DeepMatcher. +(i), (ii) Using only self- and cross-SlimFormer layers leads +to a severe decrease in matching performance, demonstrat- +ing interleaving the self- and cross-SlimFormer layers can +effectively integrate intra-/inter-image message. (iii) Removing +the Feature Transition Module results in a much lower accu- +racy (−0.66%, −0.47%, −0.33%), proving the effectiveness +of ensuring smooth transition between feature extractor and +SlimFormer in terms of context ranges. (iv) Removing Relative +Position Encoding spawns a large drop in pose estimation +accuracy (−1.36%, −1.67%, −1.38%), proving the relative +position information is crucial to distinguish similar features. +(v) Removing the Fine Matches Module results in lower +AUC values (−8.34%, −12.56%, −14.86%), indicating the +effectiveness of refining coarse matches. +Effect of the Learnable Scale Factor ξ. To validate that +layer-scale strategy can simulate the human behaviour that + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +12 +TABLE V +ABLATION STUDY WITH DIFFERENT VARIANTS OF DEEPMATCHER ON +SCANNET DATASET. +Methods +Pose estimation AUC +@5◦ +@10◦ +@20◦ +(i) w only self-SlimFormer layers +20.38 +37.71 +53.89 +(ii) w only cross-SlimFormer layers +22.33 +40.27 +56.64 +(iii) w/o Feature Transition Module +24.72 +43.91 +60.02 +(iv) w/o Relative Potition Encoding +24.02 +42.71 +58.97 +(v) w/o Fine Matches Module +17.04 +31.82 +45.49 +DeepMatcher full +25.38 +44.38 +60.35 +TABLE VI +ABLATION STUDY WITH DIFFERENT LEARNABLE SCALE FACTORS ξ ON +SCANNET DATASET. +Methods +Pose estimation AUC +@5◦ +@10◦ +@20◦ +w residual scale factor ξ +25.38 +44.38 +60.35 +w/o residual scale factor ξ +24.50 +42.91 +59.32 +humans can acquire different matching cues each time they +scan an image pair to further improve matching performance, +we remove the learnable scale factor ξ and conduct an ab- +lation experiment. +As shown in Table VI, we can observe +that introducing the residual scaling factor leads to superior +performance. +Effect of the Relative Position Encoding. Humans lever- +age the relative position information to establish the connec- +tion between objects. Therefore, the relative position encoding +is more conducive to realize elaborate scene parsing. To +prove this opinion, we implement an ablation experiment +using three structures: (i) Removing all position encoding in +all SlimFormer. (ii) Using absolute position encoding pro- +posed in LoFTR. (iii) Using relative position encoding. As +shown in Table VII, we can observe that both absolute and +relative position encoding boost AUC values, in which the +relative position encoding exhibits more superior performance +than absolute position encoding with the improvement of +(0.86%, 1.24%, 0.84%). +Effect of the Fine Matches Module. Compared with the +coarse-to-fine module used in LoFTR, FMM views the match +refinement as a combination of classification and regression +tasks. To validate the availability of FMM, we conduct an +ablation experiment. As shown in Table VIII, we can observe +that using FMM significantly achieves superior performance +with the improvement of (1.64%, 2.12%, 1.49%), proving the +rationality of predicting offset and confidence concurrently +using a network. +V. CONCLUSION +In this work, we propose DeepMatcher, a deep Transformer- +based network for local feature matching. DeepMatcher sim- +ulates human behaviors when humans match image pairs, in- +cluding: (1) Deep SlimFormer layers of the network to aggre- +gate information intra-/inter-images; (2) Layer-scale strategy +TABLE VII +ABLATION STUDY WITH DIFFERENT POSITION ENCODING ON SCANNET +DATASET. +Methods +Pose estimation AUC +@5◦ +@10◦ +@20◦ +w/o position encoding +24.02 +42.71 +58.97 +w absolute position encoding +24.52 +43.14 +59.51 +w relative position encoding +25.38 +44.38 +60.35 +TABLE VIII +ABLATION STUDY WITH DIFFERENT COARSE-TO-FINE MODULES ON +SCANNET DATASET. +Methods +Pose estimation AUC +@5◦ +@10◦ +@20◦ +Coarse-to-fine Module in LoFTR +23.74 +42.26 +58.86 +Fine Matches Module +25.38 +44.38 +60.35 +to assimilate message exchange from each layer adaptively. +Besides, relative position encoding is applied to each layer +so as to explicitly disclose relative distance information, +hence improving the representation of DeepMatcher. We also +propose Fine Matches Module to refine the coarse matches, +thus generating robust and accurate matches. Extensive exper- +iments demonstrate that DeepMatcher surpasses state-of-the- +art approaches on several benchmarks, confirming the superior +matching capability of DeepMatcher. +REFERENCES +[1] J. Chen, S. Chen, X. Chen, Y. Dai, and Y. Yang, “Csr-net: Learning adap- +tive context structure representation for robust feature correspondence,” +IEEE Transactions on Image Processing, vol. 31, pp. 3197–3210, 2022. +[2] J. Sun, Z. Shen, Y. Wang, H. Bao, and X. Zhou, “Loftr: Detector- +free local feature matching with transformers,” in Proceedings of the +IEEE/CVF Conference on Computer Vision and Pattern Recognition, +2021, pp. 8922–8931. +[3] P.-E. Sarlin, D. DeTone, T. Malisiewicz, and A. Rabinovich, “Superglue: +Learning feature matching with graph neural networks,” in Proceedings +of the IEEE/CVF conference on computer vision and pattern recognition, +2020, pp. 4938–4947. +[4] L. Zheng, G. Xiao, Z. Shi, S. Wang, and J. Ma, “Msa-net: Establishing +reliable correspondences by multiscale attention network,” IEEE Trans- +actions on Image Processing, vol. 31, pp. 4598–4608, 2022. +[5] R. Mur-Artal and J. D. Tard´os, “Orb-slam2: An open-source slam +system for monocular, stereo, and rgb-d cameras,” IEEE transactions +on robotics, vol. 33, no. 5, pp. 1255–1262, 2017. +[6] C. Campos, R. Elvira, J. J. G. Rodr´ıguez, J. M. Montiel, and J. D. +Tard´os, “Orb-slam3: An accurate open-source library for visual, visual– +inertial, and multimap slam,” IEEE Transactions on Robotics, vol. 37, +no. 6, pp. 1874–1890, 2021. +[7] J. L. Schonberger and J.-M. Frahm, “Structure-from-motion revisited,” +in Proceedings of the IEEE conference on computer vision and pattern +recognition, 2016, pp. 4104–4113. +[8] H. Cui, D. Tu, F. Tang, P. Xu, H. Liu, and S. Shen, “Vidsfm: Robust +and accurate structure-from-motion for monocular videos,” IEEE Trans- +actions on Image Processing, vol. 31, pp. 2449–2462, 2022. +[9] D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” +International journal of computer vision, vol. 60, no. 2, pp. 91–110, +2004. +[10] E. Rublee, V. Rabaud, K. Konolige, and G. Bradski, “Orb: An efficient +alternative to sift or surf,” in 2011 International conference on computer +vision. +Ieee, 2011, pp. 2564–2571. +[11] D. DeTone, T. Malisiewicz, and A. Rabinovich, “Superpoint: Self- +supervised interest point detection and description,” in Proceedings of +the IEEE Conference on Computer Vision and Pattern Recognition +(CVPR) Workshops, June 2018. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +13 +[12] M. Dusmanu, I. Rocco, T. Pajdla, M. Pollefeys, J. Sivic, A. Torii, and +T. Sattler, “D2-net: A trainable cnn for joint description and detection +of local features,” in Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition (CVPR), June 2019. +[13] J. Revaud, P. Weinzaepfel, C. De Souza, N. Pion, G. Csurka, Y. Cabon, +and M. Humenberger, “R2d2: repeatable and reliable detector and +descriptor,” in NeurIPS, 2019. +[14] M. J. Tyszkiewicz, P. Fua, and E. Trulls, “Disk: Learning local features +with policy gradient,” in NeurIPS, 2020. +[15] Y. Xia and J. Ma, “Locality-guided global-preserving optimization for +robust feature matching,” IEEE Transactions on Image Processing, +vol. 31, pp. 5093–5108, 2022. +[16] H. Chen, Z. Luo, J. Zhang, L. Zhou, X. Bai, Z. Hu, C.-L. Tai, and +L. Quan, “Learning to match features with seeded graph matching +network,” in Proceedings of the IEEE/CVF International Conference +on Computer Vision, 2021, pp. 6301–6310. +[17] Z. Kuang, J. Li, M. He, T. Wang, and Y. Zhao, “Densegap: Graph- +structured dense correspondence learning with anchor points,” arXiv +preprint arXiv:2112.06910, 2021. +[18] Y. Shi, J.-X. Cai, Y. Shavit, T.-J. Mu, W. Feng, and K. Zhang, “Clus- +tergnn: Cluster-based coarse-to-fine graph neural network for efficient +feature matching,” in Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, 2022, pp. 12 517–12 526. +[19] D. DeTone, T. Malisiewicz, and A. Rabinovich, “Superpoint: Self- +supervised interest point detection and description,” in Proceedings +of the IEEE conference on computer vision and pattern recognition +workshops, 2018, pp. 224–236. +[20] J. Bian, W.-Y. Lin, Y. Matsushita, S.-K. Yeung, T.-D. Nguyen, and M.-M. +Cheng, “Gms: Grid-based motion statistics for fast, ultra-robust feature +correspondence,” in Proceedings of the IEEE conference on computer +vision and pattern recognition, 2017, pp. 4181–4190. +[21] I. Rocco, M. Cimpoi, R. Arandjelovi´c, A. Torii, T. Pajdla, and J. Sivic, +“Neighbourhood consensus networks,” Advances in neural information +processing systems, vol. 31, 2018. +[22] I. Rocco, R. Arandjelovi´c, and J. Sivic, “Efficient neighbourhood +consensus networks via submanifold sparse convolutions,” in European +Conference on Computer Vision. +Springer, 2020, pp. 605–621. +[23] X. Li, K. Han, S. Li, and V. Prisacariu, “Dual-resolution correspondence +networks,” Advances in Neural Information Processing Systems, vol. 33, +pp. 17 346–17 357, 2020. +[24] Q. Zhou, T. Sattler, and L. Leal-Taixe, “Patch2pix: Epipolar-guided +pixel-level correspondences,” in Proceedings of the IEEE/CVF Confer- +ence on Computer Vision and Pattern Recognition, 2021, pp. 4669–4678. +[25] U. Efe, K. G. Ince, and A. Alatan, “Dfm: A performance baseline for +deep feature matching,” in Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, 2021, pp. 4284–4293. +[26] Q. Wang, J. Zhang, K. Yang, K. Peng, and R. Stiefelhagen, “Match- +former: Interleaving attention in transformers for feature matching,” +arXiv preprint arXiv:2203.09645, 2022. +[27] K. Truong Giang, S. Song, and S. Jo, “Topicfm: Robust and interpretable +feature matching with topic-assisted,” arXiv e-prints, pp. arXiv–2207, +2022. +[28] H. Chen, Z. Luo, L. Zhou, Y. Tian, M. Zhen, T. Fang, D. McKinnon, +Y. Tsin, and L. Quan, “Aspanformer: Detector-free image matching +with adaptive span transformer,” in European Conference on Computer +Vision. +Springer, 2022, pp. 20–36. +[29] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, +Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” in Advances +in neural information processing systems, 2017, pp. 5998–6008. +[30] A. Katharopoulos, A. Vyas, N. Pappas, and F. Fleuret, “Transformers +are rnns: Fast autoregressive transformers with linear attention,” in +International Conference on Machine Learning. +PMLR, 2020, pp. +5156–5165. +[31] M. Karpushin, G. Valenzise, and F. Dufaux, “Keypoint detection in rgbd +images based on an anisotropic scale space,” IEEE Transactions on +Multimedia, vol. 18, no. 9, pp. 1762–1771, 2016. +[32] V. Balntas, E. Riba, D. Ponsa, and K. Mikolajczyk, “Learning local fea- +ture descriptors with triplets and shallow convolutional neural networks.” +in Bmvc, vol. 1, no. 2, 2016, p. 3. +[33] X. Jiang, J. Ma, J. Jiang, and X. Guo, “Robust feature matching using +spatial clustering with heavy outliers,” IEEE Transactions on Image +Processing, vol. 29, pp. 736–746, 2019. +[34] M. Sun, W. Suo, P. Wang, Y. Zhang, and Q. Wu, “A proposal-free one- +stage framework for referring expression comprehension and generation +via dense cross-attention,” IEEE Transactions on Multimedia, 2022. +[35] K. Fu, Q. Zhao, and I. Y.-H. Gu, “Refinet: A deep segmentation assisted +refinement network for salient object detection,” IEEE Transactions on +Multimedia, vol. 21, no. 2, pp. 457–469, 2018. +[36] Y. Song, X. Chen, X. Wang, Y. Zhang, and J. Li, “6-dof image local- +ization from massive geo-tagged reference images,” IEEE Transactions +on Multimedia, vol. 18, no. 8, pp. 1542–1554, 2016. +[37] B. Fan, Y. Yang, W. Feng, F. Wu, J. Lu, and H. Liu, “Seeing through +darkness: Visual localization at night via weakly supervised learning of +domain invariant features,” IEEE Transactions on Multimedia, 2022. +[38] M. A. Fischler and R. C. Bolles, “Random sample consensus: a paradigm +for model fitting with applications to image analysis and automated +cartography,” Communications of the ACM, vol. 24, no. 6, pp. 381–395, +1981. +[39] V. Fragoso, P. Sen, S. Rodriguez, and M. Turk, “Evsac: accelerating +hypotheses generation by modeling matching scores with extreme value +theory,” in Proceedings of the IEEE international conference on com- +puter vision, 2013, pp. 2472–2479. +[40] D. Barath, J. Matas, and J. Noskova, “Magsac: marginalizing sample +consensus,” in Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, 2019, pp. 10 197–10 205. +[41] V. Mousavi, M. Varshosaz, F. Remondino, S. Pirasteh, and J. Li, “A +two-step descriptor-based keypoint filtering algorithm for robust image +matching,” IEEE Transactions on Geoscience and Remote Sensing, +vol. 60, pp. 1–21, 2022. +[42] V. Fragoso, C. Sweeney, P. Sen, and M. Turk, “Ansac: Adaptive non- +minimal sample and consensus,” arXiv preprint arXiv:1709.09559, 2017. +[43] S. Tang, J. Zhang, S. Zhu, and P. Tan, “Quadtree attention for vision +transformers,” arXiv preprint arXiv:2201.02767, 2022. +[44] M. Zaheer, G. Guruganesh, K. A. Dubey, J. Ainslie, C. Alberti, +S. Ontanon, P. Pham, A. Ravula, Q. Wang, L. Yang et al., “Big bird: +Transformers for longer sequences,” Advances in Neural Information +Processing Systems, vol. 33, pp. 17 283–17 297, 2020. +[45] C. Wu, F. Wu, T. Qi, Y. Huang, and X. Xie, “Fastformer: Additive +attention can be all you need,” arXiv preprint arXiv:2108.09084, 2021. +[46] T.-Y. Lin, P. Doll´ar, R. Girshick, K. He, B. Hariharan, and S. Belongie, +“Feature pyramid networks for object detection,” in Proceedings of the +IEEE conference on computer vision and pattern recognition, 2017, pp. +2117–2125. +[47] A. G. Howard, M. Zhu, B. Chen, D. Kalenichenko, W. Wang, +T. Weyand, M. Andreetto, and H. Adam, “Mobilenets: Efficient convo- +lutional neural networks for mobile vision applications,” arXiv preprint +arXiv:1704.04861, 2017. +[48] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image +recognition,” in Proceedings of the IEEE conference on computer vision +and pattern recognition, 2016, pp. 770–778. +[49] J. Su, Y. Lu, S. Pan, B. Wen, and Y. Liu, “Roformer: En- +hanced transformer with rotary position embedding,” arXiv preprint +arXiv:2104.09864, 2021. +[50] T.-Y. Lin, P. Goyal, R. Girshick, K. He, and P. Doll´ar, “Focal loss +for dense object detection,” in Proceedings of the IEEE international +conference on computer vision, 2017, pp. 2980–2988. +[51] A. Dai, A. X. Chang, M. Savva, M. Halber, T. Funkhouser, and +M. Nießner, “Scannet: Richly-annotated 3d reconstructions of indoor +scenes,” in Proceedings of the IEEE conference on computer vision and +pattern recognition, 2017, pp. 5828–5839. +[52] Z. Li and N. Snavely, “Megadepth: Learning single-view depth predic- +tion from internet photos,” in Proceedings of the IEEE Conference on +Computer Vision and Pattern Recognition, 2018, pp. 2041–2050. +[53] Z. Luo, T. Shen, L. Zhou, J. Zhang, Y. Yao, S. Li, T. Fang, and L. Quan, +“Contextdesc: Local descriptor augmentation with cross-modality con- +text,” in Proceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition, 2019, pp. 2527–2536. +[54] J. Zhang, D. Sun, Z. Luo, A. Yao, L. Zhou, T. Shen, Y. Chen, L. Quan, +and H. Liao, “Learning two-view correspondences and geometry using +order-aware network,” in Proceedings of the IEEE/CVF International +Conference on Computer Vision, 2019, pp. 5845–5854. +[55] Y. Cai, L. Li, D. Wang, X. Li, and X. Liu, “Htmatch: An efficient hybrid +transformer based graph neural network for local feature matching,” +Signal Processing, p. 108859, 2022. +[56] V. Balntas, K. Lenc, A. Vedaldi, and K. Mikolajczyk, “Hpatches: A +benchmark and evaluation of handcrafted and learned local descriptors,” +in Proceedings of the IEEE conference on computer vision and pattern +recognition, 2017, pp. 5173–5182. +[57] Z. Luo, L. Zhou, X. Bai, H. Chen, J. Zhang, Y. Yao, S. Li, T. Fang, +and L. Quan, “Aslfeat: Learning local features of accurate shape and +localization,” in Proceedings of the IEEE/CVF conference on computer +vision and pattern recognition, 2020, pp. 6589–6598. + diff --git a/h9E1T4oBgHgl3EQfMwOA/content/tmp_files/load_file.txt b/h9E1T4oBgHgl3EQfMwOA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb7abc3654b4733c3a66a024abaa744594b84135 --- /dev/null +++ b/h9E1T4oBgHgl3EQfMwOA/content/tmp_files/load_file.txt @@ -0,0 +1,1525 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf,len=1524 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 1 DeepMatcher: A Deep Transformer-based Network for Robust and Accurate Local Feature Matching Tao Xie†, Kun Dai†, Ke Wang, Ruifeng Li, Lijun Zhao Abstract—Local feature matching between images remains a challenging task, especially in the presence of significant appearance variations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=', extreme viewpoint changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In this work, we propose DeepMatcher, a deep Transformer-based network built upon our investigation of local feature matching in detector-free methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The key insight is that local feature matcher with deep layers can capture more human-intuitive and simpler-to-match features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Based on this, we propose a Slimming Transformer (SlimFormer) dedicated for DeepMatcher, which leverages vector-based attention to model relevance among all keypoints and achieves long-range context aggregation in an efficient and effective manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' A relative position encoding is applied to each SlimFormer so as to explicitly disclose relative distance information, further improving the representation of keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' A layer-scale strategy is also employed in each Slim- Former to enable the network to assimilate message exchange from the residual block adaptively, thus allowing it to simulate the human behaviour that humans can acquire different matching cues each time they scan an image pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To facilitate a better adaption of the SlimFormer, we introduce a Feature Transition Module (FTM) to ensure a smooth transition in feature scopes with different receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' By interleaving the self- and cross- SlimFormer multiple times, DeepMatcher can easily establish pixel-wise dense matches at coarse level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Finally, we perceive the match refinement as a combination of classification and regression problems and design Fine Matches Module to predict confidence and offset concurrently, thereby generating robust and accurate matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Experimentally, we show that DeepMatcher significantly outperforms the state-of-the-art methods on several benchmarks, demonstrating the superior matching capability of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='com/XT- 1997/DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Index Terms—Local feature matching, Pose Estimation, Trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' INTRODUCTION L OCAL feature matching [1]–[4] is the prerequisite for a variety of geometric computer vision applications, in- cluding Simultaneous Localization and Mapping (SLAM) [5], [6] and Structure-from-Motion (SFM) [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As a broadly acknowledged matching pipeline, detector-based matching [3], [9]–[18] is typically accomplished by (i) detecting and de- scribing a set of sparse keypoints such as SIFT [9], ORB [10], and learning-based equivalents [13], [19], (2) instituting point-to-point correspondences via nearest neighbour search This work was in part by National Natural Science Foundation of China under Grant 62176072 and 62073101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (Corresponding author: Ruifeng Li and Ke Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' †: These authors contribute equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=') Tao Xie, Kun Dai, Ke Wang, Ruifeng Li, Lijun Zhao are with State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150006, China (email: xietao1997@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 20s108237@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' wangke@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' lrf100@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' zhaolj@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tao Xie is also with SenseTime Group Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=', Beijing 100080, China (email: xietao@sensetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' LoFTR DeepMatcher DeepMatcher P:93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:423 DeepMatcher P:97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches:215 DeepMatcher P:93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:188 DeepMatcher LoFTR P:84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1% Matches:679 LoFTR P:89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches:167 LoFTR P:73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3% Matches:15 LoFTR LoFTR DeepMatcher DeepMatcher-L 624 GFLOPs 328 303 255 196 352 Matches Precision 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% DeepMatcher-L P:80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:624 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81°, Δt:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='26° LoFTR P:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:196 ΔR:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02°, Δt:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='13° DeepMatcher P:69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:352 ΔR:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='57°, Δt:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='03° Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The comparison between LoFTR and DeepMatcher under large viewpoint changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DeepMatcher families considerably outperform LoFTR with more dense and precise matches while using less GFLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' or more advanced matching algorithms [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The use of a feature detector narrows the matching search space, revealing the general efficacy of such detector-based matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Nonetheless, when dealing with image pairs with severe viewpoint variations, such pipeline struggles to build reliable correspondences since the detectors are essentially incapable of extracting repeated keypoints in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Parallel to the detector-based matching, another stream of research seeks to establish correspondences directly from original images by extracting visual descriptors on dense grids across an image, thus assuring that substantial repeat- ing keypoints could well be captured [2], [21]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Earlier detector-free matching works [21]–[25] generally depend on iterative convolution based on correlation or cost volume to identify probable neighbourhood consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Transformer [29] has recently attracted considerable interest in computer vision due to its excellent model capability and superior potentials for capturing long-range relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' On the basis of this insight, various studies base their modelling of long-range relationships on Transformer backbone [2], [26]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As a representative work, LoFTR [2] updates features by repeatedly interleaving the self- and cross-attention layers, and replace vanilla Transformer with linear Transformer [30] to achieve manageable computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' These detector-free methdos can generate repeatable keypoints in indistinct regions with poor textures or motion blur, thus yielding impressive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' After witnessing the success of detector-free methods, an intriguing issue arises: could we build a deeper yet compact local feature matcher to further improve performance while reducing computing costs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Intuitively, when individuals match images, they scan the images back and forth, and the more times they scan them, the easier it is for them to remember the easier-to-match features, which indicates that a deep lo- cal feature matcher could display superior matching ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02993v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='CV] 8 Jan 2023 L0FTR, P: 84:1% Matches: 679DeepMatcherP:9 Matches: ate ESEARN fonicL0FTR, P: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches: 167DeepMatcher, P:97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches: 215DeepMatcher, P:93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:188L0FTR,P:73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3% Matches:15ate ESEARK ronicJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 2 However, as demonstrated in LoFTR, doubling the number of Transformer layers has little effect on the results, which is contrary to expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In this work, we argue that the following obstacles hinder us from developing a deep local feature matcher for detector-free methods: (i) Typical detector-free methods begin with a convolution neural network (CNN) as the basic feature extractor, followed by Transformer layers to capture long-range relevance so that generating credible correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In terms of context ranges, there is apparently a gap between the global receptive field of Transformer and the local neighborhood of CNN, which is detrimental to subsequent stages involving deep feature interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (ii) The translation invariance of CNN causes ambiguity in scenes with recurring geometry patterns or symmetrical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Current detector-free methods utilize absolute po- sition encodings before Transformer layers to tackle this issue, while the position information would disappear as the Transformer layers grow deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, humans naturally associate items across observations by referring to not only their absolute position but also their relative position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (iii) Intuitively, the depth of the network is more prominent than width in the field of feature matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' However, as the linear Transformer layer in LoFTR goes deeper, the model fails to learn effective context aggregation from deeper layers since the linear Transformer uses a context-agnostic manner to approximate self-attention, which cannot efficaciously sim- ulate relevance among all keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To this end, we propose DeepMatcher, a deep local feature matching network that can produce more human-intuitive and simpler-to-match features for accurate correspondence with less computational complexity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Firstly, we utilize a CNN network to generate pixel tokens with enriched features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Secondly, an Feature Transition Module (FTM) is introduced to ensure a smooth transition from the locally aggregated features extracted by CNN to features with a global receptive field extracted by Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, we propose a Slimming Transformer (SlimFormer) to build deep network that strengthens long-range global context mod- elling intra-/inter-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Technically, SlimFormer leverages vector-based attention that efficiently handles pixel tokens with linear complexity for robust long-range global context aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, a relative position encoding is applied to each SlimFormer to clearly express relative distance informa- tion, boosting the network’s capacity to convey information, particularly in deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, SlimFormer utilizes a layer-scale strategy that enables the network to assimilate message exchange from the residual block adaptively, thus allowing it to simulate the human behavior that human can receive different matching information each time they scan an image pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' By interleaving the self- and cross-SlimFormer multiple times, DeepMatcher learns the discriminative features to construct dense matches at the coarse level by Coarse Matches Module (CMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ultimately, we view the match refinement as a combination of classification and regression problems and devise Fine Matches Module (FMM) to pre- dict confidence and offset concurrently, obtaining robust and accurate matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To summarize, the main contributions of this work are as follows: We propose DeepMatcher, a deep Transformer-based network for local feature matching, achieving state-of- the-art results on various benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We propose a Feature Transition Module (FTM) to ensure a smooth transition from the locally aggregated features extracted by CNN to features with a global receptive field extracted by SlimFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We propose a Slimming Transformer (SlimFormer) that integrates long-range global context aggregation, relative position encoding, and layer-scale strategy to enable DeepMatcher to be extended into dozen layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We propose Fine Matches Module (FMM) that views the match refinement as a combination of classification and regression problems to optimze coarse matches, deriving robust and accurate matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Detector-based Methods The conventional pipeline of detector-based matching sys- tems detects two sets of keypoints, describes them with high-dimensional vectors, and then implements a matching algorithm to generate matches between the two sets of key- points [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Regarding feature detection and description, there are nu- merous handcrafted methods that seek to strike a balance between accuracy and efficiency [9], [10], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' However, the handcrafted descriptors are fragile when coping with image pairs with extreme appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' With the development of deep learning [34]–[37], numerous approaches leverage elaborate convolution neural network (CNN) to ex- tract robust feature representations, hence achieving superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' SuperPoint [11] builds a large dataset of pseudo- ground truth interest point locations in real images, supervised by the interest point detector itself, as opposed to a large- scale human annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' D2-Net [12] makes the collected keypoints more stable by delaying the detection to a later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Subsequently, the aforementioned methods utilize the nearest neighbor search, followed by a robust estimator, such as RANSAC or its variants [38]–[42], to find matches between the retrieved keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Recent researches [3], [16]–[18] has interpreted local feature matching as a graph matching problem involving two sets of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' These methods utilize keypoints as nodes to construct graph neural network (GNN), employ the self- and cross- attention layers in Transformer to exchange global visual and geometric messages across nodes, and then generate the matches in accordance with soft assignment matrixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Typically, SuperGlue [3] utilizes self- and cross-attention in Transformer to integrate global context information, followed by the Sinkhorn algorithm to generate matches according to the soft assignment matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Nonetheless, the matrix multipli- cation in vanilla Transformer results in quadratic complexity with respect to the number of keypoints, making SuperGlue costly to deal with substantial keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To tackle this prob- lem, many approaches attempt to ameliorate the structure of JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 3 SuperGlue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' SGMNet [16] exploits the sparsity of graph neural network to lower the computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' ClusterGNN [18] employs a progressive clustering module adaptively to divide keypoints into different subgraphs to reduce compu- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' However, limited to inherent essence, detector-based approaches are incapable of extracting repeated keypoints when handling image pairs with large appearance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Detector-free Methods Detector-free methods exclude the feature detector and gen- erate dense matches directly from the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Earlier detector-free matching researches [21]–[25] generally utilize convolutional neural network (CNN) based on correlation or cost volume to identify probable neighbourhood consen- sus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DRC-Net [23] generates a 4D correlation tensor from the coarse-resolution features, which is refined by a learn- able neighborhood consensus module to generate matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Patch2Pix [24] proposes a weakly supervised approach to learn matches that are consistent with the epipolar geometry of image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DFM [25] uses pre-trained VGG architecture as a feature extractor and captures matches without any additional training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Although elevating the matching accuracy, these methods extract ambiguous feature representations and fail to discriminate incorrect matches owing to the limited receptive field of CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To handle this issue, LoFTR [2], the pioneering detector-free GNN method, utlizes Transformer to realize global context information exchange and extracts matches in a coarse-to-fine manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Matchformer [26] proposes a human-intuitive extract- and-match scheme that interleaves self- and cross-attention in each stage of the hierarchical encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Such a match-aware encoder releases the overloaded decoder and makes the model highly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' QuadTree [43] proposes a novel Transformer structure that builds token pyramids and computes attention in a coarse-to-fine manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, the QuadTree Transformer is integrated into LoFTR and achieves superior matching performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' TopicFM [27] applies a topic-modeling strategy to encode high-level contexts in images, which improves the robustness of matching by focusing on the same semantic areas between the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' ASpanFormer [28] proposes a Transformer-based detector-free architecture, in which the flow maps are regressed in each cross-attention phase to perform local attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' These detector-free methods are capable of generating repeatable keypoints in indistinct regions with poor textures or motion blur, thus resulting in amazing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' However, the architecture of existing detector-free methods is designed as shallow-broad, and building a deeper and more compact local feature matcher to further improve performance while reducing computational costs has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Efficient Transformer In the vanilla Transformer, the memory cost is quadratic to the length of sequences due to the matrix multiplication, which has become a bottleneck for Transformer when dealing with long sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Recently, several approaches have been proposed to improve the efficiency of Transformer [30], [44], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Linear Transformer [30] expresses self-attention as a LoFTR CNN Transformer Layer PE correlation-based refinement block DeepMatcher CNN … Transformer Layer PE PE PE PE PE PE network-based refinement block CNN Transformer Layer PE correlation-based refinement block LoFTR P:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:196 ΔR:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02°, Δt:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='13° LoFTR DeepMatcher … Transformer Layer PE PE PE PE PE network-based refinement block DeepMatcher-L P:80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:624 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81°, Δt:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='26° CNN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The comparison between LoFTR and DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DeepMatcher designs a deep-narrow Transformer layers to capture more human-intuitive and simpler-to-match features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, the position encoding (PE) is integrated to each Transformer layer to convey position information in deep layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, a network-based refinement block is proposed to extract more precise matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' linear dot product of kernel feature maps and makes use of the associativity property of matrix products to reduce the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' BigBird [44] combines local attention and global attention at certain positions and utilizes random attention on several randomly selected token pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' FastFormer [45] uses additive attention mechanism to model global contexts, achieving effective context modeling with linear complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Overall Intuitively, when humans match images, they scan the images back and forth, and the more times they scan them, the simpler it is for them to recall the easier-to-match fea- tures, which suggests that a deep local feature matcher can exhibit higher matching abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Thus, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2, we consider depth of the network is more prominent than width and present a deep Transformer-based network, namely DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 3, given the image pair IA and IB, our network produces reliable and accurate matches across images in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The matching process starts with a CNN-based encoder to extract the fine-level features ¯FA, ¯FB and coarse-level features ˆFA, ˆFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Before feeding these features to Slimming Transformer (SlimFormer), we utilize the Feature Transition Module (FTM) to guarantee a smooth tran- sition to SlimFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, we utilize SlimFormer to achieve long-range global context aggregation intra-/inter-images in an efficient and effective way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' A relative position encoding is applied on each SlimFormer to explicitly model relative distance information, hence enhancing the DeepMatcher’s ability to convey information, particularly in deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' A layer-scale strategy is also leveraged in each SlimFormer to enables the network to assimilate message exchange from the feed-forward modules adaptively, thus allowing it to simulate the human behavior that human can receive different matching information each time they scan an image pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' After interleav- ing SlimFormer by L times, the enhanced features LF seq A and LF seq B are utilized to establish coarse matches Hc, which are further optimized to fine matches Hf using Correspondence Refine Module (CRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In the following part, we introduce the details and underlying insights of each individual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 4 (a) Local Feature Extractor (b) Feature-transition Module 1×1Conv 1×1DWConv 1×1Conv 3×3DWConv 1×1Conv 5×5DWConv 1×1Conv 7×7DWConv Fine Matches (d) Coarse Matches Module Score Matrix Softmax Softmax Soft Assignment Matrix Coarse Matches 1×1Conv Img2 Seq Img2 Seq SlimFormer×L2 Seq2 Img Seq2 Img Concat 1×1Conv Max Pooling 1×1Conv 1×1Conv 1×1Conv 1×1Conv Sigmoid Confidence Offset (e) Fine Matches Module Img 2Seq Img 2Seq … c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Locality Feed-forward Network (c) Slimming Transformer Concat MLP GELU MLP ×L Vector-based Attention (c) Slimming Transformer … … Feed-forward Network Layer Norm … MLP Softmax MLP MLP Softmax MLP MLP MLP Value vector Position Encoding Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The network architecture of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DeepMatcher takes an image pair(IA, IB) as input and generates transitional features from Local Feature Extractor and Feature Transition Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, DeepMatcher interleaves Slimming Transformer by L times to perform long-range context aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Coarse Matches Module is utilized to establish coarse matches, which are optimize to fine matches by Fine Matches Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Local Feature Extractor As the first part of DeepMatcher, we use a standard con- volutional neural network (CNN) with FPN [46] to extract the coarse-level features ˆ FA, ˆ FB ∈ R ˆ C×H/8×W/8, and the fine-level features ¯FA, ¯FB ∈ R ¯ C×H/2×W/2 for the image pair IA and IB, where H and W are the height and width of the original images, ˆC, ¯C denote feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' For convenience, we denote N = H/8 × W/8 as the number of pixel tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Since each pixel in ˆFA, ˆFB represents an 8 × 8 grid in the original images IA, IB, we view the central position of all grids as the pixel coordinates PA, PB ∈ RN×2 of keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Feature Transition Module (FTM) In the subsequent steps, we construct graph neural network (GNN) and propose SlimFormer that leverages self-/cross- attention in Transformer to aggregate global context infor- mation intra-/inter-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Nevertheless, there is apparently a gap between the feature extractor and SlimFormer in terms of context ranges, which is deleterious to subsequent steps in- volving deep feature interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, representing features at multiple scales is so critical for discriminating objects or regions of varying sizes that it can ensure prominent features at various scales can be preserved for deep features aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Thus, we propose a Feature Transition Module (FTM) inserted between the local feature extractor and SlimFormer to adjust the receptive fields of the extracted features, ensuring effective deep feature interaction in SlimFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, instead of directly using (1 × 1, 3 × 3, 5 × 5, 7 × 7) convolution, FTM adopts (1 × 1, 3 × 3, 5 × 5, 7 × 7) depth-wise convolution [47] followed by 1 × 1 point-wise convolution to reduce model parameters and computation, obtaining multi-scale feature rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, we concatenate the features along channel dimension, hence enlarging the receptive fields of ˆ FA, ˆ FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Finally, we derive the updated F ftm A , F ftm B ∈ R ˆ C×H/8×W/8, which can be formulated as: FTM(F) = [C1/4 1 (DW1(F))||C1/4 1 (DW3(F))|| C1/4 1 (DW5(F))||C1/4 1 (DW7(F))], F ftm A = FTM( ˆFA), F ftm B = FTM( ˆFB), (1) where C1/4 1 means using 1×1 convolution to squeeze the chan- nel dimension to ˆC/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DW1, DW3, DW5, DW7 mean depth- wise convolution with kernel size of 1, 3, 5, 7, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [·||·] means concatenation along the channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Slimming Transformer (SlimFormer) We flatten the updated enhanced features F ftm A , F ftm B to be the input sequence for deep feature aggregation, obtaining F seq A , F seq B ∈ RN× ˆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following [3], we view keypoints with features F seq A , F seq B in image pairs as nodes to construct GNN, in which the global context aggregation intra-/inter- image is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Intuitively, more observations between images can result in more precise matches, indicating that deep feature interaction is essential for local features matching task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Nevertheless, the ablation study of LoFTR demonstrates that the matching performance has not been significantly improved with more Transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We attribute this phenomenon to the following reasons: (i) LoFTR only uti- lizes absolute position encoding before Transformer layers, where the position information would disappear when the Transformer layers grow deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, humans primarily associate objects by referring to their relative positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (ii) The linear Transformer utilized in LoFTR uses a context- agnostic manner to approximate self-attention, which cannot UJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 5 fully model relevance among all keypoints, especially in deep layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To handle this dilemma, we propose SlimFormer that leverages relative position information and global context information to boost the capability of DeepMatcher to convey abundant information, hence extracting discriminative feature representations LF seq A ,L F seq B ∈ R ˆ C×H/8×W/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Vector-based Attention (VAtt) Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Instead of using a context-agnostic manner to approximate self-attention, we convert query vector to global query contexts and leverage element-wise product to model relevance among all keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Technically, during each feature enhancement process, we uti- lize self-/cross-attention to aggregate long-range context infor- mation intra-/inter-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' For self-attention, the input features U and R are same (either (F seq A , F seq A ) or (F seq B , F seq B )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' For cross-attention, the input features U and R are different (either (F seq A , F seq B ) or (F seq B , F seq A )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Firstly, SlimFormer transforms the input features U and R into the query, key, and value vectors Q, K, V ∈ RN× ˆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Q = UWQ, K = RWK, V = RWV , (2) where WQ, WK, WV ∈ R ˆ C× ˆ C denote learnable weights for feature transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, we perform relative position encoding on query vector Q and key K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' ˜Q = DPE(Q), ˜K = DPE(K), (3) where DPE(·) means relative position encoding operation, described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Next, modeling the context information of the input fea- tures based on the interactions among ˜Q, ˜K, and V is a critical problem for Transformer-like architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In the vanilla Transformer, dot-product attention mechanism leads to quadratic complexity, making it unrealistic to establish deep Transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' A potential method to reduce the com- putational complexity is to summarize the attention matrices before modeling their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Inspired by [45], we intro- duce vector-based attention that effectively models long-range interactions among pixel tokens to alleviate this bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Instead of computing a quadratic attention map QKT that encodes all possible interactions between candidate matches, we form a compact representation of query-key interactions via vector-based attention that computes the correlation between global query vector and each key vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, we firstly leverage MLP to calculate the weight ˜Qimp ∈ R1×N of each query vector: ˜Qimp = Softmax(MLP( ˜Q)), (4) where Softmax(·) means softmax operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The global query vector ˘Q ∈ R1× ˆ C is set to be a linear combination of ˜Q: ˘Q = ˜Qimp ⊗ ˜Q (5) where ⊗ means matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, we utilize the element-wise multiplication between the global query vector ˘Q and each key vector to model their interaction, obtaining context-aware key vector ˜KQ ∈ RN× ˆ C: ˜KQ = ˘Q ⊙ ˜K, (6) where ⊙ denotes element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We utilize a similar vector-based attention to extract global context-aware key vector ˘KQ and model the interaction be- tween ˘KQ and V : ˜KQimp = Softmax(MLP( ˜KQ)) ˘KQ = ˜KQimp ⊗ ˜KQ Λ = ˘KQ ⊙ V (7) Subsequently, we employ a MLP and short-cut structure to derive the global message M ∈ RN× ˆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' M = MLP(Λ) + ˜Q (8) For convenience, we define the process of vector-based attention layer as: M = VAtt(U, R) (9) Feed-forward Network (FFN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Inspired by conventional Transformers, we employ a feed-forward network applied to M to extract discriminative features for effectively deep features aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The feed-forward network consists of two fully-connected layers and a GELU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The hidden dimension between the two fully-connected layers is extended by a scale rate γ to learn abundant feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' This process can be formulated as: FFN(U, M) = MLP1/γ(GELU(MLPγ/2([U||M]))), (10) where MLP1/γ, MLPγ/2 mean expand the channel dimen- sion by 1/γ, γ/2 times with a MLP, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [·||·] means concatenation along channel dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' GELU(·) means GELU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ultimately, we obtain enhanced message ˘ M ∈ RN× ˆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Layer Scale Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Intuitively, people obtain different message after observing images each time, which inspires us to propose a layer-scale strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, in accordance with ResNet [48], we utilize a shortcut structure to realize efficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, we design a learnable scaling factor ξ to adaptively balance original features U and enhanced message ˘ M, which is formulated as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' ˘U = U + ξ ˘ M (11) By incorporating ξ into SlimFormer, SlimFormer can easily simulate the human behaviour that humans acquire different matching cues each time they scan an image pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Relative Position Encoding (RPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The local feature ex- tractor learns strict translation invariant features, which could cause ambiguity in scenes that have repetitive geometry texture or symmetric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Previous works [2], [3], [16], [26] attach a distinctive absolute positional embedding to each keypoint, thus alleviating such ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' However, compared with absolute position, relative position is more conducive for humans to establish connections between objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Therefore, we argue that incorporating the explicit relative position de- pendency during each deep feature aggregation is essential for distinguishing identical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' However, relative position is not applicable to transformers with linear complexity as they do not explicitly calculate the quadratic complexity attention JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 6 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To this end, we employ rotary positional embedding (RoPE) [49] that leverages absolute position encoding to achieve relative position encoding without manipulating the attention matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Given a pixel token Ti and its features Fi ∈ R ˆ C, the rotary position encoding function is defined by: Pos(Ti, Fi) = Θ(Ti)Fi = � � � � � � M1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 0 0 M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 0 M ˆ C/2 � � � � � � Fi, (12) where Θ(Ti) ∈ R ˆ C× ˆ C is a block diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Each block with size of 2 × 2 is defined by: Mk = �cos iθk −sin iθk sin iθk cos iθk � , θk = 1 100002(k−1)/ ˆ C (13) where θk encodes the index of the feature channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Compared to sinusoidal encoding [2], [3], [26], rotary positional embedding has two advantages: (i) Θ(·) is an orthogonal function, the encoding only changes the feature’s direction but not the feature’s length, which could stabilize the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (ii) The dot product of two encoded features < Pos(Ti, Fi), Pos(Tj, Fj) > in self-attention of vanilla Transformer can be derived to: [Θ(Ti)Fi]T Θ(Tj)Fj = (Fi)T Θ(Tj − Ti)Fj (14) which means the relative 2D distance information can be explicitly revealed by the dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Since RoPE injects position information by rotation, which maintains the norm of hidden representations unchanged, such positional encoding can be directly applied to linear com- plexity transformers as demonstrated in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In SlimFormer, we implement this by employing rotary positional embedding into Q, K to incorporate relative position information, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 3 or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' For more details about RoPE, we encourage readers to refer to original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Self-/Cross-SlimFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In summary, the SlimFormer is formatted as: Slim(U, R) = U + ξFFN(U, VAtt(U, R)) (15) We perform L times of SlimFormer for feature enhance- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' During the l-th feature enhancement, we use self-/cross- attention mechanism to integrate intra-/inter-image informa- tion, which can be formulated as: l−1F seq A = Slim(l−1F seq A , l−1F seq A ), l−1F seq B = Slim(l−1F seq B , l−1F seq B ), lF seq A = Slim(l−1F seq A , l−1F seq B ), lF seq B = Slim(l−1F seq B , lF seq A ) (16) Ultimately, we incorporate relative position information and global context message into enhanced features LF seq A ,L F seq B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' : in Coarse Matches c B P : in Fine Matches f B P : Ground Truth Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Visualization of refinement result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The keypoints (green) in the fine matches approximate the ground-truth (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Coarse Matches Module (CMM) Given LF seq A and LF seq B , we utilize inner product of LF seq A , LF seq B to calculate the score matrix S ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' S(i, j) = ⟨LF seq A , LF seq B ⟩, (17) where ⟨·, ·⟩ means the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Subsequently, we apply softmax operator on both dimensions (denoted as dual-softmax operation) to convert the S to soft assignment matrix G ∈ RN×N: G = Softmax(S)col · Softmax(S)row, (18) where Softmax(·)col, Softmax(·)row mean performing softmax on each column and row of S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, for the i-th keypoint in IA and the j-th keypoint in IB, we regard them as a pair of predicted coarse matches if they satisfy the following two conditions: (i) The soft assignment score is higher than a predefined threshold λ: G(i, j) > λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (ii) They satisfy the mutual nearest neighbor (MNN) criteria, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=', G(i, j) is the maximum value in the corresponding row and column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ultimately, we derive the index D of anchor points in coarse matches: D = {(i, j)|(i, j) ∈ MNN(M), M(i, j) > λ} (19) Given the index D and the keypoints coordinates PA, PB, the coarse matches Hc = {(P c A, P c B)} are formulated as: Hc = {(PA(i), PB(j)) | ∀(i, j) ∈ D} (20) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fine Matches Module (FMM) After establishing coarse matches, a coarse-to-fine module is applied to refine these matches to the original picture resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' However, the coarse-to-fine module in LoFTR only predicts the offset of the coarse matches without appraising whether the predicted matches are reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To tackle this issue, we view the match refinement as a combination of classifica- tion and regression problems and design Fine Matches Module to predict confidence and offset concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 3, for each coarse match, we locate its position at fine-level feature maps and crop two sets of local image patches with the size of w ×w, obtaining local features ¯F w A , ¯F w B ∈ RK× ¯ C×w×w, where K is the number of coarse matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, we flatten ¯F w A , ¯F w B to be sequences, imple- ment SlimFormer to perform L2 times of global information passing, and rearrange the sequences into 2D feature maps, JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 7 obtaining L2 ¯F w A ,L2 ¯F w B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The feature maps are concatenated along channel dimension and fed into a network, which is comprised of two convolution layers, a max pooling layer, and four convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The network predicts the offset ∆ ∈ RK×2 of the P c B and the confidence c ∈ RK×1 of the predicted coarse matches: ¯Fmid = C1(C1(Pmax(C1(C1([L2 ¯F w A ||L2 ¯F w B ]))))), c = Sig(C1( ¯Fmid)), ∆ = C1( ¯Fmid), (21) where Pmax means global max pooling operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' C1(·) means 1 × 1 convolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [·||·] denotes concatenation along the channel dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sig(·) means sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ultimately, we obtain the fine matches Hf = {(P f A, P f B)}: Hf = {(P c A(i), P c B(i) + ∆(i)) | i ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=', K}} (22) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Loss DeepMatcher generates final dense matches according to soft assignment matrix G and offset ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Therefore, the total loss Lall of DeepMatcher comprises of matching loss Lm, regression loss Lr, and classification loss Lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lall = Lm + βLr + φLc, (23) where β and φ are weighting coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Matching Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following [2], we calculate the index Egt of the ground truth matches, which are utilized in conjunction with soft assignment matrix G to calculate matching loss Lm defined as focal loss [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lm = −[ 1 |Egt| � (i,j)∈Egt α(1 − G(i, j))ηlog G(i, j)+ 1 N − |Egt| � (i,j)/∈Egt (1 − α)G(i, j)ηlog (1 − G(i, j))], (24) where α is a weighting factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' η is a focusing parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' |Egt| means the number of ground truth matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Regression Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' For predicted matches {(P f A, P f B)}, we project P f A in the first image to second image, deriving P gt B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, the ground truth offset ∆gt is formulated as: ∆gt = P gt B − P f B (25) According to predicted offset ∆ and ground truth offset ∆gt, we define the regression loss Lr as: Lr = 1 K K � i=1 ∥∆gt(i) − ∆(i)∥2 2, (26) where K is the number of predicted matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Notably, we ignore the predicted matches with ∆gt larger than predefined threshold ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Classification Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' For the predicted matches with ground truth offset less than ψ, we regard them as positive and define the classification label as 1, while other matches are viewed as negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ultimately, we obtain the ground truth confidence cgt, while are utilized to calculate classification loss Lc together with predicted confidence c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lc = − 1 K K � i=1 � cgt(i)log c(i) + (1 − cgt(i))log (1 − c(i)) � (27) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Implementation Details Architecture details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We adopt a slightly modified ResNet- 18 with FPN for local feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We use a width of 96 for the stem layer, followed by widths of [96, 128, 192] for the next three stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We construct the FPN with levels P1 through P3 and take P3 features as the coarse-level features, P1 features as the fine-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Thus, the dimensions of fine-level and coarse-level feature maps are C = 96, ˆC = 192, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The scale rate γ in feed-forward network is set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following SuperGlue, we set the confidence threshold λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2 to obtain coarse matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, we choose w = 5 to crop local windows in fine-level feature maps for matches refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To reconcile the coarse matching loss, regression loss, and classification loss, we set both weighting coefficients β and φ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' For matching loss, we set the weighting factor α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='25 and the focusing parameter η = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' When making classification labels, we set ψ to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In this work, we elaborately design two versions of DeepMatcher that interleave SlimFormer by L = 6, 10 times for feature enhancement, resulting in DeepMatcher and DeepMatcher-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Training scheme for Scannet [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We train DeepMatcher on Scannet [51] dataset with 32 Tesla V100 GPUs for indoor local feature matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In accordance with LoFTR, we sample 200 image pairs per scene at each epoch and balance scene variants over iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We employ the AdamW solver for optimization with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The initial learning rate is set to 6 ×10−4 and will decrease by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 every 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We use gradient clipping that is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 to avoid exploding gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Training scheme for MegaDepth [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We train Deep- Matcher on MegaDepth [52] datasets with 32 Tesla V100 GPUs for outdoor local feature matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following LoFTR, we randomly sample 100 pairs from each sub-scene during each epoch of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We train DeepMatcher for 30 epochs in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We also employ the AdamW solver for optimization with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The initial learning rate is set to 8 × 10−4, with a linear learning rate warm-up in 3 epochs from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1 to the initial learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We decay the learning rate by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 every 4 epochs starting from the 4-th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Indoor Pose Estimation Typically, indoor pose estimation task is hampered by motion blur and significant viewpoint shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' There are com- monly extensive regions of low textures in indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To evaluate the performance of DeepMatcher in such situations, we conducted indoor pose estimation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We use ScanNet [51] dataset to validate the effectiveness of DeepMatcher on indoor pose estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' ScanNet consists of 1513 RGB-D sequences with RGB images JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 8 TABLE I INDOOR POSE ESTIMATION EVALUATION ON SCANNET DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' THE AUC@(5◦, 10◦, 20◦) IS REPORTED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Local features Matcher Pose estimation AUC @5◦ @10◦ @20◦ Detector-based Methods D2-Net [12] NN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='25 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='53 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='96 ContextDesc [53] ratio test [9] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='64 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='01 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='75 SuperPoint [11] NN 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='43 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='53 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='40 NN + OANet [54] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='76 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='90 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='85 SuperGlue [3] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='16 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='84 SGMNet [16] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='40 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='06 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='32 DenseGAP [17] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='01 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='07 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='66 HTMatch [55] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='11 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='42 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='23 Detector-free Methods —— LoFTR [2] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='06 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='80 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='62 QuadTree [43] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='90 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='70 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='80 MatchFormer [26] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='31 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='90 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='41 ASpanFormer [28] 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='60 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='00 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='30 DeepMatcher 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='35 DeepMatcher-L 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='32 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='25 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='49 and corresponding ground-truth poses in indoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following [3], we select 230M image pairs with overlap values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8 as the training set and 1500 image pairs as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' All images are resized to 640 × 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Evaluation Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In accordance with [2], [3], we report the area under the cumulative curve (AUC) of pose errors at the thresholds (5◦, 10◦, 20◦), where pose errors are defined as the maximum of translational and rotational errors between ground-truth poses and predicted poses by DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, given the predicted dense matches, we utilize OPENCV to calculate the essential matrix E and relative pose ˜T of image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, the pose errors ∆T are defined as the maximum of translational and rotational errors between ground-truth relative pose T = [R|t] and estimated relative pose ˜T = [ ˜R|˜t]: ∆T =max(∆t, ∆R), ∆t = arccos( ˜t · t ||˜t||2 · ||t||2 ), ∆R = arccos(tr( ˜RT R) − 1 2 ), (28) where ∆t and ∆R denote the translational error and rotational error, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' R, t is the ground-truth rotation matrix and translation vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' ˜R, ˜t mean the predicted rotation matrix and translation vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' tr(·) means the trace of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Given the pose errors of all image pairs, we plot the cumu- lative error distribution curve, whose area at three thresholds (5◦, 10◦, 20◦) are computed as AUC@(5◦, 10◦, 20◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As illustrated in Table I, we observe that the detector-free methods achieve superior performance than detector-based methods since the detector strug- gles to extract repeatable keypoints when handling image pairs with significant viewpoint change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wherein, Deep- Matcher and DeepMatcher-L outperform all cutting-edge detector-based and detector-free methods by a great mar- gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' More specifically, DeepMatcher-L surpasses detector- based method DenseGAP by (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='31%, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='18%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='83%) in TABLE II OUTDOOR POSE ESTIMATION EVALUATION ON MEGADEPTH DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' THE AUC@(5◦, 10◦, 20◦) IS REPORTED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Local features Matcher Pose estimation AUC @5◦ @10◦ @20◦ Detector-based Methods SuperPoint [11] SuperGlue [3] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='18 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='16 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='96 DenseGAP [17] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='17 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='87 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='22 ClusterGNN [18] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='19 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='54 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='33 Detector-free Methods —— DRC-Net [23] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='01 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='96 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='31 LoFTR [2] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='80 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='19 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='18 QuadTree [43] 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='60 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='50 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='20 TopicFM [27] 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='10 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='10 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='60 MatchFormer [26] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='91 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='00 ASpanFormer [28] 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='30 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='50 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='10 DeepMatcher 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='71 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='25 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='49 DeepMatcher-L 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='11 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='15 terms of AUC@(5◦, 10◦, 20◦), demonstrating the superior- ity of detector-free structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Compared with the pioneer- ing method LoFTR, DeepMatcher-L realizes superior per- formance with the improvement of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='26%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='45%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='87%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Furthermore, DeepMatcher-L outperforms the state-of-the- art detector-free method ASpanFormer by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='72%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='25%) in terms of AUC@(5◦, 10◦), proving the deep Transformer architecture is essential to extract more human-intuitive and easier-to-match features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Additionally, DeepMatcher-L only consumes 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='65% GFLOPs and achieves 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='95% inference speed boost compared with ASpanFormer, as demonstrated in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Outdoor Pose Estimation Outdoor pose estimation remains a challenging task owing to the intricate 3D geometry, extreme illumination and view- point changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To demonstrate the efficacy of DeepMatcher in overcoming these obstacles, an outdoor pose estimation experiment is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We utilize MegaDepth [52] to conduct the outdoor pose estimation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' MegaDepth contains 1M internet images from 196 different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' These images come from photo-tourism and contain challenging conditions, including large viewpoint and illumination variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following [2], [14], we select 100 image pairs each scene for training and 1500 image pairs for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Images are resized such that their longer dimensions are equal to 840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Evaluation Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We use the same evaluation metrics AUC@(5◦, 10◦, 20◦) as the indoor pose estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Table II, we can observe that DeepMatcher families surpass other methods in all evalua- tion metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, DeepMatcher-L noticeably outper- forms the cutting-edge detector-based method ClusterGNN by (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='79%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='57%, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='82%) in AUC@(5◦, 10◦, 20◦) since the detector struggles to extract repeatable keypoints in im- age pairs with extreme viewpoint change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, com- pared with the baseline approach LoFTR, DeepMatcher- L achieves superior performance with the improvement of JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 9 1 2 3 4 5 6 7 8 9 10 threshold(px) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0 MMA Overall DeepMatcher-L DeepMatcher MatchFormer LoFTR SuperGlue D2-Net R2D2 SparseNCNet 1 2 3 4 5 6 7 8 9 10 threshold(px) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0 Illumination 1 2 3 4 5 6 7 8 9 10 threshold(px) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0 Viewpoint Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Image matching evaluation on HPatches dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='18%, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='92%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='97%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, DeepMatcher-L also sur- passes the state-of-the-art detector-free method ASpanFormer by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='68%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='61%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='05%), further validating the rationality of the deep Transformer structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Image Matching As a fundamental visual task, image matching plays an important role in several applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Therefore, we conduct an image matching experiment to validate the performance of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We conduct homography estimation experiments on the HPatches dataset [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following [12], we select 108 sequences from HPatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Each sequence consists of a ground- truth homography matrix and 6 images of progressively larger illumination (52 sequences with illumination changes) or view- point changes (56 sequences with viewpoint changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Evaluation Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We adopt the generally employed mean matching accuracy (MMA) as metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=', the average proportion of correct correspondences per image pair [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, the keypoints from the i-th query image are projected into the reference image by using the provided homography matrix Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Then, the matches with reprojection errors that are lower than a predefined threshold t are deemed correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Finally, we compute the average percentage of correct matches across all image pairs and define MMA as: MMA(t) = 1 HP HP � i=1 ( �N f j=1 1(t − ||Hi(P f A,i,j) − P f B,i,j||2) N f ), (29) where HP means the number of image pairs in HPatches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' N f means the number of predicted matches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1(·) is a binary indi- cator function whose output is 1 for non-negative value and 0 otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' t is the threshold of reprojection error, varying from 1 to 10 pixels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Hi(·) means warping the keypoints in the i-th query image to reference image by ground-truth homography matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (P f A,i,j, P f B,i,j) means the pixel coordinates of the j-th match in the i-th image pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5, we can observe that Deep- Matcher families achieve superior performance than detector- free methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' MatchFormer, LoFTR, SparseNCNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Un- der varying illumination conditions, DeepMatcher yields in- ferior performance at low thresholds, while achieving out- standing performance when the threshold is larger than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, when handling image pairs with viewpoint changes, DeepMatcher exhibits extremely superior robotness compared LoFTR DeepMatcher DeepMatcher-L P:93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:423 DeepMatcher-L P:97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches:215 DeepMatcher-L P:93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:188 DeepMatcher LoFTR P:84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1% Matches:679 LoFTR P:89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches:167 LoFTR P:73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3% Matches:15 LoFTR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Visualization of the predicted matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The mismatches, whose reprojection errors are larger than 5px, are colored red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' TABLE III HOMOGRAPHY ESTIMATION EVALUATION ON HPATCHES DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Local features Matcher Overall Illumination Viewpoint CCM (ε<1/3/5 pixels) Detector-based Methods D2-Net [12] NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='71/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='66/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='95/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='12/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='49/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='67 R2D2 [13] NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='47/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='77/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='63/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='93/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='32/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='64/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='70 ASLFeat [57] NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='48/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='62/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='94/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='34/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='69/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='78 SuperPoint [11] NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='46/0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74 SuperGlue [3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='51/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='82/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='60/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='92/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='42/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='71/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81 SGMNet [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='52/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='85/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='59/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='94/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='46/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='84 ClusterGNN [18] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='52/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='84/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='61/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='93/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='44/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81 Detector-free Methods —— SparseNCNet [22] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='36/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='65/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='62/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='92/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='13/0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='30/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='64/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='76 LoFTR [2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='95/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='69/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='76 MatchFormer [26] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='75/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='95/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='37/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='68/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='78 DeepMatcher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='50/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='62/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='93/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='70/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='81 DeepMatcher-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='51/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='83/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='64/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='94/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='39/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='72/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='84 with other detector-free methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 6, we select three pairs of image pairs from HPatches dataset and visualize the matches predicted by LoFTR and DeepMatcher- L to further validate the robustness of DeepMatcher-L to viewpoint variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Homography Estimation Since the distribution and number of matches are essential to estimate reliable geometry relationship between image pairs, we conduct a homography estimation experiment to comprehensively evaluate the performance of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We assess DeepMatcher on HPatches dataset, which is widely used for homography estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Evaluation Protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Following the corner correctness met- ric (CCM) utilized in [24], we report the percentage of image pairs with average corner errors ε smaller than 1/3/5 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, based on the predicted dense matches, we use OPENCV to calculate the homography matrix ˜Hi for the i-th image pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Subsequently, four corners in the query image are projected into the reference image by using the ground-truth homography matrix Hi and the predicted homography matrix L0FTR, P: 84:1% Matches: 679DeepMatcherP:9 Matches: ate ESEARN fonicL0FTR, P: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches: 167DeepMatcher, P:97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches: 215DeepMatcher, P:93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:188L0FTR,P:73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3% Matches:15ate ESEARK ronicJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 10 LoFTR DeepMatcher SuperPoint + SuperGlue Indoor Outdoor SGMNet LoFTR DeepMatcher SuperPoint + SuperGlue Indoor Outdoor SGMNet LoFTR DeepMatcher-L SuperPoint + SuperGlue Indoor Outdoor SGMNet DeepMatcher-L P:100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0% Matches:1268 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02°, Δt:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='83° SuperGlue P:82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:117 ΔR:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='87°, Δt:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='13° LoFTR P:90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='4% Matches:1025 ΔR:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='16°, Δt:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='99° SGMNet P:85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:358 ΔR:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='21°, Δt:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='47° DeepMatcher-L P:97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches:576 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='51°, Δt:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='05° LoFTR P:91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1% Matches:492 ΔR:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='44°, Δt:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='42° SGMNet P:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches:127 ΔR:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='23°, Δt:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='46° SuperGlue P:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:56 ΔR:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38°, Δt:77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='89° DeepMatcher-L P:96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:293 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='34°, Δt:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74° LoFTR P:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:167 ΔR:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='28°, Δt:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='01° SuperGlue P:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches:168 ΔR:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='18°, Δt:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='15° SGMNet P:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0% Matches:92 ΔR:inf °, Δt:inf ° DeepMatcher-L P:86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches:203 ΔR:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98°, Δt:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='55° LoFTR P:79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches:125 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='98°, Δt:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='24° SGMNet P:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5% Matches:37 ΔR:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='92°, Δt:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='69° SuperGlue P:74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0% Matches:154 ΔR:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='41°, Δt:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='49° Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Visualization of the predicted matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The matches are colored by their reprojection errors (green indicates correct matches, and red indicates mismatches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We set the error threshold to 10 and 15 pixels for indoor and outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' L=1 L=2 L=3 L=4 L=5 L=10 L=9 L=8 L=7 L=6 L=1 L=2 L=3 L=4 L=5 L=10 L=9 L=8 L=7 L=6 DeepMatcher P:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches:358 ΔR:8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='19°, Δt:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='93° DeepMatcher P:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0% Matches:3532 ΔR:inf °, Δt:inf ° DeepMatcher P:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:747 ΔR:inf °, Δt:inf ° DeepMatcher P:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches:580 ΔR:inf °, Δt:inf ° DeepMatcher P:8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='7% Matches:447 ΔR:inf °, Δt:inf ° DeepMatcher P:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:356 ΔR:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='10°, Δt:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='03° DeepMatcher P:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:401 ΔR:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='62°, Δt:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='60° DeepMatcher P:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1% Matches:403 ΔR:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='44°, Δt:9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='13° DeepMatcher P:73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:648 ΔR:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='15°, Δt:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74° DeepMatcher P:74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1% Matches:690 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='08°, Δt:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='07° L=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10 DeepMatcher-L P:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:636 ΔR:8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='21°, Δt:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='70° DeepMatcher-L P:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:3124 ΔR:inf °, Δt:inf ° DeepMatcher-L P:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches:810 ΔR:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='05°, Δt:78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='64° DeepMatcher-L P:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:718 ΔR:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='32°, Δt:76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='97° DeepMatcher-L P:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches:640 ΔR:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='65 °, Δt:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='37 ° DeepMatcher-L P:44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:657 ΔR:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='92°, Δt:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='50° DeepMatcher-L P:62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:755 ΔR:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='01°, Δt:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='80° DeepMatcher-L P:68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:892 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='45°, Δt:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='01° DeepMatcher-L P:75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:1140 ΔR:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='06°, Δt:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='03° DeepMatcher-L P:94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1% Matches:909 ΔR:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='35°, Δt:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='23° L=1 L=2 L=3 L=4 L=5 L=6 L=7 L=8 L=9 L=10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The predicted matches of DeepMatcher-L after each SlimFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The threshold λ of the soft assignment matrix is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' ˜Hi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ultimately, we calculate average reprojection error as corner error εi, thereby obtaining the CCM: εi = � p∈Pco ||Hi(p) − ˜Hi(p)||2 4 , CCM(t) = �HP i=1 1(t − εi) HP , (30) where Pco = {(0, 0), (Wo − 1, 0), (0, Ho − 1), (Wo − 1, Ho − 1)} means the four corners coordinates of the query image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Hi(·) and ˜Hi(·) mean warping the corners in the i-th query image to reference image by ground-truth homography matrix and predicted homography matrix, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' t ∈ {1, 3, 5} means the predefined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Table III, DeepMatcher achieves the best performance among the detector-free methods under extreme viewpoint changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, DeepMatcher out- performs LoFTR and MatchFormer with the improvement of (1%, 5%) and (2%, 3%) when thresholds are set to 3, 5 pixels, repsectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Furthermore, DeepMatcher-L surpasses LoFTR and MatchFormer with the improvement of (1%, 3%, 8%) and (2%, 4%, 6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, the detector-based methods are more robust to viewpoint variations, while the detector-free methods realize superior performance when handling image pairs with extreme illumination changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' In comparison, DeepMatcher strikes a decent balance when handling image pairs with various viewpoint and illumination changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Understanding DeepMatcher Qualitative Results Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To further exhibit the capability of DeepMatcher to handle image pairs with extreme appearance settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=', sparse texture, motion blur, large viewpoint and illumination changes, we visualize the matches predicted by SGMNet, SuperGlue, LoFTR, and DeepMatcher- L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 7, we can observe that DeepMatcher-L achieves dense and accurate matching performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Visual Descriptors Enhancement Efficacy Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To validate the effectiveness of performing L times of Slim- Former for feature enhancment, we visualize the matching results of DeepMatcher after each SlimFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, we can observe that the matching precision is promoted consistently, demonstrating interleaving SlimFormer can effectively integrate intra-/inter-image information, hence extracting easier-to-match features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Efficiency Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To validate the efficiency of Deep- Matcher, we compare several cutting-edge detector-free meth- ods in terms of parameters, flops, and inference speed to determine their computational cost and storage consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We resize the input images to 640 × 480 and conduct all experiments on a single NVIDIA TITAN RTX GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' When counting runtime, we run the test code 500 times and re- port the average time to eliminate occasionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Notably, we only compare DeepMatcher families with other detector- free methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Table IV, we can observe that DeepMatcher families realize competitive inference speed with LoFTR, P: 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matchesi167DeepMatcher, P: 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='6% Matches:293SuperGlue, P: 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='3% Matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='161SuperGlue, P: 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:117SGMNet, P: 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:90SuperGlue, P: 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches:320FTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' P: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:98DeepMatcher,P:91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1 Matches:381SGMNet, P: 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='8% Matches:358 △R:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='21°At:1 47DeepMatcherP:100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0% Matches:1268 △R:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02°△t:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='83°L0FTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' P: 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='4% Matches:1025 △R:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='16°△t:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='99°SuperGlue, P: 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='9% Matches: 117 △R:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='87°Zt:6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='13L0FTR, P: 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='2% Matches125SGMNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' P:307% Matches:127 △R:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='23°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' At:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='46DeepMatcher,P:977% Matches:576 △R:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='51°△t:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='05°LoFTR, P: 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='1% Matches:49 AR:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='44°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} 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+page_content='96 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='089 O(NC) DeepMatcher-L 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='51 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='103 O(NC) Original Image Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Visualization of weights in SlimFormer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' SlimFormer emphasises keypoints at object boundaries to incorporate global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' less GFLOPs since SlimFormer leverages element-wise prod- uct to model relevance among all keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Compared with the baseline LoFTR, DeepMatcher and DeepMatcher-L only consume (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='89%, 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='46%) GFLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, compared with cutting-edge detector-free methods QuadTree, Match- Former and AspanFormer, DeepMatcher-L exhibits more effi- cient matching performance with (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='45%, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='44%, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='35%) less GFLOPs and (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='24%, 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='15%, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='95%) inference speed boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Furthermore, we also record the dominant computational complexity of the attention layers in various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Table IV, DeepMatcher achieves the minimum computational complexity of the attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Specifically, compared with the baseline LoFTR, DeepMatcher reduces the dominant computational complexity from O(NC2) to O(NC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Weight Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To explore which keypoints SlimFormer pays attention to when extracting global vectors, we visualize the weight ˜Qimp and ˜KQimp in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (7), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 9, we can observe that SlimFormer primarily pays attention to the prominent keypoints at object boundaries that involve tremendous visual and geometry infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Consequently, SlimFormer exhibits puissant capability to aggregate global context information effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2 4 6 8 10 12 14 Transformer Layers Number 18 20 22 24 26 28 AUC@5° Values DeepMatcher LoFTR 2 4 6 8 10 12 14 Transformer Layers Number 38 40 42 44 46 AUC@10° Values DeepMatcher LoFTR 2 4 6 8 10 12 14 Transformer Layers Number 54 56 58 60 62 AUC@20° Values DeepMatcher LoFTR 2 4 6 8 10 12 14 Transformer Layers Number 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='0 Paeameters (MB) DeepMatcher LoFTR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' The AUC values and parameters of DeepMatcher and LoFTR with the Transformer layers increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Deep Transformer Architecture Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To verify the opinion that deep Transformer architecture is essential to extract more human-intuitive and easier-to-match features, we record the indoor pose estimation precision with the num- ber of Transformer layers increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 10, since SlimFormer leverages vector-based attention for robust long-range global context aggregation and utilizes layer-scale strategy and relative position encoding to enhance the rep- resentation of keypoints, the AUC values of DeepMatcher are promoted consistently with the Transformer layers going deep, while the accuracy of LoFTR is constantly fluctuating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, the matching performance of DeepMatcher is sig- nificantly superior to LoFTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Moreover, the parameters of DeepMatcher and LoFTR increase linearly with the number of Transformer layers, while DeepMatcher occupies fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ablation Study Effect of the Proposed Modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To thoroughly validate the rationality of each module, we conduct indoor pose esti- mation experiments using different variants of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As illustrated in Table V, we can observe that all components contribute to the outstanding performance of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (i), (ii) Using only self- and cross-SlimFormer layers leads to a severe decrease in matching performance, demonstrat- ing interleaving the self- and cross-SlimFormer layers can effectively integrate intra-/inter-image message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (iii) Removing the Feature Transition Module results in a much lower accu- racy (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='66%, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='47%, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='33%), proving the effectiveness of ensuring smooth transition between feature extractor and SlimFormer in terms of context ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (iv) Removing Relative Position Encoding spawns a large drop in pose estimation accuracy (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='36%, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='67%, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38%), proving the relative position information is crucial to distinguish similar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (v) Removing the Fine Matches Module results in lower AUC values (−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='34%, −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='56%, −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='86%), indicating the effectiveness of refining coarse matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Effect of the Learnable Scale Factor ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To validate that layer-scale strategy can simulate the human behaviour that JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 12 TABLE V ABLATION STUDY WITH DIFFERENT VARIANTS OF DEEPMATCHER ON SCANNET DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Methods Pose estimation AUC @5◦ @10◦ @20◦ (i) w only self-SlimFormer layers 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='71 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='89 (ii) w only cross-SlimFormer layers 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='33 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='27 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='64 (iii) w/o Feature Transition Module 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='72 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='91 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02 (iv) w/o Relative Potition Encoding 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='71 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='97 (v) w/o Fine Matches Module 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='04 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='82 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='49 DeepMatcher full 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='35 TABLE VI ABLATION STUDY WITH DIFFERENT LEARNABLE SCALE FACTORS ξ ON SCANNET DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Methods Pose estimation AUC @5◦ @10◦ @20◦ w residual scale factor ξ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='35 w/o residual scale factor ξ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='50 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='91 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='32 humans can acquire different matching cues each time they scan an image pair to further improve matching performance, we remove the learnable scale factor ξ and conduct an ab- lation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Table VI, we can observe that introducing the residual scaling factor leads to superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Effect of the Relative Position Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Humans lever- age the relative position information to establish the connec- tion between objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Therefore, the relative position encoding is more conducive to realize elaborate scene parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To prove this opinion, we implement an ablation experiment using three structures: (i) Removing all position encoding in all SlimFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (ii) Using absolute position encoding pro- posed in LoFTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (iii) Using relative position encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Table VII, we can observe that both absolute and relative position encoding boost AUC values, in which the relative position encoding exhibits more superior performance than absolute position encoding with the improvement of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='86%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='24%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='84%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Effect of the Fine Matches Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Compared with the coarse-to-fine module used in LoFTR, FMM views the match refinement as a combination of classification and regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' To validate the availability of FMM, we conduct an ablation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' As shown in Table VIII, we can observe that using FMM significantly achieves superior performance with the improvement of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='64%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='12%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='49%), proving the rationality of predicting offset and confidence concurrently using a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' CONCLUSION In this work, we propose DeepMatcher, a deep Transformer- based network for local feature matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DeepMatcher sim- ulates human behaviors when humans match image pairs, in- cluding: (1) Deep SlimFormer layers of the network to aggre- gate information intra-/inter-images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' (2) Layer-scale strategy TABLE VII ABLATION STUDY WITH DIFFERENT POSITION ENCODING ON SCANNET DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Methods Pose estimation AUC @5◦ @10◦ @20◦ w/o position encoding 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='71 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='97 w absolute position encoding 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='52 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='14 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='51 w relative position encoding 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='35 TABLE VIII ABLATION STUDY WITH DIFFERENT COARSE-TO-FINE MODULES ON SCANNET DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Methods Pose estimation AUC @5◦ @10◦ @20◦ Coarse-to-fine Module in LoFTR 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='74 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='26 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='86 Fine Matches Module 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='38 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='35 to assimilate message exchange from each layer adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Besides, relative position encoding is applied to each layer so as to explicitly disclose relative distance information, hence improving the representation of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' We also propose Fine Matches Module to refine the coarse matches, thus generating robust and accurate matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Extensive exper- iments demonstrate that DeepMatcher surpasses state-of-the- art approaches on several benchmarks, confirming the superior matching capability of DeepMatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dai, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yang, “Csr-net: Learning adap- tive context structure representation for robust feature correspondence,” IEEE Transactions on Image Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 31, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 3197–3210, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Bao, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhou, “Loftr: Detector- free local feature matching with transformers,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8922–8931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sarlin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DeTone, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Malisiewicz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rabinovich, “Superglue: Learning feature matching with graph neural networks,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 4938–4947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zheng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Xiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ma, “Msa-net: Establishing reliable correspondences by multiscale attention network,” IEEE Trans- actions on Image Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 31, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 4598–4608, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Mur-Artal and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tard´os, “Orb-slam2: An open-source slam system for monocular, stereo, and rgb-d cameras,” IEEE transactions on robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1255–1262, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Campos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Elvira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rodr´ıguez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Montiel, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tard´os, “Orb-slam3: An accurate open-source library for visual, visual– inertial, and multimap slam,” IEEE Transactions on Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1874–1890, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Schonberger and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Frahm, “Structure-from-motion revisited,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 4104–4113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Cui, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Liu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shen, “Vidsfm: Robust and accurate structure-from-motion for monocular videos,” IEEE Trans- actions on Image Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 31, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2449–2462, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lowe, “Distinctive image features from scale-invariant keypoints,” International journal of computer vision, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 91–110, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rublee, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rabaud, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Konolige, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Bradski, “Orb: An efficient alternative to sift or surf,” in 2011 International conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ieee, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2564–2571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DeTone, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Malisiewicz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rabinovich, “Superpoint: Self- supervised interest point detection and description,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, AUGUST 2021 13 [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dusmanu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rocco, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pajdla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pollefeys, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sivic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Torii, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sattler, “D2-net: A trainable cnn for joint description and detection of local features,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Revaud, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Weinzaepfel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' De Souza, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pion, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Csurka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Cabon, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Humenberger, “R2d2: repeatable and reliable detector and descriptor,” in NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tyszkiewicz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fua, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Trulls, “Disk: Learning local features with policy gradient,” in NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Xia and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ma, “Locality-guided global-preserving optimization for robust feature matching,” IEEE Transactions on Image Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 31, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5093–5108, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Bai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Hu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tai, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Quan, “Learning to match features with seeded graph matching network,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 6301–6310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [17] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Kuang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' He, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhao, “Densegap: Graph- structured dense correspondence learning with anchor points,” arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='06910, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shavit, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Mu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Feng, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, “Clus- tergnn: Cluster-based coarse-to-fine graph neural network for efficient feature matching,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 12 517–12 526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' DeTone, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Malisiewicz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rabinovich, “Superpoint: Self- supervised interest point detection and description,” in Proceedings of the IEEE conference on computer vision and pattern recognition workshops, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 224–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Bian, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Matsushita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yeung, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Nguyen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Cheng, “Gms: Grid-based motion statistics for fast, ultra-robust feature correspondence,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 4181–4190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [21] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rocco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Cimpoi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Arandjelovi´c, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Torii, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pajdla, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sivic, “Neighbourhood consensus networks,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [22] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rocco, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Arandjelovi´c, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sivic, “Efficient neighbourhood consensus networks via submanifold sparse convolutions,” in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 605–621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [23] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Han, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Prisacariu, “Dual-resolution correspondence networks,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 17 346–17 357, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [24] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sattler, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Leal-Taixe, “Patch2pix: Epipolar-guided pixel-level correspondences,” in Proceedings of the IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 4669–4678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [25] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Efe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ince, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Alatan, “Dfm: A performance baseline for deep feature matching,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 4284–4293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [26] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Peng, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Stiefelhagen, “Match- former: Interleaving attention in transformers for feature matching,” arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='09645, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Truong Giang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Song, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Jo, “Topicfm: Robust and interpretable feature matching with topic-assisted,” arXiv e-prints, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' arXiv–2207, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Luo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' McKinnon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tsin, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Quan, “Aspanformer: Detector-free image matching with adaptive span transformer,” in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Springer, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 20–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Vaswani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Gomez, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Polosukhin, “Attention is all you need,” in Advances in neural information processing systems, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Katharopoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Vyas, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pappas, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fleuret, “Transformers are rnns: Fast autoregressive transformers with linear attention,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5156–5165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Karpushin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Valenzise, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dufaux, “Keypoint detection in rgbd images based on an anisotropic scale space,” IEEE Transactions on Multimedia, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1762–1771, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [32] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Balntas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Riba, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ponsa, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Mikolajczyk, “Learning local fea- ture descriptors with triplets and shallow convolutional neural networks.” in Bmvc, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2, 2016, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [33] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Jiang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Guo, “Robust feature matching using spatial clustering with heavy outliers,” IEEE Transactions on Image Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 736–746, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sun, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Suo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wu, “A proposal-free one- stage framework for referring expression comprehension and generation via dense cross-attention,” IEEE Transactions on Multimedia, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [35] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhao, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Gu, “Refinet: A deep segmentation assisted refinement network for salient object detection,” IEEE Transactions on Multimedia, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 457–469, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [36] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Song, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, “6-dof image local- ization from massive geo-tagged reference images,” IEEE Transactions on Multimedia, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1542–1554, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [37] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Feng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Liu, “Seeing through darkness: Visual localization at night via weakly supervised learning of domain invariant features,” IEEE Transactions on Multimedia, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fischler and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Communications of the ACM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 381–395, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [39] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fragoso, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Rodriguez, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Turk, “Evsac: accelerating hypotheses generation by modeling matching scores with extreme value theory,” in Proceedings of the IEEE international conference on com- puter vision, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2472–2479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [40] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Barath, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Matas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Noskova, “Magsac: marginalizing sample consensus,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 10 197–10 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [41] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Mousavi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Varshosaz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Remondino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pirasteh, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, “A two-step descriptor-based keypoint filtering algorithm for robust image matching,” IEEE Transactions on Geoscience and Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 60, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 1–21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [42] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fragoso, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sweeney, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Turk, “Ansac: Adaptive non- minimal sample and consensus,” arXiv preprint arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='09559, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [43] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Tan, “Quadtree attention for vision transformers,” arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='02767, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zaheer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Guruganesh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dubey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ainslie, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Alberti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ontanon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ravula, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=', “Big bird: Transformers for longer sequences,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 17 283–17 297, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [45] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Huang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Xie, “Fastformer: Additive attention can be all you need,” arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='09084, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [46] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Doll´ar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Girshick, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' He, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Hariharan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Belongie, “Feature pyramid networks for object detection,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2117–2125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Howard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Kalenichenko, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Weyand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Andreetto, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Adam, “Mobilenets: Efficient convo- lutional neural networks for mobile vision applications,” arXiv preprint arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='04861, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [48] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Pan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Liu, “Roformer: En- hanced transformer with rotary position embedding,” arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='09864, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [50] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Goyal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Girshick, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' He, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Doll´ar, “Focal loss for dense object detection,” in Proceedings of the IEEE international conference on computer vision, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2980–2988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Dai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Savva, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Halber, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Funkhouser, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Nießner, “Scannet: Richly-annotated 3d reconstructions of indoor scenes,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5828–5839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [52] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Snavely, “Megadepth: Learning single-view depth predic- tion from internet photos,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2041–2050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [53] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Luo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Quan, “Contextdesc: Local descriptor augmentation with cross-modality con- text,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 2527–2536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Luo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Quan, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Liao, “Learning two-view correspondences and geometry using order-aware network,” in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5845–5854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [55] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Cai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Liu, “Htmatch: An efficient hybrid transformer based graph neural network for local feature matching,” Signal Processing, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 108859, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [56] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Balntas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Lenc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Vedaldi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Mikolajczyk, “Hpatches: A benchmark and evaluation of handcrafted and learned local descriptors,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 5173–5182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' [57] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Luo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Bai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Yao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Fang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' Quan, “Aslfeat: Learning local features of accurate shape and localization,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9E1T4oBgHgl3EQfMwOA/content/2301.02993v1.pdf'} +page_content=' 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Given a categorical action of a Lie algebra, a celebrated theorem of +Chuang and Rouquier proves that the blocks corresponding to weight spaces +in the same orbit of the Weyl group are derived equivalent, proving an even +more celebrated conjecture of Brou´e for the case of the symmetric group. +In many cases, these derived equivalences are t-exact, and thus induce +equivalences of abelian categories between different blocks. We call two such +blocks “Scopes equivalent.” In this paper, we describe how Scopes equiva- +lence classes for any affine categorification can be classified by the chambers +of a finite hyperplane arrangement, which can be found through simple Lie +theoretic calculations. +We pay special attention to the largest equivalence +classes, which we call RoCK, and show how this matches with recent work of +Lyle on Rouquier blocks for Ariki-Koike algebras. We also provide Sage code +that tests whether blocks are RoCK and finds RoCK blocks for Ariki-Koike +algebras. +1. Introduction +A remarkable theorem of Chuang and Rouquier [CRb], proving a conjecture of Brou´e, +shows that any two blocks of of modules over FpSn and FpSm with the same defect group +are derived equivalent. The proof of this theorem runs through a remarkable fact about +the theory of symmetric groups: that it is best understood in terms of the representation +theory of the affine Lie algebra �slp, as pointed out in the title of [Gro]. In fact, Chuang +and Rouquier prove a much more general theorem, of which Brou´e’s conjecture is a +special case. +They use the notion of a categorical representation of sl2 (a strong sl2-categorification, +in their terminology) on a category C. This data consists of: +(1) a decomposition C ∼= � +n∈Z Cn; +(2) functors E: Cn → Cn+2 and F: Cn → Cn−2; +(3) certain special natural tranformations between compositions of these functors +which force [E], [F] to satisfy the relations of sl2 on the level of the Grothendieck +group, with Cn corresponding to the n-weight space. +1Supported by NSERC through a Discovery Grant. This research was supported in part by Perime- +ter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the +Government of Canada through the Department of Innovation, Science and Economic Development +Canada and by the Province of Ontario through the Ministry of Colleges and Universities. +1 + +RoCK blocks for affine categorical representations +Chuang and Rouquier then prove that for any categorical sl2-representation, we have +an equivalence of derived categories Db(Cn) ∼= Db(C−n) categorifying the action of the +unique element of the Weyl group. Their proof of the Brou´e conjecture for the symmet- +ric group proceeds by successively applying different such equivalences for the sl2-actions +given by i-induction and i-restriction for different i ∈ Fp. +These equivalences of derived categories are sometimes t-exact, and sometimes not. +If the Chuang-Rouquier equivalence is t-exact, then it induces an equivalence of cate- +gories Cn ∼= C−n. We call these Scopes equivalences, since in the case of the sym- +metric group, they recover the Morita equivalences described by Scopes [Sco]. Using +these equivalences, Scopes showed that only finitely different abelian categories up to +equivalence appear amongst the blocks of a given defect. In fact, “most” blocks are +all equivalent as abelian categories. The blocks in this class are called RoCK (for +Rouquier-Chuang-Kessar) or simply Rouquier. +Our purpose in this paper is to describe how, in direct analogy with Chuang and +Rouquier’s approach to the Brou´e conjecture, the theory of Scopes equivalences and +RoCK blocks extend immediately to all categorical modules over an affine Lie algebra +g. For any categorical module C over an affine Lie algebra g, we let its support be the +set of weights µ such that Cµ ̸= 0; similarly, for a g-module, we let its support be the +set of weights with non-zero weight spaces. +Let ¯g denote the corresponding finite dimensional Lie algebra, ¯h its Cartan and W, ¯W +the corresponding affine and finite Weyl groups. For each choice of C and dominant +weight λ in the support of C with stabilizer Wλ, we’ll define a finite hyperplane ar- +rangement in ¯h given by certain translates of coroot hyperplanes; the arrangement only +depends on the support of C as a set (and thus will be the same for categorifical actions +with the same support). We’ll call the chambers of this arrangement Scopes cham- +bers. For each Weyl chamber of ¯W, there is a unique Scopes chamber which contains +a translate of this Weyl chamber, which we call its RoCK chamber. +We call two weights λ and λ′ in the support of C Scopes equivalent if there is +a t-exact Chuang-Rouquier equivalence between them. +Note, this is stronger than +requiring that the blocks are equivalent as abelian categories; there are examples where +the corresponding blocks are equivalent as categories, but not via a Chuang-Rouquier +equivalence. +In analogy with Scopes’ results, we show: +Theorem A For any categorical module C over an affine Lie algebra g and any domi- +nant weight µ in its support: +(1) The orbit {wµ}w∈W of µ under the affine Weyl group is the union of finitely +many Scopes equivalence classes. +(2) These classes are in bijection with orbits of Wλ on the Scopes chambers. In +particular, the Scopes equivalence classes of a categorical module only depend +on its support. +2 + +Ben Webster +Under this bijection, certain equivalence classes (as many as # ¯W) corespond to +RoCK chambers. We call the weight spaces in these equivalence classes RoCK. +This definition is heavily inspired by work of Lyle on the case of Ariki-Koike alge- +bras [Lyl] for the parameter q a root of unity with quantum characteristic e, with the +parameters Qj all lying in the set qZ. Let wi be the multiplicity of qi in the multiset of +Qj’s. The modules over the Ariki-Koike algebras ⊕nAKn(k, q, w) -mod categorify the +simple module V (Λ) = V (w0Λ0 + · · · + we−1Λe−1) for �sle. In §3 of loc. cit., Lyle gives a +definition of a Rouquier block for an Ariki-Koike algebra; this is a purely combina- +torial property of a weight in the support of V (Λ), so we can just as easily apply it to +any other categorical module with this support. Note that while “Rouquier block” and +“RoCK block” are usually used synonymously, here we use them to distinguish Lyle’s +definition from ours. +Theorem B For any categorical representation C of g = �sle with support V (Λ), the +Scopes equivalence classes will coincide those for the Ariki-Koike algebra, and a Scopes +equivalence class is RoCK if and only if it contains a Rouquier weight. +The proof of Theorem A depends on the proof of a generalization of [Lyl, Conj. 1] +in the context of categorical representations (Lemma 3.2), confirming Lyle’s conjecture +and upgrading the decomposition equivalences of [Lyl, §3] to Scopes (and thus Morita) +equivalences. +Furthermore, the Scopes walls, which control Scopes equivalence and RoCKness, +are very conducive to computer computation. We’ve written a Sage program (available +here on CoCalc) which tests whether blocks are RoCK and construct examples of RoCK +blocks. As discussed above, these computations apply not just to Ariki-Koike algebras, +but also to other categorifications with the same support, in particular, the cyclotomic +q-Schur algebras and category O for Cherednik algebras of G(ℓ, 1, n). +2. Background +2.1. Affine Lie algebras. Let g be an affine Lie algebra, h its abstract Cartan and +W its Weyl group. Since there are multiple variations of this algebra, we should clarify +that we take the abstract Kac-Moody algebra defined by a given affine Cartan matrix. +That is, we assume that the simple coroots α∨ +1 , . . . , α∨ +n ∈ h are linearly independent, as +are the simple roots α1, . . . , αn ∈ h∗. Since the Cartan matrix of g has corank 1, this +means that the span of the coroots is codimension 1 in h and the span of the roots is +codimension 1 in h∗. +There is a unique primitive Z>0-linear combination δ = � δiαi which is non-zero, +but perpendicular to all α∨ +i ’s and similarly a unique primitive Z>0-linear combination +δ∨ = � δ∨ +i α∨ +i which is non-zero, but perpendicular to all αi. +Let ¯hR be the quotient of the R-span of α∨ +i by the R-span of δ∨. The dual ¯h∗ +R is +the quotient of the R-span of αi by the R-span of δ. The symmetrized Cartan matrix +3 + +RoCK blocks for affine categorical representations +defines an inner product on ¯hR. For each affine root α, let ¯α be its image in ¯h∗ +R. These +images are integer multiples of roots in a system corresponding to a finite dimension +simple Lie algebra ¯g. Let ¯W be the Weyl group of this finite root system. +The value of δ∨ is thus constant on the weights of any irreducible representation of +g. This invariant of a representation is called its level. The elements h∗ +R where δ∨ > 0 +is the Tits cone of g, the union of all the weights in the W-orbit of the dominant Weyl +chamber. Readers will be familiar with the actions of W on a fixed level coset +h∗ +c = {λ ∈ h∗/Cδ | δ∨(λ) = c, α∨ +i (λ) ∈ R}. +This subspace is a coset of ¯h∗ +R, and thus inherits a metric. The reflection in any real +root α of g becomes the usual geometric reflection in the hyperplane +(2.1) +Hα = {λ ∈ h∗ +c|α∨(λ) = 0}. +This hyperplane is a coset of the vanishing set of ¯α∨ in h∗ +R. +The hyperplanes Hα cut h∗ +c into chambers called alcoves. Each alcove corresponds +to a Weyl chamber in the W-orbit of the dominant Weyl chamber. In particular, we +have the dominant alcove +A = {λ ∈ h∗ +c | α∨(λ) > 0 for all positive roots α}. +The Weyl group W acts simply transitively on the set of alcoves. For any point o ∈ h∗ +c, +we can define a ¯W-action on h∗ +c by ¯w(o+h) = o+ ¯wh for h ∈ ¯h∗ +R. Following convention, +we will take o to be the unique point satisfying α∨ +1 (o) = · · · = α∨ +n−1(o) = 0. In particular, +this point is a vertex of A. The action of any element w ∈ W on h∗ +c can be factored +uniquely as product of a translation τh by an element of ¯h∗ +R, and element ¯w in ¯W. Note +that ¯wτh = τwh ¯w, so this factorization can be taken in either order. +The most important example for us will be �sle, the affine Lie algebra obtained from +sle((t)) by taking the unique central extension, and adding a loop element ∂ which acts +by t ∂ +∂t. The torus h can most easily be described as +h = {(h0, h1 . . . , he+1) ∈ Re+2 | h1 + · · · + he = 0} +with the coroots +α∨ +n = (1, −1, 0, . . . , 0, 1, 0) +α∨ +i = (0, . . . , 1, −1, 0, . . . , 0). +The full set of positive roots is thus αij;n for i, j ∈ [1, e], the vector with hi = 1, hj = −1, +and last coordinate −n, where n ≥ 0 if i < j, and n > 0 if i > j. Note that αe1;1 = αe. +We can identify the same space with h∗ via inner product, and take the roots to be +α0 = (0, −1, 0, . . . , 0, 1, −1) +αi = (0, . . . , 1, −1, 0, . . . , 0). +Note that +δ = (0, 0, . . . , 0, −1) +δ∨ = (1, 0, . . . , 0, 0) +The reduced space is defined by +¯h ∼= ¯h∗ ∼= {(h1, . . . , he) ∈ Re | h1 + · · · + he = 0} +4 + +Ben Webster +where again, the metric is the usual one induced by inner product on Re. +The action of si = sαi on h is thus given by the matrix I − αi · α∨ +i , where we view αi +as a column vector, and α∨ +i as a row vector. Thus, the reflections si = sαi for i > 0 act +by the usual permutation matrices on the coordinates hi and hi+1, and se = sαe acts by + + +1 +0 +0 +· · · +0 +0 +0 +1 +0 +0 +· · · +0 +1 +0 +0 +0 +1 +· · · +0 +0 +0 +... +... +... +... +... +... +... +0 +0 +0 +· · · +1 +0 +0 +−1 +1 +0 +· · · +0 +0 +0 +1 +−1 +0 +· · · +0 +1 +1 + + +Considering the quotient by δ, we obtain +h∗/Cδ = {(h0, h1 . . . , he) ∈ Re+1 | h1 + · · · + he = 0}, +and the induced action on this quotient is obtained by simply deleting the rightmost +column and bottom row from each matrix. Fixing ourself on the level h∗ +c, we fix the +coordinate h0 = c, so +si(c, h1, . . . , he) = +� +(c, h1, . . . , hi+1, hi, . . . , he) +i > 0 +(c, he + c, h2, . . . , he−1, h1 − c) +i = 0 +Thus, se acts by reflection in the line h1 − he = c. More generally, the different roots +of g act by reflection in the lines hi − hj = cm for m ∈ Z. +From this perspective, the Weyl group W acts by affine transformations; this can be +seen from the fact that it is generated by the copy of Se generated by s1, . . . , se−1, which +acts linearly, and by translation by vectors in the root lattice of sle, i.e. the vectors +X = {(ca1, . . . , cae) | ai ∈ Z, a1 + · · · + ae = 0}. +The fundamental alcove is the set A = {h1 ≥ h2 ≥ · · · ≥ he ≥ h1 − c}; the last +inequality follows by requiring α∨ +e (c, h1, . . . , he, ∗) = c + he − h1 ≥ 0. +For example, if e = 2, then we have that h∗ +c ∼= R is 1-dimensional, parameterized +by h1 with h2 = −h1. We have root hyperplanes at h1 − h2 = cm for m ∈ Z, or +equivalently, h1 = c m +2 . Thus, the alcoves are given by the intervals Am = [c m +2 , c m+1 +2 ], +and the fundamental alcove is defined by h2 +c ≥ h1 ≥ h2 or equivalently h1 ∈ A0. The +chambers Am with m even are the image of A0 under the translations by cm (these are +the even elements of the Weyl group) and those with m odd are the image of A0 under +a reflection at h1 = c m+1 +4 , the midpoint between these two chambers (these are all the +odd elements of the affine Weyl group). +If e = 3, then the alcoves are equilateral triangles. Around each point in the root +lattice, there are 6 triangles that touch that element, which form a hexagon. For 0, this +hexagon is defined by |hi − hj| ≤ 1 for all i, j. The translations in the root lattice act +5 + +RoCK blocks for affine categorical representations +freely transitively on the set of these hexagons. In fact, these are the Voronoi tesselation +of the root lattice. +This makes visible the factorization into a translation and an element of ¯W: a unique +translation sends each alcove to one in the hexagon around the origin, which is the tip +of one of the finite Weyl chambers, and then a unique element of ¯W sends this to +the fundamental alcove. In the image below, the fundamental alcove is colored green, +and its orbit under ¯W is colored red, and all the elements of root lattice that fit in +the picture are marked with a black dot. The other vertex points of alcoves are the +elements of the weight lattice of sl3 which don’t lie in the root lattice; the other vertices +of the fundamental alcove are ( 1 +3, 1 +3, −2 +3) and ( 2 +3, −1 +3, −1 +3): +(1, 0, −1) +(−1, 0, 1) +(1, −1, 0) +(−1, 1, 0) +(0, −1, 1) +(0, 1, −1) +2.2. Scopes chambers. Let N denote any finite set of positive roots. +The hyperplanes Hα for α ∈ N cut h∗ +c into finitely many chambers, which we call +Scopes chambers. Every Scopes chamber is defined by choosing ǫα ∈ {±1} for α ∈ N, +and considering the inequalities C = {h | ǫαα(h) ≥ 0}. +Lemma 2.1 +(1) We have w · ∆+ ∩ N = w′ · ∆+ ∩ N if and only if wA and w′A lie in the same +Scopes chamber. +(2) If wA and w′A lie in the same Scopes chamber then there is a sequence i1, . . . , ip +such that for w′ = wsi1 · · · sip, and of k = 0, . . . , k, wsi1 · · · sikA lies in the same +Scopes chamber as wA and w′A. +Proof. +(1) Note that α∨ is positive on wA if and only if w−1α∨ is positive on A, that is, +if w−1α ∈ ∆. Thus, w · ∆+ ∩ N is exactly the subset of N which is positive on +6 + +Ben Webster +wA. By definition, this subset is the same on another alcove if and only if they +are in the same Scopes equivalence class. +(2) Let us prove this by induction on the number of root hyperplanes separating wA +and w′A. When this number is 0, w = w′ and the claim is tautological. Let Hα +be a hyperplane which is a facet of wA, and separates it from w′A. Then Hw−1α +is a facet of A, and so w−1α is ±1 times a simple root αi1. This means that the +reflection across Hα can be written as sα = wsi1w−1. Thus, sαwA = wsi1A is a +chamber separated from w′A by only the hyperplanes separating wA from w′A, +excluding Hα. Note, this means that it is not separated from wA or w′A by the +hyperplane defined by an element of N, and so is in the same Scopes chamber. +By the inductive hypothesis applied to the chambers wsi1A and w′A, we can +find i2, . . . , ip such that w′ = wsi1 · · · sip and wsi1 · · ·sikA is in the same Scopes +chamber. This completes the proof. +□ +Every Scopes chamber C has an asymptotic cone ¯C = {h ∈ h | th ∈ C for t ≫ 0}. +Lemma 2.2 The asymptotic cone of a Scopes chamber is always a face of the hyper- +plane arrangement ¯α = 0 for α ∈ N. For each finite Weyl chamber, there is a unique +Scopes chamber containing it in its asymptotic cone. +Proof. The asymptotic cone of the this chamber is defined by ¯C = {h | ǫα¯α(h) ≥ 0}, +since α(th) ≥ 0 for t ≫ 0 iff ¯α(h) ≥ 0. This asymptotic cone will thus contain a Weyl +chamber if and only if ǫα = ǫβ whenever ¯α = ¯β. Given a Weyl chamber, we can define +corresponding ǫα to have the same sign as ¯α on the Weyl chamber. This will define the +only Scopes chamber with our fixed Weyl chamber in its asymptotic cone. +□ +We call a Scopes chamber C RoCK if ¯C contains a finite Weyl chamber, or equiva- +lently, C contains a translate of a finite Weyl chamber. There are at most # ¯W RoCK +chambers, one for each finite Weyl chamber; if the cone ¯C contains a Weyl chamber c, +we say that C or an alcove in C is RoCK for c. +One helpful way to think about these different RoCK chambers is to factor your +Weyl group element into a translation and an element of ¯W. As discussed above, we +can write w = τh ¯w. Since ¯w sends A to one of the other chambers adjacent to o, the +element h accounts for most of the hyperplanes separating wA from A. More precisely, +w · o = h + o, so wA must be one of the chambers having h + o as a vertex, and ¯w +only controls which of these chambers it is. In particular, if h + o is in the interior of +a Scopes chamber, then all adjacent chambers are in the same Scopes chamber. Thus, +we can conclude that when h is deep inside a Weyl chamber, that is α∨(h) ≫ 0 for +all roots α positive on the chamber, then the alcove wA will be in the corresponding +RoCK chamber. +7 + +RoCK blocks for affine categorical representations +2.3. Categorical actions. While the fundamental input of this paper is a categorical +action of a Lie algebra, we will not use this definition in a deep way, and thus will only +give the basic facts we need here. +A categorical action of a Kac-Moody algebra g is a representation of a particular +2-category U(g). This 2-category has: +(0) object set given by the weight lattice Y of g. +(1) 1-morphisms generated by symbols Ei, Fi for i ranging of the simple roots of g. +These act on weight lattice in the same way that Chevalley generators change +weights: +Ei : λ → λ + αi +Fi : λ → λ − αi +(2) For our purposes, a detailed description of the 2-morphisms is not needed. See +[Bru] for a detailed discussion of the different possible generating sets and rela- +tions. +We will use 2 basic facts about categorical actions all of which follow from examination +of the 2-morphisms: +(i) The power Fk +i is the direct sum of k! isomorphic summands, which we denote +by F(k) +i +and call the divided power functor. +(ii) For each simple root, there is a chain complex Θi : λ → siλ of 1-morphisms in +U which is invertible up to homotopy. A version of this complex was defined in +[CRb, §6.1] but we use the definition in [Cau, (3-4)] for more modern notation. +Thus, a categorical action is an assignment of a category Cλ to each element of the +weight lattice λ, a functor to each Ei and Fi, and natural transformations to each 2- +morphism. In this case, we can interpret Θi as as a functor Θi : Db(Cλ) → Db(Csiλ), +which by the invertibility must be an equivalence of categories. +We’ll discuss the relevant examples of categorical actions in Section 4 when we cover +their RoCK blocks. +3. RoCK blocks +3.1. Scopes equivalences from Chuang-Rouquier. Consider any integrable cat- +egorical module C over g, and let ν be any weight of this module. +Assume that +α∨ +i (ν) = k > 0. +Lemma 3.1 The functor F(k) +i +: Cν → Csiν is an equivalence of abelian categories (that +is, a Scopes equivalence) if and only if the category Cν+αi is trivial. +This lemma is a restatement of one special case of the [CRa, Prop. 8.4]: the Chuang- +Rouquier equivalences are perverse, and will be t-exact if and only if the perversity +function is 0 for all simples. This is equivalent to the condition that Cν+αi is trivial. +We will give a more direct proof here that doesn’t depend on the notion of a perverse +equivalence. +8 + +Ben Webster +Proof. ⇒: If F(k) +i +is an equivalence of categories, then E(k) +i +is its left and right adjoint, +and thus the inverse equivalence. Thus, for any object M ∈ Cν, we have M ∼= E(k) +i F(k) +i M. +On the other hand, if Cν+αi ̸= 0, then there is a highest weight object N in Cν+rαi for +some r > 0. The object F(r) +i N is non-zero, and +F(k) +i E(k) +i F(r) +i N ∼= F(k) +i E(k+r) +i +N⊕(k+r +k ) ∼= F(r) +i N⊕(k+r +k ) +2 +applying [Cau, Lem. 4.1] with µ = k + 2r, b = k + r, a = k. This contradicts the claim +that F(k) +i +is an equivalence. +⇐: Since Cν+αi is trivial, every object in Cν is killed by Ei, that is, it is highest weight. +This implies that the only term of the Rickard complex Θi which acts nontrivially is +F(k) +i +in homological degree 0. Thus, on Cν, the actions of Θi and the derived functor of +F(k) +i +coincide. The former is an equivalence of derived categories, and the latter is exact +in the usual t-structure. This shows that F(k) +i +is an equivalence of abelian categories as +desired. +□ +Now, we restrict to the case where ν is positive level (i.e. +δ∨(ν) > 0). +In this +case, there is a unique dominant weight µ = wν in the orbit of ν. Consider the set +Nµ = {α ∈ ∆+ | Cµ+α ̸= 0}. +The failure of the functor Θi to preserve the t-structure depends in a precise way +on the structure of the representation of the sl2 generated by Ei, Fi on the root string +ν + mαi for all m ∈ Z. Using the action of W, this is the same the structure of the +root string through µ with respect to the root wαi. In particular: +Lemma 3.2 The following are equivalent: +(i) The functor F(k) +i +: Cν → Csiν is an equivalence of abelian categories (that is, a +Scopes equivalence). +(ii) The category Cµ+wαi is trivial, i.e. wαi /∈ Nµ. +(iii) The alcoves wA and wsiA lie in the same Scopes chamber. +Proof. (1) ⇒ (2): If F(k) +i +is an equivalence of categories, then E(k) +i +is its left and right +adjoint, and thus the inverse equivalence. +Thus, for any object M ∈ Cν, we have +M ∼= E(k) +i F(k) +i M. On the other hand, if Cν+αi ̸= 0, then there is a highest weight object +N in Cν+rαi for some r > 0. The object F(r) +i N is non-zero, and +F(k) +i E(k) +i F(r) +i N ∼= F(k) +i E(k+r) +i +N⊕(k+r +k ) ∼= F(r) +i N⊕(k+r +k ) +2 +applying [Cau, Lem. 4.1] with µ = k + 2r, b = k + r, a = k. This contradicts the claim +that F(k) +i +is an equivalence. +(2) ⇒ (1): The category Cµ+wαi is trivial if and only if the category Cν+αi is. In +particular, (2) implies that every object in Cµ is killed by Ei, that is, it is highest weight. +This implies that the only term of the Rickard complex Θi which acts nontrivially is +F(k) +i +in homological degree 0. Thus, on Cν, the actions of Θi and the derived functor of +9 + +RoCK blocks for affine categorical representations +F(k) +i +coincide. The former is an equivalence of derived categories, and the latter is exact +in the usual t-structure. This shows that F(k) +i +is an equivalence of abelian categories as +desired. +(2) ⇔ (3): The only hyperplane separating wA and wsiA is Hwαi, so these lie in a +common Scopes chamber if and only if wαi /∈ Nµ. +□ +This equivalence will induce a bijection of simple modules, which of course, matches +the action of the Kashiwara operator f k +i in the crystal structure on simples. +Combining Lemma 2.1 and Lemma 3.2, we arrive at the main result of this paper. +Consider w, w′ ∈ W and as above, µ dominant and ν = w−1µ, ν′ = (w′)−1µ. +Theorem 3.3 If wA and w′A lie in the same Scopes chamber for the set Nµ, then we +have an equivalence of abelian categories Cν′ ∼= Cν. +Furthermore, some important examples of interest to us, such as Schur algebras in +positive characteristic, (cyclotomic) q-Schur algebras, and categories O for Cherednik +algebras of G(ℓ, 1, n) are highest weight categorifications in the sense of [Los]. In this +case, the Scopes equivalence sends standard filtered modules to standard filtered mod- +ules, and thus induces an equivalence of highest weight categories. For symmetric group, +Hecke, and Ariki-Koike algebras, this implies that Specht filtrations are preserved. +In all these cases, we can index simples by abacus diagrams, and Lemma 3.2 will +apply when for every object in our block, there is no way to push a bead from the +(i + 1)st runner to the ith, i.e. for every bead on the (i + 1)st, the position to its left +on the ith runner is occupied. In this case, k is the number of beads on the ith runner +where to spot to the right is empty, and the Kashiwara operator f k +i acts by pushing +these beads right, that is, by swapping the runners. As usual, when i = e, we have to +interpret all these statements as comparing the eth and first runners with a shift. +For example, in the picture below, we first show the i and (i + 1) runners of diagram +where the Kashiwara operator ei acts non-trivially, to give the second picture where it +acts non-trivially. So Lemma 3.2 applies when all abacus diagrams in your block look +like the second diagram and never like the first. As discussed above, the Kashiwara +operator f 2 +i sends the second diagram to the third by moving the two dots that have +space to move right. This has the same effect as swapping the runners. +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +Definition 3.4 We call a weight space category RoCK or a RoCK block if the +corresponding Scopes chamber is RoCK. By construction, all RoCK blocks for a given +Weyl chamber are equivalent as abelian categories. +10 + +Ben Webster +We don’t aim here to comprehensively address the question of when weight categories +are equivalent; nothing we have written above precludes the existence of an equivalence +Cν′ ∼= Cν if wA and w′A do not lie in the same Scopes chamber. There are two obvious +situations where this can happen: +(i) If Cµ+wαi is non-trivial, but the representations for the root sl2 for wαi generated +by the µ weight space is isotypic (all its highest weight vectors are of the same +weight), then some shift of the Chuang-Rouquier functor Θi is exact, and thus +induces an equivalence of abelian categories. The easiest way this can happen is +condition (2) of Lemma 3.2, but it could be that all these highest weight vectors +have weight µ + rwα for some r. In this case, every projective object P in Cν +can be written as P = F(r) +i P ′, and there is an equivalence of abelian categories +sending P �→ F(k+r) +i +P ′. +We could tighten our results a bit by defining N′ +µ to be the set of roots which +do not have this isotypic property and only considering Scopes chambers with +respect to this set. This suffers from the issue of not being only determined by +the support of the representation, and in the vast majority of cses, Nµ and N′ +µ +will be equivalent. +(ii) If µ has non-trivial stabilizer, then we can have ν′ = ν while wA and w′A do not +lie in the same Scopes chamber. This tells us that the induced autoequivalence +of Cν′ will not be exact, but this doesn’t change that the categories are exactly +equal. +Of course, the stabilizer of µ is generated by sα for α∨(µ) = 0, which can +only happen if α is simple. Let Wµ be the stabilizer; we can simplify the deter- +mination of the different blocks that show up in this case by only considering +wA in a fundamental domain of Wµ; the most natural choice is to consider the +chamber cut out by α∨ = 0 for the α such that α∨(µ) = 0 which contains the +fundamental alcove. This has the effect of requiring w to be a shortest right +coset representative for Wµ. +3.2. Irreducible support. Recall that the support of a categorical module C for g is +the set of weights µ such that Cµ ̸= 0; by analogy, for a usual linear representation V of +g, we will call the set of weights with non-zero weight spaces the support of V . Most of +the categorical modules appearing in representation theory (of which the author knows) +satisfy the following property: +(∗) The support of C is equal to the support of a simple highest weight module +V (Λ). +Note, this will happen in many cases where the representation K0 +C(C) is very much not +irreducible, but the support of one simple summand contains the support of all other +summands. For example, if we consider a tensor product VΛ ⊗ VΛ′, this is typically not +irreducible, but its support is the same as the Cartan component VΛ+Λ′. +11 + +RoCK blocks for affine categorical representations +This is a particularly nice situation since we can encode the support of C in purely +combinatorial terms. Let ≤ denote the usual root order on weights, that is, the transi- +tive closure of µ−αi ≤ µ for all µ, i. Let us recall one of the standard characterizations +of the support of a simple highest weight module: +Lemma 3.5 The following are equivalent: +(i) A weight µ is in the support of V (Λ). +(ii) There is an element w ∈ W such that wµ is dominant and wµ ≤ Λ. +(iii) For all w ∈ W, we have wµ ≤ Λ. +Thus, every µ in the support of V (Λ) is of the form +µ = Λ − +e +� +i=1 +biαi +and any dominant µ with this form is in the support by (2) ⇒ (1). Thus, given a +simple root α, we wish to determine if µ + α is in the support of C, i.e. if α ∈ Nµ. +Obviously, this can only happen if µ + α ≤ Λ, and in some corner cases, µ + α will not +be dominant, and we need to check the the dominant weight in its orbit also satisfies +w(µ + α) ≤ Λ. Note that this implies that Nµ is finite in this case. +This is easy to check for one orbit, but the reader will correctly note that the set of +positive roots is infinite, making it hard to check by hand whether this holds for each +of them. However, we can exploit the fact addition by δ commutes with every element +of W. Fix a set {β1, . . . , βr} ⊂ ∆+ of positive roots with the property that every affine +root is of the form ±βi+kδ for a unique choice of sign, i ∈ [1, r] and k ∈ Z. Note that in +twisted cases, not every element of this form is a root. So, we have that w ·(µ±βi +kδ) +is dominant if and only if w · (µ ± βi) is. For each i ∈ [1, r], we thus have integers k± +i +defined to be the largest integers such that +w(µ ± βi) + k± +i δ ≤ Λ +∀w ∈ W. +Of course, checking this for all w is equivalent to checking it only when w(µ ± βi) is +dominant by Lemma 3.5. This implies that a positive root ±βi + kδ lies in Nµ if and +only if k ≤ k± +i . However, as we noted before, there might be values of k where this is +not a positive root. +If we assume our affine Lie algebra is simply laced, that is, it is of type �A, �D or �E, then +we can simplify by choosing our βi to the positive roots in the finite type subalgebra, +so the set of positive roots can be described as +∆+ = +� +i∈[1,r] +m∈Z≥0 +{βi + mδ, −βi + (m + 1)δ} +and the resulting description of the Scopes chambers has a particularly nice form: +12 + +Ben Webster +Lemma 3.6 The set Nµ is exactly the roots of the form ±βi + mδ with m ≤ k± +i . The +walls of the Scopes chambers are defined by +β∨ +i (λ) = mδ∨(λ) for all m ∈ [−k+ +i , k− +i ]. +The combination of Lemmata 3.2 and 3.6 give us an algorithm to test whether a block +is RoCK, and to construct blocks which are RoCK for each Weyl chamber. Assume +that property (∗) holds and fix a weight µ; the steps of our algorithm are: +(i) Find the dominant element of the orbit: Let w ∈ W be the element of +minimal length such that µ′ = wµ is dominant. Also find the alcove A′ = wA. Al- +gorithmically, we can do this by checking the sign of α∨ +i (µ) for all i. If these are all +≥ 0, then µ is dominant, and we are done. Otherwise, replace µ by siµ for any i with +α∨ +i (µ) < 0. Note that this terminates since µ < siµ ≤ Λ. We can find A′ by simply +acting with si on the alcove as well. +(ii) Find the set Nµ′: For each βi, we consider µ ± βi, find the dominant element +of its W-orbit, and use this to find the bounds k± +i , which specify all the Scopes walls. +(iii) Check the sign on A′: Check the value of the ratio ρ = β∨ +i +δ∨ on any element of +A′. If −k+ +i ≤ ρ ≤ k− +i for any i then the block is not RoCK, and if this inequality does +not hold for any i, the block is RoCK. +This algorithm sounds laborious, and indeed it is to do by hand; however, it is extremely +efficient to do by computer, especially if compared to any algorithm that requires enu- +meration of the partitions in a block. In particular, its complexity does not significantly +increase as we add e-hooks to a block, whereas enumerating partitions will get much +worse. This program is publicly available for Sage here on CoCalc, and some examples +are shown later in the paper. +Let us now consider an example with e = 3. Let Λ = 2Λ1 + Λ2 + Λ3 and µ = +Λ − 2α1 − 2α2 − 3α3. Note that in this case, Wµ = {1, s3}. In this case, k− +12 = 2, k− +13 = +3, k− +23 = 2, k+ +12 = 1, k+ +13 = 1, k+ +23 = 1. +Thus, the Scopes walls are given by the red +hyperplanes below; the fundamental alcove is colored green, and the closest blocks to +the fundamental in each RoCK chamber is colored red if they are in the positive Weyl +chamber for Wµ (the one containing the fundamental alcove), and blue if they are in +the negative Weyl chamber. Thus, there are three RoCK equivalence classes of blocks +13 + +RoCK blocks for affine categorical representations +under Scopes equivalence, corresponding to the 3 red chambers. +(3.1) +4. Examples +4.1. Level 1. Let Hn(k, q) be the Hecke algebra of Sn (i.e. type A) over the field +k +for a scalar q ∈ +k, and let e be the quantum characteristic, i.e. the smallest integer +such that qe−1 + · · · + 1 = 0. In this case, we will explain how we recover the original +definition of RoCK blocks (those satisfying the conditions of [CK, Th. 2]). +Let g = �sle. The category � +n≥0 Hn(k, q) -mod is a categorical g-module, categori- +fying the simple module with highest weight Λe = (1, 0, . . . , 0). The action of Ei is +by i-restriction and Fi by i-induction. We can assign a weight to each partition ν by +letting bi be the number of boxes of content i (mod e), and considering +µν = Λe − +e +� +i=1 +biαi = (1, be − b1, b1 − b2, . . . , be−1 − be, be). +In particular, µ∅ = (1, 0, . . . , 0) is the highest weight appearing. Note that adding a +e-hook only changes this weight by adding δ, so the image of µν in h∗ +1 is determined +just by the core of µ, and there will be a unique e-core for each integral point of h∗ +1. +For example, if e = 2, then h∗ +1 is parameterized by the value of h1 as before, and the +14 + +Ben Webster +2-cores index the points in the W-orbit of h1 = 0. +∅ +h1 = 1 +h1 = 0 +h1 = 2 +h1 = −1 +h1 = −2 +Similarly, 3-cores appear index the integral points in h∗ +1 for �sl3: +∅ +The weight µν is dominant if and only if the core is empty, since no other integral weight +is in the fundamental alcove. That is, if µν = Λe − kδ = (1, 0, . . . , 0, k) for k ≥ 0, the +weight of the corresponding block. Note, the corresponding category is the principal +block over Hke(k, q). +The other prominent example of another categorical action with the same set of non- +zero weights is the module categories over the q-Schur algebra appearing endomorphisms +of the permutation modules over Hn(k, q). The blocks of this algebra are in obvious +bijection with those of Hn(k, q), and the category of modules has an induced categorical +action. Thus, the Scopes chambers are the same, and the RoCK blocks of the Schur +algebra are the same as those of the Hecke algebra. +Since any Young diagram has a box of content 0, the category CΛe−αij;m is trivial +unless m > 0; on the other hand, if m > 0 we can find a non-trivial object on this +category given by a hook with removable boxes of content i and j − 1. Thus, the set +N of roots such that Cµ+α is thus given by αij;n for n < k, since +µν + αij;n = µ∅ − αij;k−n. +Thus, by Lemma 3.6, the Scopes chambers are cut out by the hyperplanes +hi − hj = nc +n = k − 1, k − 2, . . . , −k + 1. +15 + +RoCK blocks for affine categorical representations +Note that we could also derive this from Lemma 3.5: Λe −kδ +αij;m is never dominant, +and the dominant element in its Weyl group orbit is Λe − (k − m − 1)δ, so as desired, +we must have m < k. +As discussed previously, we can describe all weights in this case in the form w−1µk +for w a shortest right coset representative of Wλ = ¯W. In the dominant finite Weyl +chamber, there is a single RoCK Scopes chamber given by the elements such that +hi ≥ hi+1 +k −1. This is the usual RoCK condition on blocks (for example, introduced +at the start of [CK, §3]). +We can visualize the Scopes chambers on blocks by drawing the corresponding core +of weight w · µ∅ over the alcove wA for w a shortest right coset representative. Our +e = 2 example then becomes: +∅ +h1 = 1.5 +h1 = 1 +h1 = 2 +h1 = 2.5 +h1 = .5 +h1 = 0 +We’ve drawn walls of the form h1 ∈ Z with solid dots and h1 ∈ Z + 1 +2 with open dots. +For a fixed k, the first k −1 alcoves are each a separate Scopes class, and then all others +are RoCK. Our e = 3 example becomes: +∅ +We’ve drawn in the walls that separate k = 2 Scopes chambers in red. As we increase +k, we add in more and more translates of each of these hyperplanes. +4.2. Ariki-Koike algebras. Now, we turn to the more interesting case of Ariki-Koike +algebras of level ℓ. +Thus, as before, we have fixed q ∈ +k \ {0} of multiplicative order e > 1, and consider +the Ariki-Koike algebra AKn( k, q, w) associated to the polynomial +f(u) = +� +i∈Z/eZ +(u − qi)wi +16 + +Ben Webster +for fixed wi ∈ Z≥0 with � wi = ℓ. +The categories ⊕nAKn(k, q, w) -mod carry a categorical action of �sle as well. +It +was proven by Ariki [Ari, Prop. 4.5] that the Grothendieck group of this category is +isomorphic to the highest weight simple VΛ for Λ = �e +i=1 wiΛi ∈ h∗ +ℓ where +Λi = (1, e − i +e +, . . . , e − i +e +� +�� +� +i times +, − i +e, . . . , − i +e +� +�� +� +e−i times +, 0). +These are fundamental weights since α∨ +i (Λj) = δij. A typical Λ will have all wi ≥ 0 +and thus will be in the interior of the fundamental alcove, and have trivial stabilizer. +These are a bit easier to write if we adopt the convention that we can denote weights +by arbitrary elements of Rn+2, which we take to be equivalent to their orthogonal +projection to the subspace +h = {(h0, h1 . . . , he+1) ∈ Re+2 | h1 + · · · + he = 0}. +In this case, we can equally well write Λi = (1, 1, . . . , 1 +� �� � +i times +, 0, . . . , 0 +� �� � +e−i times +, 0). +Ariki’s action on the Grothendieck group in fact reflects a categorical action of �sle +on ⊕nAKn(k, q, w) -mod; the fact that i-induction and i-restriction for a given i define +a strong sl2 action is proven in [CRb, §7.2.2]. The fact that this extends to a categor- +ical action of �sle is effectively equivalent to the main theorem of [BK]. We can more +systematically construct the categorical action by i-induction and i-restriction functors +using the formalism of quantum Heisenberg actions. Such an action on Ariki-Koike +algebras is defined in [BSWb, §6]. The main theorem of [BSWa] implies that this gives +us a categorical �sle-action. +We can apply the same principle to construct categorical actions of �sle on +(i) cyclotomic degenerate affine Hecke algebras in characteristic e, assuming e is +prime; +(ii) cyclotomic q-Schur algebras; +(iii) categories O of the rational Cherednik algebra of G(ℓ, 1, n) with κ = a/e as a +fraction in least terms. +In case (1), we have a degenerate Heisenberg action by [MS, Th. 6.7]; in case (3), the +quantum Heisenberg action is constructed in [BSWb, §7], and case (2) is a limiting case +of (3) (see, for example, [Web, Cor. 3.11]). The categorical action in case (3) was first +constructed by Shan [Sha], but for an unnecessarily restrictive choice of parameters and +using a different formalism for categorical actions. +All of these also have support equal to that of an irreducible representation of the +form V (Λ): +(1) The weight Λ is determined by the roots of the cyclotomic polynomial, by iden- +tifying the elements of Fp with the Dynkin diagram of �slp, as in [BK]. +17 + +RoCK blocks for affine categorical representations +(2-3) Each block in these cases is a quasi-hereditary cover of a block of the Ariki-Koike +algebra, so the we simply use the same weight combinatorics, and of course, +obtain a non-zero block if and only if the corresponding block of Ariki-Koike is +non-zero. +Everything we say will below also applies to other categorifications of �sle with the same +support. +The blocks of AKn(k, q, w) again correspond to weight spaces: by [LM], the block +of Specht module is determined by the number of bi boxes of residue i (mod e) in +the corresponding charged multi-partition, and this also determines the weight by the +formula: +µν = Λ − +e−1 +� +i=1 +biαi. +Note, when we associate an ℓ-tuple of runner diagrams to a simple as in [Lyl, §2.2] +the total number ti of beads on the ith runners of the ℓ different abacus diagrams is +given by ti = bi−1 −bi +�e +j=i wi for i = 1, . . . , e. This also allows us to write the weight +µν in the coordinates hi. We have +µν = (ℓ, t1, . . . , te, be). +Note, these coordinates are helpful for identifying the dominant weight in a given orbit: +a weight is dominant if t1 ≥ · · · ≥ te ≥ t1 − ℓ. Whereas in level 1, these were easy to +identify as the principal blocks for ranks divisible by e (i.e. the lowest rank where a +given defect group appears), in higher levels, there are many more dominant weights. +As mentioned in the introduction, [Lyl, Conj. 1] is effectively equivalent to Lemma +3.2 in this context: since Ei acts by pushing beads from the (i + 1)st runner to the ith, +the weight w−1λ+αi is not in the support of C if and only if it is never possible to push +a bead in this way. This is only the case for an abacus diagram where the weight of +the ith runner is less than the difference between the number of beads on the ith and +(i + 1)st runners; this is precisely the hypothesis of [Lyl, Conj. 1]. +To test whether a weight is RoCK, we simply apply the algorithm of Section 3.2 to +our weight. In particular, we have to calculate the statistics k±; in the type A case, we +can encode these as +k+ +ij = k− +ji = max{n | αij;n ∈ Nµ}. +Thus, the Scopes walls for Nµ will be of the form +(4.1) +hi − hj = nc +n = [−k+ +ij, k− +ij]. +While applying the full algorithm is more precise, using simpler methods, we can +give an upper bound on the set of Scopes walls that is easy to compute by hand. To +understand this more concretely, it’s useful to note that if n = 0, then αij = αi+· · · αj−1 +and if n > 0, then +αij;n = αi + · · ·αe−1 + αe + α1 + · · · + αj−1 + (n − 1)δ. +18 + +Ben Webster +That is: +Lemma 4.1 The difference between the number b′ +r of boxes of content r in a multipar- +tition of weight µ + αij;n = Λ − �e−1 +i=0 b′ +iαi compared to the number br in one of weight +µ is: +br − b′ +r = + + + + + +n + 1 +i < j, r ∈ [i, j − 1] +n − 1 +j < i, r ∈ [j, i − 1] +n +otherwise +To phrase this, it’s useful to think about the statistic +bij = +� +min({br − 1}r∈[i,j−1] ∪ {br}r /∈[i,j−1]) +i < j +min({br + 1}r∈[j,i−1] ∪ {br}r /∈[j,i−1]) +j < i +We’ll also include the restatement based on the fact that the root αij;n ∈ Nµ if and +only if the wall hi − hj = nc separates two Scopes chambers. +Corollary 4.2 We have a bound k+ +ij ≤ bij, with equality if µ + αij;n is dominant for +all i, j, n. +Proof. We have n > bij if and only if µ + αij;n ̸≤ Λ, so the corresponding weight space +is 0. If µ + αij;n is dominant, we conversely have that the weight space is not 0 if and +only if µ + αij;n ≤ Λ. +□ +The dominance condition is needed, as the level 1 examples show. +The description (4.1) allows us to easily identify a RoCK alcove for each Weyl cham- +ber: fix a dominant weight µ and a permutation σ, and consider the alcove where +hσ−1(i) − hσ−1(i+1) ≥ ui +i = 1, . . . e − 1 +hσ−1(e) − hσ−1(1) ≥ 1 − +i−1 +� +i=1 +ui +where ui = k− +σ−1(i),σ−1(i+1). This is the image of the fundamental alcove under the action +of the translation by the vector +vσ = (ℓ +e−1 +� +i=1 +ui, ℓ +e−1 +� +i=2 +ui, . . . , 0) +followed by the permutation σ−1. Note that this might not be an element of the affine +Weyl group since the sum of the entries of vσ might not be divisible by e; one can correct +this by precomposing with an appropriate power of the element of the extended affine +Weyl group υ(h1, . . . , he) = (he + 1, h1, . . . , he−1). Note that since we are considering +the projection of these vectors to the subspace h1 + · · · + he = 0, the eth power of +this transformation is trivial. This gives us a possibly different permutation σ′ and vσ′. +Note we can avoid this if we just require that ui ≥ k− +σ−1(i),σ−1(i+1) and choose these so +19 + +RoCK blocks for affine categorical representations +that the entries of vσ have sum divisible by e, though in this case, we won’t be finding +the closest RoCK block to the fundmental alcove. +Some care is needed here, because only the usual affine Weyl group, not the extended +one, acts on the weights of an arbitrary representation, so w−1µ would not make sense +for w in the extended affine Weyl group. That is, if we act by an element of the extended +affine Weyl group, the result will correspond to a block of a potentially different Ariki- +Koike algebra where we have multiplied the roots by qs for some s. +Assume we do choose u so that sum of the entries of vσ are divisible by e. In this +case, the block of weight w−1µ is a “stretching” of the block for σµ. Stretching further +can then be achieved by simply increasing the values of ui; for sufficiently large values, +this will clearly give a Rouquier block. It is easy to get mixed up by the fact that +we invert w: the alcoves wA we consider above are translates of the tip of a Weyl +chamber in a direction deep in that chamber, as shown in (3.1). After inverting, this +means we have translated far in one fixed direction to get t′ +1 < t′ +2 < · · · < t′ +e in +(t′ +1, . . . , t′ +e) = w−1(t1, . . . , te), independent of which Weyl chamber we chose. +The Weyl chamber is encoded by the permutation σ, or put differently, how we must +permute the residues of ti (mod ℓ) to match those of t′ +i. Note that the stabilizer Wλ +exactly reflects how many of these residues are the same, that is, the failure of this +permutation to be unique. In Lyle’s indexing set, this is reflected in the choice of the +vector a in [Lyl, §3.4]. +Finally, let us compare with the definition of Rouquier blocks in the sense of [Lyl, +§3.1]. According to this definition, a block is Rouquier if for every ℓ-partition corre- +sponding to a module in the block consists of Rouquier partitions (i.e. those which +correspond to an ℓ = 1 RoCK block). +Proposition 4.3 Every Rouquier block is a RoCK block in the sense of Definition 3.4, +and every RoCK block is Scopes equivalent to a Rouquier block. +Proof. Stretching a Rouquier block doesn’t change its Scopes class: If we +stretch a Rouquier block by a vector M satisfying M1 ≪ M2 ≪ · · · ≪ Me with +� Mi = 0, then [Lyl, Th. 3.20 & 3.21] constructs a decomposition equivalence between +the original block and the shifted block (which is Rouquier). This is accomplished by +studying an element of the extended affine Weyl group that sends the original block +to the stretched one, following the recipe of [Lyl, Lem. 3.8]. For a general choice of +Mi, this would be the action of an element of the extended affine Weyl group, but the +fact that � +i Mi = 0 guarantees that it is a product of reflections in the usual affine +Weyl group, i.e. the total number of times we have applied υ sums to 0, and so we can +commute them past the reflections in the finite Weyl group to get a reduced word in +the affine Weyl group. Each of these affine reflections satisfies the hypotheses of [Lyl, +Thm. 3.10]. That is, when we compare the two runners we will switch, the left one has +20 + +Ben Webster +so many more beads than the right one that it is impossible to shift a bead leftward, +i.e. the corresponding simple is highest weight for this root sl2. +Since this holds for all simples in the block, the hypothesis (2) of Lemma 3.2 holds, +and we can conclude that we never change the Scopes class of this block. +A generic stretching is RoCK: If we take any block, and stretch it by M1 ≪ +M2 ≪ · · · ≪ Me for sufficiently large parameters, then it will be RoCK by (4.1). +Rouquier ⇒ RoCK: Consider any Rouquier block. As discussed, we can stretch it +by M1 ≪ M2 ≪ · · · ≪ Me and preserve the Scopes equivalence class. This means that +its Scopes equivalence class contains a generic stretching and is RoCK. +RoCK ⇒ Rouquier: We discussed above how with appropriate choice of ui, we +can construct a RoCK block for any Weyl chamber which is Rouquier. +□ +Let us discuss the e = 2 case. In this case, Λ = w1Λ1 + w2Λ2. Since δ = α1 + α2, any +weight where we cannot remove δ is of the form µ = Λ − b2α2 or µ = Λ − b1α1. +The weight µ = Λ − b1α1 − b2α2 = (ℓ, b2 − b1 + w1 +2 , b1 − b2 − w1 +2 , b2) will be dominant +if w1/2 ≥ b1 − b2 ≥ −w2/2. In all these cases, the corresponding category is non-zero. +Note that b12 = min(b1 − 1, b2), b21 = min(b1 + 1, b2). Thus, we need only find The case +where some of these walls could potentially not count is when µ + α12;n or µ + α21;n +is not dominant. Let’s deal with the case where µ + α12;n not dominant (the other +case is similar). In this case, we must have α∨ +2 (µ) = w2 − 2b2 + 2b1 ∈ {0, 1}, and +µ + α12;n = Λ − (b1 − n − 1)α1 + (b2 − n)α2. This is not dominant, and the dominant +element of its orbit is +s2(µ + α12;n) = +� +Λ − (b1 − n − 1)α1 + (b2 − n − 1)α2 +w2 − 2b2 + 2b1 = 1 +Λ − (b1 − n − 1)α1 + (b2 − n − 2)α2 +w2 − 2b2 + 2b1 = 0 +In the former case, this shows we have no unexpected losses: we must have b2 ≥ b1, +so b12 = b1 − 1, and we find that s2(µ + α12;n) ≤ Λ for all n ≤ b12. On the other +hand, if b12 = b1 = 1 ≥ n > b2 − 2, then s2(µ + α12;n) ̸≤ Λ. This can only happen if +b1 = b2 = n + 1 and w2 = 0, in which case there is no Scopes wall at h1 − h2 = b12c. +Similarly, if b1 − b2 = w1 = 0, then h1 − h2 = −b21c is not a Scopes wall. +To summarize, the hyperplanes dividing Scopes chambers are +h1 − h2 = mc +− k+ +12 ≤ m ≤ k− +12. +k+ +12 = +� +b12 +w2 > 0 or b1 > b2 +b12 − 1 +w2 = b1 − b2 = 0 +k− +12 = +� +b21 +w1 > 0 or b2 > b1 +b21 − 1 +w1 = b2 − b1 = 0 +Note that the possible pairs of (k− +12, k+ +12) are +• (n, n), achieved if b2 = n < b1 or b1 = b2 = n + 1 and w1 = 0; +• (n − 1, n + 1), achieved if b1 = n < b2 or b1 = b2 = n and w2 = 0; +21 + +RoCK blocks for affine categorical representations +• (n, n + 1), achieved if b1 = b2 = n and w2 ̸= 0, w1 ̸= 0. +For higher rank, let us illustrate with an example of how to use our Sage program. +We’ll reproduce the example of [Lyl, §3.1]. In this case the quantum characteristic is +e = 4 and the multicharge (2, 0), and one can calculate that the 2-partition λ for this +example is +((10, 7, 6, 5, 5, 3, 3, 1, 1, 1), (16, 13, 10, 7, 7, 5, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1)). +We enter this into Sage, and use the existing functionality to find the block for this +2-partition: +sage: e=4 +sage: multicharge=(2,0) +sage: lam=PartitionTuple([[10, 7, 6, 5, 5, 3, 3, 1, 1, 1], [16, 13, 10, 7, +7, 5, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1]]) +sage: lam.block(e,multicharge) +{2: 34, 3: 32, 0: 25, 1: 32} +This tells us that µ = Λ4 + Λ2 − 32α1 − 34α2 − 32α3 − 25α4 (though note that the +values are out of order). We can now set the block equal to this value, and find the +corresponding dominant weight. The first sequence returned is a reduced word for a +minimal length w ∈ W such that wµ is dominant, so in this case, +w = s1s2s1s3s4s1s2s1s4s3s4s1s2s1s4s3s4s2s1s4s3s4s2. +sage: al={2: 34, 3: 32, 0: 25, 1: 32} +sage: find_dominant(al,e,multicharge) +[[1, 2, 1, 3, 0, 1, 2, 1, 0, 3, 0, 1, 2, 1, 0, 3, 0, 2, 1, 0, 3, 0, 2], +{2: 3, 3: 3, 0: 3, 1: 3}] +Alternatively, we can do step (1) of the algorithm from Section 3.2 directly using the +function find_dom_w_chamber, which returns the h-coordinates of a point in wA in +addition to the weight wµ: +sage: al={2: 34, 3: 32, 0: 25, 1: 32} +sage: find_dom_w_chamber(al,e,multicharge) +[[3/2, -3, 15/4, -3/4], {2: 3, 3: 3, 0: 3, 1: 3}] +The computer can now find the set Nµ using the function findN: +sage: dom=find_dom_w_chamber(al,e,multicharge)[1] +sage: dom +{2: 3, 3: 3, 0: 3, 1: 3} +sage: findN(dom,e,multicharge) +{(2, 1): 2, (3, 1): 3, (4, 1): 3, (1, 2): 2, (3, 2): 3, +(4, 2): 3, (1, 3): 2, (2, 3): 2, (4, 3): 2, (1, 4): 2, +(2, 4): 2, (3, 4): 2} +22 + +Ben Webster +The output here is a dictionary, associating the value k+ +ij = k− +ji to the key (i,j). To find +the Scopes chamber of this block, one checks the sign of all the corresponding coroots +on the weight γ = (1, 3/2, −3, 15/4, −3/4, 0), or generally on the alcove containing this +point, which is defined by the inequalities h1 ≥ h4 + 2 ≥ h2 + 4 ≥ h3 − 2 ≥ h1 − 1. As +we computed above we have {α41;3, α41;2, α41;1, α14;0, α14;1, α14;2} ⊂ Nµ, and so we have +Scopes walls at +h1 − h4 = m +m = −2, −1, 0, 1, 2, 3 +This tells us that on the Scopes chamber containing the point γ, we have 2 < h1−h4 < 3, +and this block is not RoCK. +It’s not necessary to perform these computations by hand; the full algorithm is au- +tomated by the function test_RoCK: +sage: al={2: 34, 3: 32, 0: 25, 1: 32} +sage: test_RoCK(al,e,multicharge) +False +There is also a version of this function with a verbose output that points out which +Scopes walls witness the failure to be RoCK: +sage: test_RoCK_verbose(al,e,multicharge) +The pair [1, 2] is OK since 9/2 is not between 2 and -2 . +The pair [1, 3] is OK since -9/4 is not between 3 and -2 . +The pair [2, 3] is OK since -27/4 is not between 3 and -2 . +There’s a problem with [1, 4] since 3 > 9/4 > -2 . +The pair [2, 4] is OK since -9/4 is not between 3 and -2 . +The pair [3, 4] is OK since 9/2 is not between 2 and -2 . +False +If you want to construct RoCK blocks, then the function RoCK_weight takes in an +alcove (in the form of a point in the alcove) and a fixed block, and finds the closest +alcove in the same Weyl chamber which is RoCK for the orbit of the block we fixed. +sage: Ral=RoCK_weight([3/2, -3, 15/4, -3/4],al,e,multicharge) +sage: Ral +{0: 45, 1: 42, 2: 34, 3: 42} +sage: test_RoCK_verbose(Ral,e,multicharge) +The pair [1, 2] is OK since 11/2 is not between 2 and -2 . +The pair [1, 3] is OK since -9/4 is not between 3 and -2 . +The pair [2, 3] is OK since -31/4 is not between 3 and -2 . +The pair [1, 4] is OK since 13/4 is not between 3 and -2 . +The pair [2, 4] is OK since -9/4 is not between 3 and -2 . +The pair [3, 4] is OK since 11/2 is not between 2 and -2 . +True +sage: weight_from_block(Ral,e,multicharge, 9) +[[13, 18, 1, 6], [10, 10, 9, 9]] +23 + +RoCK blocks for affine categorical representations +Note that we used the function weight_from_block to find the value of (t1, t2, t3, t4) +for this block. Finally, we can find an example of RoCK block for each Weyl chamber: +sage: all_RoCKs(al,e,multicharge) +{(3/4, 1/2, 1/4, 0): {0: 37, 1: 34, 2: 27, 3: 34}, +(3/4, 1/2, 0, 1/4): {0: 37, 1: 34, 2: 27, 3: 34}, +(3/4, 1/4, 1/2, 0): {0: 34, 1: 42, 2: 45, 3: 42}, +(3/4, 1/4, 0, 1/2): {0: 34, 1: 42, 2: 45, 3: 42}, +(3/4, 0, 1/2, 1/4): {0: 37, 1: 29, 2: 37, 3: 39}, +(3/4, 0, 1/4, 1/2): {0: 37, 1: 29, 2: 37, 3: 39}, +(1/2, 3/4, 1/4, 0): {0: 37, 1: 34, 2: 27, 3: 34}, +(1/2, 3/4, 0, 1/4): {0: 37, 1: 34, 2: 27, 3: 34}, +(1/2, 1/4, 3/4, 0): {0: 37, 1: 39, 2: 37, 3: 29}, +(1/2, 1/4, 0, 3/4): {0: 37, 1: 39, 2: 37, 3: 29}, +(1/2, 0, 3/4, 1/4): {0: 45, 1: 42, 2: 34, 3: 42}, +(1/2, 0, 1/4, 3/4): {0: 45, 1: 42, 2: 34, 3: 42}, +(1/4, 3/4, 1/2, 0): {0: 34, 1: 42, 2: 45, 3: 42}, +(1/4, 3/4, 0, 1/2): {0: 34, 1: 42, 2: 45, 3: 42}, +(1/4, 1/2, 3/4, 0): {0: 37, 1: 39, 2: 37, 3: 29}, +(1/4, 1/2, 0, 3/4): {0: 37, 1: 39, 2: 37, 3: 29}, +(1/4, 0, 3/4, 1/2): {0: 27, 1: 34, 2: 37, 3: 34}, +(1/4, 0, 1/2, 3/4): {0: 27, 1: 34, 2: 37, 3: 34}, +(0, 3/4, 1/2, 1/4): {0: 37, 1: 29, 2: 37, 3: 39}, +(0, 3/4, 1/4, 1/2): {0: 37, 1: 29, 2: 37, 3: 39}, +(0, 1/2, 3/4, 1/4): {0: 45, 1: 42, 2: 34, 3: 42}, +(0, 1/2, 1/4, 3/4): {0: 45, 1: 42, 2: 34, 3: 42}, +(0, 1/4, 3/4, 1/2): {0: 27, 1: 34, 2: 37, 3: 34}, +(0, 1/4, 1/2, 3/4): {0: 27, 1: 34, 2: 37, 3: 34}} +Note that we see each block repeated 4 times; these correspond to the orbits under the +order 4 stabilizer of the highest weight Λ generated by s1 and s3. +Now, let us consider the example of (3.1): +sage: e=3 +sage: multicharge=(0,0,1,2) +sage: al={0:3,1:2,2:2} +sage: all_RoCKs(al,e,multicharge) +{(2/3, 1/3, 0): {0: 11, 1: 18, 2: 18}, +(2/3, 0, 1/3): {0: 23, 1: 22, 2: 14}, +(1/3, 2/3, 0): {0: 23, 1: 14, 2: 22}, +(1/3, 0, 2/3): {0: 23, 1: 22, 2: 14}, +(0, 2/3, 1/3): {0: 23, 1: 14, 2: 22}, +(0, 1/3, 2/3): {0: 11, 1: 18, 2: 18}} +Again, note that the blocks appear in pairs due to the symmetry with respect to s3. +The chambers in the dominant Weyl chamber for Wµ are those with h3 ≥ h1, so the +last three entries. These can be identified with the red triangles of (3.1) by considering +24 + +Ben Webster +which simple roots are negative on each chamber: α2 on the first, α1 on the second, +both on the third. +To find the actual underlying blocks, we apply to each block the function weight_from_block +which tells us the values of ti for each block. Note that the last input simply shifts all +the values simultaneously, which we’ve done to keep the charge of each runner positive. +sage: weight_from_block({0: 3, 1: 2, 2: 2},e,multicharge,11) +[[14, 12, 10], [13, 12, 11]] +sage: weight_from_block({0: 11, 1: 18, 2: 18},e,multicharge,11) +[[6, 12, 18], [13, 12, 11]] +sage: weight_from_block({0: 23, 1: 22, 2: 14},e,multicharge,11) +[[14, 20, 2], [13, 12, 11]] +sage: weight_from_block({0: 23, 1: 14, 2: 22},e,multicharge,11) +[[22, 4, 10], [13, 12, 11]] +Thus, one example of a multipartition in the dominant block is given by ((1, 1), (2, 1), (1, 1), ∅), +whose abacus we show below: +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +The corresponding object in the different RoCK blocks above are given by +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +−−−−−−− +One can check by hand that the first of these blocks is Rouquier in the sense of [Lyl], +while the second and third become Rouquier after applying υ or υ2; note that this +changes the highest weight. +Alternatively, we can find a Rouquier block by apply- +ing Lemma 3.2 for transformations in the finite Weyl group to go to the blocks with +(t1, t2, t3) = (2, 14, 20), (4, 10, 22), respectively. +25 + +RoCK blocks for affine categorical representations +References +[Ari] +Susumu Ariki, On the decomposition numbers of the Hecke algebra of G(m,1,n), Journal of +Mathematics of Kyoto University 36, no. 4, 789–808. MR MR1443748 (98h:20012) +[BK] +Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and +Khovanov-Lauda algebras, Inventiones mathematicae 178, no. 3, 451–484. +[Bru] +Jonathan Brundan, On the definition of Kac-Moody 2-category, pp. 353–372. MR 3451390 +[BSWa] Jonathan Brundan, Alistair Savage, and Ben Webster, Heisenberg and Kac-Moody categorifi- +cation, Selecta Mathematica. New Series 26, no. 5, Paper No. 74, 62. MR 4169511 +[BSWb] +, On the definition of quantum Heisenberg category, Algebra & Number Theory 14, +no. 2, 275–321. MR 4195648 +[Cau] +Sabin Cautis, Clasp technology to knot homology via the affine Grassmannian, Mathematische +Annalen 363, no. 3-4, 1053–1115. MR 3412353 +[CK] +Joseph Chuang and Radha Kessar, Symmetric groups, wreath products, Morita equivalences, +and Brou´e’s abelian defect group conjecture, Bulletin of the London Mathematical Society 34, +no. 2, 174–185. +[CRa] +Joseph +Chuang +and +Raphael +Rouquier, +Perverse +Equivalences, +https://www.math.ucla.edu/~rouquier/papers/perverse.pdf. +[CRb] +Joseph Chuang and Rapha¨el Rouquier, Derived equivalences for symmetric groups and sl2- +categorification, Annals of Mathematics. Second Series 167, no. 1, 245–298. MR 2373155 +[Gro] +I. Grojnowski, Affine slp controls the representation theory of the symmetric group and related +Hecke algebras, arXiv:math/9907129. +[LM] +Sin´ead Lyle and Andrew Mathas, Blocks of cyclotomic Hecke algebras, Advances in Mathe- +matics 216, no. 2, 854–878. MR 2351381 +[Los] +Ivan Losev, Highest weight sl2 categorifications I: Crystals, Mathematische Zeitschrift 274, +no. 3-4, 1231–1247. MR 3078265 +[Lyl] +Sinead Lyle, Rouquier blocks for Ariki-Koike algebras, arXiv:2206.14720. +[MS] +Marco Mackaay and Alistair Savage, Degenerate cyclotomic Hecke algebras and higher level +Heisenberg categorification, Journal of Algebra 505, 150–193. +[Sco] +Joanna Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, +Journal of Algebra 142, no. 2, 441–455. MR 1127075 +[Sha] +Peng Shan, Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras, +Annales Scientifiques de l’´Ecole Normale Sup´erieure. Quatri`eme S´erie 44, no. 1, 147–182. +MR 2760196 (2012c:20009) +[Web] +Ben Webster, Rouquier’s conjecture and diagrammatic algebra, Forum of Mathematics. Sigma +5, e27, 71. MR 3732238 +26 + diff --git a/jtAzT4oBgHgl3EQfpf1N/content/tmp_files/load_file.txt b/jtAzT4oBgHgl3EQfpf1N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30ebee20b7b27e4fa5337b1bccc3505421fe349c --- /dev/null +++ b/jtAzT4oBgHgl3EQfpf1N/content/tmp_files/load_file.txt @@ -0,0 +1,1072 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf,len=1071 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='01613v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='RT] 4 Jan 2023 RoCK blocks for affine categorical representations Ben Webster1 Department of Pure Mathematics, University of Waterloo & Perimeter Institute for Theoretical Physics Waterloo, ON Email: ben.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='webster@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='ca Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Given a categorical action of a Lie algebra, a celebrated theorem of Chuang and Rouquier proves that the blocks corresponding to weight spaces in the same orbit of the Weyl group are derived equivalent, proving an even more celebrated conjecture of Brou´e for the case of the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In many cases, these derived equivalences are t-exact, and thus induce equivalences of abelian categories between different blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We call two such blocks “Scopes equivalent.” In this paper, we describe how Scopes equiva- lence classes for any affine categorification can be classified by the chambers of a finite hyperplane arrangement, which can be found through simple Lie theoretic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We pay special attention to the largest equivalence classes, which we call RoCK, and show how this matches with recent work of Lyle on Rouquier blocks for Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We also provide Sage code that tests whether blocks are RoCK and finds RoCK blocks for Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Introduction A remarkable theorem of Chuang and Rouquier [CRb], proving a conjecture of Brou´e, shows that any two blocks of of modules over FpSn and FpSm with the same defect group are derived equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The proof of this theorem runs through a remarkable fact about the theory of symmetric groups: that it is best understood in terms of the representation theory of the affine Lie algebra �slp, as pointed out in the title of [Gro].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In fact, Chuang and Rouquier prove a much more general theorem, of which Brou´e’s conjecture is a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' They use the notion of a categorical representation of sl2 (a strong sl2-categorification, in their terminology) on a category C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This data consists of: (1) a decomposition C ∼= � n∈Z Cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2) functors E: Cn → Cn+2 and F: Cn → Cn−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (3) certain special natural tranformations between compositions of these functors which force [E], [F] to satisfy the relations of sl2 on the level of the Grothendieck group, with Cn corresponding to the n-weight space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1Supported by NSERC through a Discovery Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This research was supported in part by Perime- ter Institute for Theoretical Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1 RoCK blocks for affine categorical representations Chuang and Rouquier then prove that for any categorical sl2-representation, we have an equivalence of derived categories Db(Cn) ∼= Db(C−n) categorifying the action of the unique element of the Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Their proof of the Brou´e conjecture for the symmet- ric group proceeds by successively applying different such equivalences for the sl2-actions given by i-induction and i-restriction for different i ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' These equivalences of derived categories are sometimes t-exact, and sometimes not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' If the Chuang-Rouquier equivalence is t-exact, then it induces an equivalence of cate- gories Cn ∼= C−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We call these Scopes equivalences, since in the case of the sym- metric group, they recover the Morita equivalences described by Scopes [Sco].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Using these equivalences, Scopes showed that only finitely different abelian categories up to equivalence appear amongst the blocks of a given defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In fact, “most” blocks are all equivalent as abelian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The blocks in this class are called RoCK (for Rouquier-Chuang-Kessar) or simply Rouquier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Our purpose in this paper is to describe how, in direct analogy with Chuang and Rouquier’s approach to the Brou´e conjecture, the theory of Scopes equivalences and RoCK blocks extend immediately to all categorical modules over an affine Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For any categorical module C over an affine Lie algebra g, we let its support be the set of weights µ such that Cµ ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' similarly, for a g-module, we let its support be the set of weights with non-zero weight spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let ¯g denote the corresponding finite dimensional Lie algebra, ¯h its Cartan and W, ¯W the corresponding affine and finite Weyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For each choice of C and dominant weight λ in the support of C with stabilizer Wλ, we’ll define a finite hyperplane ar- rangement in ¯h given by certain translates of coroot hyperplanes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the arrangement only depends on the support of C as a set (and thus will be the same for categorifical actions with the same support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We’ll call the chambers of this arrangement Scopes cham- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For each Weyl chamber of ¯W, there is a unique Scopes chamber which contains a translate of this Weyl chamber, which we call its RoCK chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We call two weights λ and λ′ in the support of C Scopes equivalent if there is a t-exact Chuang-Rouquier equivalence between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note, this is stronger than requiring that the blocks are equivalent as abelian categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' there are examples where the corresponding blocks are equivalent as categories, but not via a Chuang-Rouquier equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In analogy with Scopes’ results, we show: Theorem A For any categorical module C over an affine Lie algebra g and any domi- nant weight µ in its support: (1) The orbit {wµ}w∈W of µ under the affine Weyl group is the union of finitely many Scopes equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2) These classes are in bijection with orbits of Wλ on the Scopes chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, the Scopes equivalence classes of a categorical module only depend on its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2 Ben Webster Under this bijection, certain equivalence classes (as many as # ¯W) corespond to RoCK chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We call the weight spaces in these equivalence classes RoCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This definition is heavily inspired by work of Lyle on the case of Ariki-Koike alge- bras [Lyl] for the parameter q a root of unity with quantum characteristic e, with the parameters Qj all lying in the set qZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let wi be the multiplicity of qi in the multiset of Qj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The modules over the Ariki-Koike algebras ⊕nAKn(k, q, w) -mod categorify the simple module V (Λ) = V (w0Λ0 + · · · + we−1Λe−1) for �sle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In §3 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=', Lyle gives a definition of a Rouquier block for an Ariki-Koike algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' this is a purely combina- torial property of a weight in the support of V (Λ), so we can just as easily apply it to any other categorical module with this support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that while “Rouquier block” and “RoCK block” are usually used synonymously, here we use them to distinguish Lyle’s definition from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Theorem B For any categorical representation C of g = �sle with support V (Λ), the Scopes equivalence classes will coincide those for the Ariki-Koike algebra, and a Scopes equivalence class is RoCK if and only if it contains a Rouquier weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The proof of Theorem A depends on the proof of a generalization of [Lyl, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1] in the context of categorical representations (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2), confirming Lyle’s conjecture and upgrading the decomposition equivalences of [Lyl, §3] to Scopes (and thus Morita) equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Furthermore, the Scopes walls, which control Scopes equivalence and RoCKness, are very conducive to computer computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We’ve written a Sage program (available here on CoCalc) which tests whether blocks are RoCK and construct examples of RoCK blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As discussed above, these computations apply not just to Ariki-Koike algebras, but also to other categorifications with the same support, in particular, the cyclotomic q-Schur algebras and category O for Cherednik algebras of G(ℓ, 1, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Affine Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let g be an affine Lie algebra, h its abstract Cartan and W its Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Since there are multiple variations of this algebra, we should clarify that we take the abstract Kac-Moody algebra defined by a given affine Cartan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' That is, we assume that the simple coroots α∨ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , α∨ n ∈ h are linearly independent, as are the simple roots α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , αn ∈ h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Since the Cartan matrix of g has corank 1, this means that the span of the coroots is codimension 1 in h and the span of the roots is codimension 1 in h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' There is a unique primitive Z>0-linear combination δ = � δiαi which is non-zero, but perpendicular to all α∨ i ’s and similarly a unique primitive Z>0-linear combination δ∨ = � δ∨ i α∨ i which is non-zero, but perpendicular to all αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let ¯hR be the quotient of the R-span of α∨ i by the R-span of δ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The dual ¯h∗ R is the quotient of the R-span of αi by the R-span of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The symmetrized Cartan matrix 3 RoCK blocks for affine categorical representations defines an inner product on ¯hR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For each affine root α, let ¯α be its image in ¯h∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' These images are integer multiples of roots in a system corresponding to a finite dimension simple Lie algebra ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let ¯W be the Weyl group of this finite root system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The value of δ∨ is thus constant on the weights of any irreducible representation of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This invariant of a representation is called its level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The elements h∗ R where δ∨ > 0 is the Tits cone of g, the union of all the weights in the W-orbit of the dominant Weyl chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Readers will be familiar with the actions of W on a fixed level coset h∗ c = {λ ∈ h∗/Cδ | δ∨(λ) = c, α∨ i (λ) ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This subspace is a coset of ¯h∗ R, and thus inherits a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The reflection in any real root α of g becomes the usual geometric reflection in the hyperplane (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1) Hα = {λ ∈ h∗ c|α∨(λ) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This hyperplane is a coset of the vanishing set of ¯α∨ in h∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The hyperplanes Hα cut h∗ c into chambers called alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Each alcove corresponds to a Weyl chamber in the W-orbit of the dominant Weyl chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, we have the dominant alcove A = {λ ∈ h∗ c | α∨(λ) > 0 for all positive roots α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The Weyl group W acts simply transitively on the set of alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For any point o ∈ h∗ c, we can define a ¯W-action on h∗ c by ¯w(o+h) = o+ ¯wh for h ∈ ¯h∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Following convention, we will take o to be the unique point satisfying α∨ 1 (o) = · · · = α∨ n−1(o) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, this point is a vertex of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The action of any element w ∈ W on h∗ c can be factored uniquely as product of a translation τh by an element of ¯h∗ R, and element ¯w in ¯W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that ¯wτh = τwh ¯w, so this factorization can be taken in either order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The most important example for us will be �sle, the affine Lie algebra obtained from sle((t)) by taking the unique central extension, and adding a loop element ∂ which acts by t ∂ ∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The torus h can most easily be described as h = {(h0, h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he+1) ∈ Re+2 | h1 + · · · + he = 0} with the coroots α∨ n = (1, −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0, 1, 0) α∨ i = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 1, −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The full set of positive roots is thus αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n for i, j ∈ [1, e], the vector with hi = 1, hj = −1, and last coordinate −n, where n ≥ 0 if i < j, and n > 0 if i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that αe1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1 = αe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We can identify the same space with h∗ via inner product, and take the roots to be α0 = (0, −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0, 1, −1) αi = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 1, −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that δ = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0, −1) δ∨ = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0, 0) The reduced space is defined by ¯h ∼= ¯h∗ ∼= {(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he) ∈ Re | h1 + · · · + he = 0} 4 Ben Webster where again, the metric is the usual one induced by inner product on Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The action of si = sαi on h is thus given by the matrix I − αi · α∨ i , where we view αi as a column vector, and α∨ i as a row vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, the reflections si = sαi for i > 0 act by the usual permutation matrices on the coordinates hi and hi+1, and se = sαe acts by \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 0 0 · · 0 0 0 1 0 0 · · 0 1 0 0 0 1 · · 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 0 0 0 · · 1 0 0 −1 1 0 · · 0 0 0 1 −1 0 · · 0 1 1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb Considering the quotient by δ, we obtain h∗/Cδ = {(h0, h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he) ∈ Re+1 | h1 + · · · + he = 0}, and the induced action on this quotient is obtained by simply deleting the rightmost column and bottom row from each matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Fixing ourself on the level h∗ c, we fix the coordinate h0 = c, so si(c, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he) = � (c, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , hi+1, hi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he) i > 0 (c, he + c, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he−1, h1 − c) i = 0 Thus, se acts by reflection in the line h1 − he = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' More generally, the different roots of g act by reflection in the lines hi − hj = cm for m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' From this perspective, the Weyl group W acts by affine transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' this can be seen from the fact that it is generated by the copy of Se generated by s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , se−1, which acts linearly, and by translation by vectors in the root lattice of sle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the vectors X = {(ca1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , cae) | ai ∈ Z, a1 + · · · + ae = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The fundamental alcove is the set A = {h1 ≥ h2 ≥ · · · ≥ he ≥ h1 − c};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the last inequality follows by requiring α∨ e (c, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he, ∗) = c + he − h1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For example, if e = 2, then we have that h∗ c ∼= R is 1-dimensional, parameterized by h1 with h2 = −h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We have root hyperplanes at h1 − h2 = cm for m ∈ Z, or equivalently, h1 = c m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, the alcoves are given by the intervals Am = [c m 2 , c m+1 2 ], and the fundamental alcove is defined by h2 +c ≥ h1 ≥ h2 or equivalently h1 ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The chambers Am with m even are the image of A0 under the translations by cm (these are the even elements of the Weyl group) and those with m odd are the image of A0 under a reflection at h1 = c m+1 4 , the midpoint between these two chambers (these are all the odd elements of the affine Weyl group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' If e = 3, then the alcoves are equilateral triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Around each point in the root lattice, there are 6 triangles that touch that element, which form a hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For 0, this hexagon is defined by |hi − hj| ≤ 1 for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The translations in the root lattice act 5 RoCK blocks for affine categorical representations freely transitively on the set of these hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In fact, these are the Voronoi tesselation of the root lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This makes visible the factorization into a translation and an element of ¯W: a unique translation sends each alcove to one in the hexagon around the origin, which is the tip of one of the finite Weyl chambers, and then a unique element of ¯W sends this to the fundamental alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In the image below, the fundamental alcove is colored green, and its orbit under ¯W is colored red, and all the elements of root lattice that fit in the picture are marked with a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The other vertex points of alcoves are the elements of the weight lattice of sl3 which don’t lie in the root lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the other vertices of the fundamental alcove are ( 1 3, 1 3, −2 3) and ( 2 3, −1 3, −1 3): (1, 0, −1) (−1, 0, 1) (1, −1, 0) (−1, 1, 0) (0, −1, 1) (0, 1, −1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Scopes chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let N denote any finite set of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The hyperplanes Hα for α ∈ N cut h∗ c into finitely many chambers, which we call Scopes chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Every Scopes chamber is defined by choosing ǫα ∈ {±1} for α ∈ N, and considering the inequalities C = {h | ǫαα(h) ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1 (1) We have w · ∆+ ∩ N = w′ · ∆+ ∩ N if and only if wA and w′A lie in the same Scopes chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2) If wA and w′A lie in the same Scopes chamber then there is a sequence i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , ip such that for w′ = wsi1 · · · sip, and of k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , k, wsi1 · · · sikA lies in the same Scopes chamber as wA and w′A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (1) Note that α∨ is positive on wA if and only if w−1α∨ is positive on A, that is, if w−1α ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, w · ∆+ ∩ N is exactly the subset of N which is positive on 6 Ben Webster wA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' By definition, this subset is the same on another alcove if and only if they are in the same Scopes equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2) Let us prove this by induction on the number of root hyperplanes separating wA and w′A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' When this number is 0, w = w′ and the claim is tautological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let Hα be a hyperplane which is a facet of wA, and separates it from w′A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Then Hw−1α is a facet of A, and so w−1α is ±1 times a simple root αi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This means that the reflection across Hα can be written as sα = wsi1w−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, sαwA = wsi1A is a chamber separated from w′A by only the hyperplanes separating wA from w′A, excluding Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note, this means that it is not separated from wA or w′A by the hyperplane defined by an element of N, and so is in the same Scopes chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' By the inductive hypothesis applied to the chambers wsi1A and w′A, we can find i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , ip such that w′ = wsi1 · · · sip and wsi1 · · ·sikA is in the same Scopes chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' □ Every Scopes chamber C has an asymptotic cone ¯C = {h ∈ h | th ∈ C for t ≫ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 The asymptotic cone of a Scopes chamber is always a face of the hyper- plane arrangement ¯α = 0 for α ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For each finite Weyl chamber, there is a unique Scopes chamber containing it in its asymptotic cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The asymptotic cone of the this chamber is defined by ¯C = {h | ǫα¯α(h) ≥ 0}, since α(th) ≥ 0 for t ≫ 0 iff ¯α(h) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This asymptotic cone will thus contain a Weyl chamber if and only if ǫα = ǫβ whenever ¯α = ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Given a Weyl chamber, we can define corresponding ǫα to have the same sign as ¯α on the Weyl chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This will define the only Scopes chamber with our fixed Weyl chamber in its asymptotic cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' □ We call a Scopes chamber C RoCK if ¯C contains a finite Weyl chamber, or equiva- lently, C contains a translate of a finite Weyl chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' There are at most # ¯W RoCK chambers, one for each finite Weyl chamber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' if the cone ¯C contains a Weyl chamber c, we say that C or an alcove in C is RoCK for c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' One helpful way to think about these different RoCK chambers is to factor your Weyl group element into a translation and an element of ¯W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As discussed above, we can write w = τh ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Since ¯w sends A to one of the other chambers adjacent to o, the element h accounts for most of the hyperplanes separating wA from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' More precisely, w · o = h + o, so wA must be one of the chambers having h + o as a vertex, and ¯w only controls which of these chambers it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, if h + o is in the interior of a Scopes chamber, then all adjacent chambers are in the same Scopes chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, we can conclude that when h is deep inside a Weyl chamber, that is α∨(h) ≫ 0 for all roots α positive on the chamber, then the alcove wA will be in the corresponding RoCK chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 7 RoCK blocks for affine categorical representations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Categorical actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' While the fundamental input of this paper is a categorical action of a Lie algebra, we will not use this definition in a deep way, and thus will only give the basic facts we need here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' A categorical action of a Kac-Moody algebra g is a representation of a particular 2-category U(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This 2-category has: (0) object set given by the weight lattice Y of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (1) 1-morphisms generated by symbols Ei, Fi for i ranging of the simple roots of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' These act on weight lattice in the same way that Chevalley generators change weights: Ei : λ → λ + αi Fi : λ → λ − αi (2) For our purposes, a detailed description of the 2-morphisms is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' See [Bru] for a detailed discussion of the different possible generating sets and rela- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We will use 2 basic facts about categorical actions all of which follow from examination of the 2-morphisms: (i) The power Fk i is the direct sum of k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' isomorphic summands, which we denote by F(k) i and call the divided power functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (ii) For each simple root, there is a chain complex Θi : λ → siλ of 1-morphisms in U which is invertible up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' A version of this complex was defined in [CRb, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1] but we use the definition in [Cau, (3-4)] for more modern notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, a categorical action is an assignment of a category Cλ to each element of the weight lattice λ, a functor to each Ei and Fi, and natural transformations to each 2- morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, we can interpret Θi as as a functor Θi : Db(Cλ) → Db(Csiλ), which by the invertibility must be an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We’ll discuss the relevant examples of categorical actions in Section 4 when we cover their RoCK blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' RoCK blocks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Scopes equivalences from Chuang-Rouquier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Consider any integrable cat- egorical module C over g, and let ν be any weight of this module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Assume that α∨ i (ν) = k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1 The functor F(k) i : Cν → Csiν is an equivalence of abelian categories (that is, a Scopes equivalence) if and only if the category Cν+αi is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This lemma is a restatement of one special case of the [CRa, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='4]: the Chuang- Rouquier equivalences are perverse, and will be t-exact if and only if the perversity function is 0 for all simples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This is equivalent to the condition that Cν+αi is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We will give a more direct proof here that doesn’t depend on the notion of a perverse equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 8 Ben Webster Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' ⇒: If F(k) i is an equivalence of categories, then E(k) i is its left and right adjoint, and thus the inverse equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, for any object M ∈ Cν, we have M ∼= E(k) i F(k) i M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' On the other hand, if Cν+αi ̸= 0, then there is a highest weight object N in Cν+rαi for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The object F(r) i N is non-zero, and F(k) i E(k) i F(r) i N ∼= F(k) i E(k+r) i N⊕(k+r k ) ∼= F(r) i N⊕(k+r k ) 2 applying [Cau, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1] with µ = k + 2r, b = k + r, a = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This contradicts the claim that F(k) i is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' ⇐: Since Cν+αi is trivial, every object in Cν is killed by Ei, that is, it is highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This implies that the only term of the Rickard complex Θi which acts nontrivially is F(k) i in homological degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, on Cν, the actions of Θi and the derived functor of F(k) i coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The former is an equivalence of derived categories, and the latter is exact in the usual t-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This shows that F(k) i is an equivalence of abelian categories as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' □ Now, we restrict to the case where ν is positive level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' δ∨(ν) > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, there is a unique dominant weight µ = wν in the orbit of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Consider the set Nµ = {α ∈ ∆+ | Cµ+α ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The failure of the functor Θi to preserve the t-structure depends in a precise way on the structure of the representation of the sl2 generated by Ei, Fi on the root string ν + mαi for all m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Using the action of W, this is the same the structure of the root string through µ with respect to the root wαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 The following are equivalent: (i) The functor F(k) i : Cν → Csiν is an equivalence of abelian categories (that is, a Scopes equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (ii) The category Cµ+wαi is trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' wαi /∈ Nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (iii) The alcoves wA and wsiA lie in the same Scopes chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (1) ⇒ (2): If F(k) i is an equivalence of categories, then E(k) i is its left and right adjoint, and thus the inverse equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, for any object M ∈ Cν, we have M ∼= E(k) i F(k) i M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' On the other hand, if Cν+αi ̸= 0, then there is a highest weight object N in Cν+rαi for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The object F(r) i N is non-zero, and F(k) i E(k) i F(r) i N ∼= F(k) i E(k+r) i N⊕(k+r k ) ∼= F(r) i N⊕(k+r k ) 2 applying [Cau, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1] with µ = k + 2r, b = k + r, a = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This contradicts the claim that F(k) i is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2) ⇒ (1): The category Cµ+wαi is trivial if and only if the category Cν+αi is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, (2) implies that every object in Cµ is killed by Ei, that is, it is highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This implies that the only term of the Rickard complex Θi which acts nontrivially is F(k) i in homological degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, on Cν, the actions of Θi and the derived functor of 9 RoCK blocks for affine categorical representations F(k) i coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The former is an equivalence of derived categories, and the latter is exact in the usual t-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This shows that F(k) i is an equivalence of abelian categories as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2) ⇔ (3): The only hyperplane separating wA and wsiA is Hwαi, so these lie in a common Scopes chamber if and only if wαi /∈ Nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' □ This equivalence will induce a bijection of simple modules, which of course, matches the action of the Kashiwara operator f k i in the crystal structure on simples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2, we arrive at the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Consider w, w′ ∈ W and as above, µ dominant and ν = w−1µ, ν′ = (w′)−1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='3 If wA and w′A lie in the same Scopes chamber for the set Nµ, then we have an equivalence of abelian categories Cν′ ∼= Cν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Furthermore, some important examples of interest to us, such as Schur algebras in positive characteristic, (cyclotomic) q-Schur algebras, and categories O for Cherednik algebras of G(ℓ, 1, n) are highest weight categorifications in the sense of [Los].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, the Scopes equivalence sends standard filtered modules to standard filtered mod- ules, and thus induces an equivalence of highest weight categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For symmetric group, Hecke, and Ariki-Koike algebras, this implies that Specht filtrations are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In all these cases, we can index simples by abacus diagrams, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 will apply when for every object in our block, there is no way to push a bead from the (i + 1)st runner to the ith, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' for every bead on the (i + 1)st, the position to its left on the ith runner is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, k is the number of beads on the ith runner where to spot to the right is empty, and the Kashiwara operator f k i acts by pushing these beads right, that is, by swapping the runners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As usual, when i = e, we have to interpret all these statements as comparing the eth and first runners with a shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For example, in the picture below, we first show the i and (i + 1) runners of diagram where the Kashiwara operator ei acts non-trivially, to give the second picture where it acts non-trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' So Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 applies when all abacus diagrams in your block look like the second diagram and never like the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As discussed above, the Kashiwara operator f 2 i sends the second diagram to the third by moving the two dots that have space to move right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This has the same effect as swapping the runners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' −−−−−−− −−−−−−− −−−−−−− −−−−−−− −−−−−−− −−−−−−− −−−−−−− −−−−−−− −−−−−−− Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='4 We call a weight space category RoCK or a RoCK block if the corresponding Scopes chamber is RoCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' By construction, all RoCK blocks for a given Weyl chamber are equivalent as abelian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 10 Ben Webster We don’t aim here to comprehensively address the question of when weight categories are equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' nothing we have written above precludes the existence of an equivalence Cν′ ∼= Cν if wA and w′A do not lie in the same Scopes chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' There are two obvious situations where this can happen: (i) If Cµ+wαi is non-trivial, but the representations for the root sl2 for wαi generated by the µ weight space is isotypic (all its highest weight vectors are of the same weight), then some shift of the Chuang-Rouquier functor Θi is exact, and thus induces an equivalence of abelian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The easiest way this can happen is condition (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2, but it could be that all these highest weight vectors have weight µ + rwα for some r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, every projective object P in Cν can be written as P = F(r) i P ′, and there is an equivalence of abelian categories sending P �→ F(k+r) i P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We could tighten our results a bit by defining N′ µ to be the set of roots which do not have this isotypic property and only considering Scopes chambers with respect to this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This suffers from the issue of not being only determined by the support of the representation, and in the vast majority of cses, Nµ and N′ µ will be equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (ii) If µ has non-trivial stabilizer, then we can have ν′ = ν while wA and w′A do not lie in the same Scopes chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This tells us that the induced autoequivalence of Cν′ will not be exact, but this doesn’t change that the categories are exactly equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Of course, the stabilizer of µ is generated by sα for α∨(µ) = 0, which can only happen if α is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let Wµ be the stabilizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' we can simplify the deter- mination of the different blocks that show up in this case by only considering wA in a fundamental domain of Wµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the most natural choice is to consider the chamber cut out by α∨ = 0 for the α such that α∨(µ) = 0 which contains the fundamental alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This has the effect of requiring w to be a shortest right coset representative for Wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Irreducible support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Recall that the support of a categorical module C for g is the set of weights µ such that Cµ ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' by analogy, for a usual linear representation V of g, we will call the set of weights with non-zero weight spaces the support of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Most of the categorical modules appearing in representation theory (of which the author knows) satisfy the following property: (∗) The support of C is equal to the support of a simple highest weight module V (Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note, this will happen in many cases where the representation K0 C(C) is very much not irreducible, but the support of one simple summand contains the support of all other summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For example, if we consider a tensor product VΛ ⊗ VΛ′, this is typically not irreducible, but its support is the same as the Cartan component VΛ+Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 11 RoCK blocks for affine categorical representations This is a particularly nice situation since we can encode the support of C in purely combinatorial terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let ≤ denote the usual root order on weights, that is, the transi- tive closure of µ−αi ≤ µ for all µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let us recall one of the standard characterizations of the support of a simple highest weight module: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='5 The following are equivalent: (i) A weight µ is in the support of V (Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (ii) There is an element w ∈ W such that wµ is dominant and wµ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (iii) For all w ∈ W, we have wµ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, every µ in the support of V (Λ) is of the form µ = Λ − e � i=1 biαi and any dominant µ with this form is in the support by (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, given a simple root α, we wish to determine if µ + α is in the support of C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' if α ∈ Nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Obviously, this can only happen if µ + α ≤ Λ, and in some corner cases, µ + α will not be dominant, and we need to check the the dominant weight in its orbit also satisfies w(µ + α) ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that this implies that Nµ is finite in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This is easy to check for one orbit, but the reader will correctly note that the set of positive roots is infinite, making it hard to check by hand whether this holds for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' However, we can exploit the fact addition by δ commutes with every element of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Fix a set {β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , βr} ⊂ ∆+ of positive roots with the property that every affine root is of the form ±βi+kδ for a unique choice of sign, i ∈ [1, r] and k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that in twisted cases, not every element of this form is a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' So, we have that w ·(µ±βi +kδ) is dominant if and only if w · (µ ± βi) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For each i ∈ [1, r], we thus have integers k± i defined to be the largest integers such that w(µ ± βi) + k± i δ ≤ Λ ∀w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Of course, checking this for all w is equivalent to checking it only when w(µ ± βi) is dominant by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This implies that a positive root ±βi + kδ lies in Nµ if and only if k ≤ k± i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' However, as we noted before, there might be values of k where this is not a positive root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' If we assume our affine Lie algebra is simply laced, that is, it is of type �A, �D or �E, then we can simplify by choosing our βi to the positive roots in the finite type subalgebra, so the set of positive roots can be described as ∆+ = � i∈[1,r] m∈Z≥0 {βi + mδ, −βi + (m + 1)δ} and the resulting description of the Scopes chambers has a particularly nice form: 12 Ben Webster Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='6 The set Nµ is exactly the roots of the form ±βi + mδ with m ≤ k± i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The walls of the Scopes chambers are defined by β∨ i (λ) = mδ∨(λ) for all m ∈ [−k+ i , k− i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The combination of Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='6 give us an algorithm to test whether a block is RoCK, and to construct blocks which are RoCK for each Weyl chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Assume that property (∗) holds and fix a weight µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the steps of our algorithm are: (i) Find the dominant element of the orbit: Let w ∈ W be the element of minimal length such that µ′ = wµ is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Also find the alcove A′ = wA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Al- gorithmically, we can do this by checking the sign of α∨ i (µ) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' If these are all ≥ 0, then µ is dominant, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Otherwise, replace µ by siµ for any i with α∨ i (µ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that this terminates since µ < siµ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We can find A′ by simply acting with si on the alcove as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (ii) Find the set Nµ′: For each βi, we consider µ ± βi, find the dominant element of its W-orbit, and use this to find the bounds k± i , which specify all the Scopes walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (iii) Check the sign on A′: Check the value of the ratio ρ = β∨ i δ∨ on any element of A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' If −k+ i ≤ ρ ≤ k− i for any i then the block is not RoCK, and if this inequality does not hold for any i, the block is RoCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This algorithm sounds laborious, and indeed it is to do by hand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' however, it is extremely efficient to do by computer, especially if compared to any algorithm that requires enu- meration of the partitions in a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, its complexity does not significantly increase as we add e-hooks to a block, whereas enumerating partitions will get much worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This program is publicly available for Sage here on CoCalc, and some examples are shown later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let us now consider an example with e = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let Λ = 2Λ1 + Λ2 + Λ3 and µ = Λ − 2α1 − 2α2 − 3α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that in this case, Wµ = {1, s3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, k− 12 = 2, k− 13 = 3, k− 23 = 2, k+ 12 = 1, k+ 13 = 1, k+ 23 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, the Scopes walls are given by the red hyperplanes below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the fundamental alcove is colored green, and the closest blocks to the fundamental in each RoCK chamber is colored red if they are in the positive Weyl chamber for Wµ (the one containing the fundamental alcove), and blue if they are in the negative Weyl chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, there are three RoCK equivalence classes of blocks 13 RoCK blocks for affine categorical representations under Scopes equivalence, corresponding to the 3 red chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let Hn(k, q) be the Hecke algebra of Sn (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' type A) over the field k for a scalar q ∈ k, and let e be the quantum characteristic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the smallest integer such that qe−1 + · · · + 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, we will explain how we recover the original definition of RoCK blocks (those satisfying the conditions of [CK, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let g = �sle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The category � n≥0 Hn(k, q) -mod is a categorical g-module, categori- fying the simple module with highest weight Λe = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The action of Ei is by i-restriction and Fi by i-induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We can assign a weight to each partition ν by letting bi be the number of boxes of content i (mod e), and considering µν = Λe − e � i=1 biαi = (1, be − b1, b1 − b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , be−1 − be, be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, µ∅ = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0) is the highest weight appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that adding a e-hook only changes this weight by adding δ, so the image of µν in h∗ 1 is determined just by the core of µ, and there will be a unique e-core for each integral point of h∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For example, if e = 2, then h∗ 1 is parameterized by the value of h1 as before, and the 14 Ben Webster 2-cores index the points in the W-orbit of h1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' ∅ h1 = 1 h1 = 0 h1 = 2 h1 = −1 h1 = −2 Similarly, 3-cores appear index the integral points in h∗ 1 for �sl3: ∅ The weight µν is dominant if and only if the core is empty, since no other integral weight is in the fundamental alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' That is, if µν = Λe − kδ = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0, k) for k ≥ 0, the weight of the corresponding block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note, the corresponding category is the principal block over Hke(k, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The other prominent example of another categorical action with the same set of non- zero weights is the module categories over the q-Schur algebra appearing endomorphisms of the permutation modules over Hn(k, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The blocks of this algebra are in obvious bijection with those of Hn(k, q), and the category of modules has an induced categorical action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, the Scopes chambers are the same, and the RoCK blocks of the Schur algebra are the same as those of the Hecke algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Since any Young diagram has a box of content 0, the category CΛe−αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='m is trivial unless m > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' on the other hand, if m > 0 we can find a non-trivial object on this category given by a hook with removable boxes of content i and j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, the set N of roots such that Cµ+α is thus given by αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n for n < k, since µν + αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n = µ∅ − αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='k−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='6, the Scopes chambers are cut out by the hyperplanes hi − hj = nc n = k − 1, k − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , −k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 15 RoCK blocks for affine categorical representations Note that we could also derive this from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='5: Λe −kδ +αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='m is never dominant, and the dominant element in its Weyl group orbit is Λe − (k − m − 1)δ, so as desired, we must have m < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As discussed previously, we can describe all weights in this case in the form w−1µk for w a shortest right coset representative of Wλ = ¯W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In the dominant finite Weyl chamber, there is a single RoCK Scopes chamber given by the elements such that hi ≥ hi+1 +k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This is the usual RoCK condition on blocks (for example, introduced at the start of [CK, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We can visualize the Scopes chambers on blocks by drawing the corresponding core of weight w · µ∅ over the alcove wA for w a shortest right coset representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Our e = 2 example then becomes: ∅ h1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='5 h1 = 1 h1 = 2 h1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='5 h1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='5 h1 = 0 We’ve drawn walls of the form h1 ∈ Z with solid dots and h1 ∈ Z + 1 2 with open dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For a fixed k, the first k −1 alcoves are each a separate Scopes class, and then all others are RoCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Our e = 3 example becomes: ∅ We’ve drawn in the walls that separate k = 2 Scopes chambers in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As we increase k, we add in more and more translates of each of these hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Ariki-Koike algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Now, we turn to the more interesting case of Ariki-Koike algebras of level ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, as before, we have fixed q ∈ k \\ {0} of multiplicative order e > 1, and consider the Ariki-Koike algebra AKn( k, q, w) associated to the polynomial f(u) = � i∈Z/eZ (u − qi)wi 16 Ben Webster for fixed wi ∈ Z≥0 with � wi = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The categories ⊕nAKn(k, q, w) -mod carry a categorical action of �sle as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' It was proven by Ariki [Ari, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='5] that the Grothendieck group of this category is isomorphic to the highest weight simple VΛ for Λ = �e i=1 wiΛi ∈ h∗ ℓ where Λi = (1, e − i e , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , e − i e � �� � i times , − i e, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , − i e � �� � e−i times , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' These are fundamental weights since α∨ i (Λj) = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' A typical Λ will have all wi ≥ 0 and thus will be in the interior of the fundamental alcove, and have trivial stabilizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' These are a bit easier to write if we adopt the convention that we can denote weights by arbitrary elements of Rn+2, which we take to be equivalent to their orthogonal projection to the subspace h = {(h0, h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he+1) ∈ Re+2 | h1 + · · · + he = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, we can equally well write Λi = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 1 � �� � i times , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0 � �� � e−i times , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Ariki’s action on the Grothendieck group in fact reflects a categorical action of �sle on ⊕nAKn(k, q, w) -mod;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the fact that i-induction and i-restriction for a given i define a strong sl2 action is proven in [CRb, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The fact that this extends to a categor- ical action of �sle is effectively equivalent to the main theorem of [BK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We can more systematically construct the categorical action by i-induction and i-restriction functors using the formalism of quantum Heisenberg actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Such an action on Ariki-Koike algebras is defined in [BSWb, §6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The main theorem of [BSWa] implies that this gives us a categorical �sle-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We can apply the same principle to construct categorical actions of �sle on (i) cyclotomic degenerate affine Hecke algebras in characteristic e, assuming e is prime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (ii) cyclotomic q-Schur algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (iii) categories O of the rational Cherednik algebra of G(ℓ, 1, n) with κ = a/e as a fraction in least terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In case (1), we have a degenerate Heisenberg action by [MS, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' in case (3), the quantum Heisenberg action is constructed in [BSWb, §7], and case (2) is a limiting case of (3) (see, for example, [Web, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The categorical action in case (3) was first constructed by Shan [Sha], but for an unnecessarily restrictive choice of parameters and using a different formalism for categorical actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' All of these also have support equal to that of an irreducible representation of the form V (Λ): (1) The weight Λ is determined by the roots of the cyclotomic polynomial, by iden- tifying the elements of Fp with the Dynkin diagram of �slp, as in [BK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 17 RoCK blocks for affine categorical representations (2-3) Each block in these cases is a quasi-hereditary cover of a block of the Ariki-Koike algebra, so the we simply use the same weight combinatorics, and of course, obtain a non-zero block if and only if the corresponding block of Ariki-Koike is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Everything we say will below also applies to other categorifications of �sle with the same support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The blocks of AKn(k, q, w) again correspond to weight spaces: by [LM], the block of Specht module is determined by the number of bi boxes of residue i (mod e) in the corresponding charged multi-partition, and this also determines the weight by the formula: µν = Λ − e−1 � i=1 biαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note, when we associate an ℓ-tuple of runner diagrams to a simple as in [Lyl, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2] the total number ti of beads on the ith runners of the ℓ different abacus diagrams is given by ti = bi−1 −bi +�e j=i wi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This also allows us to write the weight µν in the coordinates hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We have µν = (ℓ, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , te, be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note, these coordinates are helpful for identifying the dominant weight in a given orbit: a weight is dominant if t1 ≥ · · · ≥ te ≥ t1 − ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Whereas in level 1, these were easy to identify as the principal blocks for ranks divisible by e (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the lowest rank where a given defect group appears), in higher levels, there are many more dominant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As mentioned in the introduction, [Lyl, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1] is effectively equivalent to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 in this context: since Ei acts by pushing beads from the (i + 1)st runner to the ith, the weight w−1λ+αi is not in the support of C if and only if it is never possible to push a bead in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This is only the case for an abacus diagram where the weight of the ith runner is less than the difference between the number of beads on the ith and (i + 1)st runners;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' this is precisely the hypothesis of [Lyl, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' To test whether a weight is RoCK, we simply apply the algorithm of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 to our weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In particular, we have to calculate the statistics k±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' in the type A case, we can encode these as k+ ij = k− ji = max{n | αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n ∈ Nµ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, the Scopes walls for Nµ will be of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1) hi − hj = nc n = [−k+ ij, k− ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' While applying the full algorithm is more precise, using simpler methods, we can give an upper bound on the set of Scopes walls that is easy to compute by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' To understand this more concretely, it’s useful to note that if n = 0, then αij = αi+· · · αj−1 and if n > 0, then αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n = αi + · · ·αe−1 + αe + α1 + · · · + αj−1 + (n − 1)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 18 Ben Webster That is: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1 The difference between the number b′ r of boxes of content r in a multipar- tition of weight µ + αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n = Λ − �e−1 i=0 b′ iαi compared to the number br in one of weight µ is: br − b′ r = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 n + 1 i < j, r ∈ [i, j − 1] n − 1 j < i, r ∈ [j, i − 1] n otherwise To phrase this, it’s useful to think about the statistic bij = � min({br − 1}r∈[i,j−1] ∪ {br}r /∈[i,j−1]) i < j min({br + 1}r∈[j,i−1] ∪ {br}r /∈[j,i−1]) j < i We’ll also include the restatement based on the fact that the root αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n ∈ Nµ if and only if the wall hi − hj = nc separates two Scopes chambers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 We have a bound k+ ij ≤ bij, with equality if µ + αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n is dominant for all i, j, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We have n > bij if and only if µ + αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n ̸≤ Λ, so the corresponding weight space is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' If µ + αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n is dominant, we conversely have that the weight space is not 0 if and only if µ + αij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' □ The dominance condition is needed, as the level 1 examples show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The description (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1) allows us to easily identify a RoCK alcove for each Weyl cham- ber: fix a dominant weight µ and a permutation σ, and consider the alcove where hσ−1(i) − hσ−1(i+1) ≥ ui i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' e − 1 hσ−1(e) − hσ−1(1) ≥ 1 − i−1 � i=1 ui where ui = k− σ−1(i),σ−1(i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This is the image of the fundamental alcove under the action of the translation by the vector vσ = (ℓ e−1 � i=1 ui, ℓ e−1 � i=2 ui, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , 0) followed by the permutation σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that this might not be an element of the affine Weyl group since the sum of the entries of vσ might not be divisible by e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' one can correct this by precomposing with an appropriate power of the element of the extended affine Weyl group υ(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he) = (he + 1, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , he−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that since we are considering the projection of these vectors to the subspace h1 + · · · + he = 0, the eth power of this transformation is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This gives us a possibly different permutation σ′ and vσ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note we can avoid this if we just require that ui ≥ k− σ−1(i),σ−1(i+1) and choose these so 19 RoCK blocks for affine categorical representations that the entries of vσ have sum divisible by e, though in this case, we won’t be finding the closest RoCK block to the fundmental alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Some care is needed here, because only the usual affine Weyl group, not the extended one, acts on the weights of an arbitrary representation, so w−1µ would not make sense for w in the extended affine Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' That is, if we act by an element of the extended affine Weyl group, the result will correspond to a block of a potentially different Ariki- Koike algebra where we have multiplied the roots by qs for some s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Assume we do choose u so that sum of the entries of vσ are divisible by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, the block of weight w−1µ is a “stretching” of the block for σµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Stretching further can then be achieved by simply increasing the values of ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' for sufficiently large values, this will clearly give a Rouquier block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' It is easy to get mixed up by the fact that we invert w: the alcoves wA we consider above are translates of the tip of a Weyl chamber in a direction deep in that chamber, as shown in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' After inverting, this means we have translated far in one fixed direction to get t′ 1 < t′ 2 < · · · < t′ e in (t′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , t′ e) = w−1(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' , te), independent of which Weyl chamber we chose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The Weyl chamber is encoded by the permutation σ, or put differently, how we must permute the residues of ti (mod ℓ) to match those of t′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that the stabilizer Wλ exactly reflects how many of these residues are the same, that is, the failure of this permutation to be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In Lyle’s indexing set, this is reflected in the choice of the vector a in [Lyl, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Finally, let us compare with the definition of Rouquier blocks in the sense of [Lyl, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' According to this definition, a block is Rouquier if for every ℓ-partition corre- sponding to a module in the block consists of Rouquier partitions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' those which correspond to an ℓ = 1 RoCK block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='3 Every Rouquier block is a RoCK block in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='4, and every RoCK block is Scopes equivalent to a Rouquier block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Stretching a Rouquier block doesn’t change its Scopes class: If we stretch a Rouquier block by a vector M satisfying M1 ≪ M2 ≪ · · · ≪ Me with � Mi = 0, then [Lyl, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='20 & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='21] constructs a decomposition equivalence between the original block and the shifted block (which is Rouquier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This is accomplished by studying an element of the extended affine Weyl group that sends the original block to the stretched one, following the recipe of [Lyl, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For a general choice of Mi, this would be the action of an element of the extended affine Weyl group, but the fact that � i Mi = 0 guarantees that it is a product of reflections in the usual affine Weyl group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the total number of times we have applied υ sums to 0, and so we can commute them past the reflections in the finite Weyl group to get a reduced word in the affine Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Each of these affine reflections satisfies the hypotheses of [Lyl, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' That is, when we compare the two runners we will switch, the left one has 20 Ben Webster so many more beads than the right one that it is impossible to shift a bead leftward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the corresponding simple is highest weight for this root sl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Since this holds for all simples in the block, the hypothesis (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 holds, and we can conclude that we never change the Scopes class of this block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' A generic stretching is RoCK: If we take any block, and stretch it by M1 ≪ M2 ≪ · · · ≪ Me for sufficiently large parameters, then it will be RoCK by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Rouquier ⇒ RoCK: Consider any Rouquier block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As discussed, we can stretch it by M1 ≪ M2 ≪ · · · ≪ Me and preserve the Scopes equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This means that its Scopes equivalence class contains a generic stretching and is RoCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' RoCK ⇒ Rouquier: We discussed above how with appropriate choice of ui, we can construct a RoCK block for any Weyl chamber which is Rouquier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' □ Let us discuss the e = 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, Λ = w1Λ1 + w2Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Since δ = α1 + α2, any weight where we cannot remove δ is of the form µ = Λ − b2α2 or µ = Λ − b1α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The weight µ = Λ − b1α1 − b2α2 = (ℓ, b2 − b1 + w1 2 , b1 − b2 − w1 2 , b2) will be dominant if w1/2 ≥ b1 − b2 ≥ −w2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In all these cases, the corresponding category is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that b12 = min(b1 − 1, b2), b21 = min(b1 + 1, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Thus, we need only find The case where some of these walls could potentially not count is when µ + α12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n or µ + α21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n is not dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Let’s deal with the case where µ + α12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n not dominant (the other case is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case, we must have α∨ 2 (µ) = w2 − 2b2 + 2b1 ∈ {0, 1}, and µ + α12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n = Λ − (b1 − n − 1)α1 + (b2 − n)α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This is not dominant, and the dominant element of its orbit is s2(µ + α12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n) = � Λ − (b1 − n − 1)α1 + (b2 − n − 1)α2 w2 − 2b2 + 2b1 = 1 Λ − (b1 − n − 1)α1 + (b2 − n − 2)α2 w2 − 2b2 + 2b1 = 0 In the former case, this shows we have no unexpected losses: we must have b2 ≥ b1, so b12 = b1 − 1, and we find that s2(µ + α12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n) ≤ Λ for all n ≤ b12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' On the other hand, if b12 = b1 = 1 ≥ n > b2 − 2, then s2(µ + α12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='n) ̸≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' This can only happen if b1 = b2 = n + 1 and w2 = 0, in which case there is no Scopes wall at h1 − h2 = b12c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Similarly, if b1 − b2 = w1 = 0, then h1 − h2 = −b21c is not a Scopes wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' To summarize, the hyperplanes dividing Scopes chambers are h1 − h2 = mc − k+ 12 ≤ m ≤ k− 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' k+ 12 = � b12 w2 > 0 or b1 > b2 b12 − 1 w2 = b1 − b2 = 0 k− 12 = � b21 w1 > 0 or b2 > b1 b21 − 1 w1 = b2 − b1 = 0 Note that the possible pairs of (k− 12, k+ 12) are (n, n), achieved if b2 = n < b1 or b1 = b2 = n + 1 and w1 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (n − 1, n + 1), achieved if b1 = n < b2 or b1 = b2 = n and w2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 21 RoCK blocks for affine categorical representations (n, n + 1), achieved if b1 = b2 = n and w2 ̸= 0, w1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' For higher rank, let us illustrate with an example of how to use our Sage program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We’ll reproduce the example of [Lyl, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' In this case the quantum characteristic is e = 4 and the multicharge (2, 0), and one can calculate that the 2-partition λ for this example is ((10, 7, 6, 5, 5, 3, 3, 1, 1, 1), (16, 13, 10, 7, 7, 5, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We enter this into Sage, and use the existing functionality to find the block for this 2-partition: sage: e=4 sage: multicharge=(2,0) sage: lam=PartitionTuple([[10, 7, 6, 5, 5, 3, 3, 1, 1, 1], [16, 13, 10, 7, 7, 5, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1]]) sage: lam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='block(e,multicharge) {2: 34, 3: 32, 0: 25, 1: 32} This tells us that µ = Λ4 + Λ2 − 32α1 − 34α2 − 32α3 − 25α4 (though note that the values are out of order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' We can now set the block equal to this value, and find the corresponding dominant weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The first sequence returned is a reduced word for a minimal length w ∈ W such that wµ is dominant, so in this case, w = s1s2s1s3s4s1s2s1s4s3s4s1s2s1s4s3s4s2s1s4s3s4s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' sage: al={2: 34, 3: 32, 0: 25, 1: 32} sage: find_dominant(al,e,multicharge) [[1, 2, 1, 3, 0, 1, 2, 1, 0, 3, 0, 1, 2, 1, 0, 3, 0, 2, 1, 0, 3, 0, 2], {2: 3, 3: 3, 0: 3, 1: 3}] Alternatively, we can do step (1) of the algorithm from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 directly using the function find_dom_w_chamber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' which returns the h-coordinates of a point in wA in addition to the weight wµ: sage: al={2: 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3: 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 0: 25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 32} sage: find_dom_w_chamber(al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge) [[3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' -3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 15/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' -3/4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' {2: 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3: 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 0: 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 3}] The computer can now find the set Nµ using the function findN: sage: dom=find_dom_w_chamber(al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge)[1] sage: dom {2: 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3: 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 0: 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 3} sage: findN(dom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge) {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1): 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1): 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1): 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2): 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2): 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2): 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3): 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3): 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3): 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4): 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4): 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4): 2} 22 Ben Webster The output here is a dictionary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' associating the value k+ ij = k− ji to the key (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' To find the Scopes chamber of this block, one checks the sign of all the corresponding coroots on the weight γ = (1, 3/2, −3, 15/4, −3/4, 0), or generally on the alcove containing this point, which is defined by the inequalities h1 ≥ h4 + 2 ≥ h2 + 4 ≥ h3 − 2 ≥ h1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' As we computed above we have {α41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='3, α41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2, α41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1, α14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='0, α14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1, α14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2} ⊂ Nµ, and so we have Scopes walls at h1 − h4 = m m = −2, −1, 0, 1, 2, 3 This tells us that on the Scopes chamber containing the point γ, we have 2 < h1−h4 < 3, and this block is not RoCK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' It’s not necessary to perform these computations by hand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' the full algorithm is au- tomated by the function test_RoCK: sage: al={2: 34, 3: 32, 0: 25, 1: 32} sage: test_RoCK(al,e,multicharge) False There is also a version of this function with a verbose output that points out which Scopes walls witness the failure to be RoCK: sage: test_RoCK_verbose(al,e,multicharge) The pair [1, 2] is OK since 9/2 is not between 2 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [1, 3] is OK since -9/4 is not between 3 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [2, 3] is OK since -27/4 is not between 3 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' There’s a problem with [1, 4] since 3 > 9/4 > -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [2, 4] is OK since -9/4 is not between 3 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [3, 4] is OK since 9/2 is not between 2 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' False If you want to construct RoCK blocks, then the function RoCK_weight takes in an alcove (in the form of a point in the alcove) and a fixed block, and finds the closest alcove in the same Weyl chamber which is RoCK for the orbit of the block we fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' sage: Ral=RoCK_weight([3/2, -3, 15/4, -3/4],al,e,multicharge) sage: Ral {0: 45, 1: 42, 2: 34, 3: 42} sage: test_RoCK_verbose(Ral,e,multicharge) The pair [1, 2] is OK since 11/2 is not between 2 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [1, 3] is OK since -9/4 is not between 3 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [2, 3] is OK since -31/4 is not between 3 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [1, 4] is OK since 13/4 is not between 3 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [2, 4] is OK since -9/4 is not between 3 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The pair [3, 4] is OK since 11/2 is not between 2 and -2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' True sage: weight_from_block(Ral,e,multicharge, 9) [[13, 18, 1, 6], [10, 10, 9, 9]] 23 RoCK blocks for affine categorical representations Note that we used the function weight_from_block to find the value of (t1, t2, t3, t4) for this block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' we can find an example of RoCK block for each Weyl chamber: sage: all_RoCKs(al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge) {(3/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1/2,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1/2): {0: 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2: 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3: 34},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3/4): {0: 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2: 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3: 34}} Note that we see each block repeated 4 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' these correspond to the orbits under the order 4 stabilizer of the highest weight Λ generated by s1 and s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Now, let us consider the example of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1): sage: e=3 sage: multicharge=(0,0,1,2) sage: al={0:3,1:2,2:2} sage: all_RoCKs(al,e,multicharge) {(2/3, 1/3, 0): {0: 11, 1: 18, 2: 18}, (2/3, 0, 1/3): {0: 23, 1: 22, 2: 14}, (1/3, 2/3, 0): {0: 23, 1: 14, 2: 22}, (1/3, 0, 2/3): {0: 23, 1: 22, 2: 14}, (0, 2/3, 1/3): {0: 23, 1: 14, 2: 22}, (0, 1/3, 2/3): {0: 11, 1: 18, 2: 18}} Again, note that the blocks appear in pairs due to the symmetry with respect to s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' The chambers in the dominant Weyl chamber for Wµ are those with h3 ≥ h1, so the last three entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' These can be identified with the red triangles of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='1) by considering 24 Ben Webster which simple roots are negative on each chamber: α2 on the first, α1 on the second, both on the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' To find the actual underlying blocks, we apply to each block the function weight_from_block which tells us the values of ti for each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Note that the last input simply shifts all the values simultaneously, which we’ve done to keep the charge of each runner positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' sage: weight_from_block({0: 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2: 2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='11) [[14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 11]] sage: weight_from_block({0: 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2: 18},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='11) [[6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 11]] sage: weight_from_block({0: 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2: 14},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='11) [[14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 11]] sage: weight_from_block({0: 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1: 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2: 22},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='multicharge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='11) [[22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 11]] Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' one example of a multipartition in the dominant block is given by ((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' ∅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='whose abacus we show below: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='The corresponding object in the different RoCK blocks above are given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='−−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='One can check by hand that the first of these blocks is Rouquier in the sense of [Lyl],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' while the second and third become Rouquier after applying υ or υ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' note that this changes the highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Alternatively, we can find a Rouquier block by apply- ing Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='2 for transformations in the finite Weyl group to go to the blocks with (t1, t2, t3) = (2, 14, 20), (4, 10, 22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 25 RoCK blocks for affine categorical representations References [Ari] Susumu Ariki, On the decomposition numbers of the Hecke algebra of G(m,1,n), Journal of Mathematics of Kyoto University 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 4, 789–808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR MR1443748 (98h:20012) [BK] Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Inventiones mathematicae 178, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3, 451–484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [Bru] Jonathan Brundan, On the definition of Kac-Moody 2-category, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 353–372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 3451390 [BSWa] Jonathan Brundan, Alistair Savage, and Ben Webster, Heisenberg and Kac-Moody categorifi- cation, Selecta Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' New Series 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 5, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 74, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 4169511 [BSWb] , On the definition of quantum Heisenberg category, Algebra & Number Theory 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2, 275–321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 4195648 [Cau] Sabin Cautis, Clasp technology to knot homology via the affine Grassmannian, Mathematische Annalen 363, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3-4, 1053–1115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 3412353 [CK] Joseph Chuang and Radha Kessar, Symmetric groups, wreath products, Morita equivalences, and Brou´e’s abelian defect group conjecture, Bulletin of the London Mathematical Society 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2, 174–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [CRa] Joseph Chuang and Raphael Rouquier, Perverse Equivalences, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='edu/~rouquier/papers/perverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [CRb] Joseph Chuang and Rapha¨el Rouquier, Derived equivalences for symmetric groups and sl2- categorification, Annals of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Second Series 167, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1, 245–298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 2373155 [Gro] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Grojnowski, Affine slp controls the representation theory of the symmetric group and related Hecke algebras, arXiv:math/9907129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [LM] Sin´ead Lyle and Andrew Mathas, Blocks of cyclotomic Hecke algebras, Advances in Mathe- matics 216, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2, 854–878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 2351381 [Los] Ivan Losev, Highest weight sl2 categorifications I: Crystals, Mathematische Zeitschrift 274, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 3-4, 1231–1247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 3078265 [Lyl] Sinead Lyle, Rouquier blocks for Ariki-Koike algebras, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content='14720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [MS] Marco Mackaay and Alistair Savage, Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification, Journal of Algebra 505, 150–193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' [Sco] Joanna Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, Journal of Algebra 142, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 2, 441–455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 1127075 [Sha] Peng Shan, Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras, Annales Scientifiques de l’´Ecole Normale Sup´erieure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Quatri`eme S´erie 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' 1, 147–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 2760196 (2012c:20009) [Web] Ben Webster, Rouquier’s conjecture and diagrammatic algebra, Forum of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' Sigma 5, e27, 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} +page_content=' MR 3732238 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtAzT4oBgHgl3EQfpf1N/content/2301.01613v1.pdf'} diff --git 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b/samples/content/test.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3a137ad1381578ddc4ced057aa764e2cbb51be9d Binary files /dev/null and b/samples/content/test.pdf differ diff --git a/samples/isssues_merge/langchain-ChatGLM_closed.csv b/samples/isssues_merge/langchain-ChatGLM_closed.csv new file mode 100644 index 0000000000000000000000000000000000000000..b1ca2ac9d64563278ec19323efafe70382820d5f --- /dev/null +++ b/samples/isssues_merge/langchain-ChatGLM_closed.csv @@ -0,0 +1,173 @@ +,title,file,url,detail,id +0,加油~以及一些建议,2023-03-31.0002,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/2,加油，我认为你的方向是对的。,0 +1,当前的运行环境是什么，windows还是Linux,2023-04-01.0003,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/3,当前的运行环境是什么，windows还是Linux，python是什么版本？,1 +2,请问这是在CLM基础上运行吗？,2023-04-01.0004,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/4,请问是不是需要本地安装好clm并正常运行的情况下，再按文中的步骤执行才能运行起来？,2 +3,[复现问题] 构造 prompt 时从知识库中提取的文字乱码,2023-04-01.0005,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/5,hi，我在尝试复现 README 中的效果，也使用了 ChatGLM-6B 的 README 作为输入文本，但发现从知识库中提取的文字是乱码，导致构造的 prompt 不可用。想了解如何解决这个问题。,3 +4,后面能否加入上下文对话功能？,2023-04-02.0006,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/6,目前的get_wiki_agent_answer函数中已经实现了历史消息传递的功能，后面我再确认一下是否有langchain中model调用过程中是否传递了chat_history。,4 +5,请问：纯cpu可以吗？,2023-04-03.0007,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/7,很酷的实现，极大地开拓了我的眼界！很顺利的在gpu机器上运行了,5 +6,运行报错：AttributeError: 'NoneType' object has no attribute 'message_types_by_name',2023-04-03.0008,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/8,报错：,6 +7,运行环境：GPU需要多大的？,2023-04-03.0009,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/9,如果按照THUDM/ChatGLM-6B的说法，使用的GPU大小应该在13GB左右，但运行脚本后，占用了24GB还不够。,7 +8,请问本地知识的格式是什么？,2023-04-03.0010,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/10,已测试格式包括docx、md文件中的文本信息，具体格式可以参考 [langchain文档](https://python.langchain.com/en/latest/modules/indexes/document_loaders/examples/unstructured_file.html?highlight=pdf#),8 +9,24G的显存还是爆掉了，是否支持双卡运行,2023-04-03.0011,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/11,RuntimeError: CUDA out of memory. Tried to allocate 96.00 MiB (GPU 0; 23.70 GiB total capacity; 22.18 GiB already allocated; 12.75 MiB free; 22.18 GiB reserved in total by PyTorch) If reserved memory is >> allocated memory try setting max_split_size_mb to avoid fragmentation. See documentation for Memory Management and PYTORCH_CUDA_ALLOC_CONF,9 +10,你怎么知道embeddings方式和模型训练时候的方式是一样的?,2023-04-03.0012,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/12,embedding和LLM的方式不用一致，embedding能够解决语义检索的需求就行。这个项目里用到embedding是在对本地知识建立索引和对问句转换成向量的过程。,10 +11,是否能提供本地知识文件的格式？,2023-04-04.0013,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/13,是否能提供本地知识文件的格式？,11 +12,是否可以像清华原版跑在8G一以下的卡？,2023-04-04.0016,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/16,是否可以像清华原版跑在8G一以下的卡？我的8G卡爆显存了🤣🤣🤣,12 +13,请教一下langchain协调使用向量库和chatGLM工作的,2023-04-05.0018,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/18,代码里面这段是创建问答模型的，会接入ChatGLM和本地语料的向量库，langchain回答的时候是怎么个优先顺序？先搜向量库，没有再找chatglm么？ 还是什么机制？,13 +14,在mac m2max上抛出了ValueError: 150001 is not in list这个异常,2023-04-05.0019,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/19,我把chatglm_llm.py加载模型的代码改成如下,14 +15,程序运行后一直卡住,2023-04-05.0020,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/20,感谢作者的付出，不过本人在运行时出现了问题，请大家帮助。,15 +16,问一下chat_history的逻辑,2023-04-06.0022,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/22,感谢开源。,16 +17,为什么每次运行都会loading checkpoint,2023-04-06.0023,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/23,我把这个embeding模型下载到本地后，无法正常启动。,17 +18,本地知识文件能否上传一些示例？,2023-04-06.0025,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/25,如题，怎么构造知识文件，效果更好？能否提供一个样例,18 +19,What version of you are using?,2023-04-06.0026,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/26,"Hi Panda, I saw the `pip install -r requirements` command in README, and want to confirm you are using python2 or python3? because my pip and pip3 version are all is 22.3.",19 +20,有兴趣交流本项目应用的朋友可以加一下微信群,2023-04-07.0027,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/27,![IMG_1630](https://user-images.githubusercontent.com/5668498/230533162-8b9bfcdd-249c-4efe-b066-4f9ba2ce9f23.jpeg),20 +21,本地知识越多，回答时检索的时间是否会越长,2023-04-07.0029,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/29,是的 因为需要进行向量匹配检索,21 +22,爲啥最後還是報錯 哭。。,2023-04-07.0030,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/30,Failed to import transformers.models.t5.configuration_t5 because of the following error (look up to see,22 +23,对话到第二次的时候就报错UnicodeDecodeError: 'utf-8' codec can't decode,2023-04-07.0031,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/31,对话第一次是没问题的，模型返回输出后又给到请输入你的问题，我再输入问题就报错,23 +24,用的in4的量化版本，推理的时候显示需要申请10Gb的显存,2023-04-07.0033,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/33,"File ""/root/.cache/huggingface/modules/transformers_modules/chatglm-6b-int4-qe/modeling_chatglm.py"", line 581, in forward",24 +25,使用colab运行，python3.9，提示包导入有问题,2023-04-07.0034,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/34,"from ._util import is_directory, is_path",25 +26,运行失败，Loading checkpoint未达到100%被kill了，请问下是什么原因？,2023-04-07.0035,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/35,日志如下：,26 +27,弄了个交流群，自己弄好多细节不会，大家技术讨论 加connection-image 我来拉你,2023-04-08.0036,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/36,自己搞好多不清楚的，一起来弄吧。。准备搞个部署问题的解决文档出来,27 +28,Error using the new version with langchain,2023-04-09.0043,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/43,Error with the new changes:,28 +29,程序报错torch.cuda.OutOfMemoryError如何解决？,2023-04-10.0044,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/44,报错详细信息如下：,29 +30,qa的训练数据格式是如何设置的,2023-04-10.0045,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/45,本项目不是使用微调的方式，所以并不涉及到训练过程。,30 +31,The FileType.UNK file type is not supported in partition. 解决办法,2023-04-10.0046,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/46,ValueError: Invalid file /home/yawu/Documents/langchain-ChatGLM-master/data. The FileType.UNK file type is not supported in partition.,31 +32,如何读取多个txt文档？,2023-04-10.0047,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/47,如题，请教一下如何读取多个txt文档？示例代码中只给了读一个文档的案例，这个input我换成string之后也只能指定一个文档，无法用通配符指定多个文档，也无法传入多个文件路径的列表。,32 +33,nltk package unable to either download or load local nltk_data folder,2023-04-10.0049,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/49,I'm running this project on an offline Windows Server environment so I download the Punkt and averaged_perceptron_tagger tokenizer in this directory:,33 +34,requirements.txt中需要指定langchain版本,2023-04-11.0055,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/55,langchain版本0.116下无法引入RetrievalQA，需要指定更高版本（0.136版本下无问题）,34 +35,Demo演示无法给出输出内容,2023-04-12.0059,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/59,你好，测试了项目自带新闻稿示例和自行上传的一个文本，可以加载进去，但是无法给出答案，请问属于什么情况，如何解决，谢谢。PS: 1、今天早上刚下载全部代码；2、硬件服务器满足要求；3、按操作说明正常操作。,35 +36,群人数过多无法进群，求帮忙拉进群,2023-04-12.0061,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/61,您好，您的群人数超过了200人，目前无法通过二维码加群，请问您方便加我微信拉我进群吗？万分感谢,36 +37,群人数已满，求大佬拉入群,2023-04-12.0062,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/62,已在README中更新拉群二维码,37 +38,requirements中langchain版本错误,2023-04-12.0065,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/65,langchain版本应该是0.0.12而不是0.0.120,38 +39,Linux : Searchd in,2023-04-13.0068,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/68,import nltk,39 +40,No sentence-transformers model found,2023-04-13.0069,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/69,加载不了这个模型，错误原因是找不到这个模型，但是路径是配置好了的,40 +41,Error loading punkt: ",58 +59,为啥放到方法调用会出错，这个怎么处理？,2023-04-20.0150,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/150,```python,59 +60,No sentence-transformers model found with name C:\Users\Administrator/.cache\torch\sentence_transformers\GanymedeNil_text2vec-large-chinese. Creating a new one with MEAN pooling.,2023-04-21.0154,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/154,卡在这块很久是正常现象吗,60 +61,微信群需要邀请才能加入,2023-04-21.0155,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/155,RT，给个个人联系方式白,61 +62,No sentence-transformers model found with name GanymedeNil/text2vec-large-chinese. Creating a new one with MEAN pooling,2023-04-21.0156,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/156,ls GanymedeNil/text2vec-large-chinese,62 +63,embedding会加载两次,2023-04-23.0159,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/159,你好，为什么要这样设置呢，这样会加载两次呀。,63 +64,扫二维码加的那个群，群成员满了进不去了,2023-04-23.0160,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/160,如题,64 +65,执行python3 cli_demo.py 报错AttributeError: 'NoneType' object has no attribute 'chat',2023-04-24.0163,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/163,"刚开始怀疑是内存不足问题，换成int4,int4-qe也不行，有人知道是什么原因吗",65 +66,匹配得分,2023-04-24.0167,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/167,在示例cli_demo.py中返回的匹配文本没有对应的score，可以加上这个feature吗,66 +67,大佬有计划往web_ui.py加入打字机功能吗,2023-04-25.0170,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/170,目前在载入了知识库后，单张V100 32G在回答垂直领域的问题时也需要20S以上，没有打字机逐字输出的使用体验还是比较煎熬的....,67 +68,Is it possible to use a verctorDB for the embedings?,2023-04-25.0171,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/171,"when I play, I have to load the local data again and again when to start. I wonder if it is possible to use",68 +69,请问通过lora训练官方模型得到的微调模型文件该如何加载？,2023-04-25.0173,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/173,通过lora训练的方式得到以下文件:,69 +70,from langchain.chains import RetrievalQA的代码在哪里？,2023-04-25.0174,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/174,local_doc_qa.py,70 +71,哪里有knowledge_based_chatglm.py文件？怎么找不到了？？是被替换成cli_demo.py文件了吗？,2023-04-26.0175,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/175,哪里有knowledge_based_chatglm.py文件？怎么找不到了？？是被替换成cli_demo.py文件了吗？,71 +72,AttributeError: 'Chatbot' object has no attribute 'value',2023-04-26.0177,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/177,Traceback (most recent call last):,72 +73,控制台调api.py报警告,2023-04-26.0178,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/178,"you must pass the application as an import string to enable ""reload"" or ""workers""",73 +74,如何加入群聊,2023-04-27.0183,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/183,微信群超过200人了，需要邀请，如何加入呢？,74 +75,如何将Chatglm和本地知识相结合,2023-04-27.0185,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/185,您好，我想请教一下怎么才能让知识库匹配到的文本和chatglm生成的相结合，而不是说如果没搜索到，就说根据已知信息无法回答该问题，谢谢,75 +76,一点建议,2023-04-27.0189,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/189,1.weiui的get_vector_store方法里面添加一个判断以兼容gradio版本导致的上传异常,76 +77,windows环境下，按照教程，配置好conda环境，git完项目，修改完模型路径相关内容后，运行demo报错缺少,2023-04-28.0194,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/194,报错代码如下：,77 +78,ValueError: too many values to unpack (expected 2),2023-04-28.0198,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/198,"When i tried to use the non-streaming, `ValueError: too many values to unpack (expected 2)` error came out.",78 +79,加载doc后覆盖原本知识,2023-04-28.0201,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/201,加载较大量级的私有知识库后，原本的知识会被覆盖,79 +80,自定义知识库回答效果很差,2023-04-28.0203,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/203,"请问加了自定义知识库知识库,回答效果很差，是因为数据量太小的原因么",80 +81,python310下，安装pycocotools失败，提示低版本cython，实际已安装高版本,2023-04-29.0208,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/208,RT，纯离线环境安装，依赖安装的十分艰难，最后碰到pycocotools，始终无法安装上，求教方法！,81 +82,[FEATURE] 支持 RWKV 模型（目前已有 pip package & rwkv.cpp 等等）,2023-05-01.0216,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/216,您好，我是 RWKV 的作者，介绍见：https://zhuanlan.zhihu.com/p/626083366,82 +83,[BUG] 为啥主机/服务器不联网不能正常启动服务？,2023-05-02.0220,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/220,**问题描述 / Problem Description**,83 +84,[BUG] 简洁阐述问题 / Concise description of the issue,2023-05-03.0222,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/222,**local variable 'torch' referenced before assignment**,84 +85,不支持txt文件的中文输入,2023-05-04.0235,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/235,"vs_path, _ = local_doc_qa.init_knowledge_vector_store(filepath)",85 +86,文件均未成功加载，请检查依赖包或替换为其他文件再次上传。 文件未成功加载，请重新上传文件,2023-05-05.0237,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/237,请大佬帮忙解决，谢谢！,86 +87,[BUG] 使用多卡时chatglm模型加载两次,2023-05-05.0241,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/241,chatglm_llm.py文件下第129行先加载了一次chatglm模型，第143行又加载了一次,87 +88,[BUG] similarity_search_with_score_by_vector函数返回多个doc时的score结果错误,2023-05-06.0252,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/252,**问题描述 / Problem Description**,88 +89,可以再建一个交流群吗，这个群满了进不去。,2023-05-06.0255,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/255,上午应该已经在readme里更新过了，如果不能添加可能是网页缓存问题，可以试试看直接扫描img/qr_code_12.jpg,89 +90,请问这是什么错误哇？KeyError: 'serialized_input',2023-05-06.0257,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/257,运行“python webui.py” 后这是什么错误？怎么解决啊？,90 +91,修改哪里的代码，可以再cpu上跑？,2023-05-06.0258,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/258,**问题描述 / Problem Description**,91 +92,ModuleNotFoundError: No module named 'modelscope',2023-05-07.0266,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/266,安装这个,92 +93,加载lora微调模型时，lora参数加载成功，但显示模型未成功加载？,2023-05-08.0270,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/270,什么原因呀？,93 +94,[BUG] 运行webui.py报错：name 'EMBEDDING_DEVICE' is not defined,2023-05-08.0274,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/274,解决了，我修改model_config时候把这个变量改错了,94 +95,基于ptuning训练完成，新老模型都进行了加载，但是只有新的,2023-05-08.0280,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/280,licitly passing a `revision` is encouraged when loading a model with custom code to ensure no malicious code has been contributed in a newer revision.,95 +96,[BUG] 使用chatyuan模型时，对话Error，has no attribute 'stream_chat',2023-05-08.0282,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/282,**问题描述 / Problem Description**,96 +97,chaglm调用过程中 _call提示有一个 stop,2023-05-09.0286,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/286,**功能描述 / Feature Description**,97 +98,Logger._log() got an unexpected keyword argument 'end',2023-05-10.0295,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/295,使用cli_demo的时候，加载一个普通txt文件，输入问题后，报错：“TypeError: Logger._log() got an unexpected keyword argument 'end'”,98 +99,[BUG] 请问可以解释下这个FAISS.similarity_search_with_score_by_vector = similarity_search_with_score_by_vector的目的吗,2023-05-10.0296,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/296,我不太明白这个库自己写的similarity_search_with_score_by_vector方法做的事情，因为langchain原版的similarity_search_with_score_by_vector只是search faiss之后把返回的topk句子组合起来。我觉得原版理解起来没什么问题，但是这个库里自己写的我就没太看明白多做了什么其他的事情，因为没有注释。,99 +100,[BUG] Windows下上传中文文件名文件，faiss无法生成向量数据库文件,2023-05-11.0318,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/318,**问题描述 / Problem Description**,100 +101,cli_demo中的流式输出能否接着前一答案输出?,2023-05-11.0320,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/320,现有流式输出结果样式为:,101 +102,内网部署时网页无法加载，能否增加离线静态资源,2023-05-12.0326,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/326,内网部署时网页无法加载，能否增加离线静态资源,102 +103,我想把文件字符的编码格式改为encoding='utf-8'在哪修改呢，因为会有ascii codec can't decode byte报错,2023-05-14.0360,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/360,上传中文的txt文件时报错，编码格式为utf-8,103 +104,Batches的进度条是在哪里设置的?能否关闭显示?,2023-05-15.0366,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/366,"使用cli_demo.py进行命令行测试时,每句回答前都有个Batches的进度条",104 +105,ImportError: dlopen: cannot load any more object with static TLS or Segmentation fault,2023-05-15.0368,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/368,**问题描述 / Problem Description**,105 +106,读取PDF时报错,2023-05-16.0373,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/373,在Colab上执行cli_demo.py时，在路径文件夹里放了pdf文件，在加载的过程中会显示错误，然后无法加载PDF文件,106 +107,[BUG] webui报错 InvalidURL,2023-05-16.0375,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/375,python 版本：3.8.16,107 +108,[FEATURE] 如果让回答不包含出处，应该怎么处理,2023-05-16.0380,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/380,**功能描述 / Feature Description**,108 +109,加载PDF文件时，出现 unsupported colorspace for 'png',2023-05-16.0381,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/381,**问题描述 / Problem Description**,109 +110,'ascii' codec can't encode characters in position 14-44: ordinal not in range(128) 经典bug,2023-05-16.0382,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/382,添加了知识库之后进行对话，之后再新增知识库就会出现这个问题。,110 +111,微信群人数超过200了，扫码进不去了，群主可以再创建一个新群吗,2023-05-17.0391,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/391,**功能描述 / Feature Description**,111 +112,TypeError: 'ListDocsResponse' object is not subscriptable,2023-05-17.0393,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/393,应该是用remain_docs.code和remain_docs.data吧？吗？,112 +113,[BUG] 加载chatglm模型报错：'NoneType' object has no attribute 'message_types_by_name',2023-05-17.0398,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/398,**问题描述 / Problem Description**,113 +114,[BUG] 执行 python webui.py 没有报错，但是ui界面提示 Something went wrong Expecting value: line 1 column 1 (char 0,2023-05-18.0399,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/399,**环境配置**,114 +115,启动后调用api接口正常，过一会就不断的爆出 Since the angle classifier is not initialized,2023-05-18.0404,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/404,**问题描述 / Problem Description**,115 +116,[BUG] write_check_file方法中，open函数未指定编码,2023-05-18.0408,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/408,"def write_check_file(filepath, docs):",116 +117,导入的PDF中存在图片，有大概率出现 “unsupported colorspace for 'png'”异常,2023-05-18.0409,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/409,"pix = fitz.Pixmap(doc, img[0])",117 +118,请问流程图是用什么软件画的,2023-05-18.0410,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/410,draw.io,118 +119,mac 加载模型失败,2023-05-19.0417,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/417,Explicitly passing a `revision` is encouraged when loading a model with custom code to ensure no malicious code has been contributed in a newer revision.,119 +120,使用GPU本地运行知识库问答，提问第一个问题出现异常。,2023-05-20.0419,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/419,配置文件model_config.py为：,120 +121,想加入讨论群,2023-05-20.0420,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/420,OK,121 +122,有没有直接调用LLM的API，目前只有知识库的API？,2023-05-22.0426,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/426,-------------------------------------------------------------------------------,122 +123,上传文件后出现 ERROR __init__() got an unexpected keyword argument 'autodetect_encoding',2023-05-22.0428,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/428,"上传文件后出现这个问题：ERROR 2023-05-22 11:46:19,568-1d: __init__() got an unexpected keyword argument 'autodetect_encoding'",123 +124,想问下README中用到的流程图用什么软件画的,2023-05-22.0431,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/431,**功能描述 / Feature Description**,124 +125,No matching distribution found for langchain==0.0.174,2023-05-23.0436,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/436,ERROR: Could not find a version that satisfies the requirement langchain==0.0.174 ,125 +126,[FEATURE] bing是必须的么？,2023-05-23.0437,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/437,从这个[脚步](https://github.com/imClumsyPanda/langchain-ChatGLM/blob/master/configs/model_config.py#L129)里面发现需要申请bing api，如果不申请，纯用模型推理不可吗？,126 +127,同一台环境下部署了5.22号更新的langchain-chatglm v0.1.13和之前的版本，回复速度明显变慢,2023-05-23.0442,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/442,新langchain-chatglm v0.1.13版本速度很慢,127 +128,Error reported during startup,2023-05-23.0443,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/443,Traceback (most recent call last):,128 +129,"ValueError: not enough values to unpack (expected 2, got 1)on of the issue",2023-05-24.0449,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/449,"File "".cache\huggingface\modules\transformers_modules\chatglm-6b-int4\modeling_chatglm.py"", line 1280, in chat",129 +130,[BUG] API部署，流式输出的函数，少了个question,2023-05-24.0451,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/451,**问题描述 / Problem Description**,130 +131,项目结构的简洁性保持,2023-05-24.0454,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/454,**功能描述 / Feature Description**,131 +132,项目群扫码进不去了,2023-05-24.0455,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/455,项目群扫码进不去了，是否可以加一下微信拉我进群，谢谢！微信号：daniel-0527,132 +133,请求拉我入群讨论，海硕一枚，专注于LLM等相关技术,2023-05-24.0461,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/461,**功能描述 / Feature Description**,133 +134,[BUG] chatglm-6b模型报错OSError: Error no file named pytorch_model.bin found in directory /chatGLM/model/model-6b,2023-05-26.0474,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/474,**1、简述：**,134 +135,现在本项目交流群二维码扫描不进去了，需要群主通过,2023-05-27.0478,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/478,现在本项目交流群二维码扫描不进去了，需要群主通过,135 +136,RuntimeError: Only Tensors of floating point and complex dtype can require gradients,2023-05-28.0483,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/483,刚更新了最新版本：,136 +137,"RuntimeError: ""LayerNormKernelImpl"" not implemented for 'Half'",2023-05-28.0484,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/484,"已经解决了 params 只用两个参数 {'trust_remote_code': True, 'torch_dtype': torch.float16}",137 +138,[BUG] 文件未成功加载，请重新上传文件,2023-05-31.0504,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/504,webui.py,138 +139,[BUG] bug 17 ，pdf和pdf为啥还不一样呢？为啥有的pdf能识别？有的pdf识别不了呢？,2023-05-31.0506,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/506,bug 17 ，pdf和pdf为啥还不一样呢？为啥有的pdf能识别？有的pdf识别不了呢？,139 +140,[FEATURE] 简洁阐述功能 / Concise description of the feature,2023-05-31.0513,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/513,**功能描述 / Feature Description**,140 +141,[BUG] webui.py 加载chatglm-6b-int4 失败,2023-06-02.0524,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/524,**问题描述 / Problem Description**,141 +142,[BUG] webui.py 加载chatglm-6b模型异常,2023-06-02.0525,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/525,**问题描述 / Problem Description**,142 +143,增加对chatgpt的embedding和api调用的支持,2023-06-02.0531,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/531,能否支持openai的embedding api和对话的api？,143 +144,[FEATURE] 调整模型下载的位置,2023-06-02.0537,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/537,模型默认下载到 $HOME/.cache/huggingface/，当 C 盘空间不足时无法完成模型的下载。configs/model_config.py 中也没有调整模型位置的参数。,144 +145,[BUG] langchain=0.0.174 出错,2023-06-04.0543,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/543,**问题描述 / Problem Description**,145 +146,[BUG] 更新后加载本地模型路径不正确,2023-06-05.0545,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/545,**问题描述 / Problem Description**,146 +147,SystemError: 8bit 模型需要 CUDA 支持，或者改用量化后模型！,2023-06-06.0550,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/550,"docker 部署后,启动docker,过会儿容器会自动退出,logs报错 SystemError: 8bit 模型需要 CUDA 支持，或者改用量化后模型！ [NVIDIA Container Toolkit](https://github.com/NVIDIA/nvidia-container-toolkit) 也已经安装了",147 +148,[BUG] 上传知识库超过1M报错,2023-06-06.0556,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/556,**问题描述 / Problem Description**,148 +149,打开跨域访问后仍然报错，不能请求,2023-06-06.0560,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/560,报错信息：,149 +150,dialogue_answering 里面的代码是不是没有用到？，没有看到调用,2023-06-07.0571,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/571,dialogue_answering 是干啥的,150 +151,[BUG] 响应速度极慢，应从哪里入手优化？48C/128G/8卡,2023-06-07.0573,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/573,运行环境：ubuntu20.04,151 +152,纯CPU环境下运行cli_demo时报错，提示找不到nvcuda.dll,2023-06-08.0576,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/576,本地部署环境是纯CPU，之前的版本在纯CPU环境下能正常运行，但上传本地知识库经常出现encode问题。今天重新git项目后，运行时出现如下问题，请问该如何解决。,152 +153,如何加载本地的embedding模型（text2vec-large-chinese模型文件）,2023-06-08.0582,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/582,"因为需要离线部署，所以要把模型放到本地，我修改了chains/local_doc_qa.py中的HuggingFaceEmbeddings()，在其中加了一个cache_folder的参数，保证下载的文件在cache_folder中，model_name是text2vec-large-chinese。如cache_folder='/home/xx/model/text2vec-large-chinese', model_name='text2vec-large-chinese'，这样仍然需要联网下载报错，请问大佬如何解决该问题？",153 +154,ChatGLM-6B 在另外服务器安装好了，请问如何修改model.cofnig.py 来使用它的接口呢？？,2023-06-09.0588,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/588,我本来想在这加一个api base url 但是运行web.py 发现 还是会去连huggingface 下载模型,154 +155,[BUG] raise partially initialized module 'charset_normalizer' has no attribute 'md__mypyc' when call interface `upload_file`,2023-06-10.0591,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/591,**问题描述 / Problem Description**,155 +156,[BUG] raise OSError: [Errno 101] Network is unreachable when call interface upload_file and upload .pdf files,2023-06-10.0592,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/592,**问题描述 / Problem Description**,156 +157,如果直接用vicuna作为基座大模型，需要修改的地方有哪些？,2023-06-12.0596,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/596,vicuna模型有直接转换好的没有？也就是llama转换之后的vicuna。,157 +158,[BUG] 通过cli.py调用api时抛出AttributeError: 'NoneType' object has no attribute 'get'错误,2023-06-12.0598,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/598,通过`python cli.py start api --ip localhost --port 8001` 命令调用api时，抛出：,158 +159,[BUG] 通过cli.py调用api时直接报错`langchain-ChatGLM: error: unrecognized arguments: start cli`,2023-06-12.0601,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/601,通过python cli.py start cli启动cli_demo时，报错：,159 +160,[BUG] error: unrecognized arguments: --model-dir conf/models/,2023-06-12.0602,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/602,关键字参数修改了吗？有没有文档啊？大佬,160 +161,[BUG] 上传文件全部失败,2023-06-12.0603,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/603,ERROR: Exception in ASGI application,161 +162,[BUG] config 使用 chatyuan 无法启动,2023-06-12.0604,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/604,"""chatyuan"": {",162 +163,使用fashchat api之后，后台报错APIError 如图所示,2023-06-12.0606,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/606,我按照https://github.com/imClumsyPanda/langchain-ChatGLM/blob/master/docs/fastchat.md,163 +164,[BUG] 启用上下文关联，每次embedding搜索到的内容都会比前一次多一段,2023-06-13.0613,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/613,**问题描述 / Problem Description**,164 +165,local_doc_qa.py中MyFAISS.from_documents() 这个语句看不太懂。MyFAISS类中没有这个方法，其父类FAISS和VectorStore中也只有from_texts方法[BUG] 简洁阐述问题 / Concise description of the issue,2023-06-14.0619,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/619,local_doc_qa.py中MyFAISS.from_documents() 这个语句看不太懂。MyFAISS类中没有这个方法，其父类FAISS和VectorStore中也只有from_texts方法,165 +166,[BUG] TypeError: similarity_search_with_score_by_vector() got an unexpected keyword argument 'filter',2023-06-14.0624,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/624,**问题描述 / Problem Description**,166 +167,please delete this issue,2023-06-15.0633,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/633,"sorry, incorrect submission. Please remove this issue!",167 +168,[BUG] vue前端镜像构建失败,2023-06-15.0635,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/635,**问题描述 / Problem Description**,168 +169,ChatGLM-6B模型能否回答英文问题？,2023-06-15.0640,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/640,大佬，请问一下，如果本地知识文档是英文，ChatGLM-6B模型能否回答英文问题？不能的话，有没有替代的模型推荐，期待你的回复，谢谢,169 +170,[BUG] 简洁阐述问题 / Concise description of the issue,2023-06-16.0644,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/644,**问题描述 / Problem Description**,170 +171,KeyError: 3224,2023-06-16.0645,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/645,```,171 diff --git a/samples/isssues_merge/langchain-ChatGLM_closed.jsonl b/samples/isssues_merge/langchain-ChatGLM_closed.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..fd2040e151f67fc60bc6f921fe9b357aa4c458ae --- /dev/null +++ b/samples/isssues_merge/langchain-ChatGLM_closed.jsonl @@ -0,0 +1,172 @@ +{"title": "加油~以及一些建议", "file": "2023-03-31.0002", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/2", "detail": "加油，我认为你的方向是对的。", "id": 0} +{"title": "当前的运行环境是什么，windows还是Linux", "file": "2023-04-01.0003", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/3", "detail": "当前的运行环境是什么，windows还是Linux，python是什么版本？", "id": 1} +{"title": "请问这是在CLM基础上运行吗？", "file": "2023-04-01.0004", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/4", "detail": "请问是不是需要本地安装好clm并正常运行的情况下，再按文中的步骤执行才能运行起来？", "id": 2} +{"title": "[复现问题] 构造 prompt 时从知识库中提取的文字乱码", "file": "2023-04-01.0005", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/5", "detail": "hi，我在尝试复现 README 中的效果，也使用了 ChatGLM-6B 的 README 作为输入文本，但发现从知识库中提取的文字是乱码，导致构造的 prompt 不可用。想了解如何解决这个问题。", "id": 3} +{"title": "后面能否加入上下文对话功能？", "file": "2023-04-02.0006", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/6", "detail": "目前的get_wiki_agent_answer函数中已经实现了历史消息传递的功能，后面我再确认一下是否有langchain中model调用过程中是否传递了chat_history。", "id": 4} +{"title": "请问：纯cpu可以吗？", "file": "2023-04-03.0007", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/7", "detail": "很酷的实现，极大地开拓了我的眼界！很顺利的在gpu机器上运行了", "id": 5} +{"title": "运行报错：AttributeError: 'NoneType' object has no attribute 'message_types_by_name'", "file": "2023-04-03.0008", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/8", "detail": "报错：", "id": 6} +{"title": "运行环境：GPU需要多大的？", "file": "2023-04-03.0009", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/9", "detail": "如果按照THUDM/ChatGLM-6B的说法，使用的GPU大小应该在13GB左右，但运行脚本后，占用了24GB还不够。", "id": 7} +{"title": "请问本地知识的格式是什么？", "file": "2023-04-03.0010", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/10", "detail": "已测试格式包括docx、md文件中的文本信息，具体格式可以参考 [langchain文档](https://python.langchain.com/en/latest/modules/indexes/document_loaders/examples/unstructured_file.html?highlight=pdf#)", "id": 8} +{"title": "24G的显存还是爆掉了，是否支持双卡运行", "file": "2023-04-03.0011", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/11", "detail": "RuntimeError: CUDA out of memory. Tried to allocate 96.00 MiB (GPU 0; 23.70 GiB total capacity; 22.18 GiB already allocated; 12.75 MiB free; 22.18 GiB reserved in total by PyTorch) If reserved memory is >> allocated memory try setting max_split_size_mb to avoid fragmentation. See documentation for Memory Management and PYTORCH_CUDA_ALLOC_CONF", "id": 9} +{"title": "你怎么知道embeddings方式和模型训练时候的方式是一样的?", "file": "2023-04-03.0012", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/12", "detail": "embedding和LLM的方式不用一致，embedding能够解决语义检索的需求就行。这个项目里用到embedding是在对本地知识建立索引和对问句转换成向量的过程。", "id": 10} +{"title": "是否能提供本地知识文件的格式？", "file": "2023-04-04.0013", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/13", "detail": "是否能提供本地知识文件的格式？", "id": 11} +{"title": "是否可以像清华原版跑在8G一以下的卡？", "file": "2023-04-04.0016", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/16", "detail": "是否可以像清华原版跑在8G一以下的卡？我的8G卡爆显存了🤣🤣🤣", "id": 12} +{"title": "请教一下langchain协调使用向量库和chatGLM工作的", "file": "2023-04-05.0018", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/18", "detail": "代码里面这段是创建问答模型的，会接入ChatGLM和本地语料的向量库，langchain回答的时候是怎么个优先顺序？先搜向量库，没有再找chatglm么？ 还是什么机制？", "id": 13} +{"title": "在mac m2max上抛出了ValueError: 150001 is not in list这个异常", "file": "2023-04-05.0019", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/19", "detail": "我把chatglm_llm.py加载模型的代码改成如下", "id": 14} +{"title": "程序运行后一直卡住", "file": "2023-04-05.0020", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/20", "detail": "感谢作者的付出，不过本人在运行时出现了问题，请大家帮助。", "id": 15} +{"title": "问一下chat_history的逻辑", "file": "2023-04-06.0022", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/22", "detail": "感谢开源。", "id": 16} +{"title": "为什么每次运行都会loading checkpoint", "file": "2023-04-06.0023", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/23", "detail": "我把这个embeding模型下载到本地后，无法正常启动。", "id": 17} +{"title": "本地知识文件能否上传一些示例？", "file": "2023-04-06.0025", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/25", "detail": "如题，怎么构造知识文件，效果更好？能否提供一个样例", "id": 18} +{"title": "What version of you are using?", "file": "2023-04-06.0026", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/26", "detail": "Hi Panda, I saw the `pip install -r requirements` command in README, and want to confirm you are using python2 or python3? because my pip and pip3 version are all is 22.3.", "id": 19} +{"title": "有兴趣交流本项目应用的朋友可以加一下微信群", "file": "2023-04-07.0027", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/27", "detail": "![IMG_1630](https://user-images.githubusercontent.com/5668498/230533162-8b9bfcdd-249c-4efe-b066-4f9ba2ce9f23.jpeg)", "id": 20} +{"title": "本地知识越多，回答时检索的时间是否会越长", "file": "2023-04-07.0029", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/29", "detail": "是的 因为需要进行向量匹配检索", "id": 21} +{"title": "爲啥最後還是報錯 哭。。", "file": "2023-04-07.0030", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/30", "detail": "Failed to import transformers.models.t5.configuration_t5 because of the following error (look up to see", "id": 22} +{"title": "对话到第二次的时候就报错UnicodeDecodeError: 'utf-8' codec can't decode", "file": "2023-04-07.0031", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/31", "detail": "对话第一次是没问题的，模型返回输出后又给到请输入你的问题，我再输入问题就报错", "id": 23} +{"title": "用的in4的量化版本，推理的时候显示需要申请10Gb的显存", "file": "2023-04-07.0033", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/33", "detail": "File \"/root/.cache/huggingface/modules/transformers_modules/chatglm-6b-int4-qe/modeling_chatglm.py\", line 581, in forward", "id": 24} +{"title": "使用colab运行，python3.9，提示包导入有问题", "file": "2023-04-07.0034", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/34", "detail": "from ._util import is_directory, is_path", "id": 25} +{"title": "运行失败，Loading checkpoint未达到100%被kill了，请问下是什么原因？", "file": "2023-04-07.0035", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/35", "detail": "日志如下：", "id": 26} +{"title": "弄了个交流群，自己弄好多细节不会，大家技术讨论 加connection-image 我来拉你", "file": "2023-04-08.0036", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/36", "detail": "自己搞好多不清楚的，一起来弄吧。。准备搞个部署问题的解决文档出来", "id": 27} +{"title": "Error using the new version with langchain", "file": "2023-04-09.0043", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/43", "detail": "Error with the new changes:", "id": 28} +{"title": "程序报错torch.cuda.OutOfMemoryError如何解决？", "file": "2023-04-10.0044", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/44", "detail": "报错详细信息如下：", "id": 29} +{"title": "qa的训练数据格式是如何设置的", "file": "2023-04-10.0045", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/45", "detail": "本项目不是使用微调的方式，所以并不涉及到训练过程。", "id": 30} +{"title": "The FileType.UNK file type is not supported in partition. 解决办法", "file": "2023-04-10.0046", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/46", "detail": "ValueError: Invalid file /home/yawu/Documents/langchain-ChatGLM-master/data. The FileType.UNK file type is not supported in partition.", "id": 31} +{"title": "如何读取多个txt文档？", "file": "2023-04-10.0047", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/47", "detail": "如题，请教一下如何读取多个txt文档？示例代码中只给了读一个文档的案例，这个input我换成string之后也只能指定一个文档，无法用通配符指定多个文档，也无法传入多个文件路径的列表。", "id": 32} +{"title": "nltk package unable to either download or load local nltk_data folder", "file": "2023-04-10.0049", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/49", "detail": "I'm running this project on an offline Windows Server environment so I download the Punkt and averaged_perceptron_tagger tokenizer in this directory:", "id": 33} +{"title": "requirements.txt中需要指定langchain版本", "file": "2023-04-11.0055", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/55", "detail": "langchain版本0.116下无法引入RetrievalQA，需要指定更高版本（0.136版本下无问题）", "id": 34} +{"title": "Demo演示无法给出输出内容", "file": "2023-04-12.0059", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/59", "detail": "你好，测试了项目自带新闻稿示例和自行上传的一个文本，可以加载进去，但是无法给出答案，请问属于什么情况，如何解决，谢谢。PS: 1、今天早上刚下载全部代码；2、硬件服务器满足要求；3、按操作说明正常操作。", "id": 35} +{"title": "群人数过多无法进群，求帮忙拉进群", "file": "2023-04-12.0061", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/61", "detail": "您好，您的群人数超过了200人，目前无法通过二维码加群，请问您方便加我微信拉我进群吗？万分感谢", "id": 36} +{"title": "群人数已满，求大佬拉入群", "file": "2023-04-12.0062", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/62", "detail": "已在README中更新拉群二维码", "id": 37} +{"title": "requirements中langchain版本错误", "file": "2023-04-12.0065", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/65", "detail": "langchain版本应该是0.0.12而不是0.0.120", "id": 38} +{"title": "Linux : Searchd in", "file": "2023-04-13.0068", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/68", "detail": "import nltk", "id": 39} +{"title": "No sentence-transformers model found", "file": "2023-04-13.0069", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/69", "detail": "加载不了这个模型，错误原因是找不到这个模型，但是路径是配置好了的", "id": 40} +{"title": "Error loading punkt: ", "id": 58} +{"title": "为啥放到方法调用会出错，这个怎么处理？", "file": "2023-04-20.0150", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/150", "detail": "```python", "id": 59} +{"title": "No sentence-transformers model found with name C:\\Users\\Administrator/.cache\\torch\\sentence_transformers\\GanymedeNil_text2vec-large-chinese. Creating a new one with MEAN pooling.", "file": "2023-04-21.0154", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/154", "detail": "卡在这块很久是正常现象吗", "id": 60} +{"title": "微信群需要邀请才能加入", "file": "2023-04-21.0155", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/155", "detail": "RT，给个个人联系方式白", "id": 61} +{"title": "No sentence-transformers model found with name GanymedeNil/text2vec-large-chinese. Creating a new one with MEAN pooling", "file": "2023-04-21.0156", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/156", "detail": "ls GanymedeNil/text2vec-large-chinese", "id": 62} +{"title": "embedding会加载两次", "file": "2023-04-23.0159", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/159", "detail": "你好，为什么要这样设置呢，这样会加载两次呀。", "id": 63} +{"title": "扫二维码加的那个群，群成员满了进不去了", "file": "2023-04-23.0160", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/160", "detail": "如题", "id": 64} +{"title": "执行python3 cli_demo.py 报错AttributeError: 'NoneType' object has no attribute 'chat'", "file": "2023-04-24.0163", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/163", "detail": "刚开始怀疑是内存不足问题，换成int4,int4-qe也不行，有人知道是什么原因吗", "id": 65} +{"title": "匹配得分", "file": "2023-04-24.0167", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/167", "detail": "在示例cli_demo.py中返回的匹配文本没有对应的score，可以加上这个feature吗", "id": 66} +{"title": "大佬有计划往web_ui.py加入打字机功能吗", "file": "2023-04-25.0170", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/170", "detail": "目前在载入了知识库后，单张V100 32G在回答垂直领域的问题时也需要20S以上，没有打字机逐字输出的使用体验还是比较煎熬的....", "id": 67} +{"title": "Is it possible to use a verctorDB for the embedings?", "file": "2023-04-25.0171", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/171", "detail": "when I play, I have to load the local data again and again when to start. I wonder if it is possible to use", "id": 68} +{"title": "请问通过lora训练官方模型得到的微调模型文件该如何加载？", "file": "2023-04-25.0173", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/173", "detail": "通过lora训练的方式得到以下文件:", "id": 69} +{"title": "from langchain.chains import RetrievalQA的代码在哪里？", "file": "2023-04-25.0174", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/174", "detail": "local_doc_qa.py", "id": 70} +{"title": "哪里有knowledge_based_chatglm.py文件？怎么找不到了？？是被替换成cli_demo.py文件了吗？", "file": "2023-04-26.0175", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/175", "detail": "哪里有knowledge_based_chatglm.py文件？怎么找不到了？？是被替换成cli_demo.py文件了吗？", "id": 71} +{"title": "AttributeError: 'Chatbot' object has no attribute 'value'", "file": "2023-04-26.0177", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/177", "detail": "Traceback (most recent call last):", "id": 72} +{"title": "控制台调api.py报警告", "file": "2023-04-26.0178", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/178", "detail": "you must pass the application as an import string to enable \"reload\" or \"workers\"", "id": 73} +{"title": "如何加入群聊", "file": "2023-04-27.0183", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/183", "detail": "微信群超过200人了，需要邀请，如何加入呢？", "id": 74} +{"title": "如何将Chatglm和本地知识相结合", "file": "2023-04-27.0185", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/185", "detail": "您好，我想请教一下怎么才能让知识库匹配到的文本和chatglm生成的相结合，而不是说如果没搜索到，就说根据已知信息无法回答该问题，谢谢", "id": 75} +{"title": "一点建议", "file": "2023-04-27.0189", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/189", "detail": "1.weiui的get_vector_store方法里面添加一个判断以兼容gradio版本导致的上传异常", "id": 76} +{"title": "windows环境下，按照教程，配置好conda环境，git完项目，修改完模型路径相关内容后，运行demo报错缺少", "file": "2023-04-28.0194", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/194", "detail": "报错代码如下：", "id": 77} +{"title": "ValueError: too many values to unpack (expected 2)", "file": "2023-04-28.0198", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/198", "detail": "When i tried to use the non-streaming, `ValueError: too many values to unpack (expected 2)` error came out.", "id": 78} +{"title": "加载doc后覆盖原本知识", "file": "2023-04-28.0201", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/201", "detail": "加载较大量级的私有知识库后，原本的知识会被覆盖", "id": 79} +{"title": "自定义知识库回答效果很差", "file": "2023-04-28.0203", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/203", "detail": "请问加了自定义知识库知识库,回答效果很差，是因为数据量太小的原因么", "id": 80} +{"title": "python310下，安装pycocotools失败，提示低版本cython，实际已安装高版本", "file": "2023-04-29.0208", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/208", "detail": "RT，纯离线环境安装，依赖安装的十分艰难，最后碰到pycocotools，始终无法安装上，求教方法！", "id": 81} +{"title": "[FEATURE] 支持 RWKV 模型（目前已有 pip package & rwkv.cpp 等等）", "file": "2023-05-01.0216", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/216", "detail": "您好，我是 RWKV 的作者，介绍见：https://zhuanlan.zhihu.com/p/626083366", "id": 82} +{"title": "[BUG] 为啥主机/服务器不联网不能正常启动服务？", "file": "2023-05-02.0220", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/220", "detail": "**问题描述 / Problem Description**", "id": 83} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-05-03.0222", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/222", "detail": "**local variable 'torch' referenced before assignment**", "id": 84} +{"title": "不支持txt文件的中文输入", "file": "2023-05-04.0235", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/235", "detail": "vs_path, _ = local_doc_qa.init_knowledge_vector_store(filepath)", "id": 85} +{"title": "文件均未成功加载，请检查依赖包或替换为其他文件再次上传。 文件未成功加载，请重新上传文件", "file": "2023-05-05.0237", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/237", "detail": "请大佬帮忙解决，谢谢！", "id": 86} +{"title": "[BUG] 使用多卡时chatglm模型加载两次", "file": "2023-05-05.0241", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/241", "detail": "chatglm_llm.py文件下第129行先加载了一次chatglm模型，第143行又加载了一次", "id": 87} +{"title": "[BUG] similarity_search_with_score_by_vector函数返回多个doc时的score结果错误", "file": "2023-05-06.0252", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/252", "detail": "**问题描述 / Problem Description**", "id": 88} +{"title": "可以再建一个交流群吗，这个群满了进不去。", "file": "2023-05-06.0255", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/255", "detail": "上午应该已经在readme里更新过了，如果不能添加可能是网页缓存问题，可以试试看直接扫描img/qr_code_12.jpg", "id": 89} +{"title": "请问这是什么错误哇？KeyError: 'serialized_input'", "file": "2023-05-06.0257", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/257", "detail": "运行“python webui.py” 后这是什么错误？怎么解决啊？", "id": 90} +{"title": "修改哪里的代码，可以再cpu上跑？", "file": "2023-05-06.0258", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/258", "detail": "**问题描述 / Problem Description**", "id": 91} +{"title": "ModuleNotFoundError: No module named 'modelscope'", "file": "2023-05-07.0266", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/266", "detail": "安装这个", "id": 92} +{"title": "加载lora微调模型时，lora参数加载成功，但显示模型未成功加载？", "file": "2023-05-08.0270", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/270", "detail": "什么原因呀？", "id": 93} +{"title": "[BUG] 运行webui.py报错：name 'EMBEDDING_DEVICE' is not defined", "file": "2023-05-08.0274", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/274", "detail": "解决了，我修改model_config时候把这个变量改错了", "id": 94} +{"title": "基于ptuning训练完成，新老模型都进行了加载，但是只有新的", "file": "2023-05-08.0280", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/280", "detail": "licitly passing a `revision` is encouraged when loading a model with custom code to ensure no malicious code has been contributed in a newer revision.", "id": 95} +{"title": "[BUG] 使用chatyuan模型时，对话Error，has no attribute 'stream_chat'", "file": "2023-05-08.0282", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/282", "detail": "**问题描述 / Problem Description**", "id": 96} +{"title": "chaglm调用过程中 _call提示有一个 stop", "file": "2023-05-09.0286", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/286", "detail": "**功能描述 / Feature Description**", "id": 97} +{"title": "Logger._log() got an unexpected keyword argument 'end'", "file": "2023-05-10.0295", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/295", "detail": "使用cli_demo的时候，加载一个普通txt文件，输入问题后，报错：“TypeError: Logger._log() got an unexpected keyword argument 'end'”", "id": 98} +{"title": "[BUG] 请问可以解释下这个FAISS.similarity_search_with_score_by_vector = similarity_search_with_score_by_vector的目的吗", "file": "2023-05-10.0296", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/296", "detail": "我不太明白这个库自己写的similarity_search_with_score_by_vector方法做的事情，因为langchain原版的similarity_search_with_score_by_vector只是search faiss之后把返回的topk句子组合起来。我觉得原版理解起来没什么问题，但是这个库里自己写的我就没太看明白多做了什么其他的事情，因为没有注释。", "id": 99} +{"title": "[BUG] Windows下上传中文文件名文件，faiss无法生成向量数据库文件", "file": "2023-05-11.0318", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/318", "detail": "**问题描述 / Problem Description**", "id": 100} +{"title": "cli_demo中的流式输出能否接着前一答案输出?", "file": "2023-05-11.0320", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/320", "detail": "现有流式输出结果样式为:", "id": 101} +{"title": "内网部署时网页无法加载，能否增加离线静态资源", "file": "2023-05-12.0326", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/326", "detail": "内网部署时网页无法加载，能否增加离线静态资源", "id": 102} +{"title": "我想把文件字符的编码格式改为encoding='utf-8'在哪修改呢，因为会有ascii codec can't decode byte报错", "file": "2023-05-14.0360", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/360", "detail": "上传中文的txt文件时报错，编码格式为utf-8", "id": 103} +{"title": "Batches的进度条是在哪里设置的?能否关闭显示?", "file": "2023-05-15.0366", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/366", "detail": "使用cli_demo.py进行命令行测试时,每句回答前都有个Batches的进度条", "id": 104} +{"title": "ImportError: dlopen: cannot load any more object with static TLS or Segmentation fault", "file": "2023-05-15.0368", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/368", "detail": "**问题描述 / Problem Description**", "id": 105} +{"title": "读取PDF时报错", "file": "2023-05-16.0373", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/373", "detail": "在Colab上执行cli_demo.py时，在路径文件夹里放了pdf文件，在加载的过程中会显示错误，然后无法加载PDF文件", "id": 106} +{"title": "[BUG] webui报错 InvalidURL", "file": "2023-05-16.0375", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/375", "detail": "python 版本：3.8.16", "id": 107} +{"title": "[FEATURE] 如果让回答不包含出处，应该怎么处理", "file": "2023-05-16.0380", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/380", "detail": "**功能描述 / Feature Description**", "id": 108} +{"title": "加载PDF文件时，出现 unsupported colorspace for 'png'", "file": "2023-05-16.0381", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/381", "detail": "**问题描述 / Problem Description**", "id": 109} +{"title": "'ascii' codec can't encode characters in position 14-44: ordinal not in range(128) 经典bug", "file": "2023-05-16.0382", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/382", "detail": "添加了知识库之后进行对话，之后再新增知识库就会出现这个问题。", "id": 110} +{"title": "微信群人数超过200了，扫码进不去了，群主可以再创建一个新群吗", "file": "2023-05-17.0391", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/391", "detail": "**功能描述 / Feature Description**", "id": 111} +{"title": "TypeError: 'ListDocsResponse' object is not subscriptable", "file": "2023-05-17.0393", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/393", "detail": "应该是用remain_docs.code和remain_docs.data吧？吗？", "id": 112} +{"title": "[BUG] 加载chatglm模型报错：'NoneType' object has no attribute 'message_types_by_name'", "file": "2023-05-17.0398", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/398", "detail": "**问题描述 / Problem Description**", "id": 113} +{"title": "[BUG] 执行 python webui.py 没有报错，但是ui界面提示 Something went wrong Expecting value: line 1 column 1 (char 0", "file": "2023-05-18.0399", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/399", "detail": "**环境配置**", "id": 114} +{"title": "启动后调用api接口正常，过一会就不断的爆出 Since the angle classifier is not initialized", "file": "2023-05-18.0404", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/404", "detail": "**问题描述 / Problem Description**", "id": 115} +{"title": "[BUG] write_check_file方法中，open函数未指定编码", "file": "2023-05-18.0408", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/408", "detail": "def write_check_file(filepath, docs):", "id": 116} +{"title": "导入的PDF中存在图片，有大概率出现 “unsupported colorspace for 'png'”异常", "file": "2023-05-18.0409", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/409", "detail": "pix = fitz.Pixmap(doc, img[0])", "id": 117} +{"title": "请问流程图是用什么软件画的", "file": "2023-05-18.0410", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/410", "detail": "draw.io", "id": 118} +{"title": "mac 加载模型失败", "file": "2023-05-19.0417", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/417", "detail": "Explicitly passing a `revision` is encouraged when loading a model with custom code to ensure no malicious code has been contributed in a newer revision.", "id": 119} +{"title": "使用GPU本地运行知识库问答，提问第一个问题出现异常。", "file": "2023-05-20.0419", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/419", "detail": "配置文件model_config.py为：", "id": 120} +{"title": "想加入讨论群", "file": "2023-05-20.0420", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/420", "detail": "OK", "id": 121} +{"title": "有没有直接调用LLM的API，目前只有知识库的API？", "file": "2023-05-22.0426", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/426", "detail": "-------------------------------------------------------------------------------", "id": 122} +{"title": "上传文件后出现 ERROR __init__() got an unexpected keyword argument 'autodetect_encoding'", "file": "2023-05-22.0428", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/428", "detail": "上传文件后出现这个问题：ERROR 2023-05-22 11:46:19,568-1d: __init__() got an unexpected keyword argument 'autodetect_encoding'", "id": 123} +{"title": "想问下README中用到的流程图用什么软件画的", "file": "2023-05-22.0431", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/431", "detail": "**功能描述 / Feature Description**", "id": 124} +{"title": "No matching distribution found for langchain==0.0.174", "file": "2023-05-23.0436", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/436", "detail": "ERROR: Could not find a version that satisfies the requirement langchain==0.0.174 ", "id": 125} +{"title": "[FEATURE] bing是必须的么？", "file": "2023-05-23.0437", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/437", "detail": "从这个[脚步](https://github.com/imClumsyPanda/langchain-ChatGLM/blob/master/configs/model_config.py#L129)里面发现需要申请bing api，如果不申请，纯用模型推理不可吗？", "id": 126} +{"title": "同一台环境下部署了5.22号更新的langchain-chatglm v0.1.13和之前的版本，回复速度明显变慢", "file": "2023-05-23.0442", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/442", "detail": "新langchain-chatglm v0.1.13版本速度很慢", "id": 127} +{"title": "Error reported during startup", "file": "2023-05-23.0443", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/443", "detail": "Traceback (most recent call last):", "id": 128} +{"title": "ValueError: not enough values to unpack (expected 2, got 1)on of the issue", "file": "2023-05-24.0449", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/449", "detail": "File \".cache\\huggingface\\modules\\transformers_modules\\chatglm-6b-int4\\modeling_chatglm.py\", line 1280, in chat", "id": 129} +{"title": "[BUG] API部署，流式输出的函数，少了个question", "file": "2023-05-24.0451", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/451", "detail": "**问题描述 / Problem Description**", "id": 130} +{"title": "项目结构的简洁性保持", "file": "2023-05-24.0454", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/454", "detail": "**功能描述 / Feature Description**", "id": 131} +{"title": "项目群扫码进不去了", "file": "2023-05-24.0455", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/455", "detail": "项目群扫码进不去了，是否可以加一下微信拉我进群，谢谢！微信号：daniel-0527", "id": 132} +{"title": "请求拉我入群讨论，海硕一枚，专注于LLM等相关技术", "file": "2023-05-24.0461", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/461", "detail": "**功能描述 / Feature Description**", "id": 133} +{"title": "[BUG] chatglm-6b模型报错OSError: Error no file named pytorch_model.bin found in directory /chatGLM/model/model-6b", "file": "2023-05-26.0474", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/474", "detail": "**1、简述：**", "id": 134} +{"title": "现在本项目交流群二维码扫描不进去了，需要群主通过", "file": "2023-05-27.0478", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/478", "detail": "现在本项目交流群二维码扫描不进去了，需要群主通过", "id": 135} +{"title": "RuntimeError: Only Tensors of floating point and complex dtype can require gradients", "file": "2023-05-28.0483", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/483", "detail": "刚更新了最新版本：", "id": 136} +{"title": "RuntimeError: \"LayerNormKernelImpl\" not implemented for 'Half'", "file": "2023-05-28.0484", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/484", "detail": "已经解决了 params 只用两个参数 {'trust_remote_code': True, 'torch_dtype': torch.float16}", "id": 137} +{"title": "[BUG] 文件未成功加载，请重新上传文件", "file": "2023-05-31.0504", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/504", "detail": "webui.py", "id": 138} +{"title": "[BUG] bug 17 ，pdf和pdf为啥还不一样呢？为啥有的pdf能识别？有的pdf识别不了呢？", "file": "2023-05-31.0506", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/506", "detail": "bug 17 ，pdf和pdf为啥还不一样呢？为啥有的pdf能识别？有的pdf识别不了呢？", "id": 139} +{"title": "[FEATURE] 简洁阐述功能 / Concise description of the feature", "file": "2023-05-31.0513", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/513", "detail": "**功能描述 / Feature Description**", "id": 140} +{"title": "[BUG] webui.py 加载chatglm-6b-int4 失败", "file": "2023-06-02.0524", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/524", "detail": "**问题描述 / Problem Description**", "id": 141} +{"title": "[BUG] webui.py 加载chatglm-6b模型异常", "file": "2023-06-02.0525", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/525", "detail": "**问题描述 / Problem Description**", "id": 142} +{"title": "增加对chatgpt的embedding和api调用的支持", "file": "2023-06-02.0531", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/531", "detail": "能否支持openai的embedding api和对话的api？", "id": 143} +{"title": "[FEATURE] 调整模型下载的位置", "file": "2023-06-02.0537", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/537", "detail": "模型默认下载到 $HOME/.cache/huggingface/，当 C 盘空间不足时无法完成模型的下载。configs/model_config.py 中也没有调整模型位置的参数。", "id": 144} +{"title": "[BUG] langchain=0.0.174 出错", "file": "2023-06-04.0543", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/543", "detail": "**问题描述 / Problem Description**", "id": 145} +{"title": "[BUG] 更新后加载本地模型路径不正确", "file": "2023-06-05.0545", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/545", "detail": "**问题描述 / Problem Description**", "id": 146} +{"title": "SystemError: 8bit 模型需要 CUDA 支持，或者改用量化后模型！", "file": "2023-06-06.0550", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/550", "detail": "docker 部署后,启动docker,过会儿容器会自动退出,logs报错 SystemError: 8bit 模型需要 CUDA 支持，或者改用量化后模型！ [NVIDIA Container Toolkit](https://github.com/NVIDIA/nvidia-container-toolkit) 也已经安装了", "id": 147} +{"title": "[BUG] 上传知识库超过1M报错", "file": "2023-06-06.0556", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/556", "detail": "**问题描述 / Problem Description**", "id": 148} +{"title": "打开跨域访问后仍然报错，不能请求", "file": "2023-06-06.0560", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/560", "detail": "报错信息：", "id": 149} +{"title": "dialogue_answering 里面的代码是不是没有用到？，没有看到调用", "file": "2023-06-07.0571", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/571", "detail": "dialogue_answering 是干啥的", "id": 150} +{"title": "[BUG] 响应速度极慢，应从哪里入手优化？48C/128G/8卡", "file": "2023-06-07.0573", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/573", "detail": "运行环境：ubuntu20.04", "id": 151} +{"title": "纯CPU环境下运行cli_demo时报错，提示找不到nvcuda.dll", "file": "2023-06-08.0576", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/576", "detail": "本地部署环境是纯CPU，之前的版本在纯CPU环境下能正常运行，但上传本地知识库经常出现encode问题。今天重新git项目后，运行时出现如下问题，请问该如何解决。", "id": 152} +{"title": "如何加载本地的embedding模型（text2vec-large-chinese模型文件）", "file": "2023-06-08.0582", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/582", "detail": "因为需要离线部署，所以要把模型放到本地，我修改了chains/local_doc_qa.py中的HuggingFaceEmbeddings()，在其中加了一个cache_folder的参数，保证下载的文件在cache_folder中，model_name是text2vec-large-chinese。如cache_folder='/home/xx/model/text2vec-large-chinese', model_name='text2vec-large-chinese'，这样仍然需要联网下载报错，请问大佬如何解决该问题？", "id": 153} +{"title": "ChatGLM-6B 在另外服务器安装好了，请问如何修改model.cofnig.py 来使用它的接口呢？？", "file": "2023-06-09.0588", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/588", "detail": "我本来想在这加一个api base url 但是运行web.py 发现 还是会去连huggingface 下载模型", "id": 154} +{"title": "[BUG] raise partially initialized module 'charset_normalizer' has no attribute 'md__mypyc' when call interface `upload_file`", "file": "2023-06-10.0591", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/591", "detail": "**问题描述 / Problem Description**", "id": 155} +{"title": "[BUG] raise OSError: [Errno 101] Network is unreachable when call interface upload_file and upload .pdf files", "file": "2023-06-10.0592", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/592", "detail": "**问题描述 / Problem Description**", "id": 156} +{"title": "如果直接用vicuna作为基座大模型，需要修改的地方有哪些？", "file": "2023-06-12.0596", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/596", "detail": "vicuna模型有直接转换好的没有？也就是llama转换之后的vicuna。", "id": 157} +{"title": "[BUG] 通过cli.py调用api时抛出AttributeError: 'NoneType' object has no attribute 'get'错误", "file": "2023-06-12.0598", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/598", "detail": "通过`python cli.py start api --ip localhost --port 8001` 命令调用api时，抛出：", "id": 158} +{"title": "[BUG] 通过cli.py调用api时直接报错`langchain-ChatGLM: error: unrecognized arguments: start cli`", "file": "2023-06-12.0601", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/601", "detail": "通过python cli.py start cli启动cli_demo时，报错：", "id": 159} +{"title": "[BUG] error: unrecognized arguments: --model-dir conf/models/", "file": "2023-06-12.0602", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/602", "detail": "关键字参数修改了吗？有没有文档啊？大佬", "id": 160} +{"title": "[BUG] 上传文件全部失败", "file": "2023-06-12.0603", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/603", "detail": "ERROR: Exception in ASGI application", "id": 161} +{"title": "[BUG] config 使用 chatyuan 无法启动", "file": "2023-06-12.0604", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/604", "detail": "\"chatyuan\": {", "id": 162} +{"title": "使用fashchat api之后，后台报错APIError 如图所示", "file": "2023-06-12.0606", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/606", "detail": "我按照https://github.com/imClumsyPanda/langchain-ChatGLM/blob/master/docs/fastchat.md", "id": 163} +{"title": "[BUG] 启用上下文关联，每次embedding搜索到的内容都会比前一次多一段", "file": "2023-06-13.0613", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/613", "detail": "**问题描述 / Problem Description**", "id": 164} +{"title": "local_doc_qa.py中MyFAISS.from_documents() 这个语句看不太懂。MyFAISS类中没有这个方法，其父类FAISS和VectorStore中也只有from_texts方法[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-06-14.0619", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/619", "detail": "local_doc_qa.py中MyFAISS.from_documents() 这个语句看不太懂。MyFAISS类中没有这个方法，其父类FAISS和VectorStore中也只有from_texts方法", "id": 165} +{"title": "[BUG] TypeError: similarity_search_with_score_by_vector() got an unexpected keyword argument 'filter'", "file": "2023-06-14.0624", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/624", "detail": "**问题描述 / Problem Description**", "id": 166} +{"title": "please delete this issue", "file": "2023-06-15.0633", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/633", "detail": "sorry, incorrect submission. Please remove this issue!", "id": 167} +{"title": "[BUG] vue前端镜像构建失败", "file": "2023-06-15.0635", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/635", "detail": "**问题描述 / Problem Description**", "id": 168} +{"title": "ChatGLM-6B模型能否回答英文问题？", "file": "2023-06-15.0640", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/640", "detail": "大佬，请问一下，如果本地知识文档是英文，ChatGLM-6B模型能否回答英文问题？不能的话，有没有替代的模型推荐，期待你的回复，谢谢", "id": 169} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-06-16.0644", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/644", "detail": "**问题描述 / Problem Description**", "id": 170} +{"title": "KeyError: 3224", "file": "2023-06-16.0645", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/645", "detail": "```", "id": 171} diff --git a/samples/isssues_merge/langchain-ChatGLM_open.csv b/samples/isssues_merge/langchain-ChatGLM_open.csv new file mode 100644 index 0000000000000000000000000000000000000000..56ba7ca35d78a60443a7462f3ce99b2bc5d66cf6 --- /dev/null +++ b/samples/isssues_merge/langchain-ChatGLM_open.csv @@ -0,0 +1,324 @@ +,title,file,url,detail,id +0,效果如何优化,2023-04-04.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/14,如图所示，将该项目的README.md和该项目结合后，回答效果并不理想，请问可以从哪些方面进行优化,0 +1,怎么让模型严格根据检索的数据进行回答，减少胡说八道的回答呢,2023-04-04.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/15,举个例子：,1 +2,"When I try to run the `python knowledge_based_chatglm.py`, I got this error in macOS(M1 Max, OS 13.2)",2023-04-07.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/32,```python,2 +3,萌新求教大佬怎么改成AMD显卡或者CPU？,2023-04-10.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/48,把.cuda()去掉就行,3 +4,输出answer的时间很长，是否可以把文本向量化的部分提前做好存储起来？,2023-04-10.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/50,GPU：4090 24G显存,4 +5,报错Use `repo_type` argument if needed.,2023-04-11.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/57,Traceback (most recent call last):,5 +6,无法打开gradio的页面,2023-04-11.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/58,$ python webui.py,6 +7,支持word，那word里面的图片正常显示吗？,2023-04-12.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/60,如题，刚刚从隔壁转过来的，想先了解下,7 +8,detectron2 is not installed. Cannot use the hi_res partitioning strategy. Falling back to partitioning with the fast strategy.,2023-04-12.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/63,能够正常的跑起来，在加载content文件夹中的文件时，每加载一个文件都会提示：,8 +9,cpu上运行webui，step3 asking时报错,2023-04-12.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/66,web运行，文件加载都正常，asking时报错,9 +10,建议弄一个插件系统,2023-04-13.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/67,如题弄成stable-diffusion-webui那种能装插件，再开一个存储库给使用者或插件开发，存储或下载插件。,10 +11,请教加载模型出错！？,2023-04-13.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/75,AttributeError: module 'transformers_modules.chatglm-6b.configuration_chatglm' has no attribute 'ChatGLMConfig 怎么解决呀,11 +12,从本地知识检索内容的时候，是否可以设置相似度阈值，小于这个阈值的内容不返回，即使会小于设置的VECTOR_SEARCH_TOP_K参数呢？谢谢大佬,2023-04-13.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/76,比如 问一些 你好/你是谁 等一些跟本地知识库无关的问题,12 +13,如何改成多卡推理？,2023-04-13.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/77,+1,13 +14,能否弄个懒人包，可以一键体验？,2023-04-13.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/78,能否弄个懒人包，可以一键体验？,14 +15,连续问问题会导致崩溃,2023-04-13.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/79,看上去不是爆内存的问题，连续问问题后，会出现如下报错,15 +16,AttributeError: 'NoneType' object has no attribute 'as_retriever',2023-04-14.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/86,"环境：windows 11, anaconda/python 3.8",16 +17,FileNotFoundError: Could not find module 'nvcuda.dll' (or one of its dependencies). Try using the full path with constructor syntax.,2023-04-14.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/87,请检查一下cuda或cudnn是否存在安装问题,17 +18,加载txt文件失败？,2023-04-14.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/89,![JppHrGOWFa](https://user-images.githubusercontent.com/109277248/232009383-bf7c46d1-a01e-4e0a-9de6-5b5ed3e36158.jpg),18 +19,NameError: name 'chatglm' is not defined,2023-04-14.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/90,"This share link expires in 72 hours. For free permanent hosting and GPU upgrades (NEW!), check out Spaces: https://huggingface.co/spaces",19 +20,打不开地址？,2023-04-14.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/91,报错数据如下：,20 +21,加载md文件出错,2023-04-14.00,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/98,运行 webui.py后能访问页面，上传一个md文件后，日志中有错误。等待后能加载完成，提示可以提问了，但提问没反应，日志中有错误。 具体日志如下。,21 +22,建议增加获取在线知识的能力,2023-04-15.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/101,建议增加获取在线知识的能力,22 +23,txt 未能成功加载,2023-04-15.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/103,hinese. Creating a new one with MEAN pooling.,23 +24,pdf加载失败,2023-04-15.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/105,e:\a.txt加载成功了，e:\a.pdf加载就失败，pdf文件里面前面几页是图片，后面都是文字，加载失败没有报更多错误，请问该怎么排查？,24 +25,一直停在文本加载处,2023-04-15.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/108,一直停在文本加载处,25 +26," File ""/root/.cache/huggingface/modules/transformers_modules/chatglm-6b/modeling_chatglm.py"", line 440, in forward new_tensor_shape = mixed_raw_layer.size()[:-1] + ( TypeError: torch.Size() takes an iterable of 'int' (item 2 is 'float')",2023-04-17.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/113,按照最新的代码，发现,26 +27,后续会提供前后端分离的功能吗？,2023-04-17.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/114,类似这种https://github.com/lm-sys/FastChat/tree/main/fastchat/serve,27 +28,安装依赖报错,2023-04-17.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/115,(test) C:\Users\linh\Desktop\langchain-ChatGLM-master>pip install -r requirements.txt,28 +29,问特定问题会出现爆显存,2023-04-17.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/117,正常提问没问题。,29 +30,Expecting value: line 1 column 1 (char 0),2023-04-17.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/118,运行后 第一步加载配置一直报错：,30 +31,embedding https://huggingface.co/GanymedeNil/text2vec-large-chinese/tree/main是免费的，效果比对openai的如何？,2023-04-17.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/119,-------------------------------------------------------------------------------,31 +32,这是什么错误，在Colab上运行的。,2023-04-17.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/120,libcuda.so.1: cannot open shared object file: No such file or directory,32 +33,只想用自己的lora微调后的模型进行对话，不想加载任何本地文档，该如何调整？,2023-04-18.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/121,能出一个单独的教程吗,33 +34,"租的gpu,Running on local URL: http://0.0.0.0:7860 To create a public link, set `share=True` in `launch()`. 浏览器上访问不了？？？",2023-04-18.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/122,(chatglm20230401) root@autodl-container-e82d11963c-10ece0d7:~/autodl-tmp/chatglm/langchain-ChatGLM-20230418# python3.9 webui.py,34 +35,本地部署中的报错请教,2023-04-18.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/124,"您好，在本地运行langchain-ChatGLM过程中，环境及依赖的包都已经满足条件，但是运行webui.py,报错如下（运行cli_demo.py报错类似），请问是哪里出了错呢？盼望您的回复，谢谢！",35 +36,报错。The dtype of attention mask (torch.int64) is not bool,2023-04-18.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/131,The dtype of attention mask (torch.int64) is not bool,36 +37,[求助] pip install -r requirements.txt 的时候出现以下报错。。。有大佬帮忙看看怎么搞么，下的release里面的包,2023-04-18.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/134,$ pip install -r requirements.txt,37 +38,如何提升根据问题搜索到对应知识的准确率,2023-04-19.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/136,外链知识库最大的问题在于问题是短文本，知识是中长文本。如何根据问题精准的搜索到对应的知识是个最大的问题。这类本地化项目不像百度，由无数的网页，基本上每个问题都可以找到对应的页面。,38 +39,是否可以增加向量召回的阈值设定，有些召回内容相关性太低，导致模型胡言乱语,2023-04-20.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/140,如题,39 +40,输入长度问题,2023-04-20.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/141,感谢作者支持ptuning微调模型。,40 +41,已有部署好的chatGLM-6b，如何通过接口接入？,2023-04-20.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/144,已有部署好的chatGLM-6b，如何通过接口接入，而不是重新加载一个模型；,41 +42,执行web_demo.py后，显示Killed，就退出了，是不是配置不足呢？,2023-04-20.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/146,![图片](https://user-images.githubusercontent.com/26102866/233256425-c7aab999-11d7-4de9-867b-23ef18d519e4.png),42 +43,执行python cli_demo1.py,2023-04-20.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/147,Traceback (most recent call last):,43 +44,报错：ImportError: cannot import name 'GENERATION_CONFIG_NAME' from 'transformers.utils',2023-04-20.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/149,(mychatGLM) PS D:\Users\admin3\zrh\langchain-ChatGLM> python cli_demo.py,44 +45,上传文件并加载知识库时，会不停地出现临时文件,2023-04-21.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/153,环境：ubuntu 18.04,45 +46,向知识库中添加文件后点击”上传文件并加载知识库“后Segmentation fault报错。,2023-04-23.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/161,运行服务后的提示如下：,46 +47,langchain-serve 集成,2023-04-24.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/162,Hey 我是来自 [langchain-serve](https://github.com/jina-ai/langchain-serve) 的dev！,47 +48,大佬们，wsl的ubuntu怎么配置用cuda加速，装了运行后发现是cpu在跑,2023-04-24.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/164,大佬们，wsl的ubuntu怎么配置用cuda加速，装了运行后发现是cpu在跑,48 +49,在github codespaces docker运行出错,2023-04-24.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/165,docker run -d --restart=always --name chatglm -p 7860:7860 -v /www/wwwroot/code/langchain-ChatGLM:/chatGLM chatglm,49 +50,有计划接入Moss模型嘛,2023-04-24.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/166,后续会开展测试，目前主要在优化langchain部分效果，如果有兴趣也欢迎提PR,50 +51,怎么实现 API 部署？,2023-04-24.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/168,利用 fastapi 实现 API 部署方式，具体怎么实现，有方法说明吗？,51 +52, 'NoneType' object has no attribute 'message_types_by_name'报错,2023-04-24.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/169,_HISTOGRAMPROTO = DESCRIPTOR.message_types_by_name['HistogramProto'],52 +53,能否指定自己训练的text2vector模型？,2023-04-25.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/172,请问大佬：,53 +54,关于项目支持的模型以及quantization_bit潜在的影响的问题,2023-04-26.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/176,作者您好～,54 +55,运行python3.9 api.py WARNING: You must pass the application as an import string to enable 'reload' or 'workers'.,2023-04-26.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/179,api.py文件最下面改成这样试试：,55 +56,ValidationError: 1 validation error for HuggingFaceEmbeddings model_kwargs extra fields not permitted (type=value_error.extra),2023-04-26.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/180,ValidationError: 1 validation error for HuggingFaceEmbeddings,56 +57,如果没有检索到相关性比较高的，回答“我不知道”,2023-04-26.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/181,如果通过设计system_template，让模型在搜索到的文档都不太相关的情况下回答“我不知道”,57 +58,请问如果不能联网，6B之类的文件从本地上传需要放到哪里,2023-04-26.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/182,感谢大佬的项目，很有启发~,58 +59,知识库问答--输入新的知识库名称是中文的话，会报error,2023-04-27.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/184,知识库问答--输入新的知识库名称是中文的话，会报error，选择要加载的知识库那里也不显示之前添加的知识库,59 +60,现在能通过问题匹配的相似度值，来直接返回文档中的文段，而不经过模型吗？因为有些答案在文档中，模型自己回答，不能回答文档中的答案,2023-04-27.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/186,现在能通过问题匹配的相似度值，来直接返回文档中的文段，而不经过模型吗？因为有些答案在文档中，模型自己回答，不能回答文档中的答案。也就是说，提供向量检索回答+模型回答相结合的策略。如果相似度值高于一定数值，直接返回文档中的文本，没有高于就返回模型的回答或者不知道,60 +61,"TypeError: The type of ChatGLM.callback_manager differs from the new default value; if you wish to change the type of this field, please use a type annotation",2023-04-27.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/188,"Mac 运行 python3 ./webui.py 报 TypeError: The type of ChatGLM.callback_manager differs from the new default value; if you wish to change the type of this field, please use a type annotation",61 +62,Not Enough Memory,2023-04-27.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/190,"运行命令行程序python cli_demo.py， 已经成功加载pdf文件, 报“DefaultCPUAllocator: not enough memory: you tried to allocate 458288380900 bytes”错误，请问哪里可以配置default memory",62 +63,参与开发问题,2023-04-27.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/191,1.是否需要进专门的开发群,63 +64,对话框中代码片段格式需改进,2023-04-27.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/192,最好能改进下输出代码片段的格式，目前输出的格式还不友好。,64 +65,请问未来有可能支持belle吗,2023-04-28.01,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/195,如题，谢谢大佬,65 +66,TypeError: cannot unpack non-iterable NoneType object,2023-04-28.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/200,"When i tried to change the knowledge vector store through `init_knowledge_vector_store`, the error `TypeError: cannot unpack non-iterable NoneType object` came out.",66 +67,生成结果,2023-04-28.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/202,你好，想问一下langchain+chatglm-6B，找到相似匹配的prompt，是直接返回prompt对应的答案信息，还是chatglm-6B在此基础上自己优化答案？,67 +68,在win、ubuntu下都出现这个错误：attributeerror: 't5forconditionalgeneration' object has no attribute 'stream_chat',2023-04-29.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/207,在win、ubuntu。下载完模型后，没办法修改代码以执行本地模型，每次都要重新输入路径； LLM 模型、Embedding 模型支持也都在官网下的，在其他项目（wenda）下可以使用,68 +69,[FEATURE] knowledge_based_chatglm.py: renamed or missing?,2023-04-30.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/210,"Not found. Was it renamed? Or, is it missing? How can I get it?",69 +70,sudo apt-get install -y nvidia-container-toolkit-base执行报错,2023-05-01.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/211,**问题描述 / Problem Description**,70 +71,效果不佳几乎答不上来,2023-05-01.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/212,提供了50条问答的docx文件,71 +72,有没有可能新增一个基于chatglm api调用的方式构建langchain,2023-05-02.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/218,我有两台8G GPU/40G内存的服务器，一个台做成了chatglm的api ；想基于另外一台服务器部署langchain；网上好像没有类似的代码。,72 +73,电脑是intel的集成显卡； 运行时告知我找不到nvcuda.dll，模型无法运行,2023-05-02.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/219,您好，我的电脑是intel的集成显卡，不过CPU是i5-11400 @ 2.60GHz ，内存64G；,73 +74,根据langchain官方的文档和使用模式，是否可以改Faiss为Elasticsearch？会需要做哪些额外调整？求解,2023-05-03.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/221,本人新手小白，由于业务模式的原因（有一些自己的场景和优化），希望利用Elasticsearch做这个体系内部的检索机制，不知道是否可以替换，同时，还会涉及到哪些地方的改动？或者说可能会有哪些其他影响，希望作者和大佬们不吝赐教！,74 +75,请问未来有可能支持t5吗,2023-05-04.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/224,请问可能支持基於t5的模型吗?,75 +76,[BUG] 内存溢出 / torch.cuda.OutOfMemoryError:,2023-05-04.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/229,**问题描述 / Problem Description**,76 +77,报错 No module named 'chatglm_llm',2023-05-04.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/230,明明已经安装了包，却在python里吊不出来,77 +78,能出一个api部署的描述文档吗,2023-05-04.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/233,**功能描述 / Feature Description**,78 +79,使用docs/API.md 出错,2023-05-04.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/234,使用API.md文档2种方法，出错,79 +80,加载pdf文档报错？,2023-05-05.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/238,ew one with MEAN pooling.,80 +81,上传的本地知识文件后再次上传不能显示，只显示成功了一个，别的上传成功后再次刷新就没了,2023-05-05.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/239,您好，项目有很大启发，感谢~,81 +82,创建了新的虚拟环境，安装了相关包，并且自动下载了相关的模型，但是仍旧出现：OSError: Unable to load weights from pytorch checkpoint file for,2023-05-05.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/240,![78ac8e663fdc312d0e9d78da95925c4](https://user-images.githubusercontent.com/34124260/236378728-9ea4424f-0f7f-4013-9d33-820b723de321.png),82 +83,[BUG] 数据加载不进来,2023-05-05.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/243,使用的.txt格式，utf-8编码，报以下错误,83 +84,不能读取pdf,2023-05-05.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/244,请问是webui还是cli_demo,84 +85,本地txt文件有500M，加载的时候很慢，如何提高速度？,2023-05-06.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/251,![yayRzxSYHP](https://user-images.githubusercontent.com/109277248/236592902-f5ab338d-c1e9-43dc-ae16-9df2cd3c1378.jpg),85 +86,[BUG] gradio上传知识库后刷新之后 知识库就不见了 只有重启才能看到之前的上传的知识库,2023-05-06.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/253,gradio上传知识库后刷新之后 知识库就不见了 只有重启才能看到之前的上传的知识库,86 +87,[FEATURE] 可以支持 OpenAI 的模型嘛？比如 GPT-3、GPT-3.5、GPT-4；embedding 增加 text-embedding-ada-002,2023-05-06.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/254,**功能描述 / Feature Description**,87 +88,[FEATURE] 能否增加对于milvus向量数据库的支持 / Concise description of the feature,2023-05-06.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/256,**功能描述 / Feature Description**,88 +89,CPU和GPU上跑，除了速度有区别，准确率效果回答上有区别吗？,2023-05-06.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/259,理论上没有区别,89 +90,m1，请问在生成回答时怎么看是否使用了mps or cpu？,2023-05-06.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/260,m1，请问在生成回答时怎么看是否使用了mps or cpu？,90 +91,知识库一刷新就没了,2023-05-07.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/263,知识库上传后刷新就没了,91 +92,本地部署报没有模型,2023-05-07.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/267,建议在下载llm和embedding模型至本地后在configs/model_config中写入模型本地存储路径后再运行,92 +93,[BUG] python3: can't open file 'webui.py': [Errno 2] No such file or directory,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/269,**问题描述 / Problem Description**,93 +94,模块缺失提示,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/271,因为已有自己使用的docker环境，直接启动webui.py，提示,94 +95,"运行api.py后，执行curl -X POST ""http://127.0.0.1:7861"" 报错？",2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/272,"执行curl -X POST ""http://127.0.0.1:7861"" \ -H 'Content-Type: application/json' \ -d '{""prompt"": ""你好"", ""history"": []}'，报错怎么解决",95 +96,[BUG] colab安装requirements提示protobuf版本问题？,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/273,pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.,96 +97,请问项目里面向量相似度使用了什么方法计算呀？,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/275,基本按照langchain里的FAISS.similarity_search_with_score_by_vector实现,97 +98,[BUG] 安装detectron2后，pdf无法加载,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/276,**问题描述 / Problem Description**,98 +99,[BUG] 使用ChatYuan-V2模型无法流式输出，会报错,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/277,一方面好像是ChatYuan本身不支持stream_chat，有人在clueai那边提了issue他们说还没开发，所以估计这个attribute调不起来；但是另一方面看报错好像是T5模型本身就不是decoder-only模型，所以不能流式输出吧（个人理解）,99 +100,[BUG] 无法加载text2vec模型,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/278,**问题描述 / Problem Description**,100 +101,请问能否增加网络搜索功能,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/281,请问能否增加网络搜索功能,101 +102,[FEATURE] 结构化数据sql、excel、csv啥时会支持呐。,2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/283,**功能描述 / Feature Description**,102 +103,TypeError: ChatGLM._call() got an unexpected keyword argument 'stop',2023-05-08.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/284,No sentence-transformers model found with name D:\DevProject\langchain-ChatGLM\GanymedeNil\text2vec-large-chinese. Creating a new one with MEAN pooling.,103 +104,关于api.py的一些bug和设计逻辑问题？,2023-05-09.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/285,首先冒昧的问一下，这个api.py，开发者大佬们是在自己电脑上测试后确实没问题吗？,104 +105,有没有租用的算力平台上，运行api.py后，浏览器http://localhost:7861/报错,2023-05-09.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/287,是不是租用的gpu平台上都会出现这个问题？？？,105 +106,请问一下项目中有用到文档段落切割方法吗？,2023-05-09.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/288,text_load中的文档切割方法用上了吗？在代码中看好像没有用到？,106 +107,"报错 raise ValueError(f""Knowledge base {knowledge_base_id} not found"") ValueError: Knowledge base ./vector_store not found",2023-05-09.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/289,"File ""/root/autodl-tmp/chatglm/langchain-ChatGLM-master/api.py"", line 183, in chat",107 +108,能接入vicuna模型吗,2023-05-09.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/290,目前本地已经有了vicuna模型能直接接入吗？,108 +109,[BUG] 提问公式相关问题大概率爆显存,2023-05-09.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/291,**问题描述 / Problem Description**,109 +110,安装pycocotools失败，找了好多方法都不能解决。,2023-05-10.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/292,**问题描述 / Problem Description**,110 +111,使用requirements安装，PyTorch安装的是CPU版本,2023-05-10.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/294,如题目，使用requirements安装，PyTorch安装的是CPU版本，运行程序的时候，也是使用CPU在工作。,111 +112,能不能给一个毛坯服务器的部署教程,2023-05-10.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/298,“开发部署”你当成服务器的部署教程用就行了。,112 +113, Error(s) in loading state_dict for ChatGLMForConditionalGeneration:,2023-05-10.02,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/299,运行中出现的问题，7860的端口页面显示不出来，求助。,113 +114,ChatYuan-large-v2模型加载失败,2023-05-10.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/300,**实际结果 / Actual Result**,114 +115,新增摘要功能,2023-05-10.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/303,你好，后续会考虑新增对长文本信息进行推理和语音理解功能吗？比如生成摘要,115 +116,[BUG] pip install -r requirements.txt 出错,2023-05-10.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/304,pip install langchain -i https://pypi.org/simple,116 +117,[BUG] 上传知识库文件报错,2023-05-10.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/305,![19621e29eaa547d01213bee53d81e6a](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/84606552/7f6ceb46-e494-4b0e-939c-23b585a6d9d8),117 +118,[BUG] AssertionError: Component with id 41 not a valid input component.,2023-05-10.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/306,**问题描述 / Problem Description**,118 +119,[BUG] CUDA out of memory with container deployment,2023-05-10.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/310,**问题描述 / Problem Description**,119 +120,[FEATURE] 增加微调训练功能,2023-05-11.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/311,**功能描述 / Feature Description**,120 +121,如何使用多卡部署，多个gpu,2023-05-11.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/315,"机器上有多个gpu,如何全使用了",121 +122,请问这个知识库问答，和chatglm的关系是什么,2023-05-11.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/319,这个知识库问答，哪部分关联到了chatglm，是不是没有这个chatglm，知识库问答也可单单拎出来,122 +123,[BUG] 运行的时候报错ImportError: libcudnn.so.8: cannot open shared object file: No such file or directory,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/324,**问题描述 / Problem Description**raceback (most recent call last):,123 +124,webui启动成功，但有报错,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/325,**问题描述 / Problem Description**,124 +125,切换MOSS的时候报错,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/327,danshi但是发布的源码中，,125 +126,vicuna模型是否能接入？,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/328,您好！关于MOSS模型和vicuna模型，都是AutoModelForCausalLM来加载模型的，但是稍作更改（模型路径这些）会报这个错误。这个错误的造成是什么,126 +127,你好，请问一下在阿里云CPU服务器上跑可以吗？可以的话比较理想的cpu配置是什么？,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/330,你好，请问一下在阿里云CPU服务器上跑可以吗？可以的话比较理想的cpu配置是什么？,127 +128,你好，请问8核32g的CPU可以跑多轮对话吗？,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/331,什么样的cpu配置比较好呢？我目前想部署CPU下的多轮对话？,128 +129,[BUG] 聊天内容输入超过10000个字符系统出现错误,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/332,聊天内容输入超过10000个字符系统出现错误，如下图所示：,129 +130,能增加API的多用户访问接口部署吗？,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/333,默认部署程序仅支持单用户访问，多用户则需要排队访问。测试过相关的几个Github多用户工程，但是其中一些仍然不满足要求。本节将系统介绍如何实现多用户同时访问ChatGLM的部署接口，包括http、websocket（流式输出，stream）和web页面等方式，主要目录如下所示。,130 +131,多卡部署,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/334,用单机多卡或多机多卡，fastapi部署模型，怎样提高并发,131 +132,WEBUI能否指定知识库目录？,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/335,**功能描述 / Feature Description**,132 +133,[BUG] Cannot read properties of undefined (reading 'error'),2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/336,**问题描述 / Problem Description**,133 +134,[BUG] 1 validation error for HuggingFaceEmbeddings model_kwargs extra fields not permitted.,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/337,模型加载到 100% 后出现问题：,134 +135,上传知识库需要重启能不能修复一下,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/338,挺严重的这个问题,135 +136,[BUG] 4块v100卡爆显存，在LLM会话模式也一样,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/339,**问题描述 / Problem Description**,136 +137,针对上传的文件配置不同的TextSpliter,2023-05-12.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/341,1. 目前的ChineseTextSpliter切分对英文尤其是代码文件不友好，而且限制固定长度；导致对话结果不如人意,137 +138,[FEATURE] 未来可增加Bloom系列模型吗？根据甲骨易的测试，这系列中文评测效果不错,2023-05-13.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/346,**功能描述 / Feature Description**,138 +139,[BUG] v0.1.12打包镜像后启动webui.py失败 / Concise description of the issue,2023-05-13.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/347,**问题描述 / Problem Description**,139 +140,切换MOSS模型时报错,2023-05-13.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/349,昨天问了下，说是transformers版本不对，需要4.30.0，发现没有这个版本，今天更新到4.29.1，依旧报错，错误如下,140 +141,[BUG] pdf文档加载失败,2023-05-13.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/350,**问题描述 / Problem Description**,141 +142,建议可以在后期增强一波注释，这样也有助于更多人跟进提PR,2023-05-13.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/351,知道作者和团队在疯狂更新审查代码，只是建议后续稳定后可以把核心代码进行一些注释的补充，从而能帮助更多人了解各个模块作者的思路从而提出更好的优化。,142 +143,[FEATURE] MOSS 量化版支援,2023-05-13.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/353,**功能描述 / Feature Description**,143 +144,[BUG] moss模型无法加载,2023-05-13.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/356,**问题描述 / Problem Description**,144 +145,[BUG] load_doc_qa.py 中的 load_file 函数有bug,2023-05-14.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/358,原函数为：,145 +146,[FEATURE] API模式，知识库加载优化,2023-05-14.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/359,如题，当前版本，每次调用本地知识库接口，都将加载一次知识库，是否有更好的方式？,146 +147,运行Python api.py脚本后端部署后，怎么使用curl命令调用？,2023-05-15.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/361,也就是说，我现在想做个对话机器人，想和公司的前后端联调？怎么与前后端相互调用呢？可私信，有偿解答！！！,147 +148,上传知识库需要重启能不能修复一下,2023-05-15.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/363,上传知识库需要重启能不能修复一下,148 +149,[BUG] pip install -r requirements.txt -i https://pypi.tuna.tsinghua.edu.cn/simple,2023-05-15.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/364,我的python是3.8.5的,149 +150,pip install gradio 报错,2023-05-15.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/367,大佬帮我一下,150 +151,[BUG] pip install gradio 一直卡不动,2023-05-15.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/369,![aba82742dd9d4d242181662eb5027a7](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/84606552/cd9600d9-f6e7-46b7-b1be-30ed8b99f76b),151 +152,[BUG] 简洁阐述问题 / Concise description of the issue,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/370,初次加载本地知识库成功，但提问后，就无法重写加载本地知识库,152 +153,[FEATURE] 简洁阐述功能 / Concise description of the feature,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/371,**功能描述 / Feature Description**,153 +154,在windows上，模型文件默认会安装到哪,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/372,-------------------------------------------------------------------------------,154 +155,[FEATURE] 兼顾对话管理,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/374,如何在知识库检索的情况下，兼顾对话管理？,155 +156,llm device: cpu embedding device: cpu,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/376,**问题描述 / Problem Description**,156 +157,[FEATURE] 简洁阐述功能 /文本文件的知识点之间使用什么分隔符可以分割？,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/377,**功能描述 / Feature Description**,157 +158,[BUG] 上传文件失败：PermissionError: [WinError 32] 另一个程序正在使用此文件，进程无法访问。,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/379,**问题描述 / Problem Description**,158 +159,[BUG] 执行python api.py 报错,2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/383,错误信息,159 +160,model_kwargs extra fields not permitted (type=value_error.extra),2023-05-16.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/384,"大家好，请问这个有遇到的么,？",160 +161,[BUG] 简洁阐述问题 / Concise description of the issue,2023-05-17.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/385,执行的时候出现了ls1 = [ls[0]],161 +162,[FEATURE] 性能优化,2023-05-17.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/388,**功能描述 / Feature Description**,162 +163,"[BUG] Moss模型问答，RuntimeError: probability tensor contains either inf, nan or element < 0",2023-05-17.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/390,**问题描述 / Problem Description**,163 +164,有没有人知道v100GPU的32G显存，会报错吗？支持V100GPU吗？,2023-05-17.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/392,**问题描述 / Problem Description**,164 +165,针对于编码问题比如'gbk' codec can't encode character '\xab' in position 14: illegal multibyte sequence粗浅的解决方法,2023-05-17.03,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/397,**功能描述 / Feature Description**,165 +166,Could not import sentence_transformers python package. Please install it with `pip install sentence_transformers`.,2023-05-18.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/400,**问题描述 / Problem Description**,166 +167,支持模型问答与检索问答,2023-05-18.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/401,不同的query，根据意图不一致，回答也应该不一样。,167 +168,文本分割的时候，能不能按照txt文件的每行进行分割，也就是按照换行符号\n进行分割？？？,2023-05-18.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/403,下面的代码应该怎么修改？,168 +169,local_doc_qa/local_doc_chat 接口响应是串行,2023-05-18.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/405,**问题描述 / Problem Description**,169 +170,"为什么找到出处了,但是还是无法回答该问题?",2023-05-18.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/406,![图片](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/3349611/1fc81d61-2409-4330-9065-fdda1a27c86a),170 +171,"请问下:知识库测试中的:添加单条内容,如果换成文本导入是是怎样的格式?我发现添加单条内容测试效果很好.",2023-05-18.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/412,"我发现在知识库测试中`添加单条内容`,并且勾选`禁止内容分句入库`,即使 `不开启上下文关联`的测试效果都非常好.",171 +172,[BUG] 无法配置知识库,2023-05-18.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/413,**问题描述 / Problem Description**,172 +173,[BUG] 部署在阿里PAI平台的EAS上访问页面是白屏,2023-05-19.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/414,**问题描述 / Problem Description**,173 +174,API部署后调用/local_doc_qa/local_doc_chat 返回Knowledge base samples not found,2023-05-19.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/416,入参,174 +175,[FEATURE] 上传word另存为的txt文件报 'ascii' codec can't decode byte 0xb9 in position 6: ordinal not in range(128),2023-05-20.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/421,上传word另存为的txt文件报,175 +176,创建保存的知识库刷新后没有出来，这个知识库是永久保存的吗？可以连外部的 向量知识库吗？,2023-05-21.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/422,创建保存的知识库刷新后没有出来，这个知识库是永久保存的吗？可以连外部的 向量知识库吗？,176 +177,[BUG] 用colab运行，无法加载模型，报错：'NoneType' object has no attribute 'message_types_by_name',2023-05-21.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/423,**问题描述 / Problem Description**,177 +178,请问是否需要用到向量数据库？以及什么时候需要用到向量数据库？,2023-05-21.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/424,目前用的是 text2vec ， 请问是否需要用到向量数据库？以及什么时候需要用到向量数据库？,178 +179,huggingface模型引用问题,2023-05-22.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/427,它最近似乎变成了一个Error？,179 +180,你好，加载本地txt文件出现这个killed错误，TXT文件有100M左右大小。原因是？谢谢。,2023-05-22.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/429,"",180 +181,想请问一下，关于对本地知识的管理是如何管理？例如：通过http API接口添加数据 或者 删除某条数据,2023-05-22.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/430,例如：通过http API接口添加、删除、修改 某条数据。,181 +182,[FEATURE] 双栏pdf识别问题,2023-05-22.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/432,试了一下模型，感觉对单栏pdf识别的准确性较高，但是由于使用的基本是ocr的技术，对一些双栏pdf论文识别出来有很多问题，请问有什么办法改善吗？,182 +183,部署启动小问题，小弟初学求大佬解答,2023-05-22.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/433,1.python loader/image_loader.py时，提示ModuleNotFoundError: No module named 'configs'，但是跑python webui.py还是还能跑,183 +184,能否支持检测到目录下文档有增加而去增量加载文档，不影响前台对话，其实就是支持读写分离。如果能支持查询哪些文档向量化了，删除过时文档等就更好了，谢谢。,2023-05-22.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/434,**功能描述 / Feature Description**,184 +185,[BUG] 简洁阐述问题 / windows 下cuda错误，请用https://github.com/Keith-Hon/bitsandbytes-windows.git,2023-05-22.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/435,pip install git+https://github.com/Keith-Hon/bitsandbytes-windows.git,185 +186,"[BUG] from commit 33bbb47, Required library version not found: libbitsandbytes_cuda121_nocublaslt.so. Maybe you need to compile it from source?",2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/438,**问题描述 / Problem Description**,186 +187,[BUG] 简洁阐述问题 / Concise description of the issue上传60m的txt文件报错，显示超时，请问这个能上传的文件大小有限制吗,2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/439,"ERROR 2023-05-23 11:13:09,627-1d: Timeout reached while detecting encoding for ./docs/GLM模型格式数据.txt",187 +188,[BUG] TypeError: issubclass() arg 1 must be a class,2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/440,**问题描述**,188 +189,"执行python3 webui.py后，一直提示”模型未成功加载，请到页面左上角""模型配置""选项卡中重新选择后点击""加载模型""按钮“",2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/441,**问题描述 / Problem Description**,189 +190,是否能提供网页文档得导入支持,2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/444,现在很多都是在线文档作为协作得工具，所以通过URL导入在线文档需求更大,190 +191,[BUG] history 索引问题,2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/445,在比较对话框的history和模型chat function 中的history时， 发现并不匹配，在传入 llm._call 时，history用的索引是不是有点问题，导致上一轮对话的内容并不输入给模型。,191 +192,[BUG] moss_llm没有实现,2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/447,有些方法没支持，如history_len,192 +193,请问langchain-ChatGLM如何删除一条本地知识库的数据？,2023-05-23.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/448,例如：用户刚刚提交了一条错误的数据到本地知识库中了，现在如何在本地知识库从找到，并且对此删除。,193 +194,[BUG] 简洁阐述问题 / UnboundLocalError: local variable 'resp' referenced before assignment,2023-05-24.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/450,"在最新一版的代码中， 运行api.py 出现了以上错误（UnboundLocalError: local variable 'resp' referenced before assignment）， 通过debug的方式观察到local_doc_qa.llm.generatorAnswer(prompt=question, history=history,streaming=True)可能不返回任何值。",194 +195,请问有没有 PROMPT_TEMPLATE 能让模型不回答敏感问题,2023-05-24.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/452,## PROMPT_TEMPLATE问题,195 +196,[BUG] 测试环境 Python 版本有误,2023-05-24.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/456,**问题描述 / Problem Description**,196 +197,[BUG] webui 部署后样式不正确,2023-05-24.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/458,**问题描述 / Problem Description**,197 +198,配置默认LLM模型的问题,2023-05-24.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/459,**问题描述 / Problem Description**,198 +199,[FEATURE]是时候更新一下autoDL的镜像了,2023-05-24.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/460,如题，跑了下autoDL的镜像，发现是4.27号的，git pull新版本的代码功能+老的依赖环境，各种奇奇怪怪的问题。,199 +200,[BUG] tag:0.1.13 以cpu模式下，想使用本地模型无法跑起来，各种路径参数问题,2023-05-24.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/462,-------------------------------------------------------------------------------,200 +201,[BUG] 有没有同学遇到过这个错！！！加载本地txt文件出现这个killed错误，TXT文件有100M左右大小。,2023-05-25.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/463,运行cli_demo.py。是本地的txt文件太大了吗？100M左右。,201 +202,API版本能否提供WEBSOCKET的流式接口,2023-05-25.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/464,webui 版本中，采用了WS的流式输出，整体感知反应很快,202 +203,[BUG] 安装bug记录,2023-05-25.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/465,按照[install文档](https://github.com/imClumsyPanda/langchain-ChatGLM/blob/master/docs/INSTALL.md)安装的，,203 +204,VUE的pnmp i执行失败的修复-用npm i命令即可,2023-05-25.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/466,感谢作者！非常棒的应用，用的很开心。,204 +205,请教个问题，有没有人知道cuda11.4是否支持？？？,2023-05-25.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/467,请教个问题，有没有人知道cuda11.4是否支持？？？,205 +206,请问有实现多轮问答中基于问题的搜索上下文关联么,2023-05-25.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/468,在基于知识库的多轮问答中，第一个问题讲述了一个主题，后续的问题描述没有包含这个主题的关键词，但又存在上下文的关联。如果用后续问题去搜索知识库有可能会搜索出无关的信息，从而导致大模型无法正确回答问题。请问这个项目要考虑这种情况吗？,206 +207,[BUG] 内存不足的问题,2023-05-26.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/470,我用了本地的chatglm-6b-int4模型，然后显示了内存不足（win10+32G内存+1080ti11G），一般需要多少内存才足够？这个bug应该如何解决？,207 +208,[BUG] 纯内网环境安装pycocotools失败,2023-05-26.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/472,**问题描述 / Problem Description**,208 +209,[BUG] webui.py 重新加载模型会导致 KeyError,2023-05-26.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/473,**问题描述 / Problem Description**,209 +210,chatyuan无法使用,2023-05-26.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/475,**问题描述 / Problem Description**,210 +211,[BUG] 文本分割模型AliTextSplitter存在bug，会把“.”作为分割符,2023-05-26.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/476,"阿里达摩院的语义分割模型存在bug，默认会把"".”作为分割符进行分割而不管上下文语义。是否还有其他分割符则未知。建议的修改方案：把“.”统一替换为其他字符，分割后再替换回来。或者添加其他分割模型。",211 +212,[BUG] RuntimeError: Error in faiss::FileIOReader::FileIOReader(const char*) a,2023-05-27.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/479,**问题描述 / Problem Description**,212 +213,[FEATURE] 安装，为什么conda create要额外指定路径 用-p ，而不是默认的/envs下面,2023-05-28.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/481,##**功能描述 / Feature Description**,213 +214,[小白求助] 通过Anaconda执行webui.py后，无法打开web链接,2023-05-28.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/485,在执行webui.py命令后，http://0.0.0.0:7860复制到浏览器后无法打开，显示“无法访问此网站”。,214 +215,[BUG] 使用 p-tuningv2后的模型，重新加载报错,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/486,把p-tunningv2训练完后的相关文件放到了p-tunningv2文件夹下，勾选使用p-tuningv2点重新加载模型，控制台输错错误信息：,215 +216,[小白求助] 服务器上执行webui.py后，在本地无法打开web链接,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/487,此项目执行在xxx.xx.xxx.xxx服务器上，我在webui.py上的代码为 (demo,216 +217,[FEATURE] 能不能支持VisualGLM-6B,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/488,**功能描述 / Feature Description**,217 +218,你好，问一下各位，后端api部署的时候，支持多用户同时问答吗？？？,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/489,支持多用户的话，最多支持多少用户问答？根据硬件而定吧？,218 +219,V100GPU显存占满，而利用率却为0，这是为什么？,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/491,"",219 +220,[求助] 如果在公司内部搭建产品知识库，使用INT-4模型，200人规模需要配置多少显存的服务器？,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/492,如题，计划给公司搭一个在线知识库。,220 +221,你好，请教个问题，目前问答回复需要20秒左右，如何提高速度？V10032G服务器。,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/493,**问题描述 / Problem Description**,221 +222,[FEATURE] 如何实现只匹配下文，而不要上文的结果,2023-05-29.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/494,在构建自己的知识库时，主要采用问答对的形式，那么也就是我需要的回答是在我的问题下面的内容，但是目前设置了chunk_size的值以后匹配的是上下文的内容，但我实际并不需要上文的。为了实现更完整的展示下面的答案，我只能调大chunk_size的值，但实际上上文的一半内容都是我不需要的。也就是扔了一半没用的东西给prompt，在faiss.py中我也没找到这块的一些描述，请问该如何进行修改呢？,222 +223,你好，问一下，我调用api.py部署，为什么用ip加端口可以使用postman调用，而改为域名使用postman无法调用？,2023-05-30.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/497,![5ufBSWxLyF](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/109277248/70e2fbac-5699-48d0-b0d1-3dc84fd042c2),223 +224,调用api.py中的stream_chat，返回source_documents中出现中文乱码。,2023-05-30.04,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/498,-------------------------------------------------------------------------------,224 +225,[BUG] 捉个虫，api.py中的stream_chat解析json问题,2023-05-30.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/501,**问题描述 / Problem Description**,225 +226,windows本地部署遇到了omp错误,2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/502,**问题描述 / Problem Description**,226 +227,"[BUG] bug14 ,""POST /local_doc_qa/upload_file HTTP/1.1"" 422 Unprocessable Entity",2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/503,上传的文件报错，返回错误，api.py,227 +228,你好，请教个问题，api.py部署的时候，如何改为多线程调用？谢谢,2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/505,目前的api.py脚本不支持多线程,228 +229,你好，请教一下。api.py部署的时候，能不能提供给后端流失返回结果。,2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/507,curl -X 'POST' \,229 +230,流式输出，流式接口，使用server-sent events技术。,2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/508,想这样一样，https://blog.csdn.net/weixin_43228814/article/details/130063010,230 +231,计划增加流式输出功能吗？ChatGLM模型通过api方式调用响应时间慢怎么破，Fastapi流式接口来解惑，能快速提升响应速度,2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/509,**问题描述 / Problem Description**,231 +232,[BUG] 知识库上传时发生ERROR (could not open xxx for reading: No such file or directory),2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/510,**问题描述 / Problem Description**,232 +233,api.py脚本打算增加SSE流式输出吗？,2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/511,curl调用的时候可以检测第一个字，从而提升回复的体验,233 +234,[BUG] 使用tornado实现webSocket，可以多个客户端同时连接，并且实现流式回复，但是多个客户端同时使用，答案就很乱，是模型不支持多线程吗,2023-05-31.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/512,import asyncio,234 +235,支持 chinese_alpaca_plus_lora 吗 基于llama的,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/514,支持 chinese_alpaca_plus_lora 吗 基于llama的，https://github.com/ymcui/Chinese-LLaMA-Alpaca这个项目的,235 +236,[BUG] 现在能读图片的pdf了，但是文字的pdf反而读不了了，什么情况？？？,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/515,**问题描述 / Problem Description**,236 +237,在推理的过程中卡住不动，进程无法正常结束,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/516,**问题描述 / Problem Description**,237 +238,curl调用的时候，从第二轮开始，curl如何传参可以实现多轮对话？,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/517,第一轮调用：,238 +239,建议添加api.py部署后的日志管理功能？,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/518,-------------------------------------------------------------------------------,239 +240,有大佬知道，怎么多线程部署api.py脚本吗？,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/519,api.py部署后，使用下面的请求，时间较慢，好像是单线程，如何改为多线程部署api.py：,240 +241,[BUG] 上传文件到知识库 任何格式与内容都永远失败,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/520,上传知识库的时候，传txt无法解析，就算是穿content/sample里的样例txt也无法解析，上传md、pdf等都无法加载，会持续性等待，等到了超过30分钟也不行。,241 +242,关于prompt_template的问题,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/521,请问这段prompt_template是什么意思，要怎么使用？可以给一个具体模板参考下吗？,242 +243,[BUG] 简洁阐述问题 / Concise description of the issue,2023-06-01.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/522,**问题描述 / Problem Description**,243 +244,"中文分词句号处理（关于表达金额之间的"".""）",2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/523,建议处理12.6亿元的这样的分词，最好别分成12 和6亿这样的，需要放到一起,244 +245,ImportError: cannot import name 'inference' from 'paddle',2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/526,在网上找了一圈，有说升级paddle的，我做了还是没有用，有说安装paddlepaddle的，我找了豆瓣的镜像源，但安装报错cannot detect archive format,245 +246,[BUG] webscoket 接口串行问题（/local_doc_qa/stream-chat/{knowledge_base_id}）,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/527,**问题描述 / Problem Description**,246 +247,[FEATURE] 刷新页面更新知识库列表,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/528,**功能描述以及改进方案**,247 +248,[BUG] 使用ptuning微调模型后，问答效果并不好,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/530,### 未调用ptuning,248 +249,[BUG] 多轮对话效果不佳,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/532,在进行多轮对话的时候，无论设置的history_len是多少，效果都不好。事实上我将其设置成了最大值10，但在对话中，仍然无法实现多轮对话：,249 +250,"RuntimeError: MPS backend out of memory (MPS allocated: 18.00 GB, other allocations: 4.87 MB, max allowed: 18.13 GB)",2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/533,**问题描述**,250 +251, 请大家重视这个issue！真正使用肯定是多用户并发问答，希望增加此功能！！！,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/534,这得看你有多少显卡,251 +252,在启动项目的时候如何使用到多张gpu啊？,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/535,**在启动项目的时候如何使用到多张gpu啊？**,252 +253, 使用流式输出的时候，curl调用的格式是什么？,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/536,"app.websocket(""/local_doc_qa/stream-chat/{knowledge_base_id}"")(stream_chat)中的knowledge_base_id应该填什么？？？",253 +254,使用本地 vicuna-7b模型启动错误,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/538,环境: ubuntu 22.04 cuda 12.1 没有安装nccl，使用rtx2080与m60显卡并行计算,254 +255,为什么会不调用GPU直接调用CPU呢,2023-06-02.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/539,我的阿里云配置是16G显存，用默认代码跑webui.py时提示,255 +256,上传多个文件时会互相覆盖,2023-06-03.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/541,1、在同一个知识库中上传多个文件时会互相覆盖，无法结合多个文档的知识，有大佬知道怎么解决吗？,256 +257,[BUG] ‘gcc’不是内部或外部命令/LLM对话只能持续一轮,2023-06-03.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/542,No compiled kernel found.,257 +258,以API模式启动项目却没有知识库的接口列表？,2023-06-04.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/544,请问如何获取知识库的接口列表？如果没有需要自行编写的话，可不可以提供相关的获取方式，感谢,258 +259,程序以API模式启动的时候，如何才能让接口以stream模式被调用呢？,2023-06-05.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/546,作者您好，我在以API模式进行程序启动后，我发现接口响应时间很长，怎么样才能让接口以stream模式被调用呢？我想实现像webui模式的回答那样,259 +260,关于原文中表格转为文本后数据相关度问题。,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/547,原文中表格数据转换为文本，以 （X-Y：值；...） 的格式每一行组织成一句话，但这样做后发现相关度较低，效果很差，有何好的方案吗？,260 +261,启动后LLM和知识库问答模式均只有最后一轮记录,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/548,拉取最新代码，问答时，每次页面只显示最后一次问答记录，需要修改什么参数才可以保留历史记录？,261 +262,提供system message配置，以便于让回答不要超出知识库范围,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/549,**功能描述 / Feature Description**,262 +263,[BUG] 使用p-tunningv2报错,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/551,按照readme的指示把p-tunningv2训练完后的文件放到了p-tunningv2文件夹下，勾选使用p-tuningv2点重新加载模型，控制台提示错误信息：,263 +264,[BUG] 智障，这么多问题，也好意思放出来，浪费时间,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/553,。。。,264 +265,[FEATURE] 我看代码文件中有一个ali_text_splitter.py，为什么不用他这个文本分割器了？,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/554,我看代码文件中有一个ali_text_splitter.py，为什么不用他这个文本分割器了？,265 +266,加载文档函数报错,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/557,"def load_file(filepath, sentence_size=SENTENCE_SIZE):",266 +267,参考指引安装docker后，运行cli_demo.py，提示killed,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/558,root@b3d1bd08095c:/chatGLM# python3 cli_demo.py,267 +268,注意：如果安装错误，注意这两个包的版本 wandb==0.11.0 protobuf==3.18.3,2023-06-06.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/559,Error1: 如果启动异常报错 `protobuf` 需要更新到 `protobuf==3.18.3 `,268 +269,知识库对长文的知识相关度匹配不太理想有何优化方向,2023-06-07.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/563,我们可能录入一个文章有 1W 字，里面涉及这个文章主题的很多角度问题，我们针对他提问，他相关度匹配的内容和实际我们需要的答案相差很大怎么办。,269 +270,使用stream-chat函数进行流式输出的时候，能使用curl调用吗？,2023-06-07.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/565,为什么下面这样调用会报错？？？,270 +271,有大佬实践过 并行 或者 多线程 的部署方案吗？,2023-06-07.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/566,+1,271 +272,多线程部署遇到问题？,2023-06-07.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/567,"",272 +273,[BUG] 用fastchat加载vicuna-13b模型进行知识库的问答有token的限制错误,2023-06-07.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/569,当我开启fastchat的vicuna-13b的api服务，然后config那里配置好(api本地测试过可以返回结果)，然后知识库加载好之后(知识库大概有1000多个文档，用chatGLM可以正常推理)，进行问答时出现token超过限制，就问了一句hello；,273 +274,现在的添加知识库，文件多了总是报错，也不知道自己加载了哪些文件，报错后也不知道是全部失败还是一部分成功；希望能有个加载指定文件夹作为知识库的功能,2023-06-07.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/574,**功能描述 / Feature Description**,274 +275,[BUG] moss模型本地加载报错,2023-06-08.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/577,moss模型本地加载报错：,275 +276,加载本地moss模型报错Can't instantiate abstract class MOSSLLM with abstract methods _history_len,2023-06-08.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/578,(vicuna) ps@ps[13:56:20]:/data/chat/langchain-ChatGLM2/langchain-ChatGLM-0.1.13$ python webui.py --model-dir local_models --model moss --no-remote-model,276 +277,[FEATURE] 能增加在前端页面控制prompt_template吗？或是能支持前端页面选择使用哪个prompt？,2023-06-08.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/579,目前只能在config里修改一个prompt，想在多个不同场景切换比较麻烦,277 +278,[BUG] streamlit ui的bug，在增加知识库时会报错,2023-06-08.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/580,**问题描述 / Problem Description**,278 +279,[FEATURE] webui/webui_st可以支持history吗？目前仅能一次对话,2023-06-08.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/581,试了下webui和webui_st都不支持历史对话啊，只能对话一次，不能默认开启所有history吗？,279 +280,启动python cli_demo.py --model chatglm-6b-int4-qe报错,2023-06-09.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/585,下载好模型，和相关依赖环境，之间运行`python cli_demo.py --model chatglm-6b-int4-qe`报错了：,280 +281,重新构建知识库报错,2023-06-09.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/586,**问题描述 / Problem Description**,281 +282,[FEATURE] 能否屏蔽paddle，我不需要OCR，效果差依赖环境还很复杂,2023-06-09.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/587,希望能不依赖paddle,282 +283,question ：文档向量化这个可以自己手动实现么？,2023-06-09.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/589,现有公司级数据500G+，需要使用这个功能，请问如何手动实现这个向量化，然后并加载,283 +284,view前端能进行流式的返回吗？？,2023-06-09.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/590,view前端能进行流式的返回吗？？,284 +285,"[BUG] Load parallel cpu kernel failed, using default cpu kernel code",2023-06-11.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/594,**问题描述 / Problem Description**,285 +286,[BUG] 简洁阐述问题 / Concise description of the issue,2023-06-11.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/595,**问题描述 / Problem Description**,286 +287,我在上传本地知识库时提示KeyError: 'name'错误，本地知识库都是.txt文件，文件数量大约是2000+。,2023-06-12.05,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/597,"",287 +288,model_config.py中有vicuna-13b-hf模型的配置信息，但是好像还是不可用？,2023-06-12.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/600,@dongyihua543,288 +289,"ImportError: Using SOCKS proxy, but the 'socksio' package is not installed. Make sure to install httpx using `pip install httpx[socks]`.",2023-06-12.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/605,应该代理问题，但是尝试了好多方法都解决不了，,289 +290,[BUG] similarity_search_with_score_by_vector在找不到匹配的情况下出错,2023-06-12.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/607,在设置匹配阈值 VECTOR_SEARCH_SCORE_THRESHOLD 的情况下，vectorstore会返回空，此时上述处理函数会出错,290 +291,[FEATURE] 请问如何搭建英文知识库呢,2023-06-12.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/609,**功能描述 / Feature Description**,291 +292,谁有vicuna权重？llama转换之后的,2023-06-13.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/611,**问题描述 / Problem Description**,292 +293,[FEATURE] API能实现上传文件夹的功能么？,2023-06-13.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/612,用户懒得全选所有的文件，就想上传个文件夹，请问下API能实现这个功能么？,293 +294,请问在多卡部署后，上传单个文件作为知识库，用的是单卡在生成向量还是多卡？,2023-06-13.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/614,目前我检测我本地多卡部署的，好像生成知识库向量的时候用的还是单卡,294 +295,[BUG] python webui.py提示非法指令,2023-06-13.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/615,(/data/conda-langchain [root@chatglm langchain-ChatGLM]# python webui.py,295 +296,知识库文件跨行切分问题,2023-06-13.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/616,我的知识库文件txt文件，是一行一条知识，用\n分行。,296 +297,[FEATURE] bing搜索问答有流式的API么？,2023-06-13.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/617,web端是有这个bing搜索回答，但api接口没有发现，大佬能给个提示么？,297 +298,希望出一个macos m2的安装教程,2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/620,mac m2安装，模型加载成功了，知识库文件也上传成功了，但是一问答就会报错，报错内容如下,298 +299,为【出处】提供高亮显示,2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/621,具体出处里面，对相关的内容高亮显示，不包含前后文。,299 +300,[BUG] CPU运行cli_demo.py，不回答，hang住,2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/622,没有GPU；32G内存的ubuntu机器。,300 +301,关于删除知识库里面的文档后，LLM知识库对话的时候还是会返回该被删除文档的内容,2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/623,如题，在vue前端成功执行删除知识库里面文档A.txt后，未能也在faiss索引中也删除该文档，LLM还是会返回这个A.txt的内容，并且以A.txt为出处，未能达到删除的效果,301 +302,"[BUG] 调用知识库进行问答,显存会一直叠加",2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/625,"14G的显存,调用的chatglm-6b-int8模型,进行知识库问答时,最多问答四次就会爆显存了,观察了一下显存使用情况,每一次使用就会增加一次显存,请问这样是正常的吗?是否有什么配置需要开启可以解决这个问题?例如进行一次知识库问答清空上次问题的显存?",302 +303,[BUG] web页面 重新构建数据库 失败，导致 原来的上传的数据库都没了,2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/626,web页面 重新构建数据库 失败，导致 原来的上传的数据库都没了,303 +304,在CPU上运行webui.py报错Tensor on device cpu is not on the expected device meta!,2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/627,在CPU上运行python webui.py能启动，但最后有：RuntimeError: Tensor on device cpu is not on the expected device meta!,304 +305,"OSError: [WinError 1114] 动态链接库(DLL)初始化例程失败。 Error loading ""E:\xxx\envs\langchain\lib\site-packages\torch\lib\caffe2_nvrtc.dll"" or one of its dependencies.哪位大佬知道如何解决吗？",2023-06-14.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/629,**问题描述 / Problem Description**,305 +306,[BUG] WEBUI删除知识库文档，会导致知识库问答失败,2023-06-15.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/632,如题，从知识库已有文件中选择要删除的文件，点击删除后，在问答框输入内容回车报错,306 +307,更新后的版本中，删除知识库中的文件，再提问出现error错误,2023-06-15.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/634,针对更新版本，识别到一个问题，过程如下：,307 +308,我配置好了环境，想要实现本地知识库的问答？可是它返回给我的,2023-06-15.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/637,没有总结，只有相关度的回复，但是我看演示里面表现的，回复是可以实现总结的，我去查询代码,308 +309,[BUG] NPM run dev can not successfully start the VUE frontend,2023-06-15.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/638,**问题描述 / Problem Description**,309 +310,[BUG] 简洁阐述问题 / Concise description of the issue,2023-06-15.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/639,**问题描述 / Problem Description**,310 +311,提一个模型加载的bug，我在截图中修复了，你们有空可以看一下。,2023-06-15.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/642,![model_load_bug](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/59411575/4432adc4-ccdd-45d9-aafc-5f2d1963403b),311 +312,[求助]关于设置embedding model路径的问题,2023-06-16.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/643,如题，我之前成功跑起来过一次，但因环境丢失重新配置 再运行webui就总是报错,312 +313,Lora微调后的模型可以直接使用吗,2023-06-16.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/646,看model_config.py里是有USE_LORA这个参数的，但是在cli_demo.py和webui.py这两个里面都没有用到，实际测试下来模型没有微调的效果，想问问现在这个功能实现了吗,313 +314,write_check_file在tmp_files目录下生成的load_file.txt是否需要一直保留，占用空间很大，在建完索引后能否删除,2023-06-16.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/647,**功能描述 / Feature Description**,314 +315,[BUG] /local_doc_qa/list_files?knowledge_base_id=test删除知识库bug,2023-06-16.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/649,1.新建test知识库并上传文件（在vue前端完成并检查后端发现确实生成了test文件夹以及下面的content和vec_store,315 +316,[BUG] vue webui无法加载知识库,2023-06-16.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/650,拉取了最新的代码，分别运行了后端api和前端web，点击知识库，始终只能显示simple，无法加载知识库,316 +317,不能本地加载moss模型吗？,2023-06-16.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/652,手动下载模型设置local_model_path路径依旧提示缺少文件，该如何正确配置？,317 +318,macos m2 pro docker 安装失败,2023-06-17.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/654,macos m2 pro docker 安装失败,318 +319, [BUG] mac m1 pro 运行提示 zsh: segmentation fault,2023-06-17.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/655,运行： python webui.py,319 +320,安装 requirements 报错,2023-06-17.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/656,(langchainchatglm) D:\github\langchain-ChatGLM>pip install -r requirements.txt -i https://pypi.tuna.tsinghua.edu.cn/simple/,320 +321,[BUG] AssertionError,2023-06-17.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/658,**问题描述 / Problem Description**,321 +322,[FEATURE] 支持AMD win10 本地部署吗？,2023-06-18.06,https://github.com/imClumsyPanda/langchain-ChatGLM/issues/660,**功能描述 / Feature Description**,322 diff --git a/tNAyT4oBgHgl3EQfmvgr/content/tmp_files/2301.00475v1.pdf.txt b/tNAyT4oBgHgl3EQfmvgr/content/tmp_files/2301.00475v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b56a645b15990d82d3aca19765b05ed1c011b3e --- /dev/null +++ b/tNAyT4oBgHgl3EQfmvgr/content/tmp_files/2301.00475v1.pdf.txt @@ -0,0 +1,2500 @@ +arXiv:2301.00475v1 [math.OC] 1 Jan 2023 +A Control Space Ensuring the Strong +Convergence of Continuous Approximation +for a Controlled Sweeping Process +Chadi Nour1† and Vera Zeidan2*† +1Department of Computer Science and Mathematics, Lebanese +American University, Byblos Campus, P.O. Box 36, Byblos, +Lebanon. +2Department of Mathematics, Michigan State University, East +Lansing, 48824-1027, MI, USA. +*Corresponding author(s). E-mail(s): zeidan@msu.edu; +Contributing authors: cnour@lau.edu.lb; +†These authors contributed equally to this work. +Abstract +A controlled sweeping process with prox-regular set, W 1,2-controls, +and +separable +endpoints +constraints +is +considered +in +this +paper. +Existence +of +optimal +solutions +is +established +and +local +opti- +mality +conditions +are +derived +via +strong +converging +continuous +approximations +that +entirely +reside +in +the +interior +of +the +prox- +regular +set. +Consequently, +these +results +are +expressed +in +terms +of +new +subdifferentials +for +the +original +data +that +are +strictly +smaller than the standard Clarke and Mordukhovich subdifferentials. +Keywords: Controlled sweeping process, Prox-regular sets, Necessary +optimality conditions, Local minimizers, Strong convergence, Continuous +approximations, Nonsmooth analysis +MSC Classification: 49K21 , 49K15 , 49J52 +1 + +2 +Control Space for Strong Convergence of Continuous Approximation +1 Introduction +This paper addresses the following fixed time Mayer-type optimal control +problem involving W 1,2-controlled sweeping systems +(P): Minimize g(x(0), x(1)) +over (x, u) ∈ AC([0, 1]; Rn) × W such that + + + + + + + +(D) +� +˙x(t) ∈ f(x(t), u(t)) − ∂ϕ(x(t)), a.e. t ∈ [0, 1], +x(0) ∈ C0 ⊂ dom ϕ, +x(1) ∈ C1, +where, g : Rn × Rn −→ R ∪ {∞}, f : Rn × Rm −→ Rn, ϕ: Rn −→ R ∪ {∞}, ∂ +stands for the Clarke subdifferential, C := dom ϕ is the zero-sublevel set of a +function ψ: Rn −→ R, that is, C = {x ∈ Rn : ψ(x) ≤ 0}, C0 ⊂ C, C1 ⊂ Rn, +and, for U : [0, 1] ⇒ Rm a multifunction and U := � +t∈[0,1] U(t), the set of +control functions W is defined by +W := W 1,2([0, 1]; U) = +� +u ∈ W 1,2([0, 1]; Rm) : u(t) ∈ U(t), ∀t ∈ [0, 1] +� +. (1) +Note that if (x, u) solves (D), it necessarily follows that x(t) ∈ C for all +t ∈ [0, 1]. +A pair (x, u) is admissible for (P) if x: [0, 1] −→ Rn is absolutely continu- +ous, u ∈ W, and (x, u) satisfies the perturbed and controlled sweeping process +(D), called the dynamic of (P). +An admissible pair (¯x, ¯u) for (P) is said to be a W 1,2-local minimizer (also +known as intermediate local minimizer of rank 2) if there exists δ > 0 such that +g(¯x(0), ¯x(1)) ≤ g(x(0), x(1)), +(2) +for all (x, u) admissible for (P) with ∥x− ¯x∥∞ ≤ δ, ∥ ˙x− ˙¯x∥2 +2 ≤ δ, ∥u− ¯u∥∞ ≤ δ +and ∥ ˙u − ˙¯u∥2 +2 ≤ δ. Note that if (2) is satisfied for any admissible pairs (x, u), +then (¯x, ¯u) is called a global minimizer (or an optimal solution) for (P). +Sweeping processes first appeared in the papers [28–30] by J.J. Moreau in +the context of friction and plasticity theory. Since then, these systems have +emerged in further applications such as hysteresis, ferromagnetism, electric cir- +cuits, phase transitions, economics, etc. (see e.g., [1] and its references). The +main feature of such systems is the presence in its dynamic of a normal cone to +a set C that is, the subdifferential of the indicator function of C, or more gen- +erally, the subdifferential of an extended-real-valued function. Consequently, +the dynamic is a differential inclusion with unbounded and discontinuous right- +hand side. Therefore, sweeping processes fall outside the scope of standard +differential inclusions, and hence, studying optimal control problems over this +model requires creative new techniques +In [3, 17, 20, 33, 38] (see also [18]), necessary optimality conditions in +the form of a maximum principle for optimal control problems involving +measurably-controlled sweeping processes are derived using a smooth penalty- +type approximations, called here continuous approximations. A special feature + +Control Space for Strong Convergence of Continuous Approximation +3 +of [17, 20, 33, 38], which led in [19, 32] to numerical algorithms, is the novel +exponential penalization technique that approximates the dynamic of (D) by +the following sequence of standard control systems +(Dγk) +˙x(t) = f(x(t), u(t)) − ∇Φ(x(t)) − γkeγkψ(x(t))∇ψ(x(t)), a.e. t ∈ [0, 1], +where Φ is a smooth extension to Rn of ϕ, and γk is a positive sequence that con- +verges to ∞ as k −→ ∞. In these papers, necessary optimality conditions are +developed via approximating weakly the optimal solution of (P) by a sequence +of optimal solutions for standard optimal control problems over (Dγk). That +is, the velocity sequence of the optimal state for the approximating problems +convergences weakly in L2 to the given velocity of the solution for (P). +Strong convergence of velocities is well-known to be an essential property +for numerical purposes, as pointed out in several papers, see e.g., [7, 8, 14, 17]. +In other words, it is important that the solutions of (P) be strongly approx- +imated (in the W 1,2-norm) by the solutions of approximating problems that +are computable via existing numerical algorithms. This question of strong con- +vergence of velocities was previously addressed using discrete approximations, +see for instance, [5–8, 13, 14, 16], where the authors considered optimal control +problems involving various forms of controlled sweeping processes including +the W 1,2-controls. In [6–8], this approach also served to derive necessary opti- +mality conditions phrased in terms of the weak-Pontryagin-type maximum +principle when the control space is W 1,2([0, 1]; Rm). Therein, these optimality +criteria are applied to real-life models, whose optimal controls turn out to be +W 1,2. +The main goal of the paper is motivated by the importance of approximat- +ing a solution of the sweeping process (D) by solutions of (Dγk) whose velocity +strongly converges to the velocity of the solution of (D) as described above. We +establish the validity of this result when the controls in (Dγk) are chosen to be +W 1,2. As a consequence, we embark on the study of the problem (P). We first +show that, under suitable conditions, the problem (P) admits an optimal solu- +tion (¯x, ¯u). Then, we approximate a given optimal solution (¯x, ¯u) by a sequence +of optimal solutions for standard optimal control problems over (Dγk) with +objective functions carefully formulated to guarantee the strong convergence +of the solution velocities. To our knowledge, this is a first-of-its-kind result +that uses continuous approximations, as opposed to discrete approximations, +to obtain strong convergence of velocities. Furthermore, necessary optimality +conditions are established for W 1,2-local minimizers of (P) upon taking the +limit of the optimality conditions for the corresponding approximating optimal +control problems. This latter task requires meticulous analysis. +The paper is organized as follows. Notations and some definitions from non- +smooth analysis will be given in the next section. In Section 3, we provide a +list of assumptions and some important consequences. Moreover, we present +some needed results from [33, Sections 4 & 5] including the connection between +(Dγk) and (D) under measurable controls. Section 4 is devoted to (i) show- +ing that (Dγk) strongly approximates (D) when W 1,2-controls are utilized, (ii) +establishing an existence theorem for an optimal solution of (P), (iii) con- +structing for (P) a continuous approximating sequence of standard optimal + +4 +Control Space for Strong Convergence of Continuous Approximation +control problems (Pγk), and (iv) deriving necessary optimality conditions in +the form of weak-Pontryagin-type maximum principle for W 1,2-local minimiz- +ers of (P). To maintain an easy flow of the main results, most of the proofs are +provided in Section 5, where we also establish some auxiliary results employed +in different places of the paper. +2 Preliminaries +2.1 Basic notations +In the sequel, the notations used in this paper are provided. By ∥ · ∥, ⟨·, ·⟩, +B and ¯B, we denote, respectively, the Euclidean norm, the usual inner prod- +uct, the open unit ball and the closed unit ball. An open (resp. closed) ball +of radius ρ > 0 and centered at x ∈ Rn is written as Bρ(x) (resp. ¯Bρ(x)). +For x, y ∈ Rn, [x, y] and (x, y) denote, respectively, the closed and the open +line segment joining x to y. For a set C ⊂ Rn, int C, bdry C, cl C, convC, +Cc, and C◦ designate the interior, the boundary, the closure, the convex +hull, the complement, and the polar of C, respectively. The distance from a +point x to a set C is denoted by d(x, C). For an extended-real-valued func- +tion ϕ: Rn −→ R ∪ {∞}, dom ϕ is the effective domain of ϕ and epi ϕ is its +epigraph. For a multifunction F : Rn ⇒ Rm, Gr F ⊂ Rn × Rm denotes the +graph of F, that is, Gr F := {(x, v) ∈ Rn × Rm : v ∈ F(x)}. The Lebesgue +space of p-integrable functions h: [a, b] −→ Rn is denoted by Lp([a, b]; Rn) +or simply Lp when the domain and range are clearly understood. The norms +in Lp([a, b]; Rn) and L∞([a, b]; Rn) (or C([a, b]; Rn)) are written as ∥ · ∥p +and ∥ · ∥∞, respectively. The set of all absolutely continuous functions from +an interval [a, b] to Rn will be denoted by AC([a, b]; Rn), and Mm×n([a, b]) is +the set of m × n-matrix functions on [a, b]. We say that h is a BV -function, +and we write h ∈ BV ([a, b]; Rn), if h is a function of bounded variation, +that is, V b +a (h) < ∞, where V b +a (h) is the total variation of h. We denote by +NBV [a, b] the normalized space of BV -functions on [a, b] that consists of +those BV -functions Ω such that Ω(a) = 0 and Ω is right continuous on (a, b) +(see e.g., [26, p.115]). The set C∗([a, b]; R) denotes the dual of C([a, b]; R), +equipped with the supremum norm. The norm on C∗([a, b]; R), denoted by +∥·∥T.V., is the induced norm. By Riesz representation theorem, the elements of +C∗([a, b]; R) are interpreted as belonging to M([a, b]), the set of finite signed +Radon measures on [a, b] equipped with the weak* topology. Thereby, to each +element of C∗([a, b]; R) it corresponds a unique element in NBV [a, b] related +through the Stieltjes integral and both elements have the same total varia- +tion. We denote by C⊕(a, b) the subset of C∗([a, b]; R) taking nonnegative +values on nonnegative-valued functions in C([a, b]; R). For a compact subset +S ⊂ Rd, C(S; Rn) designates the set of continuous functions from S to Rn. +We denote by W k,p([a, b]; Rn), k ∈ N and p ∈ [0, +∞], the classical Sobolev +space. Note that in this paper, the Sobolev space W 1,2([a, b]; Rn) will be con- +sidered with the norm ∥x(·)∥W 1,2 := ∥x(·)∥∞+∥ ˙x(·)∥2. Hence, the convergence +of a sequence xn strongly in the norm topology of the space W 1,2([a, b]; Rn) + +Control Space for Strong Convergence of Continuous Approximation +5 +is equivalent to the uniform convergence of xn on [a, b] and the strong conver- +gence in L2 of its derivative ˙xn. Finally, a function F : Rn → R is C1,1 if it is +Fr´echet differentiable with locally Lipschitz derivative. +2.2 +Notions in nonsmooth analysis +We begin by listing standard notions and facts for which the reader is invited +to consult the monographs [11], [27], and [36]. Let C be a nonempty and +closed subset of Rn. For x ∈ C, the proximal, the Mordukhovich (also known +as limiting) and the Clarke normal cones to C at x are denoted by N P +C (x), +N L +C(x) and NC(x), respectively. Using [11, Proposition 1.1.5(b)], we deduce +that these three normal cones enjoy an essential local property, namely, if two +closed sets in Rn are the same in a neighborhood of x, then these two sets +possess at x the same normal cone (proximal, Mordukhovich, or Clarke). An +important feature for the Mordukhovich normal cone is that the multifunction +N L +C(·) has closed values and a closed graph. On the other hand, when C is +convex then the proximal, the Mordukhovich and the Clarke normal cones to +C coincide with the well-known normal cone to convex sets. +For ρ > 0, the set C is said to be ρ-prox-regular whenever the proximal +normal inequality, see [11, Proposition 1.1.5(a)], holds for σ = +1 +2ρ, for all x ∈ C +and for all ζ unit in N P +C (x). In particular, every convex set is ρ-prox-regular +for every ρ > 0, and every compact set with a C1,1-boundary is ρ-prox-regular, +where ρ depends on the Lipschitz constant of the gradient of the boundary +parametrization. Note that, for a ρ-prox-regular set C, we have N P +C (x) = +N L +C(x) = NC(x) for all x ∈ Rn. For more information about prox-regularity, +and related properties such as positive reach, proximal smoothness, exterior +sphere condition and ϕ0-convexity, see [12, 15, 21, 31, 34]. +The following geometric properties shall be used in the rest of the paper. A +closed set A ⊂ Rn is said to be epi-Lipschitz (or wedged ) at a point x ∈ A if the +set A can be viewed near x, after application of an orthogonal matrix, as the +epigraph of a Lipschitz continuous function. If this holds for all x ∈ A, then we +simply say that A is epi-Lipschitz. This geometric definition was introduced by +Rockafellar in [35]. The epi-Lipschitz property of A at x is also characterizable +in terms of the nonemptiness of the topological interior of the Clarke tangent +cone of A at x which is also equivalent to the pointedness of the Clarke normal +cone of A at x, that is, NA(x) ∩ −NA(x) = {0}, see [9, Theorem 7.3.1]. Note +that a convex set is epi-Lipschitz if and only if it has a nonempty interior. For +more information about this property, see [9, 11, 36]. On the other hand, a set +A ⊂ Rn is quasiconvex if there exists α ≥ 0 such that any two points x, y in +A can be joined by a polygonal line γ in A satisfying l(γ) ≤ α∥x − y∥, where +l(γ) denotes the length of γ. In this paper, the quasiconvexity of C is vital +for extending our function ϕ from C to Rn while preserving special properties +(see Lemma 3.2). For more explanation about this notion, consult [4]. +Next, we recall the standard notions of proximal, Mordukhovich, and Clarke +subdifferentials, and Clarke generalized Jacobian and Hessian (see [9, 11, 27, +36]). We also enlist the nonstandard notions for subdifferentials introduced +and studied in [33, 38] that are instrumental for this paper for being strictly +smaller than their standard counterparts notions. + +6 +Control Space for Strong Convergence of Continuous Approximation +For the standard notions, given a lower semicontinuous function G: Rn −→ +R ∪ {∞}, and x ∈ dom G, the proximal, the Mordukhovich (or limiting) and +the Clarke subdifferential of G at x are denoted by ∂P G(x), ∂LG(x) and +∂G(x), respectively. From the properties of the limiting normal cone, ∂LG(·) +has a closed graph and closed values. Note that if x ∈ int (dom G) and G is +Lipschitz near x, [9, Theorem 2.5.1] yields that the Clarke subdifferential of +G at x coincides with the Clarke generalized gradient of G at x, also denoted +here by ∂G(x). If G is C1,1 near x ∈ int (dom G), ∂2G(x) denotes the Clarke +generalized Hessian of G at x. For H : Rn −→ Rn Lipschitz near x ∈ Rn, the +Clarke generalized Jacobian of H at x is denoted by ∂H(x). +For the nonstandard notions, given a lower semicontinuous function +G: Rn −→ R ∪ {∞}, S ⊂ dom G closed with int S ̸= ∅, and x ∈ cl (int S), we +define the “limiting subdifferential of G relative to int S at the point x” to be +∂L +ℓ G(x) := +� +lim +i ζi : ζi ∈ ∂P G(xi), xi ∈ int S, and xi +G +−→ x +� +. +(3) +where xi +G +−→ x signifies that xi −→ x and G(xi) −→ G(x). If dom G is +closed, int (dom G) ̸= ∅, and G is locally Lipschitz on int (dom G), then for +x ∈ cl (int (dom G)), we define the “extended Clarke generalized gradient of G +at x”, denoted by ∂ℓG(x), to be +∂ℓG(x) := conv +� +lim +i−→∞ ∇G(xi) : xi +O +−→ x and ∇G(xi) exists ∀i +� +, +(4) +where O is any full-measure subset of int (dom G). If G is C1,1 on int (dom G) +and x ∈ cl (int (dom G)), we define similarly to ∂ℓG(x) the “extended Clarke +generalized Hessian of G at x” to be +∂2 +ℓ G(x) := conv +� +lim +i−→∞ ∇2G(xi) : xi +O +−→ x and ∇2G(xi) exists ∀i +� +, +(5) +where O is any full-measure subset of int (dom G). For S ⊂ Rn closed with +int S ̸= ∅, if G is C1,1 on an open set containing S, then for x ∈ S we define +the “Clarke generalized Hessian of G relative to int S at x” to be +∂2 +ℓ G(x) := conv +� +lim +i−→∞ ∇2G(xi) : xi +O +−→ x +� +, +(6) +where O is any full-measure subset of int S. Now let S ⊂ Rn be closed with +int S ̸= ∅, and let H : Rn −→ Rn be locally Lipschitz on int S. Then for +x ∈ cl (int S), we defined the “extended Clarke generalized Jacobian of H at x” +to be +∂ℓH(x) := conv +� +lim +i−→∞ JH(xi) : xi +O +−→ x and JH(xi) exists ∀i +� +, +(7) +where O is any full-measure subset of int S and J is the Jacobian operator. +Finally, when the set defined in (4), (5), (7) or (6) is a singleton, then we shall +use the notations ∇ℓ and ∇2 +ℓ instead of ∂ℓ and ∂2 +ℓ , respectively. + +Control Space for Strong Convergence of Continuous Approximation +7 +3 Assumptions, consequences, and known +results +In this section, hypotheses on the data of (P) are introduced and some of their +important consequences are provided. We also present some needed results +from [33, Sections 4 & 5] where the connection between (Dγk) and (D) under +measurable controls is studied. We note that each result of this paper shall +require a selected group of these hypotheses. Furthermore, a local version of +(A1) is stated and used in the relevant locations of the paper. +A1: There exist M > 0 and ˜ρ > 0 such that f is M-Lipschitz on C × (U+ ˜ρ ¯B) +with ∥f(x, u)∥ ≤ M for all (x, u) ∈ C × (U + ˜ρ ¯B). +A2: The set C := dom ϕ is given by C = {x ∈ Rn : ψ(x) ≤ 0}, where +ψ : Rn −→ R. +A2.1: There exists ρ > 0 such that ψ is C1,1 on C + ρB. +A2.2: There is a constant η > 0 such that ∥∇ψ(x)∥ > 2η for all x : ψ(x) = +0. +A2.3: The function ψ is coercive, that is, lim∥x∥−→∞ ψ(x) = +∞. +A2.4: The set C has a connected interior.1 +A3: The function ϕ is globally Lipschitz on C and C1 on int C. Moreover, the +function ∇ϕ is globally Lipchitz on int C. +A4: For the sets C0, C1, and U(·) we have: +A4.1: The set C0 ⊂ C is nonempty and closed. +A4.2: The graph of U(·) is a L × B measurable set, and, for t ∈ [0, 1], U(t) +is closed, and bounded uniformly in t. +A4.3: The set C1 ⊂ Rn is nonempty and closed. +A4.4: The multifunction U(·) is lower semicontinuous. +Remark 3.1 The coercivity of ψ in (A2.3) is only assumed to obtain the boundedness +of the closed set C, and hence, this condition can be replaced by C bounded. On the +other hand, for C ⊂ Rn defined as the sub-level set of a function ψ, one can show +that: +(i) Whenever C is nonempty and compact, ψ is merely C1 on C + ρB, and (A2.2) +holds, then there exists ε > 0 such that +x ∈ C and ∥∇ψ(x)∥ ≤ η =⇒ ψ(x) < −ε. +(8) +(ii) When ψ is merely C1 on C +ρB and satisfies (A2.2)-(A2.3), by [33, Lemma 3.3], +(a) bdry C ̸= ∅ and bdry C = {x ∈ Rn : ψ(x) = 0}, +(b) int C ̸= ∅ +and int C = {x ∈ Rn : ψ(x) < 0}. +The following important properties of the compact set C were obtained in +[38, Proposition 3.1], where ψ is assumed to be C1,1 on all of Rn. However, +a slight modification in the proof of that proposition is performed in [33] to +conclude that these properties are actually valid under our assumption (A2.1). +Here and throughout the paper, ¯ +Mψ denotes an upper bound of ∥∇ψ(·)∥ on +the compact set C, and 2Mψ is a Lipschitz constant of ∇ψ(·) over the compact +set C + ρ +2 ¯B chosen large enough so that Mψ ≥ 4η +ρ . +1When ϕ has a suitable extension to Rn, as is the case for ϕ being the indicator of C, see +Remark 3.3, this condition is omitted. + +8 +Control Space for Strong Convergence of Continuous Approximation +Lemma 3.2 [33, Lemma 3.4] Under (A2.1)-(A2.3), we have the following: +(i) The nonempty set C is compact, amenable (in the sense of +[36]), epi- +Lipschitzian, C = cl (int C), and C is +η +Mψ -prox-regular. +(ii) For all x ∈ bdry C we have NC(x) = NP +C (x) = NL +C (x) = {λ∇ψ(x) : λ ≥ 0}. +(iii) If also (A2.4) holds, then int C is quasiconvex. Furthermore, if in addition (A3) +is satisfied, then there exists a function Φ ∈ C1(Rn) such that: +• Φ is bounded on Rn, and Φ(x) = ϕ(x) for all x ∈ C. +• Φ and ∇Φ are globally Lipschitz on Rn. +• For all x ∈ C we have +∂ϕ(x) = {∇Φ(x)} + NC(x). +(9) +Remark 3.3 +(i) In Lemma 3.2(iii), assumption (A2.4) is only imposed to ensure the quasicon- +vexity of C needed to obtain the extension Φ of ϕ. Consequently, when such an +extension is readily available, condition (A2.4) is discarded. This is the case, for +instance, when ϕ is constant on C. Thus, when ϕ is the indicator function of C +assumption (A2.4) is not required. +(ii) From Lemma 3.2(iii), ϕ admits a C1-extension Φ defined on Rn satisfying +equation (9), and for some K > 0, +|Φ(x)| ≤ K, ∥∇Φ(x)∥ ≤ K, and ∥∇Φ(x) − ∇Φ(y)∥ ≤ K∥x − y∥, ∀x, y ∈ Rn. +Employing (9), (D) is equivalently phrased in terms of the normal cone to C +and the extension Φ of ϕ, as follows +(D) +� +˙x(t) ∈ fΦ(x(t), u(t)) − NC(x(t)), a.e. t ∈ [0, 1], +x(0) ∈ C0 ⊂ C, +where fΦ : Rn × Rm −→ Rn is defined by +fΦ(x, u) := f(x, u) − ∇Φ(x), +∀(x, u) ∈ Rn × Rm. +(10) +Therefore, we will interchangeably use throughout this paper the original +formulation of (D) given in terms of ∂ϕ and f, and its reformulation provided +in Remark 3.3 in terms of NC(·) and the function fΦ. Note that assumptions +(A1)-(A3) imply that, for ¯ +M := M + K, fΦ satisfies the following properties: +(A1)Φ: The function fΦ is ¯ +M-Lipschitz on C × (U+ ˜ρ ¯B) with ∥fΦ(x, u)∥ ≤ ¯ +M +for all (x, u) ∈ C × (U + ˜ρ ¯B). +We define U to be +U := {u : [0, 1] → Rm : u is measurable and u(t) ∈ U(t), t ∈ [0, 1] a.e.}. +Remark 3.4 Using [38, Lemma 4.3], it is easy to see that the assumptions (A1)- +(A3) and the boundedness of C by some MC > 0 yield that any solution x of (D) +corresponding to (x0, u) ∈ C0 × U satisfies +x(t) ∈ C, ∀t ∈ [0, 1]; +∥x∥∞ ≤ MC; +and ∥ ˙x∥∞ ≤ 2 ¯ +M. +(11) +For given x(·): [0, 1] → Rn, we use the following notations throughout this +paper: I0(x) := {t ∈ [0, 1] : x(t) ∈ bdry C} and I-(x) := [0, 1] \ I0(x). + +Control Space for Strong Convergence of Continuous Approximation +9 +The next result characterizes the solutions of (D) in terms of the solutions +of a standard control system containing an extra control ξ that satisfies the +mixed control-state degenerate constraint, ξ(t)ψ(x(t)) = 0. The sufficiency part +is straightforward and was used in [38], while the necessary part follows from +applying Filippov selection theorem ([37, Theorem 2.3.13]). +Lemma 3.5 Assume that (A1-(A3) hold. Let u ∈ U and x ∈ AC([0, 1]; Rn) with +x(0) ∈ C0 and x(t) ∈ C for all t ∈ [0, 1]. Then, x is a solution for (D) correspond- +ing to the control u if and only if there exists a nonnegative measurable function ξ +supported on I0(x) such that (x, u, ξ) satisfies +˙x(t) = fΦ(x(t), u(t)) − ξ(t)∇ψ(x(t)), +t ∈ [0, 1] a.e. +(12) +In this case, the nonnegative function ξ supported in I0(x) with (x, u, ξ) satisfying +equation (12), is unique, belongs to L∞([0, 1]; R+), and + + + + + + + + + +ξ(t) = 0 +for t ∈ I-(x), +ξ(t) = ∥ ˙x(t)−fΦ(x(t),u(t))∥ +∥∇ψ(x(t))∥ +∈ +� +0, +¯ +M +2η +� +for t ∈ I0(x) a.e., +∥ξ∥∞ ≤ +¯ +M +2η . +(13) +Throughout the paper we shall employ the following notations, where η +and ¯ +M are the constants given in (A2.2) and (A1)Φ, respectively. +• (γk)k is a sequence satisfying +γk > 2 ¯ +M +η +for all k ∈ N, and γk −−−−→ +k−→∞ ∞. +(14) +• The sequence (αk)k is defined by +αk := ln +� ηγk +2 ¯ +M +� +γk +, +k ∈ N +(15) +By (14) and (15), we have that +γke−αkγk = 2 ¯ +M +η , αk > 0, αk ց and +lim +k−→∞ αk = 0. +(16) +• The sequence (ρk)k is defined by ρk := αk +η for all k ∈ N. By (16) we have +that ρk > 0 for all k ∈ N, ρk ց and +lim +k−→∞ ρk = 0. +• For k ∈ N, we define the set +C(k) := {x ∈ C : ψ(x) ≤ −αk}. +(17) +The system (Dγk) is defined as +(Dγk) +� +˙x(t) = fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)) a.e. t ∈ [0, 1], +x(0) ∈ C. + +10 +Control Space for Strong Convergence of Continuous Approximation +An important property shown in [33] is the invariance of C for the dynamic +(Dγk), see [33, Lemma 4.1]. This fact is behind disposing of the state constraint +in (Dγk), which represents a good approximation for (D) (see Theorem 3.9 +and Corollary 3.12). +Lemma 3.6 (Invariance of C and uniform convergence) Let (A1)-(A3) be satisfied. +Then, for each k, the system (Dγk) with given x(0) = cγk ∈ C and uγk ∈ U, has a +unique solution xγk ∈ W 1,2([0, 1]; Rn) such that xγk(t) ∈ C for all t ∈ [0, 1], and, +for α0 > 0 a bound of (cγk)k we have +∥xγk∥∞ ≤ α0 + +� ¯ +M2 + 2 +and +� 1 +0 +∥ ˙xγk(t)∥2dt ≤ ¯ +M2 + 2. +(18) +Hence, being equicontinuous and uniformly bounded, (xγk)k admits a subsequence +that converges uniformly to some x ∈ W 1,2([0, 1]; Rn) whose values are in C and +whose derivative ˙xγk converges weakly in L2 to ˙x. +The properties of the sets C(k) and the role of the sequence (ρk)k are +estblished in [33, Theorem 3.1 and Remark 3.6]. We enlist here the items that +deem important for this paper when constructing the initial constraint set for +the approximating problems (Pγk) and (Pγk). +Theorem 3.7 [33] Under (A2.1)-(A2.3), the following assertions hold: +(i) For all k, the set C(k) ⊂ int C and is compact, and, for k sufficiently large, +• bdry C(k) = {x ∈ Rn : ψ(x) = −αk} and int C = {x ∈ Rn : ψ(x) < −αk}; +• (C(k))k is a nondecreasing sequence whose Painlev´e-Kuratowski limit is C. +(ii) There exist ro > 0 and ¯k ∈ N such that +� +C ∩ ¯Bro(c) +� +− ρk +∇ψ(c) +∥∇ψ(c)∥ ⊂ int C(k), +∀k ≥ ¯k +and +∀c ∈ bdry C. +(19) +(iii) For c ∈ int C, there exist ˆkc ∈ N and ˆrc > 0 satisfying +¯Bˆrc(c) ⊂ int C(ˆkc) ⊂ int C(k), ∀k ≥ ˆkc. +(20) +Remark 3.8 From Theorem 3.7, it follows that for any c ∈ C, there exists a sequence +(ck)k such that, for k large enough, ck ∈ int C(k) and ck −→ c. Indeed, for c ∈ bdry C, +take ck := c − ρk +∇ψ(c) +∥∇ψ(c)∥ for all k, and for c ∈ int C, take ck = c for all k. +The following theorem will be used repeatedly in this paper. It is a special +case of [33, Theorem 4.1 & Lemma 4.2]. It provides a sufficient condition for +the uniform limit x of the solution xγk of (Dγk) to be a solution of (D), and it +connects the multiplier function ξ corresponding to x, via Lemma 3.5, to the +positive continuous penalty multiplier ξγk, associated with xγk and defined by +ξγk(·) := γkeγkψ(xγk(·)). +(21) +Theorem 3.9 ((Dγk)k & ξγk approximate (D) & ξ) Assume that (A1)-(A4.1) hold. +Let xγk be the solution of (Dγk) corresponding to (cγk, uγk), as in Lemma 3.6, and +x ∈ W 1,2([0, 1]; Rn) be its uniform limit. Then, the following statements are valid : + +Control Space for Strong Convergence of Continuous Approximation +11 +(i) The sequence (ξγk)k admits a subsequence, we do not relabel, that converges +weakly in L2 to a nonnegative function ξ ∈ L2 supported on I0(x). +(ii) If for some u ∈ U, the sequence uγk(t) a.e. t +−−−−→ u(t), then x is the unique solution +of (D) corresponding to (x0, u), and (x, u, ξ) satisfies equations (12)-(13). In +particular, ξ ∈ L∞([0, 1]; R+) and is supported on I0(x). +Remark 3.10 Note that when establishing Theorem 3.9(ii) in [33], the arguments +used to prove that (x, u, ξ) satisfies (12) are independent of having ξγk defined +through (21), and hence, this proof is valid for ξγk being any sequence of L2-functions +converging weakly in L2 to ξ. Therefore, we have that (x, u, ξ) satisfies (12) when- +ever (xj, uj, ξj)j is a sequence solving (12) with xj converging uniformly to x, uj(t) +converging pointwise a.e. to u(t), and ξj converging weakly in L2 to ξ. +The following result is extracted from [33, Theorem 5.1], in which more +properties are derived. It reveals the significance of initiating in Theorem 3.9 +the solutions xγk of (Dγk) from the subset C(k), defined in (17). +Theorem 3.11 (xγk ∈ C(k), ˙xγk & ξγk bounded) Assume (A1)-(A4.1) hold. Let +(cγk)k be a sequence such that cγk ∈ C(k), for k sufficiently large. Then there exists +ko ∈ N such that for all sequences (uγk)k in U and for all k ≥ ko, the solution xγk +of (Dγk) corresponding to (cγk, uγk) satisfies: +(i) xγk(t) ∈ C(k) ⊂ int C for all t ∈ [0, 1]. +(ii) 0 ≤ ξγk(t) ≤ 2 ¯ +M +η +for all t ∈ [0, 1]. +(iii) ∥ ˙xγk(t)∥ ≤ ¯ +M + 2 ¯ +M ¯ +Mψ +η +for a.e. t ∈ [0, 1]. +The next result is a simplified version of [33, Corollary 5.1]. It is the converse +of Theorem 3.9, as it confirms that any given solution of (D) is approximated +by a solution of (Dγk) that remains in the interior of C and enjoys all the +properties displayed in Theorem 3.11. +Corollary 3.12 (Solutions of (D) are approximated by sequences in C(k)) Assume +that (A1)-(A4.1) are satisfied. Let ¯x be the solution of (D) corresponding to +(¯x(0), ¯u) ∈ C0 × U. Consider (¯cγk)k the sequence in Remark 3.8 that converges to +c := ¯x(0), and ¯xγk the solution of (Dγk) corresponding to (¯cγk, ¯u). Then, there exists +ˆko ∈ N such that ¯xγk and its associated ¯ξγk via (21) satisfy the conclusions (i)-(iii) of +Theorem 3.11 for all k ≥ ˆko, and the following holds true: The sequence ¯xγk admits +a subsequence, we do not relabel, that converges uniformly to ¯x, the corresponding +subsequence for ¯ξγk converges weakly in L2 to some ¯ξ ∈ L∞, and (¯x, ¯u, ¯ξ) satisfies +(12)-(13). That is, ¯ξ is the unique function corresponding to (¯x, ¯u) via Lemma 3.5. +4 Main results +This section consists of the main results of this paper, namely, the strong +approximation of (D) by (Dγk) whenever the control is W 1,2 (Theorem 4.1 and +Corollary 4.2), an existence theorem for an optimal solution of (P) (Theorem + +12 +Control Space for Strong Convergence of Continuous Approximation +4.3), a strong converging continuous approximation for (P) (Theorem 4.6), and +nonsmooth necessary optimality conditions in the form of weak-Pontryagin- +type maximum principle (Theorem 4.8). +4.1 (Dγk) strongly approximates (D) with W 1,2-controls +The following theorem constitutes the backbone of this paper. It shows that, +when the underlying control space is W (defined in (1)), the velocities ˙xγk and +the functions ξγk corresponding to the approximating sequence xγk in Theorem +3.11, converge strongly in L2 to, respectively, ˙x and ξ, the functions obtained +in Theorem 3.9. The proof of this theorem is postponed to Section 5. +Theorem 4.1 (Strong convergence of the velocity sequence ˙xγk) Let the assumptions +(A1)-(A4.2) be satisfied. Consider a sequence xγk solving (Dγk) for some (cγk, uγk), +where cγk ∈ C, cγk −→ x0 ∈ C0, uγk ∈ W, and (∥ ˙uγk∥2)k is bounded. Denote by +(x, ξ) the pair in W 1,2 × L2 obtained via Lemma 3.6 and Theorem 3.9(i) such that a +subsequence (not relabeled ) of (xγk, ξγk) has xγk converging uniformly in the set C +to x and ( ˙xγk, ξγk) converging weakly in L2 to ( ˙x, ξ). Then, the following hold: +(i) There exist a subsequence (not relabeled ) of uγk, and u ∈ W such that (xγk, uγk) +converges uniformly to (x, u), and ( ˙xγk, ˙uγk, ξγk) converges weakly in L2 to +( ˙x, ˙u, ξ). The function x is the unique solution to (D) corresponding to (x0, u), +and (x, u, ξ) satisfies (12)-(13). In particular, ξ ∈ L∞ and is supported on +I0(x). +(ii) Assume that cγk +∈ C(k), for k large. Then, in addition to the conclu- +sions in Theorem 3.11, the following holds: The sequence ( ˙xγk, ξγk) is in +W 1,2([0, 1]; +Rn) × W 2,2([0, 1]; +R+), has uniform bounded variations, and +admits a subsequence, not relabeled, that converges pointwise, and hence, +strongly in L2 to ( ˙x, ξ), with ˙x ∈ BV ([0, 1]; Rn) and ξ ∈ BV ([0, 1]; R+). In +this case, (12)-(13) hold for all t ∈ [0, 1], and xγk −→ x strongly in the norm +topology of W 1,2([0, 1]; Rn). +Applying Theorem 4.1(ii) to ¯cγk, uγk := ¯u, ¯xγk, and ¯ξγk, the function +associated to ¯xγk via (21), we obtain the following corollary that shows how +the results in Corollary 3.12 are improved when W 1,2-controls are utilized. +Corollary 4.2 ((Dγk)k strongly approximates (D)) If, in addition to the assump- +tions of Corollary 3.12, we have that ¯x solves (D) for ¯u ∈ W (not only in U), then ¯ξγk, +therein, converges pointwise to ¯ξ ∈ BV ([0, 1]; R+) with ¯ξ satisfying (52), and ¯xγk, +therein, converges to ¯x strongly in the norm topology of W 1,2([0, 1]; Rn). Moreover, +(¯x, ¯u, ¯ξ) satisfies (12)-(13) for all t ∈ [0, 1], and ˙¯x ∈ BV ([0, 1]; Rn). +4.2 Existence of optimal solution for (P ) +Parallel to [6, 8, Theorems 4.1], where a discretization technique is used, the +following existence theorem of an optimal solution for the problem (P) is +established based on Corollary 4.2. + +Control Space for Strong Convergence of Continuous Approximation +13 +Theorem 4.3 (Existence of solution for (P)) Assume hypotheses (A1)-(A4.3), +g : Rn × Rn → R ∪ {∞} is lower semicontinuous, and that a minimizing sequence +(xj, uj) for (P) exists such that (∥ ˙uj∥2)j is bounded. Suppose that (P) has at least +one admissible pair (yo, vo) with (yo(0), yo(1)) ∈ dom g, then the problem (P) admits +a global optimal solution (¯x, ¯u) such that, along a subsequence, we have +xj +strongly +−−−−−−−−−−→ +W 1,2([0,1]; Rn) ¯x, uj +uniformly +−−−−−−−−→ +C([0,1]; Rm) ¯u, and ˙uj +weakly +−−−−−−−−−→ +L2([0,1]; Rm) +˙¯u. +Proof Given that (P) has an admissible pair (yo, vo) with (yo(0), yo(1)) ∈ dom g, +then inf(x,u)(P) < ∞. As g is lower semicontinuous and all admissible solutions of (P) +satisfy (x(0), x(1)) ∈ C0 × (C1 ∩ C), which is compact, we deduce that inf(x,u)(P) is +finite. On the other hand, being admissible for (P), the minimizing sequence (xj, uj)j +satisfies (D) with xj(1) ∈ C1. Hence, using that the sequence (∥ ˙uj∥2)j is bounded, +Lemma 5.1 implies the existence of (¯x, ¯u) ∈ W 1,∞([0, 1]; Rn) × W satisfying (D) +and ¯x(1) ∈ C1, with (xj, uj) converges uniformly to (¯x, ¯u), ( ˙xj)j converges strongly +in L2 to ˙¯x ∈ BV ([0, 1]; Rn), and ˙uj converges weakly in L2 to ˙¯u. Thus, (¯x, ¯u) is +admissible for (P). Owed to the lower semicontinuity of g and to (¯x, ¯u) being the +uniform limit of the minimizing sequence (xj, uj)j, the optimality of the pair (¯x, ¯u) +for the problem (P) follows readily. +□ +4.3 Continuous approximation for (P ) +On the journey of seeking for an optimal process (¯x, ¯u) of (P) a continuous +approximations consisting of optimal solutions for properly-designed standard +control problems, it is important that the convergence to (¯x, ¯u) be strong in the +norm topology of the considered space, namely, the space W 1,2([0, 1]; Rn)×W. +Corollary 4.2 already answered this question for the W 1,2-strong approxima- +tion of a solution (¯x, ¯u) of (D) by solutions of (Dγk), in which the same control +¯u is used. However, ¯u may not necessarily be optimal for approximating optimal +control problems over (Dγk). +In this subsection, we approximate the problem (P) by a certain sequence +of optimal control problems over (Dγk) with special initial and final state +endpoints constraints (C0(k) ⊂ C(k) and C1(k) in a band around C1), and with +an objective function particularly crafted so that an optimal control, uγk, exists +and has (∥ ˙uγk∥2)k uniformly bounded, and hence, the strong convergence of +the optimal state velocities shall be deduced from Theorem 4.1. The necessary +optimality conditions for (P) are then established by taking the limit of the +optimality conditions for the corresponding approximating problem. +For given δ > 0 and z ∈ C([0, 1]; Rs), we define the projection on Rs of +the closed δ-tube around z by ¯Bδ(z) := +� +t∈[0,1] +¯Bδ(z(t)). +Let (¯x, ¯u) ∈ W 1,2([0, 1]; Rn) × W be a W 1,2-local minimizer for (P) with +associated δ. We fix δo > 0 such that +δo ≤ +� +min{ˆr¯x(0), δ} +if ¯x(0) ∈ int C, +min{ro, δ} +if ¯x(0) ∈ bdry C, + +14 +Control Space for Strong Convergence of Continuous Approximation +where ro > 0 is the constant in Theorem 3.7(ii), and ˆr¯x(0) > 0 with ˆk¯x(0) ∈ N +are the constants in Theorem 3.7(iii) corresponding to c := ¯x(0). +In the remaining part below, we will assume that f satisfies the following +local version of (A1): +∃ ˜ρ > 0 such that f is Lipschitz on [C ∩ ¯Bδ(¯x)] × [(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)]. +(∗) +Note that under the assumption (∗), the function f can be extended to a +globally Lipschitz function ˜f : Rn × Rm −→ R by applying [22, Theorem 1] to +each component of f. Since in the rest of this section we only consider local +optimality notions, then, without loss of generality, we shall use the function +f instead of ˜f. Hence, when in this section f is assumed to satisfy (∗), it is +implied that f also satisfies assumption (A1). +We proceed to suitably-formulate a sequence of approximating prob- +lems (Pγk) and show that its optimal solutions strongly converges in +W 1,2([0, 1]; Rn) × W to the W 1,2-local minimizer (¯x, ¯u) of (P). This nat- +urally requires the domain of the approximating problem (Pγk) to be in +W 1,2([0, 1]; Rn) × W. The initial state constraint is taken to be x(0) ∈ C0(k), +where C0(k) is the sequence of sets defined by +C0(k) := +� +C0 ∩ ¯Bδo(¯x(0)) , ∀k ∈ N, +if ¯x(0) ∈ int C, +� +C0 ∩ ¯Bδo(¯x(0)) +� +− ρk +∇ψ(¯x(0)) +∥∇ψ(¯x(0))∥, ∀k ∈ N, +if ¯x(0) ∈ bdry C. +(22) +and the final state constraint is x(1) ∈ C1(k), where +C1(k) := +�� +C1 ∩ ¯Bδo(¯x(1)) +� +− ¯x(1) + ¯xγk(1) +� +∩ C, +k ∈ N, +in which ¯xγk is the solution of (Dγk) corresponding to (¯cγk, ¯u), where ¯cγk in +C0(k) ∩ int C(k), for k large, and is defined via Remark 3.8 for c := ¯x(0), that +is, +¯ck := +� +¯x(0), ∀k ∈ N, +if ¯x(0) ∈ int C, +¯x(0) − ρk +∇ψ(¯x(0)) +∥∇ψ(¯x(0))∥, ∀k ∈ N, +if ¯x(0) ∈ bdry C. +Note that C0(k) and C1(k) are closed, for k ∈ N. On the other hand, as +ργk −→ 0, we have ¯cγk −→ ¯x(0), Corollary 3.12 yields that the sequence ¯xγk +converges in C uniformly to ¯x, and hence, ¯xγk(1) −→ ¯x(1). Add to this that in +C0(k), ρk −→ 0, then, for a fixed ˜ρ > 0, we have that, for k sufficiently large, +Ci(k) ⊂ +� � +Ci ∩ ¯Bδ(¯x(i)) +� ++ ˜ρ ¯B +� +∩ C +� +�� +� +˜ +Ci(δ) +, for i = 0, 1, +(23) +and +lim +k→∞ C0(k) = C0 ∩ ¯Bδo(¯x(0)) & +lim +k→∞ C1(k) = C ∩ C1 ∩ ¯Bδo(¯x(1)). +(24) +Remark 4.4 Notice that we can show that, for k large enough, we have C0(k) ⊂ C(k), +and hence, by Theorem 3.11, any solution of (Dγk) corresponding to (cγk, uγk) with +cγk ∈ C0(k) and uγk ∈ U, satisfies the conditions (i)-(iii) of this theorem. Indeed: + +Control Space for Strong Convergence of Continuous Approximation +15 +• For ¯x(0) ∈ int C, use that δo ≤ ˆr¯x(0) and (20) we get +¯Bˆr¯x(0)(¯x(0)) ⊂ int C(k) ⊂ C(k), +∀k ≥ ˆk¯x(0). +This gives that +C0(k) := C0 ∩ ¯Bδo(¯x(0)) ⊂ ¯Bˆr¯x(0)(¯x(0)) ⊂ C(k), +∀k ≥ ˆk¯x(0). +• For ¯x(0) ∈ bdry C, use that δo ≤ ro and C0(k) is the nonempty set defined by the +second equation of (22), to get via (19) that +C0(k) ⊂ int C(k) ⊂ C(k), +∀k ≥ ¯k. +Remark 4.5 For c ∈ C0(k) and d ∈ C1(k), the evaluation of the normal cones +NL +C0(k)(c) and NL +C1(k)(d) in terms of NL +C0 and NL +C1, respectively, is obtained using +the local property of the limiting normal cone as the following +NL +C0(k)(c) = + + + + + + + + + + + + + +NL +C0(c), +if ¯x(0) ∈ int C, and +c ∈ Bδo(¯x(0)), +NL +C0 +� +c + ρk +∇ψ(¯x(0)) +∥∇ψ(¯x(0))∥ +� +, +if ¯x(0) ∈ bdry C, and +� +c + ρk +∇ψ(¯x(0)) +∥∇ψ(¯x(0))∥ +� +∈ Bδo(¯x(0)). +(25) +NL +C1(k)(d) = NL +C1(d + ¯x(1) − ¯xγk(1)), ∀d ∈ (int C) ∩ Bδo(¯x(1)). +(26) +We introduce the following sequence of approximating problems: +(Pγk): Minimize +J(x, y, z, u) := g(x(0), x(1)) + 1 +2 +� +∥u(0) − ¯u(0)∥2 + z(1) + ∥x(0) − ¯x(0)∥2� +over (x, y, z, u) ∈ W 1,2([0, 1]; Rn) × AC([0, 1]; R) × AC([0, 1]; R) × W +such that + + + + + + + + + + + + + + + + + + + + + +( ˜Dγk) + + +˙x(t) = fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)), t ∈ [0, 1] a.e., +˙y(t) = ∥ ˙x(t) − ˙¯x(t)∥2, t ∈ [0, 1] a.e., +˙z(t) = ∥ ˙u(t) − ˙¯u(t)∥2, t ∈ [0, 1] a.e., +(x(0), y(0), z(0)) ∈ C0(k) × {0} × {0}, +x(t) ∈ ¯Bδ(¯x(t)) and u(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), ∀t ∈ [0, 1], +(x(1), y(1), z(1)) ∈ C1(k) × [−δ, δ] × [−δ, δ]. +Note that Lemma 3.6 and the constraints on u(·) confirm that (Pγk)k is actually +equivalent to having therein (x, u) ∈ AC([0, 1]; Rn) × AC([0, 1]; Rm). +Now we are ready to state our continuous approximation result, which is +parallel to the corresponding result in [6–8, 14], where discrete approximations +are used. The proof of this approximation result is presented in Section 5. +Theorem 4.6 ((Pγk) approximates (P)) Let (¯x, ¯u) be a W 1,2-local minimizer (P) +with associated ¯ξ ∈ L∞ via Lemma 3.5. Assume that (A2)-(A4.3) hold, g is contin- +uous on ˜C0(δ) × ˜ +C1(δ), and for some ˜ρ > 0, f is Lipschitz on [C ∩ ¯Bδ(¯x)] × [(U + +˜ρ ¯B)∩ ¯Bδ(¯u)]. Then for k sufficiently large, the problem (Pγk) has an optimal solution + +16 +Control Space for Strong Convergence of Continuous Approximation +(xγk, yγk, zγk, uγk) such that, for ξγk defined in (21), we have, along a subsequence, +we do not relabel, that +(xγk, uγk) +strongly +−−−−−−→ +W 1,2×W (¯x, ¯u), (yγk, zγk) +strongly +−−−−−−−−−−−−−−→ +W 1,1([0,1]; R+×R+) (0, 0), ξγk +strongly +−−−−−−−−−→ +L2([0,1]; R+) +¯ξ, +all the conclusions of Theorem 3.11 hold, including that xγk(t) ∈ int C for all t ∈ +[0, 1], and for all k sufficiently large, +xγk(i) ∈ +� � +Ci ∩ ¯Bδo(¯x(i)) +� ++ ˜ρB +� +∩ (int C) ⊂ int ˜Ci(δ), +for i = 0, 1. +Moreover, ˙¯x ∈ BV ([0, 1]; Rn), ¯ξ ∈ BV ([0, 1]; R+), and (12)-(13) are valid at (¯x, ¯u, ¯ξ) +for all t ∈ [0, 1]. +We proceed to rewrite the problems (Pγk) as an optimal control prob- +lem with state constraints. Given (¯x, ¯u) ∈ W 1,2([0, 1]; Rn) × W a W 1,2-local +minimizer for (P), for ¯v := ˙¯u, (Pγk) is reformulated in the following way: +(Pγk): Minimize +g(x(0), x(1)) + 1 +2 +� +∥u(0) − ¯u(0)∥2 + z(1) + ∥x(0) − ¯x(0)∥2� +over +(x, y, z, u) ∈ AC([0, 1]; Rn)×AC([0, 1]; R)×AC([0, 1]; R)×AC([0, 1]; Rm) +and measurable functions v: [0, 1] −→ Rm such that + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +˙x(t) = fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)), t ∈ [0, 1] a.e., +˙u(t) = v(t), t ∈ [0, 1] a.e., +˙y(t) = ∥fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)) − ˙¯x(t)∥2, t ∈ [0, 1] a.e., +˙z(t) = ∥v(t) − ¯v(t)∥2, t ∈ [0, 1] a.e., +x(t) ∈ ¯Bδ(¯x(t)) and u(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), ∀t ∈ [0, 1], +(x(0), u(0), y(0), z(0)) ∈ C0(k) × Rm × {0} × {0}, +(x(1), u(1), y(1), z(1)) ∈ C1(k) × Rm × [−δ, δ] × [−δ, δ]. +In the following proposition we apply to the above sequence of refor- +mulated problems (Pγk), the nonsmooth Pontryagin maximum principle for +optimal control problems with multiple state constraints (see e.g., [37, page +331] and [37, p.332]). For this purpose, (x, y, z, u) is the state function in +(Pγk) and v is the control. Thus, (xγk, yγk, zγk, uγk) is the optimal state, where +(xγk, uγk) is obtained from Theorem 4.6, yγk(t) := +� t +0 ∥ ˙xγk(s) − ˙¯x(s)∥2 ds, +zγk(t) := +� t +0 ∥ ˙uγk(s) − ˙¯u(s)∥2 ds, and vγk = ˙uγk is the optimal control. Hence, +the function f(·, ·) is required to be Lipschitz near (xγk, uγk), which follows +from (∗), since xγk(t) ∈ int C and (xγk, uγk) converges uniformly to (¯x, ¯u) (see +Theorem 4.6). Furthermore, as the objective function g must be Lipschitz near +(xγk(0), xγk(1)), we introduce the following local assumption on g in which +˜C0(δ) and ˜C1(δ) are defined in (23): +∃ ˜ρ > 0 such that g is Lipschitz on ˜C0(δ) × ˜ +C1(δ). +On the other hand, the following constraint qualification property (CQ) is +required. For a given multifunction F : [0, 1] ⇒ Rm, with nonempty and closed +values, and for h ∈ C([0, 1]; F), that is, h ∈ C([0, 1]; Rm) and satisfies h(t) ∈ +F(t) for all t ∈ [0, 1], we say that F(·) satisfies the constraint qualification at +h if + +Control Space for Strong Convergence of Continuous Approximation +17 +(CQ) +conv ( ¯N L +F (t)(h(t))) is pointed for all t ∈ [0, 1]. +Here, ¯N L +F (t)(y) stands for the graphical closure at (t, y) of the multifunction +(t, y) �→ N L +F (t)(y), that is, the graph of ¯N L +F (·)(·) is the closure of the graph of +N L +F (·)(·). For more information about the (CQ) property, see Remark 5.2. +It is worth noting that in [6–8], where W 1,2-controls are employed, the +control sets U(t) are assumed to be Rm, for all t ∈ [0, 1], and hence, the (CQ) +property is trivially satisfied. +Proposition 4.7 (Maximum Principle for approximating problems (Pγk)) Let (¯x, ¯u) +be a W 1,2-local minimizer for (P). Assume that (A2)-(A4) hold, and for some ˜ρ > 0, +f is Lipschitz on [C ∩ ¯Bδ(¯x)] × [(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)] and g is Lipschitz on ˜C0(δ) × +˜ +C1(δ). Consider the optimal sequence (xγk, yγk, zγk, uγk, vγk) for (Pγk) obtained via +Theorem 4.6. If for k sufficiently large, U(·) satisfies the constraint qualification +(CQ) at uγk, then for k large enough, there exist λγk ≥ 0, pγk ∈ AC([0, 1]; Rn), +qγk ∈ AC([0, 1]; Rm), Ωγk ∈ NBV ([0, 1]; Rm), µoγk ∈ C⊕([0, 1]; Rm), and a µoγk- +integrable function βγk : [0, 1] −→ Rm such that Ωγk(t) = +� +[0,t] βγk(s)µoγk(ds), for all +t ∈ (0, 1], and: +(i) (The nontriviality condition) For all k ∈ N, we have +∥pγk(1)∥ + ∥qγk∥∞ + ∥µo +γk∥T.V. + λγk = 1; +(ii) (The adjoint equation) For a.e. t ∈ [0, 1], +� ˙pγk(t) +˙qγk(t) +� +∈ − +� +∂(x,u)fΦ(t, xγk(t), uγk(t)) +�T +pγk(t) ++ +� +γkeγkψ(xγk(t))∂2ψ(xγk(t))pγk(t) +0 +� +(27) ++ +� +γ2 +keγkψ(xγk(t))∇ψ(xγk(t))⟨∇ψ(xγk(t)), pγk(t)⟩ +0 +� +; +(iii) (The transversality equation) +(pγk(0), −pγk(1)) ∈ +λγk∂Lg(xγk(0), xγk(1))+ +�� +λγk(xγk(0)−¯x(0))+NL +C0(k)(xγk(0)) +� +×NL +C1(k)(xγk(1)) +� +, +and qγk(0) = λγk(uγk(0) − ¯u(0)), +−qγk(1) = Ωγk(1); +(iv) (The maximization condition) For a.e. t ∈ [0, 1], +max +v∈Rm +� +⟨qγk(t) + Ωγk(t), v⟩ − λγk +2 ∥v − ˙¯u(t)∥2 +� +is attained at ˙uγk(t); +(v) (The measure properties) +supp {µo +γk} ⊂ +� +t ∈ [0, 1] : (t, uγk(t)) ∈ bdry Gr +� +U(t) ∩ ¯Bδ(¯u(t)) +�� +, and +βγk(t) ∈ ∂> +u d(uγk(t), U(t) ∩ ¯Bδ(¯u(t))) +µo +γk a.e., with +∂> +u d(uγk(t), U(t) ∩ ¯Bδ(¯u(t))) ⊂ +� +conv ¯ +NL +U(t)∩ ¯ +Bδ(¯u(t))(uγk(t)) ∩ +� ¯B \ {0} +�� +. + +18 +Control Space for Strong Convergence of Continuous Approximation +4.4 Necessary optimality conditions for (P ) +The main result of this subsection is the following theorem which provides +necessary optimality conditions for the W 1,2-local minimizer, (¯x, ¯u), of (P). +The following notations are used in the statement of the theorem: +• ∂ℓϕ and ∂2 +ℓ ϕ stand, respectively, for the extended Clarke generalized gradient +and the extended Clarke generalized Hessian of ϕ defined on C via (4) & (5). +• ∂(x,u) +ℓ +f(·, ·) is the extended Clarke generalized Jacobian of f(·, ·) defined on +� +C ∩ ¯Bδ(¯x(t)) +� +× +� +(U(t) + ˜ρ ¯B) ∩ ¯Bδ(¯u(t)) +� +via (7). +• ∂2 +ℓ ψ is the Clarke generalized Hessian relative to int C of ψ, defined via (6). +• ∂L +ℓ g is the limiting subdifferential of g relative to int +� ˜C0(δ)× ˜C1(δ) +� +, defined +via (3). +Theorem 4.8 (Necessary optimality conditions for (P)) Let (¯x, ¯u) be a W 1,2-local +minimizer for (P). Let ¯ξ ∈ L∞([0, 1]; R+) be the function supported on I0(¯x) and +associated to (¯x, ¯u) via Lemma 3.5. Assume that (A2)-(A4) hold, U(·) satisfies the +constraint qualification (CQ) at ¯u, and for some ˜ρ > 0, f is Lipschitz on [C∩¯Bδ(¯x)]× +[(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)] and g is Lipschitz on ˜C0(δ) × ˜ +C1(δ). Then ˙¯x ∈ BV ([0, 1]; Rn) +and ¯ξ ∈ BV ([0, 1]; R+), and there exist λ ≥ 0, an adjoint vector ¯p ∈ BV ([0, 1]; Rn), +a finite signed Radon measure ¯ν on [0, 1] supported on I0(¯x), L∞-functions ¯ζ(·), +¯θ(·) and ¯ϑ(·) in Mn×n([0, 1]), an L∞-function ¯ω(·) in Mn×m([0, 1]), such that for +t ∈ [0, 1] a.e., +� +(¯ζ(t), ¯ω(t)), ¯θ(t), ¯ϑ(t) +� +∈ ∂(x,u) +ℓ +f(¯x(t), ¯u(t)) × ∂2 +ℓ ϕ(¯x(t)) × ∂2 +ℓ ψ(¯x(t)), +and the following hold: +(i) (The admissible equation) +(a) ˙¯x(t) = f(¯x(t), ¯u(t)) − ∇ℓ ϕ(¯x(t)) − ¯ξ(t)∇ψ(¯x(t)), ∀t ∈ [0, 1], +(b) ψ(¯x(t)) ≤ 0, ∀t ∈ [0, 1]; +(ii) (The nontriviality condition) +∥¯p(1)∥ + λ = 1; +(iii) (The adjoint equation) For any h ∈ C([0, 1]; Rn), we have +� +[0,1] +⟨h(t), d¯p(t)⟩ = +� 1 +0 +� +h(t), +� +¯θ(t) − ¯ζ(t)T� +¯p(t) +� +dt ++ +� 1 +0 +¯ξ(t) +� +h(t), ¯ϑ(t)p(t) +� +dt + +� +[0,1] +⟨h(t), ∇ψ(¯x(t))⟩d¯ν; +(iv) (The complementary slackness conditions) +(a) ¯ξ(t) = 0, ∀t ∈ I-(¯x), +(b) ¯ξ(t)⟨∇ψ(¯x(t), ¯p(t)⟩ = 0, ∀t ∈ [0, 1] a.e.; +(v) (The transversality equation) +(¯p(0), −¯p(1)) ∈ λ∂L +ℓ g(¯x(0), ¯x(1)) + +� +NL +C0(¯x(0)) × NL +C1(¯x(1)) +� +; +(vi) (The weak maximization condition) +¯ω(t)T¯p(t) ∈ conv ¯ +NL +U(t)∩ ¯ +Bδ(¯u(t))(¯u(t)), t ∈ [0, 1] a.e. +If in addition there exist εo > 0 and r > 0 such that U(t) ∩ ¯Bεo(¯u(t)) is +r-prox-regular for all t ∈ [0, 1], then we have +max +�� +¯ω(t)T¯p(t), u +� +− ∥¯ω(t)T ¯p(t)∥ +min{εo,2r} ∥u − ¯u(t)∥2 : u ∈ U(t) +� +is attained at ¯u(t) for t ∈ [0, 1] a.e. + +Control Space for Strong Convergence of Continuous Approximation +19 +Remark 4.9 Condition (vi) of Theorem 4.8 admits simplified forms when U(·) +possesses extra properties: +• If U(t) is r-prox-regular for all t ∈ [0, 1], then taking εo −→ ∞, the maximization +condition (v) reduces to +max +�� +¯ω(t)T¯p(t), u +� +− ∥¯ω(t)T ¯p(t)∥ +2r +∥u − ¯u(t)∥2 : u ∈ U(t) +� +is attained at ¯u(t) for t ∈ [0, 1] a.e. +• If U(t)∩ ¯Bεo(¯u(t)) is convex for all t ∈ [0, 1], then taking r −→ ∞, the maximization +condition (v) reduces to +max +�� +¯ω(t)T¯p(t), u +� +− ∥¯ω(t)T ¯p(t)∥ +εo +∥u − ¯u(t)∥2 : u ∈ U(t) +� +is attained at ¯u(t) for t ∈ [0, 1] a.e. +• If U(t) is convex for all t ∈ [0, 1], then taking both εo −→ ∞ and r −→ ∞, the +maximization condition (v) reduces to +max +�� +¯ω(t)T¯p(t), u +� +: u ∈ U(t) +� +is attained at ¯u(t) for t ∈ [0, 1] a.e. +Proof of Theorem 4.8. Theorem 4.6 produces a subsequence of (γk)k, we do not +relabel, and a corresponding sequence (xγk, yγk, zγk, uγk)k, with associated (ξγk)k +defined via (21), such that +• For each k, the quadruplet (xγk, yγk, zγk, uγk) is optimal for (Pγk). +• (xγk, uγk) +strongly +−−−−−−→ +W 1,2×W (¯x, ¯u), (yγk, zγk) +strongly +−−−−−−−−−−−−−−→ +W 1,1([0,1]; R+×R+) (0, 0), ξγk +strongly +−−−−−−−−−→ +L2([0,1]; R+) +¯ξ. +• ˙¯x ∈ BV ([0, 1]; Rn), ¯ξ ∈ BV ([0, 1]; R+), and (12)-(13) are valid at (¯x, ¯u, ¯ξ) for all +t ∈ [0, 1]. +• All the conclusions of Theorem 3.11 hold, including (xγk)k is uniformly Lipschitz +and xγk(t) ∈ int C for all t ∈ [0, 1]. +• For all k, we have +xγk(i) ∈ +� � +Ci ∩ ¯Bδo(¯x(i)) +� ++ ˜ρB +� +∩ (int C) ⊂ int ˜Ci(δ), +for i = 0, 1. +In order to apply Proposition 4.7, we shall show that the constraint qualification +(CQ) that holds for U(·) at ¯u, also holds true at uγk, for k large enough. Indeed, if +this is false, then, by Remark 5.2(i), there exist an increasing sequence (kn)n in N +and a sequence tn ∈ [0, 1] such that tn −→ to ∈ [0, 1] and +0 ∈ ∂> +u dU(tn, uγkn (tn)), ∀n ∈ N. +(28) +The continuity of ¯u and the uniform convergence of uγkn to ¯u yield that the +sequence (uγkn (tn))n converges to ¯u(to). Hence, using that the multifunction (t, x) �→ +∂> +u dU(t, x) has closed values and a closed graph, we conclude from (28) that +0 ∈ ∂> +u dU(to, ¯u(to)). This contradicts that the constraint qualification is satisfied +by U(·) at ¯u. Thus, for k sufficiently large, U(·) satisfies the constraint qualification +(CQ) at uγk. +Hence, by Proposition 4.7, there exist a subsequence of (γk)k, we do not rela- +bel, and corresponding sequences pγk, qγk µγk and λγk satisfying conditions (i)-(v) +therein. +Using (27), (10), and that for all t ∈ [0, 1] we have +(xγk(t), uγk(t)) ∈ int +�� +C ∩ ¯Bδ(¯x(t)) +� +× +� +(U(t)+ ˜ρ ¯B) ∩ ¯Bδ(¯u(t)) +�� +, + +20 +Control Space for Strong Convergence of Continuous Approximation +we obtain sequences ζγk, θγk and ϑγk in Mn×n([0, 1]) and ωγk in Mn×m([0, 1]) such +that, for a.e. t ∈ [0, 1], +(ζγk(t), ωγk(t)) ∈ ∂(x,u) +ℓ +f(xγk(t), uγk(t)), +(θγk(t), ϑγk(t)) ∈ ∂2 +ℓ ϕ(xγk(t)) × ∂2 +ℓ ψ(xγk(t)), +˙pγk(t) = (θγk(t) − ζγk(t))Tpγk(t) + γkeγkψ(xγk(t))ϑγk(t) pγk(t) +(29) ++ γ2 +keγkψ(xγk(t))∇ψ(xγk(t))⟨∇ψ(xγk(t)), pγk(t)⟩, and +˙qγk(t) = −(ωγk(t))Tpγk(t). +(30) +Note that for each k, the functions pγk, ˙pγk, qγk, ˙qγk, xγk, uγk and ¯u are measurable +on [0, 1], and the multifunctions ∂(x,u) +ℓ +f(·, ·), ∂2 +ℓ ϕ(·), and ∂2 +ℓ ψ(·) are measurable and +have closed graphs with nonempty, compact, and convex values. Using (A1), (A2.1), +and (A3), the Filippov measurable selection theorem (see [37, Theorem 2.3.13]) yields +that we can assume the measurability of the functions ζγk(·), θγk(·), ϑγk(·) and ωγk(·). +Moreover, these sequences are uniformly bounded in L∞, as ∥(ζγk, ωγk)∥∞ ≤ M, +∥θγk∥∞ ≤ K and ∥ϑγk∥∞ ≤ 2Mψ. +Step 1. Construction of ¯ξ, the admissible equation. +From Theorem 4.6, we have that the triplet (¯x, ¯y, ¯ξ) satisfies (12) for all t ∈ [0, 1]. +Hence, for all t ∈ [0, 1] we have +˙¯x(t) = fΦ(¯x(t), ¯u(t)) − ¯ξ(t)∇ψ(¯x(t)) = f(¯x(t), ¯u(t)) − ∇Φ(¯x(t)) − ¯ξ(t)∇ψ(¯x(t)). +Since ∇Φ(x) = ∂ℓϕ(x) = ∇ℓ ϕ(x) for all x ∈ C, we obtain that +˙¯x(t) = f(¯x(t), ¯u(t)) − ∇ℓ ϕ(¯x(t)) − ¯ξ(t)∇ψ(¯x(t)), ∀t ∈ [0, 1]. +On the other hand, since ¯x takes values in C, we have ψ(¯x(t)) ≤ 0, ∀t ∈ [0, 1]. +Step 2. Construction of ¯p, ¯ζ, ¯θ, ¯ϑ, ¯ω, ¯ν, and the adjoint equation. +For the construction of ¯p, ¯θ, ¯ϑ and ¯ν, see Steps 2-4 in the proof of [38, Theorem 5.1]. +Note that the uniform boundedness of pγk(1) established and used in Step 2 of the +proof of [38, Theorem 5.1], is easily deduced here from the nontriviality condition of +Proposition 4.7. We also note that, similarly to Step 2 of the proof of [38, Theorem +5.1], pγk has a uniformly bounded variation, and hence, Helly first theorem implies +that pγk admits a pointwise convergent subsequence whose limit ¯p is also of bounded +variation and satisfies, for some M1 > 0, the following +∥¯p∥∞ ≤ M1∥¯p(1)∥. +(31) +Using Helly second theorem we obtain that for all h ∈ C([0, 1]; Rn), +lim +k→∞ +� +[0,1] +⟨h(t), ˙pγk(t)⟩ dt = +� +[0,1] +⟨h(t), d¯p(t)⟩. +(32) +Identically to Steps 2-4 in the proof of [38, Theorem 5.1], we also have +� 1 +0 +⟨h(t), θγk(t) pγk(t)⟩ dt −→ +� 1 +0 +� +h(t), ¯θ(t) ¯p(t) +� +dt, +(33) +� 1 +0 +ξγk(t) ⟨h(t), ϑγk(t) pγk(t)⟩ dt −→ +� 1 +0 +¯ξ(t) ⟨h(t), ¯ϑ(t) ¯p(t)⟩ dt, +(34) +lim +k−→∞ +� 1 +0 +⟨h(t), ∇ψ(xγk(t))⟩ γkξγk(t) ⟨∇ψ(xγk(t)), pγk(t)⟩ dt += +� +[0,1] +⟨h(t), ∇ψ(¯x(t))⟩ d¯ν(t). +(35) + +Control Space for Strong Convergence of Continuous Approximation +21 +We proceed to construct the two functions ¯ζ and ¯ω. Note that the construction +of ζ done in Step 2 of the proof of [38, Theorem 5.1] cannot be used here, since the +closed graph hypothesis on the multifunction (x, u) �→ ∂xf(x, u) is required there, +but it is not assumed here. As the sequence (ζγk, ωγk)k is uniformly bounded in L∞, +it has a subsequence, we do not relabel, that converges weakly in L1 to some (¯ζ, ¯ω). +Using that the multifunction (x, u) �→ ∂(x,u) +ℓ +f(x, u) has closed graph with nonempty, +compact and convex values, [10, Theorem 6.39] implies that, for t ∈ [0, 1] a.e., +(¯ζ(t), ¯ω(t)) ∈ ∂(x,u) +ℓ +f(¯x(t), ¯u(t)). +Since (pγk)k is uniformly bounded in L∞ and converges pointwise to ¯p, we conclude +that +ζT +γkpγk +weakly +−−−−−→ +L1 +¯ζT¯p and ωT +γkpγk +weakly +−−−−−→ +L1 +¯ωT¯p. +(36) +Hence, for all h ∈ C([0, 1]; Rn), +� 1 +0 +⟨h(t), ζγk(t)pγk(t)⟩ dt −→ +� 1 +0 +� +h(t), ¯ζ(t)¯p(t) +� +dt. +(37) +Thus, from (29) and (32)-(37), we conclude that the adjoint equation of Theorem 4.8 +holds, and it coincides with the adjoint equation of [38, Theorem 5.1]. +Step 3. The complementary slackness conditions. +The part (a) follows from the equation (13). The part (b) follows from the uniform +boundedness of ∥γkξγk(·)⟨∇ψ(xγk(·)), pγk(·)⟩∥1 established in [38, Equation (97)]. +More details can be found in Step 6 of [33, Proof of Theorem 6.1]. +Step 4. Construction of λ and the transversality equation. +Form +the +transversality +condition +of +Proposition +4.7, +there +exist +υγk +∈ +NL +C0(k)(xγk(0)), χγk ∈ NL +C1(k)(xγk(1)) and (aγk, bγk) ∈ ∂Lg(xγk(0), xγk(1)) such +that +pγk(0) = λγkaγk + λγk(xγk(0)− ¯x(0)) + υγk, +−pγk(1) = λγkbγk + χγk, +(38) +and the following properties hold: +• ∥(aγk, bγk)∥ ≤ Lg, where Lg is the Lipschitz constant of g over ˜C0(δ)× ˜ +C1(δ), and +∥λγk∥ ≤ 1 for all k. The latter inequality gives a subsequence, we do not relabel, +such that λγk −→ λ ∈ [0, 1]. +• Due to Theorem 4.6, we have, for k large enough, +(xγk(0), xγk(1)) ∈ int ( ˜C0(δ) × ˜C1(δ)), and (xγk(0), xγk(1)) −→ (¯x(0), ¯x(1)). +• We have pγk(0) −→ ¯p(0) and pγk(1) −→ ¯p(1). +• Owing to (26), in which d := xγk(1) ∈ +� +C1(k)∩(int C)∩Bδo(¯x(1)) +� +for k sufficiently +large, we have χγk ∈ NL +C1(k)(xγk(1)) = NL +C1 (xγk(1) + ¯x(1) − ¯xγk(1)), for k large, +where, we recall that ¯xγk(1) −→ ¯x(1). +• Owing to (25), in which c := xγk(0) ∈ C0(k) ∩ Bδo(¯x(0)) for k large enough, it +follows that: +(i) If ¯x(0) ∈ int C, then for k sufficiently large +vγk ∈ NL +C0(k)(xγk(0)) = NL +C0(xγk(0)). +(ii) If ¯x(0) ∈ bdry C, using that xγk(0) −→ ¯x(0) and ρk −→ 0, then for k +sufficiently large, +� +xγk(0) + ρk +∇ψ(¯x(0)) +∥∇ψ(¯x(0))∥ +� +∈ Bδo(¯x(0)), and hence, +vγk ∈ NL +C0(k)(xγk(0)) = NL +C0 +� +xγk(0) + ρk +∇ψ(¯x(0)) +∥∇ψ(¯x(0))∥ +� +for k large. + +22 +Control Space for Strong Convergence of Continuous Approximation +Therefore, along a subsequence of (γk)k, we do not relabel, we have +λγk(aγk, bγk) −→ λ(a, b) ∈ λ∂L +ℓ g(¯x(0), ¯x(1)) and λγk(xγk(0)− ¯x(0)) −→ 0. +Thus, taking the limit as k → ∞ in (38), and using (pγk(0), pγk(1)) −→ (¯p(0), ¯p(1)), +we obtain that (vγk, χγk) must converge to some (v, χ), as all the other terms in +(38) converge. The last two bullets, stated above, yield that v ∈ NL +C0(¯x(0)) and +χ ∈ NL +C1(¯x(1)). Consequently, the limit of (38) is equivalent to +(¯p(0), −¯p(1)) ∈ λ∂L +ℓ g(¯x(0), ¯x(1)) + +� +NL +C0(¯x(0)) × NL +C1(¯x(1)) +� +; +This terminates the proof of the transversality equation. +Step 5. The weak maximization condition. +By (30), (36)(b), and the transversality equations of Proposition 4.7, we have that +˙qγk = −(ωγk)Tpγk +weakly +−−−−−→ +L1 +−(¯ω)T¯p and qγk(0) = λγk(uγk(0) − ¯u(0)). +(39) +The uniform boundedness in L∞ of the sequences (pγk)k and (ωγk)k give that +( ˙qγk)k is uniformly bounded in L∞, asserting the equicontinuity of (qγk)k. Moreover, +the nontriviality condition of Proposition 4.7 gives the uniform boundedness of the +sequence (qγk)k. Hence, by Arzel`a-Ascoli theorem, the sequence (qγk)k admits a sub- +sequence, we do not relabel, that converges uniformly to an absolutely continuous +function q satisfying q(0) = 0 (by (39)(b), where λγk −→ λ and uγk(0) −→ ¯u(0) as +k −→ ∞). Moreover, up to a subsequence, we also obtain that +˙qγk +weakly +−−−−−→ +L1 +˙q. +(40) +Hence, (39)(a) and the uniqueness of the L1-weak limit yield that +˙q(t) = −(¯ω(t))T¯p(t), t ∈ [0, 1] a.e. +(41) +We proceed to study the convergence of the sequence of NBV -functions, (Ωγk)k, +obtained in Proposition 4.7. The maximization condition (iv), therein, implies that, +for t ∈ [0, 1] a.e., +Ωγk(t) = −qγk(t) + λγk( ˙uγk(t) − ˙¯u(t)) +� +�� +� +ℓγk(t) +. +(42) +Without loss of generality, we can assume that (42) is satisfied for all t ∈ [0, 1]. +In fact, if λγk = 0, using the transversality conditions of Proposition 4.7 and that +Ωγk ∈ NBV [0, 1], we get that Ωγk(0) = −qγk(0) = 0 and Ωγk(1) = −qγk(1), and +hence, by the right continuity of Ωγk and the continuity of qγk, (42) is equivalent +to Ωγk ≡ −qγk. If, however, λγk > 0, then by modifying the values of ( ˙uγk − ˙¯u) +on the set of Lebesgue measure zero, we have (42) satisfed for all t ∈ [0, 1], and +hence, ℓγk ∈ BV [0, 1] is right continuous on (0, 1), and satisfies ℓγk(0) = qγk(0) and +ℓγk(1) = 0. Furthermore, since λγk −→ λ and ˙uγk strongly converges in L2 to ˙¯u, the +sequence (ℓγk)k strongly converges in L2 to ℓ = 0. +We claim that (Ωγk)k, considered as a sequence of continuous linear functionals +on C([0, 1]; Rm), admits a subsequence, we do not relabel, that converges weakly* to +−q. Since Ωγk satisfies (42), where the sequence of absolutely continuous functions +(qγk)k converges uniformly to q ∈ AC([0, 1]; Rm) and, by (40), ( ˙qγk)k converges +weakly in L1 to ˙q, then it is equivalent to show that the BV -sequence (ℓγk)k converges +in C∗([0, 1]; Rm) to 0. The uniform boundedness of the sequence (ℓγk)k shall follow +once we show the uniform boundedness of (Ωγk)k. For this latter, the nontriviality +condition of Proposition 4.7, implies that the sequence (µoγk)k is uniformly bounded, +and hence, it has a subsequence, we do not relabel, that converges weakly* to a + +Control Space for Strong Convergence of Continuous Approximation +23 +µo ∈ C⊕([0, 1]; Rm). Moreover, by condition (v) of Proposition 4.7 and Remark +5.2(i), ∥βγk(t)∥ ≤ 1, except on a set of µoγk-measure zero. Thus, using that +Ωγk(t) = +� +[0,t] +βγk(s)µo +γk(ds), ∀t ∈ (0, 1], and Ωγk(0) = 0, +(43) +we obtain that the sequence (Ωγk)k is uniformly bounded, and so is the sequence +(ℓγk)k. Hence, to get that the bounded BV -sequence (ℓγk)k converges weakly* to 0, +by the Banach-Steinhaus theorem in [24, p.482], it is sufficient to show that +lim +k−→∞ +� 1 +0 +⟨h(t), dℓγk(t)⟩ = 0, ∀h ∈ C1([0, 1]; Rm). +Fix h ∈ C1([0, 1]; Rm). Using an integration by parts and that Ωγk ∈ NBV, we get +� 1 +0 +⟨h(t), dℓγk(t)⟩ = ⟨h(1), ℓγk(1)⟩ − ⟨h(0), ℓγk(0)⟩ − +� 1 +0 +� +˙h(t), ℓγk(t) +� +dt += +� +−⟨h(0), qγk(0)⟩ − +� 1 +0 +� +˙h(t), ℓγk(t) +� +dt +� +−−−−−→ +k−→∞ 0, +since (ℓγk)k strongly converges in L2 to 0, and qγk(0) −→ q(0) = 0. This terminates +the proof of the claim, that is, +Ωγk +weakly* +−−−−−−−−−→ +C∗([0,1]; Rm) −q. +(44) +By [24, p. 484, #8], we also have that Ωγk(t) −→ −q(t), ∀t ∈ [0, 1]. +Now define the signed measure µγk(dt) := βγk(t)µoγk(dt). From (43) we have +Ωγk(t) = µγk[0, t], ∀t ∈ (0, 1]. +(45) +Using that µoγk +weakly* +−−−−−→ +k−→∞ +µo, and that Proposition 4.7(v) holds true, then, by applying +[37, Proposition 9.2.1] to the following data: +• Aγk(t) := ∂> +u d(uγk(t), U(t) ∩ ¯Bδ(¯u(t))) for all t ∈ [0, 1], +• A(t) := ∂> +u d(¯u(t), U(t) ∩ ¯Bδ(¯u(t))) for all t ∈ [0, 1], +• γγk := βγk, µγk := µoγk and µ0 := µo, +we obtain a Borel measurable function β : [0, 1] −→ Rm and µ ∈ C∗([0, 1]; Rm) such +that +µγk +weakly* +−−−−−→ +k−→∞ +µ, µ(dt) = β(t)µo(dt) and β(t) ∈ ∂> +u d(¯u(t), U(t) ∩ ¯Bδ(¯u(t))) µo a.e. +Since uγk converges uniformly to ¯u, and supp {µoγk} satisfies Proposition 4.7(v), we +deduce that +supp {µo} ⊂ A := +� +t ∈ [0, 1] : (t, ¯u(t)) ∈ bdry Gr +� +U(t) ∩ ¯Bδ(¯u(t)) +�� +. +(46) +Adjust β(·) on the set of µo-measure zero to arrange +t ∈ A =⇒ β(t) ∈ ∂> +u d(¯u(t), U(t) ∩ ¯Bδ(¯u(t))), +and hence, using [37, Formula (9.17)], we have +β(t) ∈ +� +conv ¯ +NL +U(t)∩ ¯ +Bδ(¯u(t))(¯u(t)) ∩ +� ¯B \ {0} +�� +, ∀t ∈ A. +(47) +Thus, by (44) and (45), we obtain that −dq(t) = µ(dt) = β(t)µo(dt). Using (41), we +arrive to +− dq(t) = (¯ω(t))T¯p(t)dt = β(t)µo(dt). +(48) + +24 +Control Space for Strong Convergence of Continuous Approximation +Next, we decompose µo(dt) = m(t)dt+µs(dt) for some nonnegative L1-function m(·) +and some nonnegative Borel measure µs totally singular with respect to Lebesgue +measure. Clearly m(t) = 0, for all t ∈ Ac, and hence, (47) implies that +β(t)m(t) ∈ conv ¯ +NL +U(t)∩ ¯ +Bδ(¯u(t))(¯u(t)), ∀t ∈ [0, 1]. +Using (48) we get that (¯ω(t))T¯p(t)dt = β(t)m(t)dt + β(t)µs(dt). This gives that +(¯ω(t))T¯p(t) = β(t)m(t), for t ∈ [0, 1] a.e. Therefore, +¯ω(t)T¯p(t) ∈ conv ¯ +NL +U(t)∩ ¯ +Bδ(¯u(t))(¯u(t)), ∀t ∈ [0, 1] a.e. +(49) +We proceed to prove that the “In addition” part of the weak maximization con- +dition. We assume the existence of εo > 0 and r > 0 such that U(t) ∩ ¯Bεo(¯u(t)) is +r-prox-regular for all t ∈ [0, 1]. From (49) and using Lemma 5.3, we obtain that for +all t ∈ [0, 1] a.e. +� +¯ω(t)T¯p(t), u − ¯u(t) +� +≤ ∥¯ω(t)T ¯p(t)∥ +min{εo,2r} ∥u − ¯u(t)∥2, for all u ∈ U(t). +Therefore, for a.e. t ∈ [0, 1], +� +ω(t)Tp(t), u +� +− ∥¯ω(t)T ¯p(t)∥ +min{εo,2r} ∥u − ¯u(t)∥2 ≤ +� +¯ω(t)T¯p(t), ¯u(t) +� +, for all u ∈ U(t). +This terminates the proof of the weak maximization condition. +Step 6. The nontriviality condition. +It is sufficient to prove its equivalent condition: ∥¯p(1)∥ + λ ̸= 0. Taking the limit as +k −→ ∞ in the nontriviality condition of Proposition 4.7, and using the convergence +of ¯pγk(1) to ¯p(1), the uniform convergence of qγk to q, the weak* convergence of µoγk +to µo, and the convergence of λγk to λ, we get that +1 = ∥¯p(1)∥ + ∥q∥∞ + ∥µo∥T.V. + λ. +(50) +We argue by contradiction. If ¯p(1) = 0 and λ = 0, by (31) we obtain that ¯p = 0. +Hence (41) yields that ˙q(t) = 0 for a.e. t ∈ [0, 1]. This gives that +β(t)µo(dt) = −dq(t) = 0 and q(t) = q(0) + +� t +0 +˙q(τ)dτ = 0, ∀t ∈ [0, 1]. +(51) +Since, by (46) and (48), supp {µo} ⊂ A and β(t) ̸= 0 for all t ∈ A, the first equation +of (51) yields that µo = 0. Therefore, ∥¯p(1)∥ + ∥q∥∞ + ∥µo∥T.V. + λ = 0 which +contradicts (50). This terminates the proof of the nontriviality condition, and then, +the proof of the conditions (i)-(vi) of Theorem 4.8 is completed. +□ +5 Appendix +In this section, we present the proofs of Theorems 4.1 and 4.6, and we establish +auxilliary results that are used in different places of the paper. We begin by +the proof of Theorem 4.1. +Proof of Theorem 4.1. (i): Having a uniform bounded derivative in L2([0, 1]; Rm), +the W 1,2-sequence uγk is equicontinuous. Since, by (A4.2), the compact sets U(t) +are uniformly bounded, then uγk is uniformly bounded in C([0, 1]; Rm), and hence, +Arzel`a-Ascoli theorem asserts that uγk admits a subsequence, we do not relabel, that +converges uniformly to an absolutely continuous function u with u(t) ∈ U(t) for all +t ∈ [0, 1]. As ˙uγk is uniformly bounded in L2([0, 1]; Rm), then, up to a subsequence, +it is weakly convergent in L2. The boundedness of (uγk(0))k then yields that the L2- +weak limit of ˙uγk is ˙u, and whence, u ∈ W. The fact that x is the unique solution +to (D) corresponding to (x0, u), and the proceeding statements of this part, follow + +Control Space for Strong Convergence of Continuous Approximation +25 +immediately from Theorem 3.9(ii). +(ii): Now, assume that cγk ∈ C(k) for k ≥ ko, where ko is the rank in Theorem 3.11. +Let us first show that (ξγk)k has uniform bounded variations. Since ψ and ∇ψ +are Lipschitz on C and xγk is Lipschitz for k ≥ ko, we deduce that, for k ≥ ko, the +function ξγk(·)∇ψ(xγk(·)), where ξγk is defined in (21), is Lipschitz continuous on +[0, 1]. Similarly, the Lipschitz property on C ×(U+ ˜ρ ¯B) of f(·, ·) (and then of fΦ(·, ·)) +and the fact that (xγk, uγk) is in W 1,∞ × W 1,2, yield that fΦ(xγk(·), uγk(·)) is in +W 1,2([0, 1]; Rn). Hence, +ζγk(t) := d +dtfΦ(xγk(t), uγk(t)), +exists for almost all t ∈ [0, 1]. By writing fΦ = (f1 +Φ, · · · , fn +Φ) +T, and using that xγk(t) ∈ +int C (for all t ∈ [0, 1]), and uγk(t) ∈ U(t) ⊂ U (for t ∈ [0, 1] a.e.), it follows from +the proof of [39, Theorem 2.1], that +ζi +γk(t) ∈ ⟨∂fi +Φ(xγk(t), uγk(t)), ( ˙xγk(t), ˙uγk(t))⟩, t ∈ [0, 1] a.e., ∀i = 1, · · · , n. +Since (∥ ˙uγk∥2)k is assumed to be bounded, and, by Theorem 3.11, (∥ ˙xγk∥∞)k is +bounded, then the sequence (∥ζγk∥2)k is bounded by some Mζ > 0 that depends on +¯ +M, ¯ +Mψ, η, and the bound of (∥ ˙uγk∥2)k. +As fΦ(xγk(·), uγk(·)) ∈ W 1,2([0, 1]; Rn), the right hand side of (Dγk) yields that +˙xγk is in W 1,2([0, 1]; Rn), and so is the function |⟨∇ψ(xγk(·)), ˙xγk(·)⟩|. This also +implies that ξγk ∈ W 2,2([0, 1]; R+), due to +˙ξγk(t) = γ2 +keγkψ(xγk(t))⟨∇ψ(xγk(t)), ˙xγk(t)⟩. +Next, calculating ¨xγk through (Dγk) in terms of ζγk and ˙xγk, and using the fact that +for h ∈ AC([0, 1]; R) we have +d +dt|h(t)| = +� d +dth(t) +� +sign(h(t)) a.e. t ∈ (0, 1), 2 +it follows that there exist measurable functions ϑ1γk and ϑ2γk whose values at t are in +∂2ψ(xγk(t)), for almost all t ∈ [0, 1], such that, for t ∈ [0, 1] a.e., we have +d +dt|⟨∇ψ(xγk(t)), ˙xγk(t)⟩| += +�� +ϑ1 +γk(t) ˙xγk(t), ˙xγk(t) +� ++ ⟨∇ψ(xγk(t)), ¨xγk(t)⟩ +� +α(t) +� +�� +� +sign(⟨∇ψ(xγk(t)), ˙xγk(t)⟩) += +�� +ϑ1 +γk(t) ˙xγk(t), ˙xγk(t) +� ++ ⟨∇ψ(xγk(t)), ζγk(t) − ξγk(t)ϑ2 +γk(t) ˙xγk(t)⟩ +� +α(t) +−⟨γkξγk(t)⟨∇ψ(xγk(t)), ˙xγk(t)⟩ +� +�� +� +˙ξγk (t) +∇ψ(xγk(t)), ∇ψ(xγk(t))⟩α(t) += +�� +ϑ1 +γk(t) ˙xγk(t), ˙xγk(t) +� ++ +� +∇ψ(xγk(t)), ζγk(t) − ξγk(t)ϑ2 +γk(t) ˙xγk(t) +�� +α(t) +−γkξγk(t) |⟨∇ψ(xγk(t)), ˙xγk(t)⟩| ∥∇ψ(xγk(t))∥2. +Integrating both sides on [0, 1] and using the boundedness of (∥ ˙xγk∥∞)k and +(∥ξγk∥∞)k (by Theorem 3.11), and assumption (A2.1), we get the existence of a +constant ˜ +M1 depending on ¯ +M, Mψ, ¯ +Mψ, η, and Mζ such that +� 1 +0 +| ˙ξγk(t)|∥∇ψ(xγk(t))∥2dt ≤ ˜ +M1. +2The function sign: R −→ R is defined by: sign(x) = +x +|x| for x ̸= 0, and 0 for x = 0. + +26 +Control Space for Strong Convergence of Continuous Approximation +Using (8) and assumption (A2.2), it follows that +� 1 +0 +| ˙ξγk(t)|dt = +� 1 +0 +γ2 +keγkψ(xγk(t))|⟨∇ψ(xγk(t)), ˙xγk(t)⟩|dt += +� +{t:∥∇ψ(xγk(t))∥≤η} +γ2 +keγkψ(xγk(t))|⟨∇ψ(xγk(t)), ˙xγk(t)⟩|dt ++ +� +{t:∥∇ψ(xγk(t))∥>η} +γ2 +keγkψ(xγk(t))|⟨∇ψ(xγk(t)), ˙xγk(t)⟩|∥∇ψ(xγk(t))∥2 +∥∇ψ(xγk(t))∥2 dt +≤ η +� +¯ +M + 2 ¯ +M ¯ +Mψ +η +� +γ2 +ke−γkε + +˜ +M1 +η2 ≤ η +� +¯ +M + 2 ¯ +M ¯ +Mψ +η +� ++ +˜ +M1 +η2 =: ˜ +M2, +for k sufficiently large, where ˜ +M2 depends on the given constants, ¯ +M, ¯ +Mψ, Mψ, η, +and on the bound of (∥ ˙uγk∥2)k. Therefore, the sequence ξγk satisfies, for k sufficiently +large, V 1 +0 (ξγk) ≤ ˜ +M2. +On the other hand, by Theorem 3.11, ∥ξγk∥∞ ≤ +2 ¯ +M +η +for all k ≥ ko. Hence, +by Helly first theorem, ξγk admits a pointwise convergent subsequence, we do not +relabel, whose limit is some function ˜ξ ∈ BV ([0, 1]; R+) with ∥˜ξ∥∞ ≤ +2 ¯ +M +η +and +V 1 +0 (˜ξ) ≤ ˜ +M2. Being pointwise convergent to ˜ξ and uniformly bounded in L∞, ξγk +strongly converges in L2 to ˜ξ. However, by part(i) of this theorem, ξγk converges +weakly in L2 to ξ, hence, ˜ξ = ξ. Thus, ξγk converges pointwise and strongly in L2 to +ξ, and ξ ∈ BV ([0, 1]; R+) with +V 1 +0 (ξ) ≤ ˜ +M2. +(52) +As f is M-Lipschitz on C × (U + ˜ρ ¯B), u ∈ W, ∇ψ is Lipschitz, and ξ ∈ BV , then +equation (12), which is satisfied by (x, u, ξ), now holds for all t ∈ [0, 1]. This yields +that (13) is also valid for all t ∈ [0, 1], and that ˙x ∈ BV ([0, 1]; Rn). +It remains to show that ˙xγk has uniform bounded variations and converges point- +wise and strongly in L2 to ˙x ∈ BV ([0, 1]; Rn). Since ξγk(·)∇ψ(xγk(·)) is Lipschitz, +uγk ∈ W, and f is M-Lipschitz on C × (U + ˜ρ ¯B), then (Dγk) holds for all t ∈ [0, 1], +that is, +˙xγk(t) = fΦ(xγk(t), uγk(t)) − ξγk(t)∇ψ(xγk(t)), +∀ t ∈ [0, 1]. +Hence, using part(i) of this theorem, the continuity of fΦ(·, ·), that the sequence +(xγk, uγk, ξγk)k has uniform bounded variations and converges pointwise to (x, u, ξ), +and that (x, u, ξ) satisfies (12) for all t ∈ [0, 1], we obtain that the sequence ˙xγk is of +bounded variations and converges pointwise to ˙x ∈ BV ([0, 1]; Rn). Since (∥ ˙xγk∥∞)k +is bounded, we conclude that the sequence ˙xγk also converges strongly in L2 to ˙x. +Therefore, xγk converges strongly in the norm topology of W 1,2 to x. +□ +We proceed to present the proof of our approximation result, namely, +Theorem 4.6. +Proof of Theorem 4.6. We consider k large enough so that C0(k) ⊂ ˜C0(δ) and C1(k) ⊂ +˜C1(δ), see (23). By Corollary 4.2, ¯xγk −→ ¯x strongly in W 1,2, and hence, for k +sufficiently large, ¯xγk(t) ∈ ¯Bδ(¯x(t)) for all t ∈ [0, 1], and ¯yγk(1) ∈ [−δ, δ], where +¯yγk(t) := +� t +0 +∥ ˙¯xγk(s) − ˙¯x(s)∥2ds. +Thus, the triplet state (¯xγk, ¯yγk, ¯zγk := 0) solves (Dγk) for ((¯ck, 0, 0), ¯u), with +¯xγk(t) +∈ +¯Bδ(¯x(t)) and +¯u(t) +∈ +U(t) ∩ ¯Bδ(¯u(t)), for +all +t +∈ +[0, 1], and +(¯xγk(1), ¯yγk(1), ¯zγk(1) = 0) ∈ C1(k) × [−δ, δ] × [−δ, δ]. Therefore, for k sufficiently + +Control Space for Strong Convergence of Continuous Approximation +27 +large, (¯xγk, ¯yγk, 0, ¯u) is an admissible quadruplet for (Pγk). Using the continuity of +g on ˜C0(δ) × ˜C1(δ) and the definition of J(x, u, z, u), we obtain that J(x, u, z, u) is +bounded from below. Hence, for k large enough, inf(Pγk) is finite. +Fix k sufficiently large so that inf(Pγk) is finite. Let (xnγk, ynγk, znγk, unγk)n ∈ +W 1,2([0, 1]; Rn) × AC([0, 1]; R) × AC([0, 1]; R) × W be a minimizing sequence for +(Pγk), that is, the sequence is admissible for (Pγk) and +lim +n−→∞ J(xn +γk, yn +γk, zn +γk, un +γk) = inf(Pγk). +(53) +Since for each n, xnγk solves (Dγk) for (xnγk(0), unγk), and (xnγk(0))n ∈ C0(k) ⊂ C, then, +by (18), we have that the sequence (xnγk)n is uniformly bounded in C([0, 1]; Rn) and +the sequence ( ˙xnγk)n is uniformly bounded in L2. On the other hand, from (A4.2), +we have that sets U(t) are compact and uniformly bounded, then, the sequence +(unγk)n, which is in W, is uniformly bounded in C([0, 1]; Rm). Moreover, its derivative +sequence, ( ˙unγk)n, must be uniformly bounded in L2. Indeed, if this is not true, then +there exists a subsequence of ˙unγk, we do not relabel, such that +lim +n−→∞ ∥ ˙unγk∥2 = ∞. +Using that g is bounded on ˜C0(δ) × ˜C1(δ), it follows that +J(xn +γk, yn +γk, zn +γk, un +γk) ≥ +min +(c1,c2)∈ ˜ +C0(δ)× ˜ +C1(δ) +g(c1, c2) + 1 +2zn +γk(1) += +min +(c1,c2)∈ ˜ +C0(δ)× ˜ +C1(δ) +g(c1, c2) + 1 +2∥ ˙un +γk − ˙¯u∥2 +2 +and hence, +lim +n−→∞ J(xnγk, ynγk, znγk, unγk) = ∞, contradicting (53). Thus, also ( ˙unγk)n is +uniformly bounded in L2. Therefore, by Arzel`a-Ascoli theorem, along a subsequence +(we do not relabel), the sequence (xnγk, unγk)n converges uniformly to a pair (xγk, uγk) +and the sequence ( ˙xnγk, ˙unγk)n converges weakly in L2 to the pair ( ˙xγk, ˙uγk). Hence, +(xγk, uγk) ∈ W 1,2([0, 1]; Rn) × W. Moreover, the following two inequalities hold +∥ ˙xγk − ˙¯x∥2 +2 ≤ lim inf +n−→∞ ∥ ˙xn +γk − ˙¯x∥2 +2 and ∥ ˙uγk − ˙¯u∥2 +2 ≤ lim inf +n−→∞ ∥ ˙un +γk − ˙¯u∥2 +2. +(54) +Since C0(k), C1(k), ¯Bδ(¯x(t)) and U(t)∩ ¯Bδ(¯u(t)) are closed for all t ∈ [0, 1], and from +the uniform convergence, as n −→ ∞, of the sequence (xnγk, unγk) to (xγk, uγk), we +get that the inclusions xγk(0) ∈ C0(k) and xγk(1) ∈ C1(k), and xγk(t) ∈ ¯Bδ(¯x(t)), +and uγk(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), for all t ∈ [0, 1]. To prove that xγk is the solution of +(Dγk) corresponding to (xγk(0), uγk), we first use that xnγk is the solution of (Dγk) +for (xnγk(0), unγk), that is, for t ∈ [0, 1], +xn +γk(t) = xn +γk(0) + +� t +0 +� +fΦ(xn +γk(s), un +γk(s)) − γkeγkψ(xn +γk(s))∇ψ(xn +γk(s)) +� +ds. +Using that (xnγk(t), unγk(t)) ∈ [C ∩ ¯Bδ(¯x(t))] × [U(t) ∩ ¯Bδ(¯u(t))], fΦ is Lipschitz on +[C ∩ ¯Bδ(¯x)] × [(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)], and (xnγk, unγk) converges uniformly to (xγk, uγk), +then, upon taking the limit, as n −→ ∞, in the last equation we conclude that +(xγk, uγk) satisfies the same equation, that is, +˙xγk(t) = fΦ(xγk(t), uγk(t)) − γkeγkψ(xγk(t))∇ψ(xγk(t)), t ∈ [0, 1] a.e. +We define for all t ∈ [0, 1], +yγk(t) := +� t +0 +∥ ˙xγk(τ) − ˙¯x(τ)∥2dτ and zγk(t) := +� t +0 +∥ ˙uγk(τ) − ˙¯u(τ)∥2dτ. +Clearly we have: +• yγk ∈ AC([0, 1]; R), +˙yγk(t) = ∥ ˙xγk(t) − ˙¯x(t)∥2, t ∈ [0, 1] a.e., and yγk(0) = 0. +• zγk ∈ AC([0, 1]; R), +˙zγk(t) = ∥ ˙uγk(t) − ˙¯u(t)∥2, t ∈ [0, 1] a.e., and zγk(0) = 0. + +28 +Control Space for Strong Convergence of Continuous Approximation +Moreover, since ∥ ˙xnγk − ˙¯x∥2 +2 = ynγk(1) ∈ [−δ, δ] and ∥ ˙unγk − ˙¯u∥2 +2 = znγk(1) ∈ [−δ, δ], the +two inequalities of (54) yield that +yγk(1) ∈ [−δ, δ] and zγk(1) ∈ [−δ, δ]. +(55) +Hence, (xγk, yγk, zγk, uγk) is admissible for (Pγk). Now using (53) and the second +inequality of (54), it follows that +inf(Pγk) = +lim +n−→∞ J(xn +γk, yn +γk, zn +γk, un +γk) += +lim +n−→∞ +� +g(xn +γk(0), xn +γk(1)) + 1 +2 +� +∥un +γk(0) − ¯u(0)∥2 + zn +γk(1) + ∥xn +γk(0) − ¯x(0)∥2�� += g(xγk(0), xγk(1))+ 1 +2∥uγk(0) − ¯u(0)∥2 + 1 +2 lim inf +n−→∞ ∥ ˙un +γk − ˙¯u∥2 +2 + 1 +2∥xγk(0) − ¯x(0)∥2 +≥ g(xγk(0), xγk(1)) + 1 +2∥uγk(0) − ¯u(0)∥2 + 1 +2∥ ˙uγk − ˙¯u∥2 +2 + 1 +2∥xγk(0) − ¯x(0)∥2 += J(xγk, yγk, zγk, uγk). +Therefore, for each k, large enough, (xγk, yγk, zγk, uγk) is optimal for (Pγk). +As Remark 4.4 asserts that, for k large, C0(k) ⊂ C(k) ⊂ C, then, Lemma 3.6 +and Theorem 3.9(i) yield that the sequence (xγk, ξγk)k, where ξγk is given via (21), +admits a subsequence, not relabled, having (xγk)k converging uniformly to some +x ∈ W 1,2([0, 1]; Rn) with images in C, ( ˙xγk, ξγk)k converging weakly in L2 to ( ˙x, ξ) +and ξ supported on I0(x). +Now, consider the sequence (uγk)k, which is in W. It has a uniformly bounded +derivative in L2. In fact, the admissibility of (¯xγk, ¯yγk, 0, ¯u), and the optimality of +(xγk, yγk, zγk, uγk) for (Pγk), imply that +J(xγk, yγk, zγk, uγk) ≤ g(¯xγk(0), ¯xγk(1)) + 1 +2∥¯xγk(0) − ¯x(0)∥2. +(56) +This, together with the continuity of g on ˜C0(δ)× ˜C1(δ), the uniform boundedness of +the sequences (xγk)k and (¯xγk)k, and the boundedness of U(0), imply that for some +ˆ +M > 0 we have that +∥ ˙uγk − ˙¯u∥2 +2 ≤ 2 (g(¯xγk(0), ¯xγk(1)) − g(xγk(0), xγk(1)) + ∥¯xγk(0) − ¯x(0)∥2 +− ∥uγk(0) − ¯u(0)∥2 − ∥xγk(0) − ¯x(0)∥2 ≤ ˆ +M. +Therefore, (uγk)k has uniformly bounded derivative in L2. Now since in addition +we have that xγk(0) ∈ C0(k) ⊂ C(k), we are in a position to apply Theorem 4.1. +We obtain a subsequence (not relabeled) of uγk, and u ∈ W such that (xγk, uγk) +converges uniformly to (x, u), ˙uγk converges weakly in L2 to ˙u, all the conclusions of +Theorem 3.11 hold including that xγk(t) ∈ int C for all t ∈ [0, 1], ( ˙xγk, ξγk) converges +strongly in L2 to ( ˙x, ξ), ˙x ∈ BV ([0, 1]; Rn), ξ ∈ BV ([0, 1]; R+), and, for all t ∈ [0, 1], +(x, u, ξ) satisfies (12)-(13) and x uniquely solves (D) for u, that is, +� +˙x(t) = fΦ(x(t), u(t)) − ξ(t)∇ψ(x(t)) ∈ f(x(t), u(t)) − ∂ϕ(x(t)), ∀ t ∈ [0, 1], +x(0) ∈ C0 ∩ ¯Bδo(¯x(0)). +Moreover, we have +∥ ˙u − ˙¯u∥2 +2 ≤ lim inf +k−→∞ ∥ ˙uγk − ˙¯u∥2 +2. +(57) +We shall show that (x, u) is admissible for (P). Since ˙xγk converges strongly in L2 +to ˙x, and using (55) and (57), we have: +• ∥ ˙x − ˙¯x∥2 +2 = +lim +k−→∞ ∥ ˙xγk − ˙¯x∥2 +2 = +lim +k−→∞ yγk(1) +(55) +∈ [−δ, δ]. +• ∥ ˙u − ˙¯u∥2 +2 +(57) +≤ lim inf +k−→∞ ∥ ˙uγk − ˙¯u∥2 +2 = lim inf +k−→∞ zγk(1) +(55) +∈ [−δ, δ]. + +Control Space for Strong Convergence of Continuous Approximation +29 +Hence, ∥ ˙x − ˙¯x∥2 +2 ≤ δ and ∥ ˙u − ˙¯u∥2 +2 ≤ δ. Since xγk(1) ∈ C1(k), (24)(b) implies +that x(1) ∈ C1 ∩ ¯Bδo(¯x(0)). Furthermore, the two inclusions xγk(t) ∈ ¯Bδ(¯x(t)) and +uγk(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), for all t ∈ [0, 1], together with the uniform convergence +of (xγk, uγk) to (x, u), give that x(t) ∈ ¯Bδ(¯x(t)) and u(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), for all +t ∈ [0, 1]. Therefore, (x, u) is admissible for (P). Hence, the optimality of (¯x, ¯u) for +(P) yields that +g(¯x(0), ¯x(1)) ≤ g(x(0), x(1)). +(58) +Now, the uniform convergence of ¯xγk to ¯x, (56), (58), the continuity of g, and the +convergence of xγk(0) to x(0), imply that +u(0) = ¯u(0) and lim inf +k−→∞ +� +∥ ˙uγk − ˙¯u∥2 +2 +� += 0, and +(59) +x(0) = ¯x(0) and g(¯x(0), ¯x(1)) = g(x(0), x(1)). +(60) +The equality (59) gives the existence of a subsequence of uγk, we do not relabel, such +that ˙uγk converges strongly in L2 to ˙¯u. It results that uγk converges uniformly to +¯u, and hence, u = ¯u. Consequently, uγk converges strongly in W to ¯u. Moreover, as +u = ¯u, the functions x and ¯x solve the dynamic (D) with the same control ¯u and +initial condition, see (60), hence, by the uniqueness of the solution of (D) we have +x = ¯x. Using Lemma 3.5, we obtain that also ξ = ¯ξ. Therefore, +xγk +uniformly +−−−−−−−−→ +C([0,1]; Rn) ¯x and ( ˙xγk, ξγk) +strongly +−−−−−−−−−−−−→ +L2([0,1]; Rn×R+) ( ˙¯x, ¯ξ). +This yields that (yγk, zγk) −→ (0, 0) in the strong topology of W 1,1([0, 1]; R+ × R+). +Since xγk(1) ∈ +�� +C1 ∩ ¯Bδo(¯x(1)) +� +− ¯x(1) + ¯xγk(1) +� +∩(int C) and ¯xγk(1) converges +to ¯x(1), it follows that xγk(1) ∈ +� � +C1 ∩ ¯Bδo(¯x(1)) +� ++ ˜ρB +� +∩ (int C), for k sufficiently +large. On the other hand, the definition of C0(k) and the convergence of ρk to 0 yield +that, for k large enough, xγk(0) ∈ +� � +C0 ∩ ¯Bδo(¯x(0)) +� ++ ˜ρB +� +∩(int C). This terminates +the proof of Theorem 4.6. +□ +In the next lemma, a compactness result is derived for the solutions of (D), +where the controls are restricted to be in W and x(1) ∈ C1. The equivalence +between (D) and equations (12)-(13) is employed. +Lemma 5.1 (Compact trajectories and controls for (D)) Assume that (A1)-(A4.3) +hold. Let (xj, uj)j be a sequence in W 1,∞ × W satisfying (D) with xj(1) ∈ C1, for +all j ∈ N, and (∥ ˙uj∥2)j be bounded. Consider (ξj)j the corresponding sequence in +L∞([0, 1]; R+) obtained via Lemma 3.5, that is, (xj, uj, ξj) satisfies (12)-(13), for +all j. Then there exist a subsequence of (xj, uj, ξj)j, we do not relabel, and (x, u, ξ) ∈ +W 1,∞([0, 1]; Rn) × W × L∞([0, 1]; R+) such that (xj, uj)j converges uniformly to +(x, u), ( ˙xj, ξj)j now converges pointwise to ( ˙x, ξ) ∈ BV ([0, 1]; Rn) × BV ([0, 1]; R+), +˙uj converge weakly in L2 to ˙u, ξ, and (x, u, ξ) satisfies (12)-(13) with x(1) ∈ C1. In +particular, (x, u) is admissible for (P) and (xj)j converges to x strongly in the norm +topology of W 1,2([0, 1]; Rn). +Proof Using (11) in Remark 3.4 for the sequence (xj)j, the boundedness of (∥ ˙uj∥2)j, +that uj(t)) ∈ U(t) for all t ∈ [0, 1], and that the sets U(t) are compact and uniformly +bounded, by (A4.2), then Arzela-Ascoli’s theorem produces a subsequence, we do +not relabel, of (xj, uj)j, that converges uniformly to an absolutely continuous pair +(x, u) with (x(t), u(t) ∈ C × U(t) for all t ∈ [0, 1], and ( ˙xj, ˙uj)j converging weakly in +L2 to ( ˙x, ˙u). As for all j ∈ N, xj(0) ∈ C0 and xj(1) ∈ C1, then (A4.1) and (A4.3) + +30 +Control Space for Strong Convergence of Continuous Approximation +yield that x(0) ∈ C0 and x(1) ∈ C1. Using Corollary 4.2, we obtain ∥ξj∥∞ ≤ +¯ +M +2η , +ξj ∈ BV ([0, 1]; R+), and V 1 +0 (ξj) ≤ ˜ +M2, where ˜ +M2 depends on the uniform bound of +(∥ ˙uj∥2)j. By Helly’s first theorem, (ξj)j convergence pointwise to ξ ∈ BV ([0, 1]; R+). +On the other hand, Corollary 4.2 also gives that (12) holds for all t ∈ [0, 1], that is, +˙xj(t) = fΦ(xj(t), uj(t)) − ξj(t)∇ψ(xj(t)), ∀t ∈ [0, 1]. +(61) +Thus, upon taking the pointwise limit as j −→ ∞ in (61), it follows that ( ˙xj)j con- +verges pointwise to its L2-limit ˙x, and hence, ˙x ∈ BV ([0, 1]; Rn). As (xj, uj) solves +(D), (11) yields that (∥ ˙xj∥∞)j is uniformly bounded, and hence, ( ˙xj)j converges to +˙x strongly in L2. +We now show that ξ(t) is supported in I0(x). Let t ∈ I-(x) be fixed, that is, +x(t) ∈ int C. Since (xj)j converges uniformly to x, then we can find δo > 0 and jo ∈ N +such that, for all s ∈ (t−δ, t+δ)∩[0, 1] and for all j ≥ jo, we have xj(s) ∈ int C, and +hence, as ξj satisfies (13), ξj(s) = 0. Thus, ξj(s) −→ 0 for s ∈ (t − δo, t + δo) ∩ [0, 1], +and whence, ξ(t) = 0, proving that ξ is supported in I0(x). Therefore, applying +Lemma 3.5 to (x, u, ξ), we conclude that (x, u) solves (D), ξ ∈ L∞([0, 1]; R+) and +(x, u, ξ) satisfies (13). +□ +In the following remark, we provide important information about the +constraint qualification property (CQ). +Remark 5.2 +(i) Let F : [0, 1] ⇒ Rm be a lower semicontinuous multifunction with closed and +nonempty values. For dF (t, x) := d(x, F(t)), we have from [25, Proposition +2.3] that for t ∈ [0, 1] and x ∈ F(t), conv ( ¯NL +F (t)(x)) is pointed if and only if +0 ̸∈ ∂> +x dF (t, x). The notion of ∂> +x g(t, x) is introduced for a general function +g(t, x) by Clarke in [9, p.121]. For g(t, x) := dF (t, x), it is shown in [25, Corollary +2.2] that +∂> +x dF (t, x) = +conv +� +ζ : ζ = +lim +i−→∞ ζi, ∥ζi∥ = 1, ζi ∈ NP +F (ti)(xi) and (ti, xi) Gr F +−−−→ (t, x) +� +, +where (ti, xi) +Gr F +−−−→ (t, x) signifies that (ti, xi) −→ (t, x) with xi ∈ F(ti) for +all i. Therefore, for h ∈ C([0, 1]; F), we have that F satisfies the constraint +qualification (CQ) at h if and only if 0 ̸∈ ∂> +x dF (t, h(t)) for all t ∈ [0, 1]. Note +that the multifunction (t, x) �→ ∂> +x dF (t, x) is uniformly bounded with compact +and convex values, and has a closed graph. +(ii) Using the proximal normal inequality, one can easily extend the arguments in +the proof of [25, Proposition 2.3(d)], to show that if the lower semicontinu- +ous multifunction F has closed and r-prox-regular values, for some r > 0, (as +opposed to convex), then conv ( ¯ +NL +F (t)(·)) = NP +F (t)(·) = NL +F (t)(·) = NF (t)(·), +and this cone is pointed at x ∈ F(t) if and only if F(t) is epi-lipschitz at x, +see [9, Theorem 7.3.1] and [36, Exercise 9.42]. Hence, a lower semicontinuous +multifunction F : [0, 1] ⇒ Rm with values that are closed and r-prox-regular, +satisfies the constraint qualification (CQ) at h ∈ C([0, 1]; F) if and only if F(t) +is epi-Lipschitz at h(t), for all t ∈ [0, 1]. +(iii) If F(t) = F for all t ∈ [0, 1], where F is closed, then conv ( ¯NL +F (t)(·)) = NF (·), +and this cone is pointed at x ∈ F if and only if F is epi-Lipschitz at x. Hence, +a constant multifunction F satisfies the constraint qualification (CQ) at h ∈ +C([0, 1]; F) if and only if F is epi-Lipschitz at h(t) for all t ∈ [0, 1]. + +Control Space for Strong Convergence of Continuous Approximation +31 +We terminate this section by the following technical lemma used in the +proof of the “In addition” part of the weak maximization condition of Theorem +4.8. The proof of the lemma follows from the local property of the normal cones, +the proximal normal inequality, and the fact that the proximal, Mordukhovich, +and Clarke normal cones coincide in our setting. +Lemma 5.3 Let F : [0, 1] ⇒ Rm be a lower semicontinuous multifunction with closed +and nonempty values and let h ∈ C([0, 1]; F). If there exist εo > 0 and r > 0 such +that F(t) ∩ ¯Bεo(h(t)) is r-prox-regular for all t ∈ [0, 1], then for any δ > 0 we have +conv ( ¯NL +F (t)∩ ¯ +Bδ(h(t))(h(t))) = NP +F (t)∩ ¯ +Bεo (h(t))(h(t)), ∀t ∈ [0, 1]. +Moreover, for all t ∈ [0, 1] and for all ζ ∈ conv ( ¯ +NL +F (t)∩ ¯ +Bδ(h(t))(h(t))), we have +⟨ζ, v − h(t)⟩ ≤ +∥ζ∥ +min{εo,2r}∥v − h(t)∥2, ∀v ∈ F(t). +References +[1] Adam, L., Outrata, J.: On optimal control of a sweeping process coupled +with an ordinary differential equation, Discrete Contin. Dyn. Syst. B 19, +2709–2738 (2014) +[2] Aubin, +J.-P.: +Viability +Theory, +Modern +Birkh¨auser +Classics, +Birkh¨auser/Springer, Basel-Boston-Berlin (2009) +[3] Brokate, M., Krejˇc´ı, P.: Optimal control of ODE systems involving a +rate independent variational inequality, Discrete and continuous dynamical +systems series B 18, 331–348 (2013) +[4] Brudnyi, A., Brudnyi, Y.: Methods of Geometric Analysis in Exten- +sion and Trace Problems, Volume 1, Monographs in Mathematics, 102. +Birkh¨auser/Springer Basel AG, Basel (2012). +[5] Cao, T.H., Colombo, G., Mordukhovich, B., Nguyen, D.: Optimization of +Fully Controlled Sweeping Processes, Journal of Differential Equations 295 +138–186 (2021) +[6] Cao, T.H., Mordukhovich, B.: Optimal control of a perturbed sweeping +process via discrete approximations, Discrete Contin. Dyn. Syst. Ser. B 21, +3331–3358 (2016) +[7] Cao, T.H., Mordukhovich, B.: Optimality conditions for a controlled sweep- +ing process with applications to the crowd motion model, Disc. Cont. Dyn. +Syst. Ser. B 22, 267–306 (2017) +[8] Cao, T.H., Mordukhovich, B.: Optimal control of a nonconvex perturbed +sweeping process, Journal of Differential Equations 266, 1003–1050 (2019) +[9] Clarke, F.H.: Optimization and Nonsmooth Analysis, John Wiley, New +York (1983) + +32 +Control Space for Strong Convergence of Continuous Approximation +[10] Clarke, F.H.: Functional Analysis, Calculus of Variations and Optimal +Control, Graduate Texts in Mathematics, 264. Springer, London (2013) +[11] Clarke, F.H., Yu. Ledyaev, Yu., Stern, R.J., Wolenski, P.R.: Nons- +mooth Analysis and Control Theory, Graduate Texts in Mathematics, 178, +Springer-Verlag, New York (1998) +[12] Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and the +lower-C2 property, J. Convex Analysis 2, 117–144 (1995) +[13] Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal +control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst. Ser. +B 19, 117–159 (2012) +[14] Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal +control of the sweeping process over polyhedral controlled sets, Journal of +Differential Equations 260 (2016) no. 4, 3397–3447. +[15] Colombo, G., Marigonda, A.: Differentiability properties for a class of +non-convex functions, Calc. Var. 25, 1–31 (2005) +[16] Colombo, G., Mordukhovich, B., Nguyen, D.: Optimization of a perturbed +sweeping process by constrained discontinuous controls, SIAM Journal on +Control and Optimization, 58 no. 4, 2678–2709 (2020) +[17] de Pinho, M.d.R., Ferreira, M.M.A., Smirnov, G.V.: Optimal Control +Involving Sweeping Processes, Set-Valued Var. Anal. 27, no. 2, 523–548 +(2019) +[18] de Pinho, M.d.R., Ferreira, M.M.A., Smirnov, G.V.: Correction to: Opti- +mal Control Involving Sweeping Processes, Set-Valued Var. Anal. 27, +1025–1027 (2019) +[19] de Pinho, M.d.R., Ferreira, M.M.A., Smirnov, G.V.: Optimal Control with +Sweeping Processes: Numerical Method, J Optim Theory Appl 185, 845– +858 (2020) +[20] de Pinho, M.d.R., Ferreira, M.M.A., Smirnov, G.V.: Necessary condi- +tions for optimal control problems with sweeping systems and end point +constraints, Optimization, 71:11, 3363–3381 (2021) +[21] Federer, H.: Curvature measures, Trans. Amer. Math. Soc. 93, 418–491 +(1959) +[22] Hiriart-Urruty, J.-B.: Extension of Lipschitz functions, Journal of Mathe- +matical Analysis and Applications, 77, 539–554 (1980) +[23] Hoehener, D.: Variational approach to second-order optimality conditions +for control problems with pure state constraints, SIAM J.Control Optim. +50, no.3, 1139–1173 (2012) + +Control Space for Strong Convergence of Continuous Approximation +33 +[24] Vladimir Kadets, A course in functional analysis and measure theory, +Translated from the 2006 Russian edition by Andrei Iacob, Universitext. +Springer, Cham (2018) +[25] Loewen, P.D., Rockafellar, R.T.: The Adjoint Arc in Nonsmooth Opti- +mization, Transactions of the American Mathematical Society 325, no. 1, +39–72 (1991) +[26] Luenberger, D.G: Optimization by vector space methods, John Wiley & +Sons, Inc. (1969) +[27] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, +I: Basic Theory, Springer, Berlin (2006) +[28] Moreau, J.J.: Rafle par un convexe variable, I, Trav. Semin. d’Anal. +Convexe, Montpellier 1, Expos´e 15, 36 pp. (1971) +[29] Moreau, J.J.: Rafle par un convexe variable, II, Trav. Semin. d’Anal. +Convexe, Montpellier 2, Expos´e 3, 43 pp. (1972) +[30] Moreau, J.J.: Evolution problem associated with a moving convex set in +a Hilbert space, Journal of Differential Equations 26, 347–374 (1977) +[31] Nour, C., Stern, R.J., Takche, J.: Proximal smoothness and the exterior +sphere condition, J. Convex Anal., 16(2), 501–514 (2009) +[32] Nour, C., Zeidan, V.: Numerical solution for a controlled nonconvex +sweeping process, IEEE Control Systems Letters, vol. 6, 1190–1195 (2022) +[33] Nour, C., Zeidan, V.: Optimal control of nonconvex sweeping pro- +cesses with separable endpoints: Nonsmooth maximum principle for local +minimizers, Journal of Differential Equations, Volume 318, 113–168 (2022). +[34] Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of +distance functions, Trans. Amer. Math. Soc. 352, 5231–5249 (2000) +[35] Rockafellar, R.T.: Clarke’s tangent cones and the boundaries of closed +sets in Rn, Nonlinear Anal. Theor. Meth. Appl. 3, 145–154 (1979) +[36] Rockafellar, R.T., Wets, R.J.-B.: Variational analysis, Grundlehren der +Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin (1998) +[37] Vinter, R.B.: Optimal Control, Birkh¨auser, Systems and Control: Foun- +dations and Applications, Boston (2000) +[38] Zeidan, V., Nour, C., Saoud, H.: A nonsmooth maximum principle for a +controlled nonconvex sweeping process, Journal of Differential Equations, +Volume 269, Issue 11, 9531–9582 (2021) + +34 +Control Space for Strong Convergence of Continuous Approximation +[39] Zeidan, V.: A modified Hamilton-Jacobi approach in the generalized +problem of Bolza, Appl. Math.Optim, no. 11, 97–109 (1984) + diff --git a/tNAyT4oBgHgl3EQfmvgr/content/tmp_files/load_file.txt b/tNAyT4oBgHgl3EQfmvgr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc2966e5cf4c48b635140aa4be8c7d30106cdbcb --- /dev/null +++ b/tNAyT4oBgHgl3EQfmvgr/content/tmp_files/load_file.txt @@ -0,0 +1,1176 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf,len=1175 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='00475v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='OC] 1 Jan 2023 A Control Space Ensuring the Strong Convergence of Continuous Approximation for a Controlled Sweeping Process Chadi Nour1† and Vera Zeidan2*† 1Department of Computer Science and Mathematics, Lebanese American University, Byblos Campus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Box 36, Byblos, Lebanon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 2Department of Mathematics, Michigan State University, East Lansing, 48824-1027, MI, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' E-mail(s): zeidan@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Contributing authors: cnour@lau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='lb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Abstract A controlled sweeping process with prox-regular set, W 1,2-controls, and separable endpoints constraints is considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Existence of optimal solutions is established and local opti- mality conditions are derived via strong converging continuous approximations that entirely reside in the interior of the prox- regular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consequently, these results are expressed in terms of new subdifferentials for the original data that are strictly smaller than the standard Clarke and Mordukhovich subdifferentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Keywords: Controlled sweeping process, Prox-regular sets, Necessary optimality conditions, Local minimizers, Strong convergence, Continuous approximations, Nonsmooth analysis MSC Classification: 49K21 , 49K15 , 49J52 1 2 Control Space for Strong Convergence of Continuous Approximation 1 Introduction This paper addresses the following fixed time Mayer-type optimal control problem involving W 1,2-controlled sweeping systems (P): Minimize g(x(0), x(1)) over (x, u) ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W such that \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (D) � ˙x(t) ∈ f(x(t), u(t)) − ∂ϕ(x(t)), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], x(0) ∈ C0 ⊂ dom ϕ, x(1) ∈ C1, where, g : Rn × Rn −→ R ∪ {∞}, f : Rn × Rm −→ Rn, ϕ: Rn −→ R ∪ {∞}, ∂ stands for the Clarke subdifferential, C := dom ϕ is the zero-sublevel set of a function ψ: Rn −→ R, that is, C = {x ∈ Rn : ψ(x) ≤ 0}, C0 ⊂ C, C1 ⊂ Rn, and, for U : [0, 1] ⇒ Rm a multifunction and U := � t∈[0,1] U(t), the set of control functions W is defined by W := W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' U) = � u ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) : u(t) ∈ U(t), ∀t ∈ [0, 1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (1) Note that if (x, u) solves (D), it necessarily follows that x(t) ∈ C for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A pair (x, u) is admissible for (P) if x: [0, 1] −→ Rn is absolutely continu- ous, u ∈ W, and (x, u) satisfies the perturbed and controlled sweeping process (D), called the dynamic of (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' An admissible pair (¯x, ¯u) for (P) is said to be a W 1,2-local minimizer (also known as intermediate local minimizer of rank 2) if there exists δ > 0 such that g(¯x(0), ¯x(1)) ≤ g(x(0), x(1)), (2) for all (x, u) admissible for (P) with ∥x− ¯x∥∞ ≤ δ, ∥ ˙x− ˙¯x∥2 2 ≤ δ, ∥u− ¯u∥∞ ≤ δ and ∥ ˙u − ˙¯u∥2 2 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that if (2) is satisfied for any admissible pairs (x, u), then (¯x, ¯u) is called a global minimizer (or an optimal solution) for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Sweeping processes first appeared in the papers [28–30] by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreau in the context of friction and plasticity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since then, these systems have emerged in further applications such as hysteresis, ferromagnetism, electric cir- cuits, phase transitions, economics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', [1] and its references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The main feature of such systems is the presence in its dynamic of a normal cone to a set C that is, the subdifferential of the indicator function of C, or more gen- erally, the subdifferential of an extended-real-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consequently, the dynamic is a differential inclusion with unbounded and discontinuous right- hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, sweeping processes fall outside the scope of standard differential inclusions, and hence, studying optimal control problems over this model requires creative new techniques In [3, 17, 20, 33, 38] (see also [18]), necessary optimality conditions in the form of a maximum principle for optimal control problems involving measurably-controlled sweeping processes are derived using a smooth penalty- type approximations, called here continuous approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A special feature Control Space for Strong Convergence of Continuous Approximation 3 of [17, 20, 33, 38], which led in [19, 32] to numerical algorithms, is the novel exponential penalization technique that approximates the dynamic of (D) by the following sequence of standard control systems (Dγk) ˙x(t) = f(x(t), u(t)) − ∇Φ(x(t)) − γkeγkψ(x(t))∇ψ(x(t)), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], where Φ is a smooth extension to Rn of ϕ, and γk is a positive sequence that con- verges to ∞ as k −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In these papers, necessary optimality conditions are developed via approximating weakly the optimal solution of (P) by a sequence of optimal solutions for standard optimal control problems over (Dγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' That is, the velocity sequence of the optimal state for the approximating problems convergences weakly in L2 to the given velocity of the solution for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Strong convergence of velocities is well-known to be an essential property for numerical purposes, as pointed out in several papers, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', [7, 8, 14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In other words, it is important that the solutions of (P) be strongly approx- imated (in the W 1,2-norm) by the solutions of approximating problems that are computable via existing numerical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This question of strong con- vergence of velocities was previously addressed using discrete approximations, see for instance, [5–8, 13, 14, 16], where the authors considered optimal control problems involving various forms of controlled sweeping processes including the W 1,2-controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In [6–8], this approach also served to derive necessary opti- mality conditions phrased in terms of the weak-Pontryagin-type maximum principle when the control space is W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therein, these optimality criteria are applied to real-life models, whose optimal controls turn out to be W 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The main goal of the paper is motivated by the importance of approximat- ing a solution of the sweeping process (D) by solutions of (Dγk) whose velocity strongly converges to the velocity of the solution of (D) as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We establish the validity of this result when the controls in (Dγk) are chosen to be W 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As a consequence, we embark on the study of the problem (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We first show that, under suitable conditions, the problem (P) admits an optimal solu- tion (¯x, ¯u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then, we approximate a given optimal solution (¯x, ¯u) by a sequence of optimal solutions for standard optimal control problems over (Dγk) with objective functions carefully formulated to guarantee the strong convergence of the solution velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' To our knowledge, this is a first-of-its-kind result that uses continuous approximations, as opposed to discrete approximations, to obtain strong convergence of velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Furthermore, necessary optimality conditions are established for W 1,2-local minimizers of (P) upon taking the limit of the optimality conditions for the corresponding approximating optimal control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This latter task requires meticulous analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Notations and some definitions from non- smooth analysis will be given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In Section 3, we provide a list of assumptions and some important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, we present some needed results from [33, Sections 4 & 5] including the connection between (Dγk) and (D) under measurable controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Section 4 is devoted to (i) show- ing that (Dγk) strongly approximates (D) when W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2-controls are utilized,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) establishing an existence theorem for an optimal solution of (P),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iii) con- structing for (P) a continuous approximating sequence of standard optimal 4 Control Space for Strong Convergence of Continuous Approximation control problems (Pγk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' and (iv) deriving necessary optimality conditions in the form of weak-Pontryagin-type maximum principle for W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2-local minimiz- ers of (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' To maintain an easy flow of the main results, most of the proofs are provided in Section 5, where we also establish some auxiliary results employed in different places of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 Basic notations In the sequel, the notations used in this paper are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' By ∥ · ∥, ⟨·, ·⟩, B and ¯B, we denote, respectively, the Euclidean norm, the usual inner prod- uct, the open unit ball and the closed unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' An open (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' closed) ball of radius ρ > 0 and centered at x ∈ Rn is written as Bρ(x) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ¯Bρ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For x, y ∈ Rn, [x, y] and (x, y) denote, respectively, the closed and the open line segment joining x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For a set C ⊂ Rn, int C, bdry C, cl C, convC, Cc, and C◦ designate the interior, the boundary, the closure, the convex hull, the complement, and the polar of C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The distance from a point x to a set C is denoted by d(x, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For an extended-real-valued func- tion ϕ: Rn −→ R ∪ {∞}, dom ϕ is the effective domain of ϕ and epi ϕ is its epigraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For a multifunction F : Rn ⇒ Rm, Gr F ⊂ Rn × Rm denotes the graph of F, that is, Gr F := {(x, v) ∈ Rn × Rm : v ∈ F(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The Lebesgue space of p-integrable functions h: [a, b] −→ Rn is denoted by Lp([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) or simply Lp when the domain and range are clearly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The norms in Lp([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) and L∞([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) (or C([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn)) are written as ∥ · ∥p and ∥ · ∥∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The set of all absolutely continuous functions from an interval [a, b] to Rn will be denoted by AC([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), and Mm×n([a, b]) is the set of m × n-matrix functions on [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We say that h is a BV -function, and we write h ∈ BV ([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), if h is a function of bounded variation, that is, V b a (h) < ∞, where V b a (h) is the total variation of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We denote by NBV [a, b] the normalized space of BV -functions on [a, b] that consists of those BV -functions Ω such that Ω(a) = 0 and Ω is right continuous on (a, b) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', [26, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='115]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The set C∗([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) denotes the dual of C([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R), equipped with the supremum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The norm on C∗([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R), denoted by ∥·∥T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', is the induced norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' By Riesz representation theorem, the elements of C∗([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) are interpreted as belonging to M([a, b]), the set of finite signed Radon measures on [a, b] equipped with the weak* topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thereby, to each element of C∗([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) it corresponds a unique element in NBV [a, b] related through the Stieltjes integral and both elements have the same total varia- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We denote by C⊕(a, b) the subset of C∗([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) taking nonnegative values on nonnegative-valued functions in C([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For a compact subset S ⊂ Rd, C(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) designates the set of continuous functions from S to Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We denote by W k,p([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), k ∈ N and p ∈ [0, +∞], the classical Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that in this paper, the Sobolev space W 1,2([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) will be con- sidered with the norm ∥x(·)∥W 1,2 := ∥x(·)∥∞+∥ ˙x(·)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, the convergence of a sequence xn strongly in the norm topology of the space W 1,2([a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) Control Space for Strong Convergence of Continuous Approximation 5 is equivalent to the uniform convergence of xn on [a, b] and the strong conver- gence in L2 of its derivative ˙xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Finally, a function F : Rn → R is C1,1 if it is Fr´echet differentiable with locally Lipschitz derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2 Notions in nonsmooth analysis We begin by listing standard notions and facts for which the reader is invited to consult the monographs [11], [27], and [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let C be a nonempty and closed subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For x ∈ C, the proximal, the Mordukhovich (also known as limiting) and the Clarke normal cones to C at x are denoted by N P C (x), N L C(x) and NC(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using [11, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5(b)], we deduce that these three normal cones enjoy an essential local property, namely, if two closed sets in Rn are the same in a neighborhood of x, then these two sets possess at x the same normal cone (proximal, Mordukhovich, or Clarke).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' An important feature for the Mordukhovich normal cone is that the multifunction N L C(·) has closed values and a closed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, when C is convex then the proximal, the Mordukhovich and the Clarke normal cones to C coincide with the well-known normal cone to convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For ρ > 0, the set C is said to be ρ-prox-regular whenever the proximal normal inequality, see [11, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5(a)], holds for σ = 1 2ρ, for all x ∈ C and for all ζ unit in N P C (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In particular, every convex set is ρ-prox-regular for every ρ > 0, and every compact set with a C1,1-boundary is ρ-prox-regular, where ρ depends on the Lipschitz constant of the gradient of the boundary parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that, for a ρ-prox-regular set C, we have N P C (x) = N L C(x) = NC(x) for all x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For more information about prox-regularity, and related properties such as positive reach, proximal smoothness, exterior sphere condition and ϕ0-convexity, see [12, 15, 21, 31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The following geometric properties shall be used in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A closed set A ⊂ Rn is said to be epi-Lipschitz (or wedged ) at a point x ∈ A if the set A can be viewed near x, after application of an orthogonal matrix, as the epigraph of a Lipschitz continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If this holds for all x ∈ A, then we simply say that A is epi-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This geometric definition was introduced by Rockafellar in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The epi-Lipschitz property of A at x is also characterizable in terms of the nonemptiness of the topological interior of the Clarke tangent cone of A at x which is also equivalent to the pointedness of the Clarke normal cone of A at x, that is, NA(x) ∩ −NA(x) = {0}, see [9, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that a convex set is epi-Lipschitz if and only if it has a nonempty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For more information about this property, see [9, 11, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, a set A ⊂ Rn is quasiconvex if there exists α ≥ 0 such that any two points x, y in A can be joined by a polygonal line γ in A satisfying l(γ) ≤ α∥x − y∥, where l(γ) denotes the length of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In this paper, the quasiconvexity of C is vital for extending our function ϕ from C to Rn while preserving special properties (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For more explanation about this notion, consult [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Next, we recall the standard notions of proximal, Mordukhovich, and Clarke subdifferentials, and Clarke generalized Jacobian and Hessian (see [9, 11, 27, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We also enlist the nonstandard notions for subdifferentials introduced and studied in [33, 38] that are instrumental for this paper for being strictly smaller than their standard counterparts notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 6 Control Space for Strong Convergence of Continuous Approximation For the standard notions, given a lower semicontinuous function G: Rn −→ R ∪ {∞}, and x ∈ dom G, the proximal, the Mordukhovich (or limiting) and the Clarke subdifferential of G at x are denoted by ∂P G(x), ∂LG(x) and ∂G(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' From the properties of the limiting normal cone, ∂LG(·) has a closed graph and closed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that if x ∈ int (dom G) and G is Lipschitz near x, [9, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1] yields that the Clarke subdifferential of G at x coincides with the Clarke generalized gradient of G at x, also denoted here by ∂G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If G is C1,1 near x ∈ int (dom G), ∂2G(x) denotes the Clarke generalized Hessian of G at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For H : Rn −→ Rn Lipschitz near x ∈ Rn, the Clarke generalized Jacobian of H at x is denoted by ∂H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For the nonstandard notions, given a lower semicontinuous function G: Rn −→ R ∪ {∞}, S ⊂ dom G closed with int S ̸= ∅, and x ∈ cl (int S), we define the “limiting subdifferential of G relative to int S at the point x” to be ∂L ℓ G(x) := � lim i ζi : ζi ∈ ∂P G(xi), xi ∈ int S, and xi G −→ x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (3) where xi G −→ x signifies that xi −→ x and G(xi) −→ G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If dom G is closed, int (dom G) ̸= ∅, and G is locally Lipschitz on int (dom G), then for x ∈ cl (int (dom G)), we define the “extended Clarke generalized gradient of G at x”, denoted by ∂ℓG(x), to be ∂ℓG(x) := conv � lim i−→∞ ∇G(xi) : xi O −→ x and ∇G(xi) exists ∀i � , (4) where O is any full-measure subset of int (dom G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If G is C1,1 on int (dom G) and x ∈ cl (int (dom G)), we define similarly to ∂ℓG(x) the “extended Clarke generalized Hessian of G at x” to be ∂2 ℓ G(x) := conv � lim i−→∞ ∇2G(xi) : xi O −→ x and ∇2G(xi) exists ∀i � , (5) where O is any full-measure subset of int (dom G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For S ⊂ Rn closed with int S ̸= ∅, if G is C1,1 on an open set containing S, then for x ∈ S we define the “Clarke generalized Hessian of G relative to int S at x” to be ∂2 ℓ G(x) := conv � lim i−→∞ ∇2G(xi) : xi O −→ x � , (6) where O is any full-measure subset of int S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Now let S ⊂ Rn be closed with int S ̸= ∅, and let H : Rn −→ Rn be locally Lipschitz on int S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then for x ∈ cl (int S), we defined the “extended Clarke generalized Jacobian of H at x” to be ∂ℓH(x) := conv � lim i−→∞ JH(xi) : xi O −→ x and JH(xi) exists ∀i � , (7) where O is any full-measure subset of int S and J is the Jacobian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Finally, when the set defined in (4), (5), (7) or (6) is a singleton, then we shall use the notations ∇ℓ and ∇2 ℓ instead of ∂ℓ and ∂2 ℓ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Control Space for Strong Convergence of Continuous Approximation 7 3 Assumptions, consequences, and known results In this section, hypotheses on the data of (P) are introduced and some of their important consequences are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We also present some needed results from [33, Sections 4 & 5] where the connection between (Dγk) and (D) under measurable controls is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We note that each result of this paper shall require a selected group of these hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Furthermore, a local version of (A1) is stated and used in the relevant locations of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A1: There exist M > 0 and ˜ρ > 0 such that f is M-Lipschitz on C × (U+ ˜ρ ¯B) with ∥f(x, u)∥ ≤ M for all (x, u) ∈ C × (U + ˜ρ ¯B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A2: The set C := dom ϕ is given by C = {x ∈ Rn : ψ(x) ≤ 0}, where ψ : Rn −→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1: There exists ρ > 0 such that ψ is C1,1 on C + ρB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2: There is a constant η > 0 such that ∥∇ψ(x)∥ > 2η for all x : ψ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3: The function ψ is coercive, that is, lim∥x∥−→∞ ψ(x) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4: The set C has a connected interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 A3: The function ϕ is globally Lipschitz on C and C1 on int C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, the function ∇ϕ is globally Lipchitz on int C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A4: For the sets C0, C1, and U(·) we have: A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1: The set C0 ⊂ C is nonempty and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2: The graph of U(·) is a L × B measurable set, and, for t ∈ [0, 1], U(t) is closed, and bounded uniformly in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3: The set C1 ⊂ Rn is nonempty and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4: The multifunction U(·) is lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 The coercivity of ψ in (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3) is only assumed to obtain the boundedness of the closed set C, and hence, this condition can be replaced by C bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, for C ⊂ Rn defined as the sub-level set of a function ψ, one can show that: (i) Whenever C is nonempty and compact, ψ is merely C1 on C + ρB, and (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2) holds, then there exists ε > 0 such that x ∈ C and ∥∇ψ(x)∥ ≤ η =⇒ ψ(x) < −ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (8) (ii) When ψ is merely C1 on C +ρB and satisfies (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2)-(A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3), by [33, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3], (a) bdry C ̸= ∅ and bdry C = {x ∈ Rn : ψ(x) = 0}, (b) int C ̸= ∅ and int C = {x ∈ Rn : ψ(x) < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The following important properties of the compact set C were obtained in [38, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1], where ψ is assumed to be C1,1 on all of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' However, a slight modification in the proof of that proposition is performed in [33] to conclude that these properties are actually valid under our assumption (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Here and throughout the paper, ¯ Mψ denotes an upper bound of ∥∇ψ(·)∥ on the compact set C, and 2Mψ is a Lipschitz constant of ∇ψ(·) over the compact set C + ρ 2 ¯B chosen large enough so that Mψ ≥ 4η ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 1When ϕ has a suitable extension to Rn, as is the case for ϕ being the indicator of C, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3, this condition is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 8 Control Space for Strong Convergence of Continuous Approximation Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2 [33, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4] Under (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1)-(A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3), we have the following: (i) The nonempty set C is compact, amenable (in the sense of [36]), epi- Lipschitzian, C = cl (int C), and C is η Mψ -prox-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) For all x ∈ bdry C we have NC(x) = NP C (x) = NL C (x) = {λ∇ψ(x) : λ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iii) If also (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4) holds, then int C is quasiconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Furthermore, if in addition (A3) is satisfied, then there exists a function Φ ∈ C1(Rn) such that: Φ is bounded on Rn, and Φ(x) = ϕ(x) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Φ and ∇Φ are globally Lipschitz on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For all x ∈ C we have ∂ϕ(x) = {∇Φ(x)} + NC(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (9) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3 (i) In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2(iii), assumption (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4) is only imposed to ensure the quasicon- vexity of C needed to obtain the extension Φ of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consequently, when such an extension is readily available, condition (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4) is discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This is the case, for instance, when ϕ is constant on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, when ϕ is the indicator function of C assumption (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4) is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2(iii), ϕ admits a C1-extension Φ defined on Rn satisfying equation (9), and for some K > 0, |Φ(x)| ≤ K, ∥∇Φ(x)∥ ≤ K, and ∥∇Φ(x) − ∇Φ(y)∥ ≤ K∥x − y∥, ∀x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Employing (9), (D) is equivalently phrased in terms of the normal cone to C and the extension Φ of ϕ, as follows (D) � ˙x(t) ∈ fΦ(x(t), u(t)) − NC(x(t)), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], x(0) ∈ C0 ⊂ C, where fΦ : Rn × Rm −→ Rn is defined by fΦ(x, u) := f(x, u) − ∇Φ(x), ∀(x, u) ∈ Rn × Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (10) Therefore, we will interchangeably use throughout this paper the original formulation of (D) given in terms of ∂ϕ and f, and its reformulation provided in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3 in terms of NC(·) and the function fΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that assumptions (A1)-(A3) imply that, for ¯ M := M + K, fΦ satisfies the following properties: (A1)Φ: The function fΦ is ¯ M-Lipschitz on C × (U+ ˜ρ ¯B) with ∥fΦ(x, u)∥ ≤ ¯ M for all (x, u) ∈ C × (U + ˜ρ ¯B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We define U to be U := {u : [0, 1] → Rm : u is measurable and u(t) ∈ U(t), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4 Using [38, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3], it is easy to see that the assumptions (A1)- (A3) and the boundedness of C by some MC > 0 yield that any solution x of (D) corresponding to (x0, u) ∈ C0 × U satisfies x(t) ∈ C, ∀t ∈ [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ∥x∥∞ ≤ MC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' and ∥ ˙x∥∞ ≤ 2 ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (11) For given x(·): [0, 1] → Rn, we use the following notations throughout this paper: I0(x) := {t ∈ [0, 1] : x(t) ∈ bdry C} and I-(x) := [0, 1] \\ I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Control Space for Strong Convergence of Continuous Approximation 9 The next result characterizes the solutions of (D) in terms of the solutions of a standard control system containing an extra control ξ that satisfies the mixed control-state degenerate constraint, ξ(t)ψ(x(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The sufficiency part is straightforward and was used in [38], while the necessary part follows from applying Filippov selection theorem ([37, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5 Assume that (A1-(A3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let u ∈ U and x ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) with x(0) ∈ C0 and x(t) ∈ C for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then, x is a solution for (D) correspond- ing to the control u if and only if there exists a nonnegative measurable function ξ supported on I0(x) such that (x, u, ξ) satisfies ˙x(t) = fΦ(x(t), u(t)) − ξ(t)∇ψ(x(t)), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (12) In this case, the nonnegative function ξ supported in I0(x) with (x, u, ξ) satisfying equation (12), is unique, belongs to L∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), and \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ξ(t) = 0 for t ∈ I-(x), ξ(t) = ∥ ˙x(t)−fΦ(x(t),u(t))∥ ∥∇ψ(x(t))∥ ∈ � 0, ¯ M 2η � for t ∈ I0(x) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', ∥ξ∥∞ ≤ ¯ M 2η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (13) Throughout the paper we shall employ the following notations, where η and ¯ M are the constants given in (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2) and (A1)Φ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (γk)k is a sequence satisfying γk > 2 ¯ M η for all k ∈ N, and γk −−−−→ k−→∞ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (14) The sequence (αk)k is defined by αk := ln � ηγk 2 ¯ M � γk , k ∈ N (15) By (14) and (15), we have that γke−αkγk = 2 ¯ M η , αk > 0, αk ց and lim k−→∞ αk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (16) The sequence (ρk)k is defined by ρk := αk η for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' By (16) we have that ρk > 0 for all k ∈ N, ρk ց and lim k−→∞ ρk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For k ∈ N, we define the set C(k) := {x ∈ C : ψ(x) ≤ −αk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (17) The system (Dγk) is defined as (Dγk) � ˙x(t) = fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], x(0) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 10 Control Space for Strong Convergence of Continuous Approximation An important property shown in [33] is the invariance of C for the dynamic (Dγk), see [33, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This fact is behind disposing of the state constraint in (Dγk), which represents a good approximation for (D) (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6 (Invariance of C and uniform convergence) Let (A1)-(A3) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then, for each k, the system (Dγk) with given x(0) = cγk ∈ C and uγk ∈ U, has a unique solution xγk ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) such that xγk(t) ∈ C for all t ∈ [0, 1], and, for α0 > 0 a bound of (cγk)k we have ∥xγk∥∞ ≤ α0 + � ¯ M2 + 2 and � 1 0 ∥ ˙xγk(t)∥2dt ≤ ¯ M2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (18) Hence, being equicontinuous and uniformly bounded, (xγk)k admits a subsequence that converges uniformly to some x ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) whose values are in C and whose derivative ˙xγk converges weakly in L2 to ˙x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The properties of the sets C(k) and the role of the sequence (ρk)k are estblished in [33, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We enlist here the items that deem important for this paper when constructing the initial constraint set for the approximating problems (Pγk) and (Pγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7 [33] Under (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1)-(A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3), the following assertions hold: (i) For all k, the set C(k) ⊂ int C and is compact, and, for k sufficiently large, bdry C(k) = {x ∈ Rn : ψ(x) = −αk} and int C = {x ∈ Rn : ψ(x) < −αk};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (C(k))k is a nondecreasing sequence whose Painlev´e-Kuratowski limit is C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) There exist ro > 0 and ¯k ∈ N such that � C ∩ ¯Bro(c) � − ρk ∇ψ(c) ∥∇ψ(c)∥ ⊂ int C(k), ∀k ≥ ¯k and ∀c ∈ bdry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (19) (iii) For c ∈ int C, there exist ˆkc ∈ N and ˆrc > 0 satisfying ¯Bˆrc(c) ⊂ int C(ˆkc) ⊂ int C(k), ∀k ≥ ˆkc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (20) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8 From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7, it follows that for any c ∈ C, there exists a sequence (ck)k such that, for k large enough, ck ∈ int C(k) and ck −→ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Indeed, for c ∈ bdry C, take ck := c − ρk ∇ψ(c) ∥∇ψ(c)∥ for all k, and for c ∈ int C, take ck = c for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The following theorem will be used repeatedly in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It is a special case of [33, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 & Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It provides a sufficient condition for the uniform limit x of the solution xγk of (Dγk) to be a solution of (D), and it connects the multiplier function ξ corresponding to x, via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5, to the positive continuous penalty multiplier ξγk, associated with xγk and defined by ξγk(·) := γkeγkψ(xγk(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (21) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9 ((Dγk)k & ξγk approximate (D) & ξ) Assume that (A1)-(A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let xγk be the solution of (Dγk) corresponding to (cγk, uγk), as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6, and x ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) be its uniform limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then, the following statements are valid : Control Space for Strong Convergence of Continuous Approximation 11 (i) The sequence (ξγk)k admits a subsequence, we do not relabel, that converges weakly in L2 to a nonnegative function ξ ∈ L2 supported on I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) If for some u ∈ U, the sequence uγk(t) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t −−−−→ u(t), then x is the unique solution of (D) corresponding to (x0, u), and (x, u, ξ) satisfies equations (12)-(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In particular, ξ ∈ L∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) and is supported on I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='10 Note that when establishing Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9(ii) in [33], the arguments used to prove that (x, u, ξ) satisfies (12) are independent of having ξγk defined through (21), and hence, this proof is valid for ξγk being any sequence of L2-functions converging weakly in L2 to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, we have that (x, u, ξ) satisfies (12) when- ever (xj, uj, ξj)j is a sequence solving (12) with xj converging uniformly to x, uj(t) converging pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' to u(t), and ξj converging weakly in L2 to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The following result is extracted from [33, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1], in which more properties are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It reveals the significance of initiating in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9 the solutions xγk of (Dγk) from the subset C(k), defined in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11 (xγk ∈ C(k), ˙xγk & ξγk bounded) Assume (A1)-(A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let (cγk)k be a sequence such that cγk ∈ C(k), for k sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then there exists ko ∈ N such that for all sequences (uγk)k in U and for all k ≥ ko, the solution xγk of (Dγk) corresponding to (cγk, uγk) satisfies: (i) xγk(t) ∈ C(k) ⊂ int C for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) 0 ≤ ξγk(t) ≤ 2 ¯ M η for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iii) ∥ ˙xγk(t)∥ ≤ ¯ M + 2 ¯ M ¯ Mψ η for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The next result is a simplified version of [33, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It is the converse of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9, as it confirms that any given solution of (D) is approximated by a solution of (Dγk) that remains in the interior of C and enjoys all the properties displayed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='12 (Solutions of (D) are approximated by sequences in C(k)) Assume that (A1)-(A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let ¯x be the solution of (D) corresponding to (¯x(0), ¯u) ∈ C0 × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consider (¯cγk)k the sequence in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8 that converges to c := ¯x(0), and ¯xγk the solution of (Dγk) corresponding to (¯cγk, ¯u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then, there exists ˆko ∈ N such that ¯xγk and its associated ¯ξγk via (21) satisfy the conclusions (i)-(iii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11 for all k ≥ ˆko, and the following holds true: The sequence ¯xγk admits a subsequence, we do not relabel, that converges uniformly to ¯x, the corresponding subsequence for ¯ξγk converges weakly in L2 to some ¯ξ ∈ L∞, and (¯x, ¯u, ¯ξ) satisfies (12)-(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' That is, ¯ξ is the unique function corresponding to (¯x, ¯u) via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 4 Main results This section consists of the main results of this paper, namely, the strong approximation of (D) by (Dγk) whenever the control is W 1,2 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2), an existence theorem for an optimal solution of (P) (Theorem 12 Control Space for Strong Convergence of Continuous Approximation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3), a strong converging continuous approximation for (P) (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6), and nonsmooth necessary optimality conditions in the form of weak-Pontryagin- type maximum principle (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 (Dγk) strongly approximates (D) with W 1,2-controls The following theorem constitutes the backbone of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It shows that, when the underlying control space is W (defined in (1)), the velocities ˙xγk and the functions ξγk corresponding to the approximating sequence xγk in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11, converge strongly in L2 to, respectively, ˙x and ξ, the functions obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The proof of this theorem is postponed to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 (Strong convergence of the velocity sequence ˙xγk) Let the assumptions (A1)-(A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consider a sequence xγk solving (Dγk) for some (cγk, uγk), where cγk ∈ C, cγk −→ x0 ∈ C0, uγk ∈ W, and (∥ ˙uγk∥2)k is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Denote by (x, ξ) the pair in W 1,2 × L2 obtained via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9(i) such that a subsequence (not relabeled ) of (xγk, ξγk) has xγk converging uniformly in the set C to x and ( ˙xγk, ξγk) converging weakly in L2 to ( ˙x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then, the following hold: (i) There exist a subsequence (not relabeled ) of uγk, and u ∈ W such that (xγk, uγk) converges uniformly to (x, u), and ( ˙xγk, ˙uγk, ξγk) converges weakly in L2 to ( ˙x, ˙u, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The function x is the unique solution to (D) corresponding to (x0, u), and (x, u, ξ) satisfies (12)-(13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In particular, ξ ∈ L∞ and is supported on I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) Assume that cγk ∈ C(k), for k large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then, in addition to the conclu- sions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11, the following holds: The sequence ( ˙xγk, ξγk) is in W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W 2,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), has uniform bounded variations, and admits a subsequence, not relabeled, that converges pointwise, and hence, strongly in L2 to ( ˙x, ξ), with ˙x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) and ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In this case, (12)-(13) hold for all t ∈ [0, 1], and xγk −→ x strongly in the norm topology of W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1(ii) to ¯cγk, uγk := ¯u, ¯xγk, and ¯ξγk, the function associated to ¯xγk via (21), we obtain the following corollary that shows how the results in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='12 are improved when W 1,2-controls are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2 ((Dγk)k strongly approximates (D)) If, in addition to the assump- tions of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='12, we have that ¯x solves (D) for ¯u ∈ W (not only in U), then ¯ξγk, therein, converges pointwise to ¯ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) with ¯ξ satisfying (52), and ¯xγk, therein, converges to ¯x strongly in the norm topology of W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, (¯x, ¯u, ¯ξ) satisfies (12)-(13) for all t ∈ [0, 1], and ˙¯x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2 Existence of optimal solution for (P ) Parallel to [6, 8, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1], where a discretization technique is used, the following existence theorem of an optimal solution for the problem (P) is established based on Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Control Space for Strong Convergence of Continuous Approximation 13 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3 (Existence of solution for (P)) Assume hypotheses (A1)-(A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3), g : Rn × Rn → R ∪ {∞} is lower semicontinuous, and that a minimizing sequence (xj, uj) for (P) exists such that (∥ ˙uj∥2)j is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Suppose that (P) has at least one admissible pair (yo, vo) with (yo(0), yo(1)) ∈ dom g, then the problem (P) admits a global optimal solution (¯x, ¯u) such that, along a subsequence, we have xj strongly −−−−−−−−−−→ W 1,2([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) ¯x, uj uniformly −−−−−−−−→ C([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) ¯u, and ˙uj weakly −−−−−−−−−→ L2([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) ˙¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Proof Given that (P) has an admissible pair (yo, vo) with (yo(0), yo(1)) ∈ dom g, then inf(x,u)(P) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As g is lower semicontinuous and all admissible solutions of (P) satisfy (x(0), x(1)) ∈ C0 × (C1 ∩ C), which is compact, we deduce that inf(x,u)(P) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, being admissible for (P), the minimizing sequence (xj, uj)j satisfies (D) with xj(1) ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, using that the sequence (∥ ˙uj∥2)j is bounded, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 implies the existence of (¯x, ¯u) ∈ W 1,∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W satisfying (D) and ¯x(1) ∈ C1, with (xj, uj) converges uniformly to (¯x, ¯u), ( ˙xj)j converges strongly in L2 to ˙¯x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), and ˙uj converges weakly in L2 to ˙¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, (¯x, ¯u) is admissible for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Owed to the lower semicontinuity of g and to (¯x, ¯u) being the uniform limit of the minimizing sequence (xj, uj)j, the optimality of the pair (¯x, ¯u) for the problem (P) follows readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3 Continuous approximation for (P ) On the journey of seeking for an optimal process (¯x, ¯u) of (P) a continuous approximations consisting of optimal solutions for properly-designed standard control problems, it is important that the convergence to (¯x, ¯u) be strong in the norm topology of the considered space, namely, the space W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn)×W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2 already answered this question for the W 1,2-strong approxima- tion of a solution (¯x, ¯u) of (D) by solutions of (Dγk), in which the same control ¯u is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' However, ¯u may not necessarily be optimal for approximating optimal control problems over (Dγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In this subsection, we approximate the problem (P) by a certain sequence of optimal control problems over (Dγk) with special initial and final state endpoints constraints (C0(k) ⊂ C(k) and C1(k) in a band around C1), and with an objective function particularly crafted so that an optimal control, uγk, exists and has (∥ ˙uγk∥2)k uniformly bounded, and hence, the strong convergence of the optimal state velocities shall be deduced from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The necessary optimality conditions for (P) are then established by taking the limit of the optimality conditions for the corresponding approximating problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For given δ > 0 and z ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rs), we define the projection on Rs of the closed δ-tube around z by ¯Bδ(z) := � t∈[0,1] ¯Bδ(z(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let (¯x, ¯u) ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W be a W 1,2-local minimizer for (P) with associated δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We fix δo > 0 such that δo ≤ � min{ˆr¯x(0), δ} if ¯x(0) ∈ int C, min{ro, δ} if ¯x(0) ∈ bdry C, 14 Control Space for Strong Convergence of Continuous Approximation where ro > 0 is the constant in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7(ii), and ˆr¯x(0) > 0 with ˆk¯x(0) ∈ N are the constants in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7(iii) corresponding to c := ¯x(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In the remaining part below, we will assume that f satisfies the following local version of (A1): ∃ ˜ρ > 0 such that f is Lipschitz on [C ∩ ¯Bδ(¯x)] × [(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (∗) Note that under the assumption (∗), the function f can be extended to a globally Lipschitz function ˜f : Rn × Rm −→ R by applying [22, Theorem 1] to each component of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since in the rest of this section we only consider local optimality notions, then, without loss of generality, we shall use the function f instead of ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, when in this section f is assumed to satisfy (∗), it is implied that f also satisfies assumption (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We proceed to suitably-formulate a sequence of approximating prob- lems (Pγk) and show that its optimal solutions strongly converges in W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W to the W 1,2-local minimizer (¯x, ¯u) of (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This nat- urally requires the domain of the approximating problem (Pγk) to be in W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The initial state constraint is taken to be x(0) ∈ C0(k), where C0(k) is the sequence of sets defined by C0(k) := � C0 ∩ ¯Bδo(¯x(0)) , ∀k ∈ N, if ¯x(0) ∈ int C, � C0 ∩ ¯Bδo(¯x(0)) � − ρk ∇ψ(¯x(0)) ∥∇ψ(¯x(0))∥, ∀k ∈ N, if ¯x(0) ∈ bdry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (22) and the final state constraint is x(1) ∈ C1(k), where C1(k) := �� C1 ∩ ¯Bδo(¯x(1)) � − ¯x(1) + ¯xγk(1) � ∩ C, k ∈ N, in which ¯xγk is the solution of (Dγk) corresponding to (¯cγk, ¯u), where ¯cγk in C0(k) ∩ int C(k), for k large, and is defined via Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8 for c := ¯x(0), that is, ¯ck := � ¯x(0), ∀k ∈ N, if ¯x(0) ∈ int C, ¯x(0) − ρk ∇ψ(¯x(0)) ∥∇ψ(¯x(0))∥, ∀k ∈ N, if ¯x(0) ∈ bdry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that C0(k) and C1(k) are closed, for k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, as ργk −→ 0, we have ¯cγk −→ ¯x(0), Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='12 yields that the sequence ¯xγk converges in C uniformly to ¯x, and hence, ¯xγk(1) −→ ¯x(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Add to this that in C0(k), ρk −→ 0, then, for a fixed ˜ρ > 0, we have that, for k sufficiently large, Ci(k) ⊂ � � Ci ∩ ¯Bδ(¯x(i)) � + ˜ρ ¯B � ∩ C � �� � ˜ Ci(δ) , for i = 0, 1, (23) and lim k→∞ C0(k) = C0 ∩ ¯Bδo(¯x(0)) & lim k→∞ C1(k) = C ∩ C1 ∩ ¯Bδo(¯x(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (24) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4 Notice that we can show that, for k large enough, we have C0(k) ⊂ C(k), and hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11, any solution of (Dγk) corresponding to (cγk, uγk) with cγk ∈ C0(k) and uγk ∈ U, satisfies the conditions (i)-(iii) of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Indeed: Control Space for Strong Convergence of Continuous Approximation 15 For ¯x(0) ∈ int C, use that δo ≤ ˆr¯x(0) and (20) we get ¯Bˆr¯x(0)(¯x(0)) ⊂ int C(k) ⊂ C(k), ∀k ≥ ˆk¯x(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This gives that C0(k) := C0 ∩ ¯Bδo(¯x(0)) ⊂ ¯Bˆr¯x(0)(¯x(0)) ⊂ C(k), ∀k ≥ ˆk¯x(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For ¯x(0) ∈ bdry C, use that δo ≤ ro and C0(k) is the nonempty set defined by the second equation of (22), to get via (19) that C0(k) ⊂ int C(k) ⊂ C(k), ∀k ≥ ¯k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5 For c ∈ C0(k) and d ∈ C1(k), the evaluation of the normal cones NL C0(k)(c) and NL C1(k)(d) in terms of NL C0 and NL C1, respectively, is obtained using the local property of the limiting normal cone as the following NL C0(k)(c) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 NL C0(c), if ¯x(0) ∈ int C, and c ∈ Bδo(¯x(0)), NL C0 � c + ρk ∇ψ(¯x(0)) ∥∇ψ(¯x(0))∥ � , if ¯x(0) ∈ bdry C, and � c + ρk ∇ψ(¯x(0)) ∥∇ψ(¯x(0))∥ � ∈ Bδo(¯x(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (25) NL C1(k)(d) = NL C1(d + ¯x(1) − ¯xγk(1)), ∀d ∈ (int C) ∩ Bδo(¯x(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (26) We introduce the following sequence of approximating problems: (Pγk): Minimize J(x, y, z, u) := g(x(0), x(1)) + 1 2 � ∥u(0) − ¯u(0)∥2 + z(1) + ∥x(0) − ¯x(0)∥2� over (x, y, z, u) ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) × AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) × W such that \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ( ˜Dγk) \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 ˙x(t) = fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', ˙y(t) = ∥ ˙x(t) − ˙¯x(t)∥2, t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', ˙z(t) = ∥ ˙u(t) − ˙¯u(t)∥2, t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', (x(0), y(0), z(0)) ∈ C0(k) × {0} × {0}, x(t) ∈ ¯Bδ(¯x(t)) and u(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), ∀t ∈ [0, 1], (x(1), y(1), z(1)) ∈ C1(k) × [−δ, δ] × [−δ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6 and the constraints on u(·) confirm that (Pγk)k is actually equivalent to having therein (x, u) ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Now we are ready to state our continuous approximation result, which is parallel to the corresponding result in [6–8, 14], where discrete approximations are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The proof of this approximation result is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6 ((Pγk) approximates (P)) Let (¯x, ¯u) be a W 1,2-local minimizer (P) with associated ¯ξ ∈ L∞ via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Assume that (A2)-(A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3) hold, g is contin- uous on ˜C0(δ) × ˜ C1(δ), and for some ˜ρ > 0, f is Lipschitz on [C ∩ ¯Bδ(¯x)] × [(U + ˜ρ ¯B)∩ ¯Bδ(¯u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then for k sufficiently large, the problem (Pγk) has an optimal solution 16 Control Space for Strong Convergence of Continuous Approximation (xγk, yγk, zγk, uγk) such that, for ξγk defined in (21), we have, along a subsequence, we do not relabel, that (xγk, uγk) strongly −−−−−−→ W 1,2×W (¯x, ¯u), (yγk, zγk) strongly −−−−−−−−−−−−−−→ W 1,1([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+×R+) (0, 0), ξγk strongly −−−−−−−−−→ L2([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) ¯ξ, all the conclusions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11 hold, including that xγk(t) ∈ int C for all t ∈ [0, 1], and for all k sufficiently large, xγk(i) ∈ � � Ci ∩ ¯Bδo(¯x(i)) � + ˜ρB � ∩ (int C) ⊂ int ˜Ci(δ), for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, ˙¯x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), ¯ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), and (12)-(13) are valid at (¯x, ¯u, ¯ξ) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We proceed to rewrite the problems (Pγk) as an optimal control prob- lem with state constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Given (¯x, ¯u) ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W a W 1,2-local minimizer for (P), for ¯v := ˙¯u, (Pγk) is reformulated in the following way: (Pγk): Minimize g(x(0), x(1)) + 1 2 � ∥u(0) − ¯u(0)∥2 + z(1) + ∥x(0) − ¯x(0)∥2� over (x, y, z, u) ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn)×AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R)×AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R)×AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) and measurable functions v: [0, 1] −→ Rm such that \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˙x(t) = fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', ˙u(t) = v(t), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', ˙y(t) = ∥fΦ(x(t), u(t)) − γkeγkψ(x(t))∇ψ(x(t)) − ˙¯x(t)∥2, t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', ˙z(t) = ∥v(t) − ¯v(t)∥2, t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', x(t) ∈ ¯Bδ(¯x(t)) and u(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), ∀t ∈ [0, 1], (x(0), u(0), y(0), z(0)) ∈ C0(k) × Rm × {0} × {0}, (x(1), u(1), y(1), z(1)) ∈ C1(k) × Rm × [−δ, δ] × [−δ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In the following proposition we apply to the above sequence of refor- mulated problems (Pγk), the nonsmooth Pontryagin maximum principle for optimal control problems with multiple state constraints (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', [37, page 331] and [37, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='332]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For this purpose, (x, y, z, u) is the state function in (Pγk) and v is the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, (xγk, yγk, zγk, uγk) is the optimal state, where (xγk, uγk) is obtained from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6, yγk(t) := � t 0 ∥ ˙xγk(s) − ˙¯x(s)∥2 ds, zγk(t) := � t 0 ∥ ˙uγk(s) − ˙¯u(s)∥2 ds, and vγk = ˙uγk is the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, the function f(·, ·) is required to be Lipschitz near (xγk, uγk), which follows from (∗), since xγk(t) ∈ int C and (xγk, uγk) converges uniformly to (¯x, ¯u) (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Furthermore, as the objective function g must be Lipschitz near (xγk(0), xγk(1)), we introduce the following local assumption on g in which ˜C0(δ) and ˜C1(δ) are defined in (23): ∃ ˜ρ > 0 such that g is Lipschitz on ˜C0(δ) × ˜ C1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, the following constraint qualification property (CQ) is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For a given multifunction F : [0, 1] ⇒ Rm, with nonempty and closed values, and for h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' F), that is, h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) and satisfies h(t) ∈ F(t) for all t ∈ [0, 1], we say that F(·) satisfies the constraint qualification at h if Control Space for Strong Convergence of Continuous Approximation 17 (CQ) conv ( ¯N L F (t)(h(t))) is pointed for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Here, ¯N L F (t)(y) stands for the graphical closure at (t, y) of the multifunction (t, y) �→ N L F (t)(y), that is, the graph of ¯N L F (·)(·) is the closure of the graph of N L F (·)(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For more information about the (CQ) property, see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It is worth noting that in [6–8], where W 1,2-controls are employed, the control sets U(t) are assumed to be Rm, for all t ∈ [0, 1], and hence, the (CQ) property is trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7 (Maximum Principle for approximating problems (Pγk)) Let (¯x, ¯u) be a W 1,2-local minimizer for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Assume that (A2)-(A4) hold, and for some ˜ρ > 0, f is Lipschitz on [C ∩ ¯Bδ(¯x)] × [(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)] and g is Lipschitz on ˜C0(δ) × ˜ C1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consider the optimal sequence (xγk, yγk, zγk, uγk, vγk) for (Pγk) obtained via Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If for k sufficiently large, U(·) satisfies the constraint qualification (CQ) at uγk, then for k large enough, there exist λγk ≥ 0, pγk ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), qγk ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm), Ωγk ∈ NBV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm), µoγk ∈ C⊕([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm), and a µoγk- integrable function βγk : [0, 1] −→ Rm such that Ωγk(t) = � [0,t] βγk(s)µoγk(ds), for all t ∈ (0, 1], and: (i) (The nontriviality condition) For all k ∈ N, we have ∥pγk(1)∥ + ∥qγk∥∞ + ∥µo γk∥T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' + λγk = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) (The adjoint equation) For a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], � ˙pγk(t) ˙qγk(t) � ∈ − � ∂(x,u)fΦ(t, xγk(t), uγk(t)) �T pγk(t) + � γkeγkψ(xγk(t))∂2ψ(xγk(t))pγk(t) 0 � (27) + � γ2 keγkψ(xγk(t))∇ψ(xγk(t))⟨∇ψ(xγk(t)), pγk(t)⟩ 0 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iii) (The transversality equation) (pγk(0), −pγk(1)) ∈ λγk∂Lg(xγk(0), xγk(1))+ �� λγk(xγk(0)−¯x(0))+NL C0(k)(xγk(0)) � ×NL C1(k)(xγk(1)) � , and qγk(0) = λγk(uγk(0) − ¯u(0)), −qγk(1) = Ωγk(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iv) (The maximization condition) For a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], max v∈Rm � ⟨qγk(t) + Ωγk(t), v⟩ − λγk 2 ∥v − ˙¯u(t)∥2 � is attained at ˙uγk(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (v) (The measure properties) supp {µo γk} ⊂ � t ∈ [0, 1] : (t, uγk(t)) ∈ bdry Gr � U(t) ∩ ¯Bδ(¯u(t)) �� , and βγk(t) ∈ ∂> u d(uγk(t), U(t) ∩ ¯Bδ(¯u(t))) µo γk a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', with ∂> u d(uγk(t), U(t) ∩ ¯Bδ(¯u(t))) ⊂ � conv ¯ NL U(t)∩ ¯ Bδ(¯u(t))(uγk(t)) ∩ � ¯B \\ {0} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 18 Control Space for Strong Convergence of Continuous Approximation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4 Necessary optimality conditions for (P ) The main result of this subsection is the following theorem which provides necessary optimality conditions for the W 1,2-local minimizer, (¯x, ¯u), of (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The following notations are used in the statement of the theorem: ∂ℓϕ and ∂2 ℓ ϕ stand, respectively, for the extended Clarke generalized gradient and the extended Clarke generalized Hessian of ϕ defined on C via (4) & (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ∂(x,u) ℓ f(·, ·) is the extended Clarke generalized Jacobian of f(·, ·) defined on � C ∩ ¯Bδ(¯x(t)) � × � (U(t) + ˜ρ ¯B) ∩ ¯Bδ(¯u(t)) � via (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ∂2 ℓ ψ is the Clarke generalized Hessian relative to int C of ψ, defined via (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ∂L ℓ g is the limiting subdifferential of g relative to int � ˜C0(δ)× ˜C1(δ) � , defined via (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8 (Necessary optimality conditions for (P)) Let (¯x, ¯u) be a W 1,2-local minimizer for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let ¯ξ ∈ L∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) be the function supported on I0(¯x) and associated to (¯x, ¯u) via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Assume that (A2)-(A4) hold, U(·) satisfies the constraint qualification (CQ) at ¯u, and for some ˜ρ > 0, f is Lipschitz on [C∩¯Bδ(¯x)]× [(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)] and g is Lipschitz on ˜C0(δ) × ˜ C1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then ˙¯x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) and ¯ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), and there exist λ ≥ 0, an adjoint vector ¯p ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), a finite signed Radon measure ¯ν on [0, 1] supported on I0(¯x), L∞-functions ¯ζ(·), ¯θ(·) and ¯ϑ(·) in Mn×n([0, 1]), an L∞-function ¯ω(·) in Mn×m([0, 1]), such that for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', � (¯ζ(t), ¯ω(t)), ¯θ(t), ¯ϑ(t) � ∈ ∂(x,u) ℓ f(¯x(t), ¯u(t)) × ∂2 ℓ ϕ(¯x(t)) × ∂2 ℓ ψ(¯x(t)), and the following hold: (i) (The admissible equation) (a) ˙¯x(t) = f(¯x(t), ¯u(t)) − ∇ℓ ϕ(¯x(t)) − ¯ξ(t)∇ψ(¯x(t)), ∀t ∈ [0, 1], (b) ψ(¯x(t)) ≤ 0, ∀t ∈ [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) (The nontriviality condition) ∥¯p(1)∥ + λ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iii) (The adjoint equation) For any h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), we have � [0,1] ⟨h(t), d¯p(t)⟩ = � 1 0 � h(t), � ¯θ(t) − ¯ζ(t)T� ¯p(t) � dt + � 1 0 ¯ξ(t) � h(t), ¯ϑ(t)p(t) � dt + � [0,1] ⟨h(t), ∇ψ(¯x(t))⟩d¯ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iv) (The complementary slackness conditions) (a) ¯ξ(t) = 0, ∀t ∈ I-(¯x), (b) ¯ξ(t)⟨∇ψ(¯x(t), ¯p(t)⟩ = 0, ∀t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (v) (The transversality equation) (¯p(0), −¯p(1)) ∈ λ∂L ℓ g(¯x(0), ¯x(1)) + � NL C0(¯x(0)) × NL C1(¯x(1)) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (vi) (The weak maximization condition) ¯ω(t)T¯p(t) ∈ conv ¯ NL U(t)∩ ¯ Bδ(¯u(t))(¯u(t)), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If in addition there exist εo > 0 and r > 0 such that U(t) ∩ ¯Bεo(¯u(t)) is r-prox-regular for all t ∈ [0, 1], then we have max �� ¯ω(t)T¯p(t), u � − ∥¯ω(t)T ¯p(t)∥ min{εo,2r} ∥u − ¯u(t)∥2 : u ∈ U(t) � is attained at ¯u(t) for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Control Space for Strong Convergence of Continuous Approximation 19 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9 Condition (vi) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8 admits simplified forms when U(·) possesses extra properties: If U(t) is r-prox-regular for all t ∈ [0, 1], then taking εo −→ ∞, the maximization condition (v) reduces to max �� ¯ω(t)T¯p(t), u � − ∥¯ω(t)T ¯p(t)∥ 2r ∥u − ¯u(t)∥2 : u ∈ U(t) � is attained at ¯u(t) for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If U(t)∩ ¯Bεo(¯u(t)) is convex for all t ∈ [0, 1], then taking r −→ ∞, the maximization condition (v) reduces to max �� ¯ω(t)T¯p(t), u � − ∥¯ω(t)T ¯p(t)∥ εo ∥u − ¯u(t)∥2 : u ∈ U(t) � is attained at ¯u(t) for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If U(t) is convex for all t ∈ [0, 1], then taking both εo −→ ∞ and r −→ ∞, the maximization condition (v) reduces to max �� ¯ω(t)T¯p(t), u � : u ∈ U(t) � is attained at ¯u(t) for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6 produces a subsequence of (γk)k, we do not relabel, and a corresponding sequence (xγk, yγk, zγk, uγk)k, with associated (ξγk)k defined via (21), such that For each k, the quadruplet (xγk, yγk, zγk, uγk) is optimal for (Pγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (xγk, uγk) strongly −−−−−−→ W 1,2×W (¯x, ¯u), (yγk, zγk) strongly −−−−−−−−−−−−−−→ W 1,1([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+×R+) (0, 0), ξγk strongly −−−−−−−−−→ L2([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) ¯ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ˙¯x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), ¯ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), and (12)-(13) are valid at (¯x, ¯u, ¯ξ) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' All the conclusions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11 hold, including (xγk)k is uniformly Lipschitz and xγk(t) ∈ int C for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For all k, we have xγk(i) ∈ � � Ci ∩ ¯Bδo(¯x(i)) � + ˜ρB � ∩ (int C) ⊂ int ˜Ci(δ), for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In order to apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7, we shall show that the constraint qualification (CQ) that holds for U(·) at ¯u, also holds true at uγk, for k large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Indeed, if this is false, then, by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2(i), there exist an increasing sequence (kn)n in N and a sequence tn ∈ [0, 1] such that tn −→ to ∈ [0, 1] and 0 ∈ ∂> u dU(tn, uγkn (tn)), ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (28) The continuity of ¯u and the uniform convergence of uγkn to ¯u yield that the sequence (uγkn (tn))n converges to ¯u(to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, using that the multifunction (t, x) �→ ∂> u dU(t, x) has closed values and a closed graph, we conclude from (28) that 0 ∈ ∂> u dU(to, ¯u(to)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This contradicts that the constraint qualification is satisfied by U(·) at ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, for k sufficiently large, U(·) satisfies the constraint qualification (CQ) at uγk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7, there exist a subsequence of (γk)k, we do not rela- bel, and corresponding sequences pγk, qγk µγk and λγk satisfying conditions (i)-(v) therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using (27), (10), and that for all t ∈ [0, 1] we have (xγk(t), uγk(t)) ∈ int �� C ∩ ¯Bδ(¯x(t)) � × � (U(t)+ ˜ρ ¯B) ∩ ¯Bδ(¯u(t)) �� , 20 Control Space for Strong Convergence of Continuous Approximation we obtain sequences ζγk, θγk and ϑγk in Mn×n([0, 1]) and ωγk in Mn×m([0, 1]) such that, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], (ζγk(t), ωγk(t)) ∈ ∂(x,u) ℓ f(xγk(t), uγk(t)), (θγk(t), ϑγk(t)) ∈ ∂2 ℓ ϕ(xγk(t)) × ∂2 ℓ ψ(xγk(t)), ˙pγk(t) = (θγk(t) − ζγk(t))Tpγk(t) + γkeγkψ(xγk(t))ϑγk(t) pγk(t) (29) + γ2 keγkψ(xγk(t))∇ψ(xγk(t))⟨∇ψ(xγk(t)), pγk(t)⟩, and ˙qγk(t) = −(ωγk(t))Tpγk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (30) Note that for each k, the functions pγk, ˙pγk, qγk, ˙qγk, xγk, uγk and ¯u are measurable on [0, 1], and the multifunctions ∂(x,u) ℓ f(·, ·), ∂2 ℓ ϕ(·), and ∂2 ℓ ψ(·) are measurable and have closed graphs with nonempty, compact, and convex values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using (A1), (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1), and (A3), the Filippov measurable selection theorem (see [37, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='13]) yields that we can assume the measurability of the functions ζγk(·), θγk(·), ϑγk(·) and ωγk(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, these sequences are uniformly bounded in L∞, as ∥(ζγk, ωγk)∥∞ ≤ M, ∥θγk∥∞ ≤ K and ∥ϑγk∥∞ ≤ 2Mψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Construction of ¯ξ, the admissible equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6, we have that the triplet (¯x, ¯y, ¯ξ) satisfies (12) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, for all t ∈ [0, 1] we have ˙¯x(t) = fΦ(¯x(t), ¯u(t)) − ¯ξ(t)∇ψ(¯x(t)) = f(¯x(t), ¯u(t)) − ∇Φ(¯x(t)) − ¯ξ(t)∇ψ(¯x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since ∇Φ(x) = ∂ℓϕ(x) = ∇ℓ ϕ(x) for all x ∈ C, we obtain that ˙¯x(t) = f(¯x(t), ¯u(t)) − ∇ℓ ϕ(¯x(t)) − ¯ξ(t)∇ψ(¯x(t)), ∀t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, since ¯x takes values in C, we have ψ(¯x(t)) ≤ 0, ∀t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Construction of ¯p, ¯ζ, ¯θ, ¯ϑ, ¯ω, ¯ν, and the adjoint equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For the construction of ¯p, ¯θ, ¯ϑ and ¯ν, see Steps 2-4 in the proof of [38, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that the uniform boundedness of pγk(1) established and used in Step 2 of the proof of [38, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1], is easily deduced here from the nontriviality condition of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We also note that, similarly to Step 2 of the proof of [38, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1], pγk has a uniformly bounded variation, and hence, Helly first theorem implies that pγk admits a pointwise convergent subsequence whose limit ¯p is also of bounded variation and satisfies, for some M1 > 0, the following ∥¯p∥∞ ≤ M1∥¯p(1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (31) Using Helly second theorem we obtain that for all h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), lim k→∞ � [0,1] ⟨h(t), ˙pγk(t)⟩ dt = � [0,1] ⟨h(t), d¯p(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (32) Identically to Steps 2-4 in the proof of [38, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1], we also have � 1 0 ⟨h(t), θγk(t) pγk(t)⟩ dt −→ � 1 0 � h(t), ¯θ(t) ¯p(t) � dt, (33) � 1 0 ξγk(t) ⟨h(t), ϑγk(t) pγk(t)⟩ dt −→ � 1 0 ¯ξ(t) ⟨h(t), ¯ϑ(t) ¯p(t)⟩ dt, (34) lim k−→∞ � 1 0 ⟨h(t), ∇ψ(xγk(t))⟩ γkξγk(t) ⟨∇ψ(xγk(t)), pγk(t)⟩ dt = � [0,1] ⟨h(t), ∇ψ(¯x(t))⟩ d¯ν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (35) Control Space for Strong Convergence of Continuous Approximation 21 We proceed to construct the two functions ¯ζ and ¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that the construction of ζ done in Step 2 of the proof of [38, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1] cannot be used here, since the closed graph hypothesis on the multifunction (x, u) �→ ∂xf(x, u) is required there, but it is not assumed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As the sequence (ζγk, ωγk)k is uniformly bounded in L∞, it has a subsequence, we do not relabel, that converges weakly in L1 to some (¯ζ, ¯ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using that the multifunction (x, u) �→ ∂(x,u) ℓ f(x, u) has closed graph with nonempty, compact and convex values, [10, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='39] implies that, for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', (¯ζ(t), ¯ω(t)) ∈ ∂(x,u) ℓ f(¯x(t), ¯u(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since (pγk)k is uniformly bounded in L∞ and converges pointwise to ¯p, we conclude that ζT γkpγk weakly −−−−−→ L1 ¯ζT¯p and ωT γkpγk weakly −−−−−→ L1 ¯ωT¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (36) Hence, for all h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), � 1 0 ⟨h(t), ζγk(t)pγk(t)⟩ dt −→ � 1 0 � h(t), ¯ζ(t)¯p(t) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (37) Thus, from (29) and (32)-(37), we conclude that the adjoint equation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8 holds, and it coincides with the adjoint equation of [38, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The complementary slackness conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The part (a) follows from the equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The part (b) follows from the uniform boundedness of ∥γkξγk(·)⟨∇ψ(xγk(·)), pγk(·)⟩∥1 established in [38, Equation (97)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' More details can be found in Step 6 of [33, Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Construction of λ and the transversality equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Form the transversality condition of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7, there exist υγk ∈ NL C0(k)(xγk(0)), χγk ∈ NL C1(k)(xγk(1)) and (aγk, bγk) ∈ ∂Lg(xγk(0), xγk(1)) such that pγk(0) = λγkaγk + λγk(xγk(0)− ¯x(0)) + υγk, −pγk(1) = λγkbγk + χγk, (38) and the following properties hold: ∥(aγk, bγk)∥ ≤ Lg, where Lg is the Lipschitz constant of g over ˜C0(δ)× ˜ C1(δ), and ∥λγk∥ ≤ 1 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The latter inequality gives a subsequence, we do not relabel, such that λγk −→ λ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Due to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6, we have, for k large enough, (xγk(0), xγk(1)) ∈ int ( ˜C0(δ) × ˜C1(δ)), and (xγk(0), xγk(1)) −→ (¯x(0), ¯x(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We have pγk(0) −→ ¯p(0) and pγk(1) −→ ¯p(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Owing to (26), in which d := xγk(1) ∈ � C1(k)∩(int C)∩Bδo(¯x(1)) � for k sufficiently large, we have χγk ∈ NL C1(k)(xγk(1)) = NL C1 (xγk(1) + ¯x(1) − ¯xγk(1)), for k large, where, we recall that ¯xγk(1) −→ ¯x(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Owing to (25), in which c := xγk(0) ∈ C0(k) ∩ Bδo(¯x(0)) for k large enough, it follows that: (i) If ¯x(0) ∈ int C, then for k sufficiently large vγk ∈ NL C0(k)(xγk(0)) = NL C0(xγk(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) If ¯x(0) ∈ bdry C, using that xγk(0) −→ ¯x(0) and ρk −→ 0, then for k sufficiently large, � xγk(0) + ρk ∇ψ(¯x(0)) ∥∇ψ(¯x(0))∥ � ∈ Bδo(¯x(0)), and hence, vγk ∈ NL C0(k)(xγk(0)) = NL C0 � xγk(0) + ρk ∇ψ(¯x(0)) ∥∇ψ(¯x(0))∥ � for k large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 22 Control Space for Strong Convergence of Continuous Approximation Therefore, along a subsequence of (γk)k, we do not relabel, we have λγk(aγk, bγk) −→ λ(a, b) ∈ λ∂L ℓ g(¯x(0), ¯x(1)) and λγk(xγk(0)− ¯x(0)) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, taking the limit as k → ∞ in (38), and using (pγk(0), pγk(1)) −→ (¯p(0), ¯p(1)), we obtain that (vγk, χγk) must converge to some (v, χ), as all the other terms in (38) converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The last two bullets, stated above, yield that v ∈ NL C0(¯x(0)) and χ ∈ NL C1(¯x(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consequently, the limit of (38) is equivalent to (¯p(0), −¯p(1)) ∈ λ∂L ℓ g(¯x(0), ¯x(1)) + � NL C0(¯x(0)) × NL C1(¯x(1)) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This terminates the proof of the transversality equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The weak maximization condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' By (30), (36)(b), and the transversality equations of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7, we have that ˙qγk = −(ωγk)Tpγk weakly −−−−−→ L1 −(¯ω)T¯p and qγk(0) = λγk(uγk(0) − ¯u(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (39) The uniform boundedness in L∞ of the sequences (pγk)k and (ωγk)k give that ( ˙qγk)k is uniformly bounded in L∞, asserting the equicontinuity of (qγk)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, the nontriviality condition of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7 gives the uniform boundedness of the sequence (qγk)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, by Arzel`a-Ascoli theorem, the sequence (qγk)k admits a sub- sequence, we do not relabel, that converges uniformly to an absolutely continuous function q satisfying q(0) = 0 (by (39)(b), where λγk −→ λ and uγk(0) −→ ¯u(0) as k −→ ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, up to a subsequence, we also obtain that ˙qγk weakly −−−−−→ L1 ˙q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (40) Hence, (39)(a) and the uniqueness of the L1-weak limit yield that ˙q(t) = −(¯ω(t))T¯p(t), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (41) We proceed to study the convergence of the sequence of NBV -functions, (Ωγk)k, obtained in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The maximization condition (iv), therein, implies that, for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Ωγk(t) = −qγk(t) + λγk( ˙uγk(t) − ˙¯u(t)) � �� � ℓγk(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (42) Without loss of generality, we can assume that (42) is satisfied for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In fact, if λγk = 0, using the transversality conditions of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7 and that Ωγk ∈ NBV [0, 1], we get that Ωγk(0) = −qγk(0) = 0 and Ωγk(1) = −qγk(1), and hence, by the right continuity of Ωγk and the continuity of qγk, (42) is equivalent to Ωγk ≡ −qγk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If, however, λγk > 0, then by modifying the values of ( ˙uγk − ˙¯u) on the set of Lebesgue measure zero, we have (42) satisfed for all t ∈ [0, 1], and hence, ℓγk ∈ BV [0, 1] is right continuous on (0, 1), and satisfies ℓγk(0) = qγk(0) and ℓγk(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Furthermore, since λγk −→ λ and ˙uγk strongly converges in L2 to ˙¯u, the sequence (ℓγk)k strongly converges in L2 to ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We claim that (Ωγk)k, considered as a sequence of continuous linear functionals on C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm), admits a subsequence, we do not relabel, that converges weakly* to −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since Ωγk satisfies (42), where the sequence of absolutely continuous functions (qγk)k converges uniformly to q ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) and, by (40), ( ˙qγk)k converges weakly in L1 to ˙q, then it is equivalent to show that the BV -sequence (ℓγk)k converges in C∗([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The uniform boundedness of the sequence (ℓγk)k shall follow once we show the uniform boundedness of (Ωγk)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For this latter, the nontriviality condition of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7, implies that the sequence (µoγk)k is uniformly bounded, and hence, it has a subsequence, we do not relabel, that converges weakly* to a Control Space for Strong Convergence of Continuous Approximation 23 µo ∈ C⊕([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, by condition (v) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2(i), ∥βγk(t)∥ ≤ 1, except on a set of µoγk-measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, using that Ωγk(t) = � [0,t] βγk(s)µo γk(ds), ∀t ∈ (0, 1], and Ωγk(0) = 0, (43) we obtain that the sequence (Ωγk)k is uniformly bounded, and so is the sequence (ℓγk)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, to get that the bounded BV -sequence (ℓγk)k converges weakly* to 0, by the Banach-Steinhaus theorem in [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='482], it is sufficient to show that lim k−→∞ � 1 0 ⟨h(t), dℓγk(t)⟩ = 0, ∀h ∈ C1([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Fix h ∈ C1([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using an integration by parts and that Ωγk ∈ NBV, we get � 1 0 ⟨h(t), dℓγk(t)⟩ = ⟨h(1), ℓγk(1)⟩ − ⟨h(0), ℓγk(0)⟩ − � 1 0 � ˙h(t), ℓγk(t) � dt = � −⟨h(0), qγk(0)⟩ − � 1 0 � ˙h(t), ℓγk(t) � dt � −−−−−→ k−→∞ 0, since (ℓγk)k strongly converges in L2 to 0, and qγk(0) −→ q(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This terminates the proof of the claim, that is, Ωγk weakly* −−−−−−−−−→ C∗([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (44) By [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 484, #8], we also have that Ωγk(t) −→ −q(t), ∀t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Now define the signed measure µγk(dt) := βγk(t)µoγk(dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' From (43) we have Ωγk(t) = µγk[0, t], ∀t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (45) Using that µoγk weakly* −−−−−→ k−→∞ µo, and that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7(v) holds true, then, by applying [37, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1] to the following data: Aγk(t) := ∂> u d(uγk(t), U(t) ∩ ¯Bδ(¯u(t))) for all t ∈ [0, 1], A(t) := ∂> u d(¯u(t), U(t) ∩ ¯Bδ(¯u(t))) for all t ∈ [0, 1], γγk := βγk, µγk := µoγk and µ0 := µo, we obtain a Borel measurable function β : [0, 1] −→ Rm and µ ∈ C∗([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm) such that µγk weakly* −−−−−→ k−→∞ µ, µ(dt) = β(t)µo(dt) and β(t) ∈ ∂> u d(¯u(t), U(t) ∩ ¯Bδ(¯u(t))) µo a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since uγk converges uniformly to ¯u, and supp {µoγk} satisfies Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7(v), we deduce that supp {µo} ⊂ A := � t ∈ [0, 1] : (t, ¯u(t)) ∈ bdry Gr � U(t) ∩ ¯Bδ(¯u(t)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (46) Adjust β(·) on the set of µo-measure zero to arrange t ∈ A =⇒ β(t) ∈ ∂> u d(¯u(t), U(t) ∩ ¯Bδ(¯u(t))), and hence, using [37, Formula (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='17)], we have β(t) ∈ � conv ¯ NL U(t)∩ ¯ Bδ(¯u(t))(¯u(t)) ∩ � ¯B \\ {0} �� , ∀t ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (47) Thus, by (44) and (45), we obtain that −dq(t) = µ(dt) = β(t)µo(dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using (41), we arrive to − dq(t) = (¯ω(t))T¯p(t)dt = β(t)µo(dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (48) 24 Control Space for Strong Convergence of Continuous Approximation Next, we decompose µo(dt) = m(t)dt+µs(dt) for some nonnegative L1-function m(·) and some nonnegative Borel measure µs totally singular with respect to Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Clearly m(t) = 0, for all t ∈ Ac, and hence, (47) implies that β(t)m(t) ∈ conv ¯ NL U(t)∩ ¯ Bδ(¯u(t))(¯u(t)), ∀t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using (48) we get that (¯ω(t))T¯p(t)dt = β(t)m(t)dt + β(t)µs(dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This gives that (¯ω(t))T¯p(t) = β(t)m(t), for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, ¯ω(t)T¯p(t) ∈ conv ¯ NL U(t)∩ ¯ Bδ(¯u(t))(¯u(t)), ∀t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (49) We proceed to prove that the “In addition” part of the weak maximization con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We assume the existence of εo > 0 and r > 0 such that U(t) ∩ ¯Bεo(¯u(t)) is r-prox-regular for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' From (49) and using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3, we obtain that for all t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' � ¯ω(t)T¯p(t), u − ¯u(t) � ≤ ∥¯ω(t)T ¯p(t)∥ min{εo,2r} ∥u − ¯u(t)∥2, for all u ∈ U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1], � ω(t)Tp(t), u � − ∥¯ω(t)T ¯p(t)∥ min{εo,2r} ∥u − ¯u(t)∥2 ≤ � ¯ω(t)T¯p(t), ¯u(t) � , for all u ∈ U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This terminates the proof of the weak maximization condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The nontriviality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It is sufficient to prove its equivalent condition: ∥¯p(1)∥ + λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Taking the limit as k −→ ∞ in the nontriviality condition of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='7, and using the convergence of ¯pγk(1) to ¯p(1), the uniform convergence of qγk to q, the weak* convergence of µoγk to µo, and the convergence of λγk to λ, we get that 1 = ∥¯p(1)∥ + ∥q∥∞ + ∥µo∥T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (50) We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If ¯p(1) = 0 and λ = 0, by (31) we obtain that ¯p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence (41) yields that ˙q(t) = 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This gives that β(t)µo(dt) = −dq(t) = 0 and q(t) = q(0) + � t 0 ˙q(τ)dτ = 0, ∀t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (51) Since, by (46) and (48), supp {µo} ⊂ A and β(t) ̸= 0 for all t ∈ A, the first equation of (51) yields that µo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, ∥¯p(1)∥ + ∥q∥∞ + ∥µo∥T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' + λ = 0 which contradicts (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This terminates the proof of the nontriviality condition, and then, the proof of the conditions (i)-(vi) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' □ 5 Appendix In this section, we present the proofs of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6, and we establish auxilliary results that are used in different places of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We begin by the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (i): Having a uniform bounded derivative in L2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm), the W 1,2-sequence uγk is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since, by (A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2), the compact sets U(t) are uniformly bounded, then uγk is uniformly bounded in C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm), and hence, Arzel`a-Ascoli theorem asserts that uγk admits a subsequence, we do not relabel, that converges uniformly to an absolutely continuous function u with u(t) ∈ U(t) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As ˙uγk is uniformly bounded in L2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm), then, up to a subsequence, it is weakly convergent in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The boundedness of (uγk(0))k then yields that the L2- weak limit of ˙uγk is ˙u, and whence, u ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The fact that x is the unique solution to (D) corresponding to (x0, u), and the proceeding statements of this part, follow Control Space for Strong Convergence of Continuous Approximation 25 immediately from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii): Now, assume that cγk ∈ C(k) for k ≥ ko, where ko is the rank in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let us first show that (ξγk)k has uniform bounded variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since ψ and ∇ψ are Lipschitz on C and xγk is Lipschitz for k ≥ ko, we deduce that, for k ≥ ko, the function ξγk(·)∇ψ(xγk(·)), where ξγk is defined in (21), is Lipschitz continuous on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Similarly, the Lipschitz property on C ×(U+ ˜ρ ¯B) of f(·, ·) (and then of fΦ(·, ·)) and the fact that (xγk, uγk) is in W 1,∞ × W 1,2, yield that fΦ(xγk(·), uγk(·)) is in W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, ζγk(t) := d dtfΦ(xγk(t), uγk(t)), exists for almost all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' By writing fΦ = (f1 Φ, · · · , fn Φ) T, and using that xγk(t) ∈ int C (for all t ∈ [0, 1]), and uγk(t) ∈ U(t) ⊂ U (for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ), it follows from the proof of [39, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1], that ζi γk(t) ∈ ⟨∂fi Φ(xγk(t), uγk(t)), ( ˙xγk(t), ˙uγk(t))⟩, t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', ∀i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since (∥ ˙uγk∥2)k is assumed to be bounded, and, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11, (∥ ˙xγk∥∞)k is bounded, then the sequence (∥ζγk∥2)k is bounded by some Mζ > 0 that depends on ¯ M, ¯ Mψ, η, and the bound of (∥ ˙uγk∥2)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As fΦ(xγk(·), uγk(·)) ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), the right hand side of (Dγk) yields that ˙xγk is in W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), and so is the function |⟨∇ψ(xγk(·)), ˙xγk(·)⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This also implies that ξγk ∈ W 2,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), due to ˙ξγk(t) = γ2 keγkψ(xγk(t))⟨∇ψ(xγk(t)), ˙xγk(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Next, calculating ¨xγk through (Dγk) in terms of ζγk and ˙xγk, and using the fact that for h ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) we have d dt|h(t)| = � d dth(t) � sign(h(t)) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' t ∈ (0, 1), 2 it follows that there exist measurable functions ϑ1γk and ϑ2γk whose values at t are in ∂2ψ(xγk(t)), for almost all t ∈ [0, 1], such that, for t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', we have d dt|⟨∇ψ(xγk(t)), ˙xγk(t)⟩| = �� ϑ1 γk(t) ˙xγk(t), ˙xγk(t) � + ⟨∇ψ(xγk(t)), ¨xγk(t)⟩ � α(t) � �� � sign(⟨∇ψ(xγk(t)), ˙xγk(t)⟩) = �� ϑ1 γk(t) ˙xγk(t), ˙xγk(t) � + ⟨∇ψ(xγk(t)), ζγk(t) − ξγk(t)ϑ2 γk(t) ˙xγk(t)⟩ � α(t) −⟨γkξγk(t)⟨∇ψ(xγk(t)), ˙xγk(t)⟩ � �� � ˙ξγk (t) ∇ψ(xγk(t)), ∇ψ(xγk(t))⟩α(t) = �� ϑ1 γk(t) ˙xγk(t), ˙xγk(t) � + � ∇ψ(xγk(t)), ζγk(t) − ξγk(t)ϑ2 γk(t) ˙xγk(t) �� α(t) −γkξγk(t) |⟨∇ψ(xγk(t)), ˙xγk(t)⟩| ∥∇ψ(xγk(t))∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Integrating both sides on [0, 1] and using the boundedness of (∥ ˙xγk∥∞)k and (∥ξγk∥∞)k (by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11), and assumption (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1), we get the existence of a constant ˜ M1 depending on ¯ M, Mψ, ¯ Mψ, η, and Mζ such that � 1 0 | ˙ξγk(t)|∥∇ψ(xγk(t))∥2dt ≤ ˜ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 2The function sign: R −→ R is defined by: sign(x) = x |x| for x ̸= 0, and 0 for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 26 Control Space for Strong Convergence of Continuous Approximation Using (8) and assumption (A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' it follows that � 1 0 | ˙ξγk(t)|dt = � 1 0 γ2 keγkψ(xγk(t))|⟨∇ψ(xγk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ˙xγk(t)⟩|dt = � {t:∥∇ψ(xγk(t))∥≤η} γ2 keγkψ(xγk(t))|⟨∇ψ(xγk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ˙xγk(t)⟩|dt + � {t:∥∇ψ(xγk(t))∥>η} γ2 keγkψ(xγk(t))|⟨∇ψ(xγk(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ˙xγk(t)⟩|∥∇ψ(xγk(t))∥2 ∥∇ψ(xγk(t))∥2 dt ≤ η � ¯ M + 2 ¯ M ¯ Mψ η � γ2 ke−γkε + ˜ M1 η2 ≤ η � ¯ M + 2 ¯ M ¯ Mψ η � + ˜ M1 η2 =: ˜ M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' for k sufficiently large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' where ˜ M2 depends on the given constants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ¯ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ¯ Mψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Mψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' and on the bound of (∥ ˙uγk∥2)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, the sequence ξγk satisfies, for k sufficiently large, V 1 0 (ξγk) ≤ ˜ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11, ∥ξγk∥∞ ≤ 2 ¯ M η for all k ≥ ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, by Helly first theorem, ξγk admits a pointwise convergent subsequence, we do not relabel, whose limit is some function ˜ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) with ∥˜ξ∥∞ ≤ 2 ¯ M η and V 1 0 (˜ξ) ≤ ˜ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Being pointwise convergent to ˜ξ and uniformly bounded in L∞, ξγk strongly converges in L2 to ˜ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' However, by part(i) of this theorem, ξγk converges weakly in L2 to ξ, hence, ˜ξ = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, ξγk converges pointwise and strongly in L2 to ξ, and ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) with V 1 0 (ξ) ≤ ˜ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (52) As f is M-Lipschitz on C × (U + ˜ρ ¯B), u ∈ W, ∇ψ is Lipschitz, and ξ ∈ BV , then equation (12), which is satisfied by (x, u, ξ), now holds for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This yields that (13) is also valid for all t ∈ [0, 1], and that ˙x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It remains to show that ˙xγk has uniform bounded variations and converges point- wise and strongly in L2 to ˙x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since ξγk(·)∇ψ(xγk(·)) is Lipschitz, uγk ∈ W, and f is M-Lipschitz on C × (U + ˜ρ ¯B), then (Dγk) holds for all t ∈ [0, 1], that is, ˙xγk(t) = fΦ(xγk(t), uγk(t)) − ξγk(t)∇ψ(xγk(t)), ∀ t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, using part(i) of this theorem, the continuity of fΦ(·, ·), that the sequence (xγk, uγk, ξγk)k has uniform bounded variations and converges pointwise to (x, u, ξ), and that (x, u, ξ) satisfies (12) for all t ∈ [0, 1], we obtain that the sequence ˙xγk is of bounded variations and converges pointwise to ˙x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since (∥ ˙xγk∥∞)k is bounded, we conclude that the sequence ˙xγk also converges strongly in L2 to ˙x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, xγk converges strongly in the norm topology of W 1,2 to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' □ We proceed to present the proof of our approximation result, namely, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We consider k large enough so that C0(k) ⊂ ˜C0(δ) and C1(k) ⊂ ˜C1(δ), see (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2, ¯xγk −→ ¯x strongly in W 1,2, and hence, for k sufficiently large, ¯xγk(t) ∈ ¯Bδ(¯x(t)) for all t ∈ [0, 1], and ¯yγk(1) ∈ [−δ, δ], where ¯yγk(t) := � t 0 ∥ ˙¯xγk(s) − ˙¯x(s)∥2ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, the triplet state (¯xγk, ¯yγk, ¯zγk := 0) solves (Dγk) for ((¯ck, 0, 0), ¯u), with ¯xγk(t) ∈ ¯Bδ(¯x(t)) and ¯u(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), for all t ∈ [0, 1], and (¯xγk(1), ¯yγk(1), ¯zγk(1) = 0) ∈ C1(k) × [−δ, δ] × [−δ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, for k sufficiently Control Space for Strong Convergence of Continuous Approximation 27 large, (¯xγk, ¯yγk, 0, ¯u) is an admissible quadruplet for (Pγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using the continuity of g on ˜C0(δ) × ˜C1(δ) and the definition of J(x, u, z, u), we obtain that J(x, u, z, u) is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, for k large enough, inf(Pγk) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Fix k sufficiently large so that inf(Pγk) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let (xnγk, ynγk, znγk, unγk)n ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) × AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R) × W be a minimizing sequence for (Pγk), that is, the sequence is admissible for (Pγk) and lim n−→∞ J(xn γk, yn γk, zn γk, un γk) = inf(Pγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (53) Since for each n, xnγk solves (Dγk) for (xnγk(0), unγk), and (xnγk(0))n ∈ C0(k) ⊂ C, then, by (18), we have that the sequence (xnγk)n is uniformly bounded in C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) and the sequence ( ˙xnγk)n is uniformly bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, from (A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2), we have that sets U(t) are compact and uniformly bounded, then, the sequence (unγk)n, which is in W, is uniformly bounded in C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, its derivative sequence, ( ˙unγk)n, must be uniformly bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Indeed, if this is not true, then there exists a subsequence of ˙unγk, we do not relabel, such that lim n−→∞ ∥ ˙unγk∥2 = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using that g is bounded on ˜C0(δ) × ˜C1(δ), it follows that J(xn γk, yn γk, zn γk, un γk) ≥ min (c1,c2)∈ ˜ C0(δ)× ˜ C1(δ) g(c1, c2) + 1 2zn γk(1) = min (c1,c2)∈ ˜ C0(δ)× ˜ C1(δ) g(c1, c2) + 1 2∥ ˙un γk − ˙¯u∥2 2 and hence, lim n−→∞ J(xnγk, ynγk, znγk, unγk) = ∞, contradicting (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, also ( ˙unγk)n is uniformly bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, by Arzel`a-Ascoli theorem, along a subsequence (we do not relabel), the sequence (xnγk, unγk)n converges uniformly to a pair (xγk, uγk) and the sequence ( ˙xnγk, ˙unγk)n converges weakly in L2 to the pair ( ˙xγk, ˙uγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, (xγk, uγk) ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, the following two inequalities hold ∥ ˙xγk − ˙¯x∥2 2 ≤ lim inf n−→∞ ∥ ˙xn γk − ˙¯x∥2 2 and ∥ ˙uγk − ˙¯u∥2 2 ≤ lim inf n−→∞ ∥ ˙un γk − ˙¯u∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (54) Since C0(k), C1(k), ¯Bδ(¯x(t)) and U(t)∩ ¯Bδ(¯u(t)) are closed for all t ∈ [0, 1], and from the uniform convergence, as n −→ ∞, of the sequence (xnγk, unγk) to (xγk, uγk), we get that the inclusions xγk(0) ∈ C0(k) and xγk(1) ∈ C1(k), and xγk(t) ∈ ¯Bδ(¯x(t)), and uγk(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' To prove that xγk is the solution of (Dγk) corresponding to (xγk(0), uγk), we first use that xnγk is the solution of (Dγk) for (xnγk(0), unγk), that is, for t ∈ [0, 1], xn γk(t) = xn γk(0) + � t 0 � fΦ(xn γk(s), un γk(s)) − γkeγkψ(xn γk(s))∇ψ(xn γk(s)) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using that (xnγk(t), unγk(t)) ∈ [C ∩ ¯Bδ(¯x(t))] × [U(t) ∩ ¯Bδ(¯u(t))], fΦ is Lipschitz on [C ∩ ¯Bδ(¯x)] × [(U + ˜ρ ¯B) ∩ ¯Bδ(¯u)], and (xnγk, unγk) converges uniformly to (xγk, uγk), then, upon taking the limit, as n −→ ∞, in the last equation we conclude that (xγk, uγk) satisfies the same equation, that is, ˙xγk(t) = fΦ(xγk(t), uγk(t)) − γkeγkψ(xγk(t))∇ψ(xγk(t)), t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We define for all t ∈ [0, 1], yγk(t) := � t 0 ∥ ˙xγk(τ) − ˙¯x(τ)∥2dτ and zγk(t) := � t 0 ∥ ˙uγk(τ) − ˙¯u(τ)∥2dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Clearly we have: yγk ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R), ˙yγk(t) = ∥ ˙xγk(t) − ˙¯x(t)∥2, t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', and yγk(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' zγk ∈ AC([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R), ˙zγk(t) = ∥ ˙uγk(t) − ˙¯u(t)∥2, t ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', and zγk(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 28 Control Space for Strong Convergence of Continuous Approximation Moreover, since ∥ ˙xnγk − ˙¯x∥2 2 = ynγk(1) ∈ [−δ, δ] and ∥ ˙unγk − ˙¯u∥2 2 = znγk(1) ∈ [−δ, δ], the two inequalities of (54) yield that yγk(1) ∈ [−δ, δ] and zγk(1) ∈ [−δ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (55) Hence, (xγk, yγk, zγk, uγk) is admissible for (Pγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Now using (53) and the second inequality of (54), it follows that inf(Pγk) = lim n−→∞ J(xn γk, yn γk, zn γk, un γk) = lim n−→∞ � g(xn γk(0), xn γk(1)) + 1 2 � ∥un γk(0) − ¯u(0)∥2 + zn γk(1) + ∥xn γk(0) − ¯x(0)∥2�� = g(xγk(0), xγk(1))+ 1 2∥uγk(0) − ¯u(0)∥2 + 1 2 lim inf n−→∞ ∥ ˙un γk − ˙¯u∥2 2 + 1 2∥xγk(0) − ¯x(0)∥2 ≥ g(xγk(0), xγk(1)) + 1 2∥uγk(0) − ¯u(0)∥2 + 1 2∥ ˙uγk − ˙¯u∥2 2 + 1 2∥xγk(0) − ¯x(0)∥2 = J(xγk, yγk, zγk, uγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, for each k, large enough, (xγk, yγk, zγk, uγk) is optimal for (Pγk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4 asserts that, for k large, C0(k) ⊂ C(k) ⊂ C, then, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='9(i) yield that the sequence (xγk, ξγk)k, where ξγk is given via (21), admits a subsequence, not relabled, having (xγk)k converging uniformly to some x ∈ W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) with images in C, ( ˙xγk, ξγk)k converging weakly in L2 to ( ˙x, ξ) and ξ supported on I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Now, consider the sequence (uγk)k, which is in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It has a uniformly bounded derivative in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In fact, the admissibility of (¯xγk, ¯yγk, 0, ¯u), and the optimality of (xγk, yγk, zγk, uγk) for (Pγk), imply that J(xγk, yγk, zγk, uγk) ≤ g(¯xγk(0), ¯xγk(1)) + 1 2∥¯xγk(0) − ¯x(0)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (56) This, together with the continuity of g on ˜C0(δ)× ˜C1(δ), the uniform boundedness of the sequences (xγk)k and (¯xγk)k, and the boundedness of U(0), imply that for some ˆ M > 0 we have that ∥ ˙uγk − ˙¯u∥2 2 ≤ 2 (g(¯xγk(0), ¯xγk(1)) − g(xγk(0), xγk(1)) + ∥¯xγk(0) − ¯x(0)∥2 − ∥uγk(0) − ¯u(0)∥2 − ∥xγk(0) − ¯x(0)∥2 ≤ ˆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, (uγk)k has uniformly bounded derivative in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Now since in addition we have that xγk(0) ∈ C0(k) ⊂ C(k), we are in a position to apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We obtain a subsequence (not relabeled) of uγk, and u ∈ W such that (xγk, uγk) converges uniformly to (x, u), ˙uγk converges weakly in L2 to ˙u, all the conclusions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='11 hold including that xγk(t) ∈ int C for all t ∈ [0, 1], ( ˙xγk, ξγk) converges strongly in L2 to ( ˙x, ξ), ˙x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn), ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), and, for all t ∈ [0, 1], (x, u, ξ) satisfies (12)-(13) and x uniquely solves (D) for u, that is, � ˙x(t) = fΦ(x(t), u(t)) − ξ(t)∇ψ(x(t)) ∈ f(x(t), u(t)) − ∂ϕ(x(t)), ∀ t ∈ [0, 1], x(0) ∈ C0 ∩ ¯Bδo(¯x(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, we have ∥ ˙u − ˙¯u∥2 2 ≤ lim inf k−→∞ ∥ ˙uγk − ˙¯u∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (57) We shall show that (x, u) is admissible for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since ˙xγk converges strongly in L2 to ˙x, and using (55) and (57), we have: ∥ ˙x − ˙¯x∥2 2 = lim k−→∞ ∥ ˙xγk − ˙¯x∥2 2 = lim k−→∞ yγk(1) (55) ∈ [−δ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' ∥ ˙u − ˙¯u∥2 2 (57) ≤ lim inf k−→∞ ∥ ˙uγk − ˙¯u∥2 2 = lim inf k−→∞ zγk(1) (55) ∈ [−δ, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Control Space for Strong Convergence of Continuous Approximation 29 Hence, ∥ ˙x − ˙¯x∥2 2 ≤ δ and ∥ ˙u − ˙¯u∥2 2 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since xγk(1) ∈ C1(k), (24)(b) implies that x(1) ∈ C1 ∩ ¯Bδo(¯x(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Furthermore, the two inclusions xγk(t) ∈ ¯Bδ(¯x(t)) and uγk(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), for all t ∈ [0, 1], together with the uniform convergence of (xγk, uγk) to (x, u), give that x(t) ∈ ¯Bδ(¯x(t)) and u(t) ∈ U(t) ∩ ¯Bδ(¯u(t)), for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, (x, u) is admissible for (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, the optimality of (¯x, ¯u) for (P) yields that g(¯x(0), ¯x(1)) ≤ g(x(0), x(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (58) Now, the uniform convergence of ¯xγk to ¯x, (56), (58), the continuity of g, and the convergence of xγk(0) to x(0), imply that u(0) = ¯u(0) and lim inf k−→∞ � ∥ ˙uγk − ˙¯u∥2 2 � = 0, and (59) x(0) = ¯x(0) and g(¯x(0), ¯x(1)) = g(x(0), x(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (60) The equality (59) gives the existence of a subsequence of uγk, we do not relabel, such that ˙uγk converges strongly in L2 to ˙¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' It results that uγk converges uniformly to ¯u, and hence, u = ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consequently, uγk converges strongly in W to ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, as u = ¯u, the functions x and ¯x solve the dynamic (D) with the same control ¯u and initial condition, see (60), hence, by the uniqueness of the solution of (D) we have x = ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5, we obtain that also ξ = ¯ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, xγk uniformly −−−−−−−−→ C([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) ¯x and ( ˙xγk, ξγk) strongly −−−−−−−−−−−−→ L2([0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn×R+) ( ˙¯x, ¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This yields that (yγk, zγk) −→ (0, 0) in the strong topology of W 1,1([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+ × R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since xγk(1) ∈ �� C1 ∩ ¯Bδo(¯x(1)) � − ¯x(1) + ¯xγk(1) � ∩(int C) and ¯xγk(1) converges to ¯x(1), it follows that xγk(1) ∈ � � C1 ∩ ¯Bδo(¯x(1)) � + ˜ρB � ∩ (int C), for k sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, the definition of C0(k) and the convergence of ρk to 0 yield that, for k large enough, xγk(0) ∈ � � C0 ∩ ¯Bδo(¯x(0)) � + ˜ρB � ∩(int C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' This terminates the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' □ In the next lemma, a compactness result is derived for the solutions of (D), where the controls are restricted to be in W and x(1) ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The equivalence between (D) and equations (12)-(13) is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1 (Compact trajectories and controls for (D)) Assume that (A1)-(A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let (xj, uj)j be a sequence in W 1,∞ × W satisfying (D) with xj(1) ∈ C1, for all j ∈ N, and (∥ ˙uj∥2)j be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Consider (ξj)j the corresponding sequence in L∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) obtained via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5, that is, (xj, uj, ξj) satisfies (12)-(13), for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Then there exist a subsequence of (xj, uj, ξj)j, we do not relabel, and (x, u, ξ) ∈ W 1,∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × W × L∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) such that (xj, uj)j converges uniformly to (x, u), ( ˙xj, ξj)j now converges pointwise to ( ˙x, ξ) ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn) × BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), ˙uj converge weakly in L2 to ˙u, ξ, and (x, u, ξ) satisfies (12)-(13) with x(1) ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' In particular, (x, u) is admissible for (P) and (xj)j converges to x strongly in the norm topology of W 1,2([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Proof Using (11) in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='4 for the sequence (xj)j, the boundedness of (∥ ˙uj∥2)j, that uj(t)) ∈ U(t) for all t ∈ [0, 1], and that the sets U(t) are compact and uniformly bounded, by (A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2), then Arzela-Ascoli’s theorem produces a subsequence, we do not relabel, of (xj, uj)j, that converges uniformly to an absolutely continuous pair (x, u) with (x(t), u(t) ∈ C × U(t) for all t ∈ [0, 1], and ( ˙xj, ˙uj)j converging weakly in L2 to ( ˙x, ˙u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As for all j ∈ N, xj(0) ∈ C0 and xj(1) ∈ C1, then (A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1) and (A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3) 30 Control Space for Strong Convergence of Continuous Approximation yield that x(0) ∈ C0 and x(1) ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2, we obtain ∥ξj∥∞ ≤ ¯ M 2η , ξj ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+), and V 1 0 (ξj) ≤ ˜ M2, where ˜ M2 depends on the uniform bound of (∥ ˙uj∥2)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' By Helly’s first theorem, (ξj)j convergence pointwise to ξ ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' On the other hand, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2 also gives that (12) holds for all t ∈ [0, 1], that is, ˙xj(t) = fΦ(xj(t), uj(t)) − ξj(t)∇ψ(xj(t)), ∀t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (61) Thus, upon taking the pointwise limit as j −→ ∞ in (61), it follows that ( ˙xj)j con- verges pointwise to its L2-limit ˙x, and hence, ˙x ∈ BV ([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' As (xj, uj) solves (D), (11) yields that (∥ ˙xj∥∞)j is uniformly bounded, and hence, ( ˙xj)j converges to ˙x strongly in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' We now show that ξ(t) is supported in I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Let t ∈ I-(x) be fixed, that is, x(t) ∈ int C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Since (xj)j converges uniformly to x, then we can find δo > 0 and jo ∈ N such that, for all s ∈ (t−δ, t+δ)∩[0, 1] and for all j ≥ jo, we have xj(s) ∈ int C, and hence, as ξj satisfies (13), ξj(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Thus, ξj(s) −→ 0 for s ∈ (t − δo, t + δo) ∩ [0, 1], and whence, ξ(t) = 0, proving that ξ is supported in I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='5 to (x, u, ξ), we conclude that (x, u) solves (D), ξ ∈ L∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' R+) and (x, u, ξ) satisfies (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' □ In the following remark, we provide important information about the constraint qualification property (CQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2 (i) Let F : [0, 1] ⇒ Rm be a lower semicontinuous multifunction with closed and nonempty values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For dF (t, x) := d(x, F(t)), we have from [25, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3] that for t ∈ [0, 1] and x ∈ F(t), conv ( ¯NL F (t)(x)) is pointed if and only if 0 ̸∈ ∂> x dF (t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The notion of ∂> x g(t, x) is introduced for a general function g(t, x) by Clarke in [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' For g(t, x) := dF (t, x), it is shown in [25, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='2] that ∂> x dF (t, x) = conv � ζ : ζ = lim i−→∞ ζi, ∥ζi∥ = 1, ζi ∈ NP F (ti)(xi) and (ti, xi) Gr F −−−→ (t, x) � , where (ti, xi) Gr F −−−→ (t, x) signifies that (ti, xi) −→ (t, x) with xi ∈ F(ti) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Therefore, for h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' F), we have that F satisfies the constraint qualification (CQ) at h if and only if 0 ̸∈ ∂> x dF (t, h(t)) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Note that the multifunction (t, x) �→ ∂> x dF (t, x) is uniformly bounded with compact and convex values, and has a closed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (ii) Using the proximal normal inequality, one can easily extend the arguments in the proof of [25, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3(d)], to show that if the lower semicontinu- ous multifunction F has closed and r-prox-regular values, for some r > 0, (as opposed to convex), then conv ( ¯ NL F (t)(·)) = NP F (t)(·) = NL F (t)(·) = NF (t)(·), and this cone is pointed at x ∈ F(t) if and only if F(t) is epi-lipschitz at x, see [9, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='1] and [36, Exercise 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, a lower semicontinuous multifunction F : [0, 1] ⇒ Rm with values that are closed and r-prox-regular, satisfies the constraint qualification (CQ) at h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' F) if and only if F(t) is epi-Lipschitz at h(t), for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (iii) If F(t) = F for all t ∈ [0, 1], where F is closed, then conv ( ¯NL F (t)(·)) = NF (·), and this cone is pointed at x ∈ F if and only if F is epi-Lipschitz at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Hence, a constant multifunction F satisfies the constraint qualification (CQ) at h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' F) if and only if F is epi-Lipschitz at h(t) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Control Space for Strong Convergence of Continuous Approximation 31 We terminate this section by the following technical lemma used in the proof of the “In addition” part of the weak maximization condition of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' The proof of the lemma follows from the local property of the normal cones, the proximal normal inequality, and the fact that the proximal, Mordukhovich, and Clarke normal cones coincide in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3 Let F : [0, 1] ⇒ Rm be a lower semicontinuous multifunction with closed and nonempty values and let h ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' If there exist εo > 0 and r > 0 such that F(t) ∩ ¯Bεo(h(t)) is r-prox-regular for all t ∈ [0, 1], then for any δ > 0 we have conv ( ¯NL F (t)∩ ¯ Bδ(h(t))(h(t))) = NP F (t)∩ ¯ Bεo (h(t))(h(t)), ∀t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Moreover, for all t ∈ [0, 1] and for all ζ ∈ conv ( ¯ NL F (t)∩ ¯ Bδ(h(t))(h(t))), we have ⟨ζ, v − h(t)⟩ ≤ ∥ζ∥ min{εo,2r}∥v − h(t)∥2, ∀v ∈ F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' References [1] Adam, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Outrata, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' B 19, 2709–2738 (2014) [2] Aubin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Viability Theory, Modern Birkh¨auser Classics, Birkh¨auser/Springer, Basel-Boston-Berlin (2009) [3] Brokate, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Krejˇc´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Optimal control of ODE systems involving a rate independent variational inequality, Discrete and continuous dynamical systems series B 18, 331–348 (2013) [4] Brudnyi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Brudnyi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Methods of Geometric Analysis in Exten- sion and Trace Problems, Volume 1, Monographs in Mathematics, 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Birkh¨auser/Springer Basel AG, Basel (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' [5] Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Colombo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Optimization of Fully Controlled Sweeping Processes, Journal of Differential Equations 295 138–186 (2021) [6] Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Optimal control of a perturbed sweeping process via discrete approximations, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' B 21, 3331–3358 (2016) [7] Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Optimality conditions for a controlled sweep- ing process with applications to the crowd motion model, Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' B 22, 267–306 (2017) [8] Cao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Optimal control of a nonconvex perturbed sweeping process, Journal of Differential Equations 266, 1003–1050 (2019) [9] Clarke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Optimization and Nonsmooth Analysis, John Wiley, New York (1983) 32 Control Space for Strong Convergence of Continuous Approximation [10] Clarke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Springer, London (2013) [11] Clarke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Ledyaev, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Stern, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Wolenski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Nons- mooth Analysis and Control Theory, Graduate Texts in Mathematics, 178, Springer-Verlag, New York (1998) [12] Clarke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Stern, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Wolenski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Proximal smoothness and the lower-C2 property, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Convex Analysis 2, 117–144 (1995) [13] Colombo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Henrion, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Hoang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Optimal control of the sweeping process, Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Discrete Impuls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' B 19, 117–159 (2012) [14] Colombo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Henrion, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Hoang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Optimal control of the sweeping process over polyhedral controlled sets, Journal of Differential Equations 260 (2016) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 4, 3397–3447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' [15] Colombo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Marigonda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Differentiability properties for a class of non-convex functions, Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 25, 1–31 (2005) [16] Colombo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Optimization of a perturbed sweeping process by constrained discontinuous controls, SIAM Journal on Control and Optimization, 58 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 4, 2678–2709 (2020) [17] de Pinho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Smirnov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Optimal Control Involving Sweeping Processes, Set-Valued Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 2, 523–548 (2019) [18] de Pinho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Smirnov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Correction to: Opti- mal Control Involving Sweeping Processes, Set-Valued Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 27, 1025–1027 (2019) [19] de Pinho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Smirnov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Optimal Control with Sweeping Processes: Numerical Method, J Optim Theory Appl 185, 845– 858 (2020) [20] de Pinho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Ferreira, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Smirnov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Necessary condi- tions for optimal control problems with sweeping systems and end point constraints, Optimization, 71:11, 3363–3381 (2021) [21] Federer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Curvature measures, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 93, 418–491 (1959) [22] Hiriart-Urruty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Extension of Lipschitz functions, Journal of Mathe- matical Analysis and Applications, 77, 539–554 (1980) [23] Hoehener, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Variational approach to second-order optimality conditions for control problems with pure state constraints, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='Control Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 50, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='3, 1139–1173 (2012) Control Space for Strong Convergence of Continuous Approximation 33 [24] Vladimir Kadets, A course in functional analysis and measure theory, Translated from the 2006 Russian edition by Andrei Iacob, Universitext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Springer, Cham (2018) [25] Loewen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Rockafellar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': The Adjoint Arc in Nonsmooth Opti- mization, Transactions of the American Mathematical Society 325, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 1, 39–72 (1991) [26] Luenberger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='G: Optimization by vector space methods, John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (1969) [27] Mordukhovich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin (2006) [28] Moreau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Rafle par un convexe variable, I, Trav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Semin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' d’Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Convexe, Montpellier 1, Expos´e 15, 36 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (1971) [29] Moreau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Rafle par un convexe variable, II, Trav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Semin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' d’Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Convexe, Montpellier 2, Expos´e 3, 43 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' (1972) [30] Moreau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Evolution problem associated with a moving convex set in a Hilbert space, Journal of Differential Equations 26, 347–374 (1977) [31] Nour, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Stern, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Takche, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Proximal smoothness and the exterior sphere condition, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Convex Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', 16(2), 501–514 (2009) [32] Nour, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Zeidan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Numerical solution for a controlled nonconvex sweeping process, IEEE Control Systems Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 6, 1190–1195 (2022) [33] Nour, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Zeidan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Optimal control of nonconvex sweeping pro- cesses with separable endpoints: Nonsmooth maximum principle for local minimizers, Journal of Differential Equations, Volume 318, 113–168 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' [34] Poliquin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Rockafellar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Thibault, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Local differentiability of distance functions, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 352, 5231–5249 (2000) [35] Rockafellar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': Clarke’s tangent cones and the boundaries of closed sets in Rn, Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 3, 145–154 (1979) [36] Rockafellar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Wets, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Variational analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin (1998) [37] Vinter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' : Optimal Control, Birkh¨auser, Systems and Control: Foun- dations and Applications, Boston (2000) [38] Zeidan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Nour, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=', Saoud, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': A nonsmooth maximum principle for a controlled nonconvex sweeping process, Journal of Differential Equations, Volume 269, Issue 11, 9531–9582 (2021) 34 Control Space for Strong Convergence of Continuous Approximation [39] Zeidan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=': A modified Hamilton-Jacobi approach in the generalized problem of Bolza, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content='Optim, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} +page_content=' 11, 97–109 (1984)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf'} diff --git a/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf b/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..22258a715be331e848169d67c782b06f601086d0 Binary files /dev/null and b/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf differ diff --git a/v9AzT4oBgHgl3EQfQPsE/content/tmp_files/2301.01195v1.pdf.txt b/v9AzT4oBgHgl3EQfQPsE/content/tmp_files/2301.01195v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33dda940f74b352e86cf9411c4101c91a4bebe55 --- /dev/null +++ b/v9AzT4oBgHgl3EQfQPsE/content/tmp_files/2301.01195v1.pdf.txt @@ -0,0 +1,37 @@ +arXiv:2301.01195v1 [math.PR] 2 Jan 2023 +Integral Functionals of Probability Measures +that Depend Only on the Mean +Daniel W. Stroock +January 4, 2023 +In this note I prove the following statement: +Assume that f : R −→ R is a continuous function with the property that +there is a function F such that +� +f dµ = F +�� +y µ(dy) +� +for all compactly supported probability measures µ on R. Then f is an affine +function. +Proof: Let ρ be any non-negative, C∞, compactly supported, even function +with Lebesgue integral 1, and set ρǫ(x) = ǫ−1ρ(ǫ−1x) for ǫ > 0. Then +� +ρǫ(x − y) dy = x for all ǫ > 0 and x ∈ R, +and so ρǫ ∗ f = F for all ǫ > 0. Hence f = F ∈ C∞. Next, +f(a + h) + f(a − h) +2 += +� +f d +�δa+h + δa−h +2 +� += f(a) for all a ∈ R and h > 0. +Hence +f ′′(a) = lim +hց0 +f(a + h) + f(a − h) − 2f(a) +h2 += 0 for all a ∈ R. +1 + diff --git a/v9AzT4oBgHgl3EQfQPsE/content/tmp_files/load_file.txt b/v9AzT4oBgHgl3EQfQPsE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae6ea3f42df3fe1e126a908ac703b00940ffbce9 --- /dev/null +++ b/v9AzT4oBgHgl3EQfQPsE/content/tmp_files/load_file.txt @@ -0,0 +1,12 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf,len=11 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content='01195v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content='PR] 2 Jan 2023 Integral Functionals of Probability Measures that Depend Only on the Mean Daniel W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' Stroock January 4, 2023 In this note I prove the following statement: Assume that f : R −→ R is a continuous function with the property that there is a function F such that � f dµ = F �� y µ(dy) � for all compactly supported probability measures µ on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' Then f is an affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' Proof: Let ρ be any non-negative, C∞, compactly supported, even function with Lebesgue integral 1, and set ρǫ(x) = ǫ−1ρ(ǫ−1x) for ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' Then � ρǫ(x − y) dy = x for all ǫ > 0 and x ∈ R, and so ρǫ ∗ f = F for all ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' Hence f = F ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' Next, f(a + h) + f(a − h) 2 = � f d �δa+h + δa−h 2 � = f(a) for all a ∈ R and h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' Hence f ′′(a) = lim hց0 f(a + h) + f(a − h) − 2f(a) h2 = 0 for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} +page_content=' 1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AzT4oBgHgl3EQfQPsE/content/2301.01195v1.pdf'} diff --git a/v9AzT4oBgHgl3EQfQPsE/vector_store/index.faiss b/v9AzT4oBgHgl3EQfQPsE/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..978b00fd899dc2e513846be100b8b9e227b05d1f Binary files /dev/null and b/v9AzT4oBgHgl3EQfQPsE/vector_store/index.faiss differ diff --git a/vNE1T4oBgHgl3EQfkQSg/vector_store/index.faiss b/vNE1T4oBgHgl3EQfkQSg/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c2678135d25b15798f358393e006c3104600480f --- /dev/null +++ b/vNE1T4oBgHgl3EQfkQSg/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3b147fb0f1c4346f940d337f491c20e52bd57cd1a9a178e644652c8df67337ec +size 4522029 diff --git a/vNE_T4oBgHgl3EQf-hyD/content/tmp_files/2301.08387v1.pdf.txt b/vNE_T4oBgHgl3EQf-hyD/content/tmp_files/2301.08387v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..417dd4822a02fd43ee9a9267b51f44b85bbe1949 --- /dev/null +++ b/vNE_T4oBgHgl3EQf-hyD/content/tmp_files/2301.08387v1.pdf.txt @@ -0,0 +1,782 @@ +Occlusion Reasoning for Skeleton Extraction +of Self-Occluded Tree Canopies +Chung Hee Kim and George Kantor +Abstract— In this work, we present a method to extract +the skeleton of a self-occluded tree canopy by estimating the +unobserved structures of the tree. A tree skeleton compactly +describes the topological structure and contains useful infor- +mation such as branch geometry, positions and hierarchy. This +can be critical to planning contact interactions for agricultural +manipulation, yet is difficult to gain due to occlusion by +leaves, fruits and other branches. Our method uses an instance +segmentation network to detect visible trunk, branches, and +twigs. Then, based on the observed tree structures, we build +a custom 3D likelihood map in the form of an occupancy +grid to hypothesize on the presence of occluded skeletons +through a series of minimum cost path searches. We show +that our method outperforms baseline methods in highly +occluded scenes, demonstrated through a set of experiments on +a synthetic tree dataset. Qualitative results are also presented +on a real tree dataset collected from the field. +I. INTRODUCTION +With increasing global population and labor shortages, +modern agriculture is adopting new technologies to enhance +sustainability and profitability. There is growing effort in +developing robotic solutions to automate repetitive and la- +borious tasks that often require complex and delicate in- +teraction with crops. Harvesting and pruning, for example, +may require the agent to manipulate crops by pushing aside +leaves or branches to reach occluded regions before picking +or cutting. Understanding the unstructured and cluttered task +environment is critical to automating such challenging tasks. +In this work, we address the perception problem, where +robots must be able to sense and understand the complex task +environment prior to crop interaction. We are particularly +interested in extracting the skeleton of a self-occluded tree +canopy by estimating the unobserved parts of the tree. A tree +skeleton is useful as it describes the topological structure +and contains useful information such as branch dimensions, +positions, and hierarchy. This knowledge is helpful in pheno- +typing crop characteristics for growth assessment and can be +critical for planning contact interactions, such as determining +optimal pruning locations or generating trajectories to pick +fruits by pulling a branch into its workspace. +One of the main challenges faced during the extraction of +the tree skeleton from sensor data (typically in the form of +point clouds obtained from laser scanners or depth cameras) +stems from noise and occlusion, which is exacerbated by the +presence of foliage as depicted in Fig. 1(a). Although there +exists prior works that address skeletonization of occluded +tree canopies, they are tailored towards sparse point clouds +C. H. Kim and G. Kantor are with Carnegie Mellon University, Pittsburgh +PA, USA {chunghek, kantor}@andrew.cmu.edu +(a) +(b) +Fig. 1. +(a) The 3D reconstruction (middle) and skeletonization (right) of +a heavily occluded tree canopy (left). (b) Tree crops are tightly organized +in rows in industrial orchard settings. +obtained from terrestrial laser scanners (TLS) and can be +ineffective when applied to dense point clouds acquired +from RGB-D or stereo cameras. Furthermore, it is generally +assumed in prior works that the tree point cloud to be +skeletonized is pre-registered from a 360 degree scan, which +is difficult to obtain in industrial orchard settings where trees +are tightly organized in rows as shown in Fig. 1(b). +We present a novel skeletonzation method that extracts +tree skeletons from one-sided views of the tree canopy +by using a depth camera attached to a robot arm. Our +method particularly addresses the challenge of skeletonizing +heavily occluded tree canopies by observing that branches in +nature generally extend linearly. We use this as a heuristic +assumption to approximate a 3D likelihood map in the form +of an occupancy grid to predict presence of occluded branch +structures by searching for minimum cost paths. We validate +the effectiveness of our approach by presenting quantitative +assessment on a synthetic tree dataset with known ground +truth, and present qualitative results from a real tree dataset +collected at an apple orchard. +II. RELATED WORK +A skeleton can be defined as a curve expressing the shape +of an object that is consistent with the topological structure +as well as the connectivity of the original object shape. +Many methods have been proposed to extract skeletons from +unordered 3D point clouds. A skeleton can be extracted by +measuring the L1-median to determine local centers of the +point cloud [1]. A Laplacian-based contraction method is +proposed in [2], which collapses the input point cloud into +a minimal volume using an iterative Laplacian smoothing +process. In [3], a skeleton is extracted by defining a new +feature representation called rotational symmetry axis. The +aforementioned methods work well with limited noise and +occlusion, while we are particularly interested in skeletoniz- +ing point cloud of botanical trees which are often noisy and +arXiv:2301.08387v1 [cs.RO] 20 Jan 2023 + +FROFig. 2. +Our system pipeline for tree skeletonization. It takes as input a series of RGB-D images of a self-occluded tree from multiple viewpoints and +outputs the underlying tree skeleton. +occluded. +There is particular interest in extracting the skeleton of a +tree canopy due to implications in automating agricultural +tasks [4] such as pruning [5], harvesting [6], or phenotyping +to estimate crop characteristics for growth assessment [7]. +Prior knowledge about tree structures are commonly used +to extract tree skeletons, such as branching properties [8], +cylindrical shape priors [9] and upright offshoots [10]. Tree +skeletons are commonly represented as graphs composed of +vertices and edges; [11] and [12] uses graph-based methods +to extract tree skeletons. A geometry-based method is used to +fit cylinders on the point cloud in a hierarchical data structure +to encapsulate parent-child relations of branches [13]. +While the above mentioned tree skeletonization meth- +ods are tailored for trees without leaves, we address a +more challenging problem in which tree canopies are self- +occluded by leaves, fruits, or other branches. To address the +problem of missing data, [14] proposes an iterative data- +completion method to recover data for 3D tree modelling. +Tree point clouds with leaves are often first separated into +branches and leaves before skeleton extraction [15], while +[16] directly generates visually convincing tree skeletons +based on optimization driven by biological priors. Several +works improve skeleton connectivity by joining disconnected +skeletal structures through geometry-based methods [17], +[18]; we use it as a baseline method and quantitatively show +that our method outperforms [17] in occluded scenes. Despite +addressing the issue of occlusion, these methods are built for +sparse TLS point clouds and are inapplicable to dense point +clouds. More importantly, it is assumed that the tree point +cloud to be skeletonized is pre-registered from a 360 degree +scan, which may be unpractical in industrial orchard setting +where trees are organized in rows. The key contributions of +this paper are: +• A novel skeletonzation method that extracts tree skele- +tons from one-sided views of the tree canopy; +• A method to reason about the unobserved structures of +the tree to predict the presence of occluded skeletons; +• Experimental results on a synthetic dataset quantita- +tively compared against existing baselines, as well as +qualitative results on a real tree dataset collected from +the field. +III. METHODOLOGY +A. System Overview +Our tree skeletonization method aims to improve the +knowledge of occluded regions in the tree that is often self- +occluded by foliage, fruits, or other branches. Driven by a +heuristic assumption on tree structures, we approximate a 3D +likelihood map in the form of an occupancy grid that stores +information of visible as well as occluded structures of a tree +canopy to extract its skeleton. Fig. 2 shows an overview of +our framework. The system takes as input a series of color +and depth images from varying viewpoints obtained from +a camera attached to a robot arm with known poses. The +images are passed onto an instance segmentation network to +generate a semantic point cloud of branch clusters (Sec. III- +B). The occupancy probability distribution of the 3D likeli- +hood map is updated based on the observed branch clusters +over a sequence of images (Sec. III-C). The final skeleton is +generated by joining disconnected branch skeletons through +a series of minimum cost path searches in the likelihood map +(Sec. III-D). +B. Semantic Point Cloud Acquisition +We first acquire a 3D point cloud semantically labeled +with visible parts of the tree trunk, branches, and twigs +(collectively referred to as branches from this point on) that +are not occluded. This is achieved by projecting branch seg- +mentation masks in 2D color images on to the 3D point cloud +obtained from the depth image. The points corresponding to +branches in the semantic point cloud are clustered based on +the projected masks. +Our branch segmentation is based on the Mask R-CNN +[19] instance segmentation network with a Feature Pyramid +Network and ResNet50 backbone. The model takes as input +1440×1080 images and outputs instance segmentation masks +as well as its confidence scores ranging from 0 to 1. The +network was trained on 130 manually labeled images (105 +real images and 25 synthetic images) for 2000 iterations +on an NVIDIA GeForce RTX 3070. Sample segmentation +results are depicted in Fig. 3(a). In order to perform line- +fitting on the point clusters (further described in Sec. III-C), +we deliberately labeled each branch instances in the training +images to be a slender polygon such that the projected + +For each viewpoint: +Color Image +Instance + Final Tree Skeleton +Task Setting +Depth Map +Accumulated Edges +Initial Skeleton(a) +(b) +Fig. 3. +(a) Instance segmentation results on a real tree image (left) and a +synthetic tree image (right). (b) Tree forks (left) or angled branch instances +(right) are labeled as multiple branch instances in the training dataset. +branch point cloud cluster in 3D space is also piece-wise +slender. For example, a tree fork or a branch instance with +a significant change in direction is labeled as two or more +branch instances instead of a single instance as shown in +Fig. 3(b). +C. Skeleton Occupancy Likelihood Map +We now propose our novel approach to hypothesize on +the presence of occluded branches by approximating a 3D +likelihood map in the form of an occupancy grid, where the +i-th grid voxel mi stores the skeleton occupancy probability +pℓ(mi) ∈ [0, 1]. The approximation is driven by a heuristic +assumption: Given a visible branch instance, it is likely that +the branch structure extends longer along its growth direction +based on the observation that branches in nature generally +grow straight. We model this as individually observed prob- +ability distributions po with an ellipsoidal contour obtained +from the branch point cloud clusters as follows. +(a) +(b) +(c) +(d) +PC1 +PC2 +cp1 +cp2 +cp3 +cp4 +1.0 +0.0 +pℓ +po +cp2 +cp3 +dl +dr +l +r +Fig. 4. +(a) A single instance of a segmented branch component with +a confidence score of 0.88. (b) The branch instance is projected into 3D +space as a dense point cloud cluster. We compute the principal components +(PCi) and fit a B-spline curve with four control points (cpi) on the point +cloud cluster. (c) po is obtained from each line segment, (d) which updates +pℓ using equation (2). +A 3-dimensional B-Spline curve of degree-one with four +control points is approximated on a point cloud cluster using +the least squares method [20]. This results in three connected +line segments that best represent the skeletal geometry of +the point cloud cluster as shown in Fig. 4(b), which are +Observed +Branch Instances +Accumulated +Line Segments +Joint 3D +Likelihood Map +−→ +−→ +Fig. 5. +Visualization of the overall joint 3D likelihood map (right) +obtained from accumulated line segments (middle), which were extracted +from observed branch clusters over the sequence of images (left). +accumulated over the sequence of images (Fig. 5). The +radius of the branch instance is also estimated by performing +principal component analysis on the point cloud cluster +(see Fig. 4(b)), where the distance between the maxima +and minima points along the second principal component +is approximated to be the diameter of the branch instance. +The probability distribution of the joint likelihood map pℓ is +updated by po based on the following rules: +1) po of the voxels containing the line segment is equal to +the mask confidence score obtained from the instance +segmentation network. +2) po of the voxels in the proximity of the line segment +diminishes in the axial and radial directions along an +ellipsoidal contour (Fig. 4(c)): +po = c − c +k +��dl +l +�2 ++ +�dr +r +�2 +(1) +where c is the mask confidence score, l is the line +segment length, and r is the estimated radius. dl and +dr is the axial and radial distance of the voxel from +the line segment, and k is a parameter that controls +the rate at which po decreases as dl and dr increases. +For example, a large k results in a small decreasing +rate, effectively enlarging the size of the ellipsoid. We +empirically set k = 3 in our experiments. +3) The voxel mi of the 3D likelihood map is updated by +the observed occupancy probability po according to the +following equation (Fig. 4(d)): +pℓ(mi) = 1 − [1 − pℓ(mi)][1 − po(mi)] +(2) +The resulting joint likelihood map updated by po from all +line segments is depicted in Fig. 5. The 3D likelihood map +effectively has higher occupancy probability in voxels that +are directly intersected by a line segment, as well as in voxels +that are jointly influenced by more than one line segment. +D. Skeleton Extraction +The final tree skeleton GT = (VT , ET ) is an undirected +acyclic graph with vertices VT and edges ET representing +the topological structure of the tree canopy. This is obtained +by post-processing the accumulated line segments with the +joint likelihood map as follows. +We first obtain an initial tree skeleton Ginit = (Vinit, Einit) +by consolidating the accumulated line segments. To do so, + +branch +branch +branch +branch +branch +branch +ch088% +10979all line segments are converted into a set of vertices by +sampling points equally spaced along the line (in practice, +we set the spacing to be the voxel size of the likelihood map). +The vertices are consolidated into a skeletal curve using the +Laplacian smoothing method which repositions each vertex +to the mean of its k-nearest neighbors: +vi = 1 +k +N +� +j=1 +vj +(3) +where k is the number of neighborhood vertices, vj is the +position of neighbor vertex j, and vi is the new position of +vertex i. Laplacian smoothing is iteratively applied until the +total change in vertex position per iteration converges below +a threshold, which results in a set of well-refined skeletal +vertices Vinit. The edges Einit are initialized by building a +Euclidean minimum spanning tree from Vinit with edges that +are longer than the voxel size removed. +(a) +(b) +gi +Minimum +cost path +Gℓ +Fig. 6. +(a) The initial skeleton Ginit shown as the curves colored black +overlapped with the weighted likelihood graph Gℓ. Ginit is composed of +disjoint subgraphs gi due to occlusion. (b) We search for minimum cost +paths in Gℓ to join disconnected subgraphs gi. +Due to occlusion by leaves and branches, Ginit = {gi} +is composed of disjoint skeletal subgraphs gi of a single +tree. In order to predict the occluded parts of the tree, we +join maximally connected subgraphs in Ginit by searching for +minimum cost paths in the 3D likelihood map as illustrated +in Fig. 6 and summarized in Algorithm 1. The likelihood +map is first converted into a weighted graph Gℓ by adding +undirected edges between all 3 × 3 × 3 adjacent nodes (Line +2, Algorithm 1). The cost of an edge connecting vertex u +and v is set to be the negative log of the average occupancy +probability: +c(euv) = − log +�pℓ(mu) + pℓ(mv) +2 +� +(4) +which results in low cost for edges between nodes with high +occupancy probability. Using the weighted likelihood graph +Gℓ, the minimum cost path between all pairwise subgraph +combinations in GT (initialized with Ginit) are computed +(Line 5-7, Algorithm 1). The path with the minimum cost +is added to GT (Line 9-10, Algorithm 1). This process is +repeated until no more paths are found resulting in our final +skeleton output GT , composed of skeletal curves that were +directly observed, as well as predicted skeletal curves that +were unobservable due to occlusion. +Algorithm 1 Likelihood Map Path Search +Input: Initial Skeleton Ginit, Likelihood Map pℓ +Output: Tree Skeleton GT +1: +GT ← Ginit +2: +Gℓ ← WeightedUndirectedGraph(pℓ) +3: +Repeat +4: +P ← EmptyArray() +5: +for all pairwise subgraph combination (gu, gv) ∈ GT do +6: +ρ ← DijkstrasMinPath(gu, gv, Gℓ) +7: +P.append(ρ) +8: +end for +9: +ρmin ← MinCostPath(P) +10: +GT ← JoinSubgraphs(GT , ρmin) +11: Until no more paths are found +(a) +(b) +Foliage Density Level +1⃝ +2⃝ +3⃝ +4⃝ +Fig. 7. +(a) Synthetic meshes of an oak tree (top row), apple tree (middle +row), and a walnut tree (bottom row) with varying foliage density increasing +from left to right. Trees in the same row have the same structural topology. +Our synthetic dataset consists of ten unique topology per tree species. (b) +The robot is controlled to take images of the tree canopy from 10 different +viewpoints in the Gazebo simulation environment. +IV. EXPERIMENTS +We evaluate our proposed tree skeletonization method +through quantitative evaluation and visual assessment. For +the former, we collected a synthetic tree mesh dataset with +known ground truth skeleton and propose metrics to measure +the precision of the skeleton as well as the effectiveness +of predicting unseen branch skeletons. Visual assessment is +presented for the simulated dataset as well as a real tree +dataset collected from an apple orchard. +A. Experiments on Synthetic Trees +Experiments in simulation were performed on three differ- +ent types of synthetic trees including oak, apple, and walnut +trees. For each species, ten different structural topologies +with four levels of varying foliage density were generated +as depicted in Fig. 7(a), totalling to 120 trees. The synthetic +tree meshes were created in SpeedTree1 which were imported +into a Gazebo2 simulation environment to model an RGB-D +camera attached to a UR5 robot arm. For each tree, the robot +is controlled to collect images from 10 different viewpoints +as shown in Fig. 7(b), which are sequentially passed onto +our skeletonization pipeline. +We propose the following metrics to assess the correctness +of the constructed skeleton as well as the effectiveness of our +approach: +1www.speedtree.com +2www.gazebosim.org + +Tree Mesh +Ground Truth +Skeleton +Observed +Branches +MST +FTSEM +Our Method +Volume +Recovered +Fig. 8. +Experiment results of synthetic trees including oak (top row), apple (middle row), and walnut (bottom row) trees with foliage density level 3⃝. We +compare our method against MST-based and FTSEM-based methods. The extracted skeleton is color coded: TP, TPocc, and FP vertices are colored blue, +green, and red, respectively. The right-most column depicts the skeleton extracted from our method with recovered volume based on the radius information +of the skeletal vertices. +• Skeleton Precision and Recall: For each synthetic tree, +vertices of the output skeleton with a corresponding +point in the ground truth skeleton within a 0.02m +radius are labeled as true positives (TP), while vertices +without correspondence are labeled as false positives +(FP). Vertices of the ground truth skeleton without a +corresponding point in the output skeleton are labeled +as false negatives (FN). +• Occluded Skeleton Ratio (OSR): To quantify the +effectiveness of our approach with regards to predicting +occluded branch skeletons correctly, we measure the +percentage of true positive vertices that were obtained +from the minimum cost paths (TPocc) versus the total +number of vertices in the constructed skeleton: +OSR = +TPocc +TP + FP +(5) +We compare our results to those obtained from a minimum +spanning tree (MST) based method [16], [21] as well as +a method presented in FTSEM [17]. For the MST-based +method, our process of connecting disjoint subgraphs via +searching for minimum cost paths in the likelihood map (de- +scribed in Sec. III-D) is replaced by returning the Euclidean +minimum spanning tree built from Vinit as the final skeleton. +For FTSEM, we use the breakpoint connection method [17] +to connect disjoint subgraphs by checking for distance and +angle conditions between vectors computed from edges in +subgraph pairs. Since the code for FTSEM is not open +sourced, we implement it ourselves based on the available +details. +The results are summarized in Table I and visualized +in Fig. 8. The precision, recall, and OSR of our method +averaged over all 120 synthetic trees is 0.98, 0.59 and 0.11, +TABLE I +EXPERIMENT RESULTS OF TREE SKELETONIZATION +ON SYNTHETIC TREES. +Tree +Type +FD +MST +FTSEM +Ours +P +R +OSR +P +R +OSR +P +R +OSR +Oak +1 +0.99 +0.88 +0.05 +0.97 +0.86 +0.03 +0.99 +0.91 +0.04 +2 +0.96 +0.78 +0.08 +0.93 +0.73 +0.04 +0.98 +0.82 +0.08 +3 +0.89 +0.51 +0.14 +0.88 +0.46 +0.11 +0.96 +0.54 +0.15 +4 +0.86 +0.29 +0.17 +0.90 +0.27 +0.16 +0.98 +0.32 +0.19 +Apple +1 +0.94 +0.71 +0.14 +0.91 +0.64 +0.07 +0.98 +0.73 +0.12 +2 +0.95 +0.62 +0.16 +0.94 +0.51 +0.05 +0.99 +0.65 +0.13 +3 +0.95 +0.49 +0.14 +0.91 +0.40 +0.05 +0.99 +0.52 +0.14 +4 +0.94 +0.39 +0.15 +0.91 +0.32 +0.06 +0.97 +0.42 +0.16 +Walnut +1 +0.95 +0.70 +0.10 +0.95 +0.65 +0.07 +0.99 +0.72 +0.08 +2 +0.93 +0.64 +0.12 +0.93 +0.56 +0.06 +0.98 +0.66 +0.10 +3 +0.86 +0.48 +0.15 +0.84 +0.39 +0.10 +0.96 +0.49 +0.16 +4 +0.80 +0.34 +0.16 +0.84 +0.29 +0.13 +0.97 +0.34 +0.17 +* FD: Foliage Density, P: Precision, R: Recall +* Each score is averaged over 10 trees belonging to the tree type and +foliage density. +respectively. Our method outperforms both baselines in terms +of precision and recall as it joins disconnected skeletons only +when a path in the likelihood map exists. In contrast, the +MST-based method greedily joins all disconnected skeletons. +Although this results in higher OSR in less occluded scenes +(Table I, foliage density 1 & 2), the relative performance of +MST deteriorates with increasing foliage density (Table I, +foliage density 3 & 4). As a result, our method outperforms +MST in terms of precision, recall and OSR where there is +high occlusion. +B. Experiments on Real Trees +Experiments on real trees were performed on a dataset +that consists of 7 apple trees (Fig. 9) collected from the + +Fig. 9. +Experiment results of seven real apple trees. The first row shows the images of the apple tree. The region bounded by the red dashed square is +imaged by the robot with a stereo camera, corresponding to the 3D reconstructed point cloud depicted in the second row. The result of our skeletonization +method is shown in the third row with recovered volume by replacing skeleton vertices with spheres of corresponding radius. +(a) +(b) +Stereo Camera +UR5 Robot Arm +Fig. 10. +(a) Our hardware setup for collecting real world data at the UMass +Cold Spring Orchard, featuring a stereo camera attached to the UR5 robot +arm. (b) The robot motion path is shown by the red arrows, where it collects +70 stereo image pairs of each tree canopy at equally distributed waypoints +throughout the motion path. +University of Massachusetts Amherst Cold Spring Orchard. +For each tree, stereo images were collected from 70 different +viewpoints using a flash stereo camera [22] attached to the +UR5 robot arm (Fig. 10(a)). The robot was controlled to +follow a motion path as depicted in Fig. 10(b), where the +viewpoints are set to be at equally distributed waypoints +throughout the motion path. Due to the robot’s limited +workspace, we were able to capture a range of approximately +1.5 meters in height for each tree canopy. +Our results are visualized in Fig. 9. The OSR averaged +over all seven trees is 0.14, consistent with the results +obtained from the synthetic dataset. As it is difficult to obtain +the ground truth skeleton for real trees, we visually compare +the extracted skeleton with the 3D reconstructed tree using +known camera poses from the robot. Despite considerable +occlusion and noise evident in the reconstructed point cloud, +the extracted skeleton is topologically correct and shows +good correspondence. +The quality of the skeleton obtained from our pipeline is +contingent on the sufficiency and correctness of the detected +branches. We expect that a control policy to collect images +from optimal viewpoints [23] (rather than fixed viewpoints as +in our experiments) to perceive sufficient amount of branches +will further improve our proposed skeletonization pipeline. +V. CONCLUSION +Our tree skeletonization method outperforms the baselines +in situations with highly occluded canopies by accurately +estimating unobserved skeletons. As future work, we plan +to improve the algorithm runtime in addition to conducting +a more rigorous evaluation on real tree datasets. Other pos- +sible directions for future work include next-best viewpoint +optimization to increase the information of occluded regions +in the tree canopy. We are also interested in estimating the +dynamics of the tree in response to external forces to plan for +contact interactions. Ultimately, the digitized model of a tree +crop in the form of a skeleton presents promising direction to +developing safe and robust agricultural robotic manipulation. +ACKNOWLEDGMENT +This work was supported in part by NSF/USDA-NIFA +Cyber Physical Systems 2020-67021-31531, NSF Robust +Intelligence 1956163, and NSF/USDA-NIFA AIIRA AI Re- +search Institute 2021-67021-35329. The authors would like +to thank Daniel Cooley, Paul O’Connor, Jon Clements, Harry +Freeman and Abhisesh Silwal for their help in field data +collection at the University of Massachusetts Amherst Cold +Spring Orchard, and Hung-Jui Huang for helpful discussions. + +FRCREFERENCES +[1] H. Huang, S. Wu, D. Cohen-Or, M. Gong, H. Zhang, G. Li, and +B. Chen, “L1-medial skeleton of point cloud.,” ACM Trans. Graph., +vol. 32, no. 4, pp. 65–1, 2013. +[2] J. Cao, A. Tagliasacchi, M. Olson, H. Zhang, and Z. Su, “Point cloud +skeletons via laplacian based contraction,” in 2010 Shape Modeling +International Conference, pp. 187–197, IEEE, 2010. +[3] A. Tagliasacchi, H. Zhang, and D. Cohen-Or, “Curve skeleton extrac- +tion from incomplete point cloud,” in ACM SIGGRAPH 2009 papers, +pp. 1–9, 2009. +[4] F. Yandun, A. Silwal, and G. Kantor, “Visual 3d reconstruction +and dynamic simulation of fruit trees for robotic manipulation,” in +Proceedings of the IEEE/CVF Conference on Computer Vision and +Pattern Recognition Workshops, pp. 54–55, 2020. +[5] A. You, H. Kolano, N. Parayil, C. Grimm, and J. R. Davidson, +“Precision fruit tree pruning using a learned hybrid vision/interaction +controller,” in 2022 International Conference on Robotics and Au- +tomation (ICRA), pp. 2280–2286, IEEE, 2022. +[6] D. Sep´uLveda, R. Fern´andez, E. Navas, M. Armada, and P. Gonz´alez- +De-Santos, “Robotic aubergine harvesting using dual-arm manipula- +tion,” IEEE Access, vol. 8, pp. 121889–121904, 2020. +[7] C. Zhang, G. Yang, Y. Jiang, B. Xu, X. Li, Y. Zhu, L. Lei, R. Chen, +Z. Dong, and H. Yang, “Apple tree branch information extraction from +terrestrial laser scanning and backpack-lidar,” Remote Sensing, vol. 12, +no. 21, p. 3592, 2020. +[8] D. Zhang, N. Xie, S. Liang, and J. Jia, “3d tree skeletonization from +multiple images based on pyrlk optical flow,” Pattern Recognition +Letters, vol. 76, pp. 49–58, 2016. Special Issue on Skeletonization +and its Application. +[9] L. Fu, J. Liu, J. Zhou, M. Zhang, and Y. Lin, “Tree skeletonization +for raw point cloud exploiting cylindrical shape prior,” IEEE Access, +vol. 8, pp. 27327–27341, 2020. +[10] A. You, C. Grimm, A. Silwal, and J. R. Davidson, “Semantics-guided +skeletonization of upright fruiting offshoot trees for robotic pruning,” +Computers and Electronics in Agriculture, vol. 192, p. 106622, 2022. +[11] S. Delagrange, C. Jauvin, and P. Rochon, “Pypetree: A tool for +reconstructing tree perennial tissues from point clouds,” Sensors, +vol. 14, no. 3, pp. 4271–4289, 2014. +[12] A. Bucksch, R. Lindenbergh, and M. Menenti, “Skeltre,” The Visual +Computer, vol. 26, no. 10, pp. 1283–1300, 2010. +[13] J. Hackenberg, C. Morhart, J. Sheppard, H. Spiecker, and M. Disney, +“Highly accurate tree models derived from terrestrial laser scan data: +A method description,” Forests, vol. 5, no. 5, pp. 1069–1105, 2014. +[14] J. Mei, L. Zhang, S. Wu, Z. Wang, and L. Zhang, “3d tree modeling +from incomplete point clouds via optimization and l 1-mst,” Interna- +tional Journal of Geographical Information Science, vol. 31, no. 5, +pp. 999–1021, 2017. +[15] L. Gao, D. Zhang, N. Li, and L. Chen, “Force field driven skeleton +extraction method for point cloud trees,” Earth Science Informatics, +vol. 12, no. 2, pp. 161–171, 2019. +[16] Y. Livny, F. Yan, M. Olson, B. Chen, H. Zhang, and J. El-Sana, “Au- +tomatic reconstruction of tree skeletal structures from point clouds,” +in ACM SIGGRAPH Asia 2010 papers, pp. 1–8, 2010. +[17] J. Sun, P. Wang, R. Li, M. Zhou, and Y. Wu, “Fast tree skeleton +extraction using voxel thinning based on tree point cloud,” Remote +Sensing, vol. 14, no. 11, p. 2558, 2022. +[18] M. Bremer, M. Rutzinger, and V. Wichmann, “Derivation of tree +skeletons and error assessment using lidar point cloud data of varying +quality,” ISPRS Journal of Photogrammetry and Remote Sensing, +vol. 80, pp. 39–50, 2013. +[19] K. He, G. Gkioxari, P. Doll´ar, and R. Girshick, “Mask r-cnn,” in +Proceedings of the IEEE international conference on computer vision, +pp. 2961–2969, 2017. +[20] L. Piegl and W. Tiller, The NURBS book. Springer Science & Business +Media, 1996. +[21] B. Gorte and N. Pfeifer, “Structuring laser-scanned trees using 3d +mathematical morphology,” International Archives of Photogrammetry +and Remote Sensing, vol. 35, no. B5, pp. 929–933, 2004. +[22] A. Silwal, T. Parhar, F. Yandun, H. Baweja, and G. Kantor, “A robust +illumination-invariant camera system for agricultural applications,” in +2021 IEEE/RSJ International Conference on Intelligent Robots and +Systems (IROS), pp. 3292–3298, 2021. +[23] X. Zeng, T. Zaenker, and M. Bennewitz, “Deep reinforcement learning +for next-best-view planning in agricultural applications,” in 2022 Inter- +national Conference on Robotics and Automation (ICRA), pp. 2323– +2329, IEEE, 2022. + diff --git a/vNE_T4oBgHgl3EQf-hyD/content/tmp_files/load_file.txt b/vNE_T4oBgHgl3EQf-hyD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a423b3f2d557230f296a9b298c0d50e2fd430de9 --- /dev/null +++ b/vNE_T4oBgHgl3EQf-hyD/content/tmp_files/load_file.txt @@ -0,0 +1,513 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf,len=512 +page_content='Occlusion Reasoning for Skeleton Extraction of Self-Occluded Tree Canopies Chung Hee Kim and George Kantor Abstract— In this work, we present a method to extract the skeleton of a self-occluded tree canopy by estimating the unobserved structures of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A tree skeleton compactly describes the topological structure and contains useful infor- mation such as branch geometry, positions and hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' This can be critical to planning contact interactions for agricultural manipulation, yet is difficult to gain due to occlusion by leaves, fruits and other branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Our method uses an instance segmentation network to detect visible trunk, branches, and twigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Then, based on the observed tree structures, we build a custom 3D likelihood map in the form of an occupancy grid to hypothesize on the presence of occluded skeletons through a series of minimum cost path searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We show that our method outperforms baseline methods in highly occluded scenes, demonstrated through a set of experiments on a synthetic tree dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Qualitative results are also presented on a real tree dataset collected from the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' INTRODUCTION With increasing global population and labor shortages, modern agriculture is adopting new technologies to enhance sustainability and profitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' There is growing effort in developing robotic solutions to automate repetitive and la- borious tasks that often require complex and delicate in- teraction with crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Harvesting and pruning, for example, may require the agent to manipulate crops by pushing aside leaves or branches to reach occluded regions before picking or cutting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Understanding the unstructured and cluttered task environment is critical to automating such challenging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' In this work, we address the perception problem, where robots must be able to sense and understand the complex task environment prior to crop interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We are particularly interested in extracting the skeleton of a self-occluded tree canopy by estimating the unobserved parts of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A tree skeleton is useful as it describes the topological structure and contains useful information such as branch dimensions, positions, and hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' This knowledge is helpful in pheno- typing crop characteristics for growth assessment and can be critical for planning contact interactions, such as determining optimal pruning locations or generating trajectories to pick fruits by pulling a branch into its workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' One of the main challenges faced during the extraction of the tree skeleton from sensor data (typically in the form of point clouds obtained from laser scanners or depth cameras) stems from noise and occlusion, which is exacerbated by the presence of foliage as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Although there exists prior works that address skeletonization of occluded tree canopies, they are tailored towards sparse point clouds C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Kim and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Kantor are with Carnegie Mellon University, Pittsburgh PA, USA {chunghek, kantor}@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='edu (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) The 3D reconstruction (middle) and skeletonization (right) of a heavily occluded tree canopy (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (b) Tree crops are tightly organized in rows in industrial orchard settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' obtained from terrestrial laser scanners (TLS) and can be ineffective when applied to dense point clouds acquired from RGB-D or stereo cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Furthermore, it is generally assumed in prior works that the tree point cloud to be skeletonized is pre-registered from a 360 degree scan, which is difficult to obtain in industrial orchard settings where trees are tightly organized in rows as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We present a novel skeletonzation method that extracts tree skeletons from one-sided views of the tree canopy by using a depth camera attached to a robot arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Our method particularly addresses the challenge of skeletonizing heavily occluded tree canopies by observing that branches in nature generally extend linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We use this as a heuristic assumption to approximate a 3D likelihood map in the form of an occupancy grid to predict presence of occluded branch structures by searching for minimum cost paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We validate the effectiveness of our approach by presenting quantitative assessment on a synthetic tree dataset with known ground truth, and present qualitative results from a real tree dataset collected at an apple orchard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' RELATED WORK A skeleton can be defined as a curve expressing the shape of an object that is consistent with the topological structure as well as the connectivity of the original object shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Many methods have been proposed to extract skeletons from unordered 3D point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A skeleton can be extracted by measuring the L1-median to determine local centers of the point cloud [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A Laplacian-based contraction method is proposed in [2], which collapses the input point cloud into a minimal volume using an iterative Laplacian smoothing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' In [3], a skeleton is extracted by defining a new feature representation called rotational symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The aforementioned methods work well with limited noise and occlusion, while we are particularly interested in skeletoniz- ing point cloud of botanical trees which are often noisy and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='08387v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='RO] 20 Jan 2023 FROFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Our system pipeline for tree skeletonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' It takes as input a series of RGB-D images of a self-occluded tree from multiple viewpoints and outputs the underlying tree skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' There is particular interest in extracting the skeleton of a tree canopy due to implications in automating agricultural tasks [4] such as pruning [5], harvesting [6], or phenotyping to estimate crop characteristics for growth assessment [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Prior knowledge about tree structures are commonly used to extract tree skeletons, such as branching properties [8], cylindrical shape priors [9] and upright offshoots [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Tree skeletons are commonly represented as graphs composed of vertices and edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [11] and [12] uses graph-based methods to extract tree skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A geometry-based method is used to fit cylinders on the point cloud in a hierarchical data structure to encapsulate parent-child relations of branches [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' While the above mentioned tree skeletonization meth- ods are tailored for trees without leaves, we address a more challenging problem in which tree canopies are self- occluded by leaves, fruits, or other branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' To address the problem of missing data, [14] proposes an iterative data- completion method to recover data for 3D tree modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Tree point clouds with leaves are often first separated into branches and leaves before skeleton extraction [15], while [16] directly generates visually convincing tree skeletons based on optimization driven by biological priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Several works improve skeleton connectivity by joining disconnected skeletal structures through geometry-based methods [17], [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' we use it as a baseline method and quantitatively show that our method outperforms [17] in occluded scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Despite addressing the issue of occlusion, these methods are built for sparse TLS point clouds and are inapplicable to dense point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' More importantly, it is assumed that the tree point cloud to be skeletonized is pre-registered from a 360 degree scan, which may be unpractical in industrial orchard setting where trees are organized in rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The key contributions of this paper are: A novel skeletonzation method that extracts tree skele- tons from one-sided views of the tree canopy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A method to reason about the unobserved structures of the tree to predict the presence of occluded skeletons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Experimental results on a synthetic dataset quantita- tively compared against existing baselines, as well as qualitative results on a real tree dataset collected from the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' System Overview Our tree skeletonization method aims to improve the knowledge of occluded regions in the tree that is often self- occluded by foliage, fruits, or other branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Driven by a heuristic assumption on tree structures, we approximate a 3D likelihood map in the form of an occupancy grid that stores information of visible as well as occluded structures of a tree canopy to extract its skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2 shows an overview of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The system takes as input a series of color and depth images from varying viewpoints obtained from a camera attached to a robot arm with known poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The images are passed onto an instance segmentation network to generate a semantic point cloud of branch clusters (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' III- B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The occupancy probability distribution of the 3D likeli- hood map is updated based on the observed branch clusters over a sequence of images (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The final skeleton is generated by joining disconnected branch skeletons through a series of minimum cost path searches in the likelihood map (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' III-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Semantic Point Cloud Acquisition We first acquire a 3D point cloud semantically labeled with visible parts of the tree trunk, branches, and twigs (collectively referred to as branches from this point on) that are not occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' This is achieved by projecting branch seg- mentation masks in 2D color images on to the 3D point cloud obtained from the depth image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The points corresponding to branches in the semantic point cloud are clustered based on the projected masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Our branch segmentation is based on the Mask R-CNN [19] instance segmentation network with a Feature Pyramid Network and ResNet50 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The model takes as input 1440×1080 images and outputs instance segmentation masks as well as its confidence scores ranging from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The network was trained on 130 manually labeled images (105 real images and 25 synthetic images) for 2000 iterations on an NVIDIA GeForce RTX 3070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Sample segmentation results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' In order to perform line- fitting on the point clusters (further described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' III-C), we deliberately labeled each branch instances in the training images to be a slender polygon such that the projected For each viewpoint: Color Image Instance Final Tree Skeleton Task Setting Depth Map Accumulated Edges Initial Skeleton(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) Instance segmentation results on a real tree image (left) and a synthetic tree image (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (b) Tree forks (left) or angled branch instances (right) are labeled as multiple branch instances in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' branch point cloud cluster in 3D space is also piece-wise slender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For example, a tree fork or a branch instance with a significant change in direction is labeled as two or more branch instances instead of a single instance as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Skeleton Occupancy Likelihood Map We now propose our novel approach to hypothesize on the presence of occluded branches by approximating a 3D likelihood map in the form of an occupancy grid, where the i-th grid voxel mi stores the skeleton occupancy probability pℓ(mi) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The approximation is driven by a heuristic assumption: Given a visible branch instance, it is likely that the branch structure extends longer along its growth direction based on the observation that branches in nature generally grow straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We model this as individually observed prob- ability distributions po with an ellipsoidal contour obtained from the branch point cloud clusters as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) (b) (c) (d) PC1 PC2 cp1 cp2 cp3 cp4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='0 pℓ po cp2 cp3 dl dr l r Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) A single instance of a segmented branch component with a confidence score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (b) The branch instance is projected into 3D space as a dense point cloud cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We compute the principal components (PCi) and fit a B-spline curve with four control points (cpi) on the point cloud cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (c) po is obtained from each line segment, (d) which updates pℓ using equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A 3-dimensional B-Spline curve of degree-one with four control points is approximated on a point cloud cluster using the least squares method [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' This results in three connected line segments that best represent the skeletal geometry of the point cloud cluster as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 4(b), which are Observed Branch Instances Accumulated Line Segments Joint 3D Likelihood Map −→ −→ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Visualization of the overall joint 3D likelihood map (right) obtained from accumulated line segments (middle), which were extracted from observed branch clusters over the sequence of images (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' accumulated over the sequence of images (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The radius of the branch instance is also estimated by performing principal component analysis on the point cloud cluster (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 4(b)), where the distance between the maxima and minima points along the second principal component is approximated to be the diameter of the branch instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The probability distribution of the joint likelihood map pℓ is updated by po based on the following rules: 1) po of the voxels containing the line segment is equal to the mask confidence score obtained from the instance segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2) po of the voxels in the proximity of the line segment diminishes in the axial and radial directions along an ellipsoidal contour (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 4(c)): po = c − c k ��dl l �2 + �dr r �2 (1) where c is the mask confidence score, l is the line segment length, and r is the estimated radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' dl and dr is the axial and radial distance of the voxel from the line segment, and k is a parameter that controls the rate at which po decreases as dl and dr increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For example, a large k results in a small decreasing rate, effectively enlarging the size of the ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We empirically set k = 3 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 3) The voxel mi of the 3D likelihood map is updated by the observed occupancy probability po according to the following equation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 4(d)): pℓ(mi) = 1 − [1 − pℓ(mi)][1 − po(mi)] (2) The resulting joint likelihood map updated by po from all line segments is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The 3D likelihood map effectively has higher occupancy probability in voxels that are directly intersected by a line segment, as well as in voxels that are jointly influenced by more than one line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Skeleton Extraction The final tree skeleton GT = (VT , ET ) is an undirected acyclic graph with vertices VT and edges ET representing the topological structure of the tree canopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' This is obtained by post-processing the accumulated line segments with the joint likelihood map as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We first obtain an initial tree skeleton Ginit = (Vinit, Einit) by consolidating the accumulated line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' To do so, branch branch branch branch branch branch ch088% 10979all line segments are converted into a set of vertices by sampling points equally spaced along the line (in practice, we set the spacing to be the voxel size of the likelihood map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The vertices are consolidated into a skeletal curve using the Laplacian smoothing method which repositions each vertex to the mean of its k-nearest neighbors: vi = 1 k N � j=1 vj (3) where k is the number of neighborhood vertices, vj is the position of neighbor vertex j, and vi is the new position of vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Laplacian smoothing is iteratively applied until the total change in vertex position per iteration converges below a threshold, which results in a set of well-refined skeletal vertices Vinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The edges Einit are initialized by building a Euclidean minimum spanning tree from Vinit with edges that are longer than the voxel size removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) (b) gi Minimum cost path Gℓ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) The initial skeleton Ginit shown as the curves colored black overlapped with the weighted likelihood graph Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Ginit is composed of disjoint subgraphs gi due to occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (b) We search for minimum cost paths in Gℓ to join disconnected subgraphs gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Due to occlusion by leaves and branches, Ginit = {gi} is composed of disjoint skeletal subgraphs gi of a single tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' In order to predict the occluded parts of the tree, we join maximally connected subgraphs in Ginit by searching for minimum cost paths in the 3D likelihood map as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 6 and summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The likelihood map is first converted into a weighted graph Gℓ by adding undirected edges between all 3 × 3 × 3 adjacent nodes (Line 2, Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The cost of an edge connecting vertex u and v is set to be the negative log of the average occupancy probability: c(euv) = − log �pℓ(mu) + pℓ(mv) 2 � (4) which results in low cost for edges between nodes with high occupancy probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Using the weighted likelihood graph Gℓ, the minimum cost path between all pairwise subgraph combinations in GT (initialized with Ginit) are computed (Line 5-7, Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The path with the minimum cost is added to GT (Line 9-10, Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' This process is repeated until no more paths are found resulting in our final skeleton output GT , composed of skeletal curves that were directly observed, as well as predicted skeletal curves that were unobservable due to occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Algorithm 1 Likelihood Map Path Search Input: Initial Skeleton Ginit, Likelihood Map pℓ Output: Tree Skeleton GT 1: GT ← Ginit 2: Gℓ ← WeightedUndirectedGraph(pℓ) 3: Repeat 4: P ← EmptyArray() 5: for all pairwise subgraph combination (gu, gv) ∈ GT do 6: ρ ← DijkstrasMinPath(gu, gv, Gℓ) 7: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='append(ρ) 8: end for 9: ρmin ← MinCostPath(P) 10: GT ← JoinSubgraphs(GT , ρmin) 11: Until no more paths are found (a) (b) Foliage Density Level 1⃝ 2⃝ 3⃝ 4⃝ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) Synthetic meshes of an oak tree (top row), apple tree (middle row), and a walnut tree (bottom row) with varying foliage density increasing from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Trees in the same row have the same structural topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Our synthetic dataset consists of ten unique topology per tree species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (b) The robot is controlled to take images of the tree canopy from 10 different viewpoints in the Gazebo simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' EXPERIMENTS We evaluate our proposed tree skeletonization method through quantitative evaluation and visual assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For the former, we collected a synthetic tree mesh dataset with known ground truth skeleton and propose metrics to measure the precision of the skeleton as well as the effectiveness of predicting unseen branch skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Visual assessment is presented for the simulated dataset as well as a real tree dataset collected from an apple orchard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Experiments on Synthetic Trees Experiments in simulation were performed on three differ- ent types of synthetic trees including oak, apple, and walnut trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For each species, ten different structural topologies with four levels of varying foliage density were generated as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 7(a), totalling to 120 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The synthetic tree meshes were created in SpeedTree1 which were imported into a Gazebo2 simulation environment to model an RGB-D camera attached to a UR5 robot arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For each tree, the robot is controlled to collect images from 10 different viewpoints as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 7(b), which are sequentially passed onto our skeletonization pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We propose the following metrics to assess the correctness of the constructed skeleton as well as the effectiveness of our approach: 1www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='speedtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='com 2www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='gazebosim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='org Tree Mesh Ground Truth Skeleton Observed Branches MST FTSEM Our Method Volume Recovered Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Experiment results of synthetic trees including oak (top row), apple (middle row), and walnut (bottom row) trees with foliage density level 3⃝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We compare our method against MST-based and FTSEM-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The extracted skeleton is color coded: TP, TPocc, and FP vertices are colored blue, green, and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The right-most column depicts the skeleton extracted from our method with recovered volume based on the radius information of the skeletal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Skeleton Precision and Recall: For each synthetic tree, vertices of the output skeleton with a corresponding point in the ground truth skeleton within a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='02m radius are labeled as true positives (TP), while vertices without correspondence are labeled as false positives (FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Vertices of the ground truth skeleton without a corresponding point in the output skeleton are labeled as false negatives (FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Occluded Skeleton Ratio (OSR): To quantify the effectiveness of our approach with regards to predicting occluded branch skeletons correctly, we measure the percentage of true positive vertices that were obtained from the minimum cost paths (TPocc) versus the total number of vertices in the constructed skeleton: OSR = TPocc TP + FP (5) We compare our results to those obtained from a minimum spanning tree (MST) based method [16], [21] as well as a method presented in FTSEM [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For the MST-based method, our process of connecting disjoint subgraphs via searching for minimum cost paths in the likelihood map (de- scribed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' III-D) is replaced by returning the Euclidean minimum spanning tree built from Vinit as the final skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For FTSEM, we use the breakpoint connection method [17] to connect disjoint subgraphs by checking for distance and angle conditions between vectors computed from edges in subgraph pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Since the code for FTSEM is not open sourced, we implement it ourselves based on the available details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The results are summarized in Table I and visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The precision, recall, and OSR of our method averaged over all 120 synthetic trees is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='59 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='11, TABLE I EXPERIMENT RESULTS OF TREE SKELETONIZATION ON SYNTHETIC TREES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Tree Type FD MST FTSEM Ours P R OSR P R OSR P R OSR Oak 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='17 FD: Foliage Density, P: Precision, R: Recall Each score is averaged over 10 trees belonging to the tree type and foliage density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Our method outperforms both baselines in terms of precision and recall as it joins disconnected skeletons only when a path in the likelihood map exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' In contrast, the MST-based method greedily joins all disconnected skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Although this results in higher OSR in less occluded scenes (Table I, foliage density 1 & 2), the relative performance of MST deteriorates with increasing foliage density (Table I, foliage density 3 & 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' As a result, our method outperforms MST in terms of precision, recall and OSR where there is high occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Experiments on Real Trees Experiments on real trees were performed on a dataset that consists of 7 apple trees (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 9) collected from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Experiment results of seven real apple trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The first row shows the images of the apple tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The region bounded by the red dashed square is imaged by the robot with a stereo camera, corresponding to the 3D reconstructed point cloud depicted in the second row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The result of our skeletonization method is shown in the third row with recovered volume by replacing skeleton vertices with spheres of corresponding radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) (b) Stereo Camera UR5 Robot Arm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (a) Our hardware setup for collecting real world data at the UMass Cold Spring Orchard, featuring a stereo camera attached to the UR5 robot arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' (b) The robot motion path is shown by the red arrows, where it collects 70 stereo image pairs of each tree canopy at equally distributed waypoints throughout the motion path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' University of Massachusetts Amherst Cold Spring Orchard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' For each tree, stereo images were collected from 70 different viewpoints using a flash stereo camera [22] attached to the UR5 robot arm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 10(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The robot was controlled to follow a motion path as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 10(b), where the viewpoints are set to be at equally distributed waypoints throughout the motion path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Due to the robot’s limited workspace, we were able to capture a range of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='5 meters in height for each tree canopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Our results are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The OSR averaged over all seven trees is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content='14, consistent with the results obtained from the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' As it is difficult to obtain the ground truth skeleton for real trees, we visually compare the extracted skeleton with the 3D reconstructed tree using known camera poses from the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Despite considerable occlusion and noise evident in the reconstructed point cloud, the extracted skeleton is topologically correct and shows good correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The quality of the skeleton obtained from our pipeline is contingent on the sufficiency and correctness of the detected branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We expect that a control policy to collect images from optimal viewpoints [23] (rather than fixed viewpoints as in our experiments) to perceive sufficient amount of branches will further improve our proposed skeletonization pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' CONCLUSION Our tree skeletonization method outperforms the baselines in situations with highly occluded canopies by accurately estimating unobserved skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' As future work, we plan to improve the algorithm runtime in addition to conducting a more rigorous evaluation on real tree datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Other pos- sible directions for future work include next-best viewpoint optimization to increase the information of occluded regions in the tree canopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' We are also interested in estimating the dynamics of the tree in response to external forces to plan for contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Ultimately, the digitized model of a tree crop in the form of a skeleton presents promising direction to developing safe and robust agricultural robotic manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported in part by NSF/USDA-NIFA Cyber Physical Systems 2020-67021-31531, NSF Robust Intelligence 1956163, and NSF/USDA-NIFA AIIRA AI Re- search Institute 2021-67021-35329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' The authors would like to thank Daniel Cooley, Paul O’Connor, Jon Clements, Harry Freeman and Abhisesh Silwal for their help in field data collection at the University of Massachusetts Amherst Cold Spring Orchard, and Hung-Jui Huang for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' FRCREFERENCES [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Cohen-Or, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Gong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Li, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Chen, “L1-medial skeleton of point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=',” ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 65–1, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Cao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Tagliasacchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Olson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Su, “Point cloud skeletons via laplacian based contraction,” in 2010 Shape Modeling International Conference, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 187–197, IEEE, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Tagliasacchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Cohen-Or, “Curve skeleton extrac- tion from incomplete point cloud,” in ACM SIGGRAPH 2009 papers, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 1–9, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Yandun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Silwal, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Kantor, “Visual 3d reconstruction and dynamic simulation of fruit trees for robotic manipulation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 54–55, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' You, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Kolano, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Parayil, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Grimm, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Davidson, “Precision fruit tree pruning using a learned hybrid vision/interaction controller,” in 2022 International Conference on Robotics and Au- tomation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2280–2286, IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Sep´uLveda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Fern´andez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Navas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Armada, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Gonz´alez- De-Santos, “Robotic aubergine harvesting using dual-arm manipula- tion,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 121889–121904, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Jiang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Lei, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Dong, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Yang, “Apple tree branch information extraction from terrestrial laser scanning and backpack-lidar,” Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 3592, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Liang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Jia, “3d tree skeletonization from multiple images based on pyrlk optical flow,” Pattern Recognition Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 76, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 49–58, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Special Issue on Skeletonization and its Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Fu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Lin, “Tree skeletonization for raw point cloud exploiting cylindrical shape prior,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 27327–27341, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' You, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Grimm, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Silwal, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Davidson, “Semantics-guided skeletonization of upright fruiting offshoot trees for robotic pruning,” Computers and Electronics in Agriculture, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 192, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 106622, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Delagrange, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Jauvin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Rochon, “Pypetree: A tool for reconstructing tree perennial tissues from point clouds,” Sensors, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 4271–4289, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Bucksch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Lindenbergh, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Menenti, “Skeltre,” The Visual Computer, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 1283–1300, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Hackenberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Morhart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Sheppard, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Spiecker, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Disney, “Highly accurate tree models derived from terrestrial laser scan data: A method description,” Forests, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 1069–1105, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Mei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Wang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, “3d tree modeling from incomplete point clouds via optimization and l 1-mst,” Interna- tional Journal of Geographical Information Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 999–1021, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Gao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Li, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Chen, “Force field driven skeleton extraction method for point cloud trees,” Earth Science Informatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 161–171, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [16] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Livny, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Yan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Olson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' El-Sana, “Au- tomatic reconstruction of tree skeletal structures from point clouds,” in ACM SIGGRAPH Asia 2010 papers, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 1–8, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Sun, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zhou, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Wu, “Fast tree skeleton extraction using voxel thinning based on tree point cloud,” Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2558, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Bremer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Rutzinger, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Wichmann, “Derivation of tree skeletons and error assessment using lidar point cloud data of varying quality,” ISPRS Journal of Photogrammetry and Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 80, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 39–50, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Gkioxari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Doll´ar, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Girshick, “Mask r-cnn,” in Proceedings of the IEEE international conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2961–2969, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Piegl and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Tiller, The NURBS book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Springer Science & Business Media, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Gorte and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Pfeifer, “Structuring laser-scanned trees using 3d mathematical morphology,” International Archives of Photogrammetry and Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' B5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 929–933, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Silwal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Parhar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Yandun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Baweja, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Kantor, “A robust illumination-invariant camera system for agricultural applications,” in 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 3292–3298, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' [23] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zeng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Zaenker, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' Bennewitz, “Deep reinforcement learning for next-best-view planning in agricultural applications,” in 2022 Inter- national Conference on Robotics and Automation (ICRA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} +page_content=' 2323– 2329, IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE_T4oBgHgl3EQf-hyD/content/2301.08387v1.pdf'} diff --git a/vdA0T4oBgHgl3EQfL__A/content/tmp_files/2301.02127v1.pdf.txt b/vdA0T4oBgHgl3EQfL__A/content/tmp_files/2301.02127v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a60c2096e117aee3f4a0b512602d99bbbd5eb6cd --- /dev/null +++ b/vdA0T4oBgHgl3EQfL__A/content/tmp_files/2301.02127v1.pdf.txt @@ -0,0 +1,2068 @@ +Generalized Dicke model and gauge-invariant master equations for two atoms in +ultrastrongly-coupled cavity quantum electrodynamics +Kamran Akbari,1, ∗ Will Salmon,1, ∗ Franco Nori,2, 3, 4 and Stephen Hughes1 +1Department of Physics, Engineering Physics and Astronomy, +Queen’s University, Kingston ON K7L 3N6, Canada +2Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan +3Quantum Computing Center, RiKEN, Wakoshi, Saitama, 351-0198, Japan +4Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA +(Dated: January 6, 2023) +We study a generalization of the well-known Dicke model, using two dissimilar atoms in the +regime of ultrastrongly coupled cavity quantum electrodynamics. Our theory uses gauge invariant +master equations, which yields consistent results in either of the standard multipolar and Coulomb +gauges, including system-bath interactions for open cavity systems. We first show how a second +atom can be treated as a sensor atom to measure the output spectrum from a single atom in the +ultrastrong-coupling regime, and compare results with the quantum regression theorem, explaining +when they can be different. We then focus on the case where the second atom is also ultrastrongly +coupled to the cavity, but with different parameters from those of the first atom, which introduces +complex coupling effects and additional resonances and spectral features. In particular, we show +multiple resonances in the cavity spectra that are visible off-resonance, which cannot be seen when +the second atom is on-resonance with the rest of the system. We also observe clear anti-crossing +features particularly pronounced for when the second atom tunes through resonance. +I. +INTRODUCTION +Recent progress in the strong and ultrastrong (USC) +regimes of light-matter interaction has opened up sig- +nificant advances in theoretical and experimental re- +search in quantum optical systems [1–9]. These strong +coupling regimes allow one to coherently exchange ex- +citations between matter and light, enabling break- +throughs in fundamental quantum experiments and tech- +nologies [1, 4, 5, 10, 11]. +In particular, USC exploits the nature of counter- +rotating wave physics and pondermotive forces [4, 5], and +pushes one toward a non-perturbative regime where the +light and matter excitations must be treated on an equal +footing, i.e., as joint/dressed states [12], where even the +ground state can contain virtual photons. These features +make the USC regime responsible for many intriguing +phenomena including the formation of high-quality quasi- +particle (e.g. exciton-polariton or plasmon-photon) col- +lective modes, hybrid and entangled states with higher +degrees of controllablity [8, 9, 13–19]. +The intricate interactions between quantized cavity +modes and quantum emitters can be modeled in the +framework of cavity quantum electrodynamics (cavity- +QED), where atoms and atom-like structures (e.g., quan- +tum dots, molecules, superconducting circuits) interact +with a (dominant) single quantized cavity mode [20–22]. +Traditionally, strong coupling occurs when the cavity- +emitter rate, g, exceeds any dissipation rate κ (of the +cavity) or γ (decay of the emitter) [23–29], while the +∗ These two authors contributed equally to this work: +kam- +ran.akbari@queensu.ca, will.salmon@queensu.ca +USC regime is characterized not only by the lower rates +of decoherence, but also when the atom-cavity coupling +strength, g, becomes a significant fraction of the bare +energies, ω0, of the system, commonly quantified as +g > 0.1ω0 [4, 5]. Additionally, the hybridization of quan- +tum states with different numbers of excitations in the +USC regime results in a population of virtual photons in +a dissipative system’s ground state, also with significant +loss (i.e., not even in the strong coupling regime) [30]. +The profound applicability of cavity-QED and its ease +of modeling are derived from truncating the full emitter +problem to a two-level system (TLS), which is typically +coupled to a single quantized cavity mode. +However, +the truncation of the Hilbert space, in either the ma- +terial and/or photonic part, causes problems for gauge +invariance when working in the USC regime [12, 31–33]. +Recently, many of these issues have been partly fixed +for the standard quantum Rabi model (QRM) Hamilto- +nian [31, 34, 35], and extended recently to ensure that +dissipation and input/output is also included in a gauge +invariant way [34]. More general quantization for arbi- +trary media, and the USC regime, has been reported as +well [33]. +A. +Gauge invariance +In the dipole gauge (specifically, the dipole approxi- +mation in the multipolar gauge), the QRM describes the +TLS-cavity system via the Hamiltonian [12, 31] (in units +of ℏ = 1): +ˆHD +QR = ωca†a + ωa +2 σz + igD(a† − a)σx, +(1) +arXiv:2301.02127v1 [quant-ph] 5 Jan 2023 + +2 +up to a constant (1ωcη2), where ωc is the cavity transi- +tion frequency, a (a†) is the cavity photon annihilation +(creation) operator, ωa is the TLS transition frequency, +σz = σ+σ− − σ−σ+ and σx = σ+ + σ−, with σ+ = |e⟩⟨g| +(σ− = |g⟩⟨e|) the atomic raising (lowering) operator; +also, gD is the atom-cavity coupling in the dipole gauge +(gD ∝ √ωc), and η = gD/ωc is the normalized coupling +parameter. +We can neglect terms proportional to the +identity as these do not affect the system dynamics; they +simply introduce an offset in the ground state energy, +which we can normalize to any value. Equation (1) re- +duces to the Jaynes-Cummings model (JCM) in the ro- +tating wave approximation (RWA) as [36, 37] +ˆHD +JC = ωca†a + ωa +2 σz + igD(a†σ− − aσ+). +(2) +When the system is subjected to matter truncation, +ˆHD +QR produces the correct eigenenergies, but the funda- +mental electric field operator [34, 38] ˆE ∝ −i(a′† − a′), +where a′ = a + iησx, which can be derived from several +different viewpoints [31, 34, 38]. For example, in the re- +stricted TLS subspace, one can transform the Coulomb +gauge operators to the dipole gauge operators, through +the projected unitary transform [31] U = exp[−iη(a + +a†)σx], so that a′ → UaU† = a + iησx [38]. These trans- +formed operators then must be used when computing +cavity field observables and for deriving master equa- +tions. +In the Coulomb gauge, the standard system Hamilto- +nian for the QRM is [12, 31, 39] +ˆHC,naive +QR += ωca†a+(ωa/2)σz +gC(a+a†)σy +D(a+a†)2, +(3) +where σy = i(σ− − σ+), gC = gDωa/ωc, and D = +(gC)2/ωa is the ponderomotive coupling strength [40]. +Unfortunately, this “naive” system Hamiltonian is wrong +(which is why we use this name in the superscript) as +it does not produce the correct eigenenergies in the USC +regime [12], and breaks gauge invariance. The breakdown +of gauge invariance here can be seen as a formation of a +potential nonlocality due to the truncation of the matter +Hilbert space [31, 41]. Instead, by applying a proper uni- +tary gauge transformation (i.e., the spin rotation along +the x-axis) to the dipole gauge-independent QRM model +Hamiltonian, the correct gauge-fixed Coulomb QRM +Hamiltonian reads [31] +ˆHC +QR = ωca†a ++ ωa +2 +� +σz cos[2(a + a†)η] + σy sin[2(a + a†)η] +� +, +(4) +which produces identical eigenenergies to ˆHD +QR. +B. +Gauge-invariant generalized master equation +For realistic cavities, one must also account for dissi- +pation/losses and photon input-output channels. Gen- +erally, open-system cavity-QED problems are introduced +by subjecting the atom and the cavity attached to general +baths, as an open quantum system. In such situations, +a master equation description is widely used, leading to +the understanding of the cavity spectra and other desired +observables [42–45]. +Commonly, the bare-state master equation formalism, +where the joint basis states are constructed from the bare +light states and the bare matter states before light-matter +interaction, is often used in open system cavity-QED, +yielding the standard Lindblad master equation. How- +ever, the bare-state master equation formalism uses the +wrong states in the USC regime (including the ground +state, which is now an entangled state of photons and +matter) and it has been shown that one needs a dressed- +state approach to avoid unphysical transitions [46]. More +generally, one also needs a “generalized” master equation +(GME) approach to account for frequency-dependent +baths and non-secular effects [44]. +Beyond these de- +tails, in the USC regime, such approaches are typically +gauge relative, and again one must use a corrected a′ for +cavity mode operators with the dipole gauge or use the +corrected Coulomb gauge Hamiltonian. Although such +studies have so far assumed very simple models for the +system-bath interactions, these approaches do produce +gauge-independent results [34, 35]. +An advantage of using a GME approach is that realistic +observables can be computed, such as the cavity-emitted +spectra, typically using the quantum regression theo- +rem [34, 47]. However, gauge-independent GMEs have +so far only been applied to the case of one atom/TLS, +and we can also expect a significant impact when ap- +plied to multiple atoms. In this regard, the Dicke model +is a fundamental model of quantum optics describing the +light-matter interaction where a cavity mode is coupled +with a set of identical TLSs [48, 49]. The model is known +to be an established description for a class of intriguing +phenomena in cavity-QED such as superradiant phase +transitions and quantum chaos [49–61]. +C. +Dicke model in the USC regime +The Dicke model has also been investigated in the USC +regime [62–69]. In the study of effective light-matter in- +teractions in a circuit QED system, coupled symmetri- +cally to multiple superconducting qubits, Ref. [63] stud- +ied a microscopic model Hamiltonian that not only de- +scribes the usual collective qubit-photon coupling but +also the effect of direct qubit-qubit interactions. Vari- +ous other works in the USC regime have been presented +on related coupling effects, mainly at the thermodynamic +limit, using simple system Hamiltonians [70, 71]. In these +extended Dicke models, similar to the previous stud- +ies on the Dicke model or even the Hopfield model in +the USC regime [64–67], the atoms are degenerate (i.e., +they share the same coupling coefficient and resonant fre- +quency). Recent studies also include gauge-invariant sys- + +3 +tem Hamiltonian models [67], or even discuss more ex- +otic schemes of the Dicke model, such as the anisotropic +or nonequilibrium models in which the counter- and co- +rotating terms have different coupling strengths, but the +two atoms are still identical [68, 69]. +D. +Generalized Dicke model +It is desirable to explore a more general two-atom case +where the TLS parameters can be different, and a natu- +ral extension to investigate is a system of two dissimilar +atoms, namely, the generalized Dicke model (GDM), in +the limit of two atoms. From a practical viewpoint, one +must also include realistic dissipation and input-output +channels to the system. In this paper, we present such +a study, using gauge-independent master equations valid +for exploring USC dynamics. Ultimately, this GDM is +a more realistic scenario for studying how atoms inter- +act in the USC regime, as it is practically impossible to +experimentally produce two identical effective TLSs for +experimental systems [72–75]. Coupling with two differ- +ent atoms also leads to new coupling regimes that are not +accessible with identical atoms. +We study the two-atom GDM by introducing a dis- +parate second atom to a general one-atom-cavity USC +problem, using a gauge-invariant GME description. We +exploit this model in two different ways: (i) we first in- +troduce a second TLS as a weakly coupled sensor atom +for the cavity-emitted spectrum [sensor atom approach, +Fig. 1(a)], and show that it produces qualitatively sim- +ilar spectra to that computed with the quantum regres- +sion theorem, though only with certain types of bath +coupling; we also confirm that these sensing atom re- +sults are identical in both the dipole gauge and Coulomb +gauge, as they must be; (ii) we then focus on the main +topic where the second atom is now also treated as an +ultrastrongly coupled atom, distinct from the first atom +[GDM, Fig. 1(b)], and demonstrate several new spectral +features that emerge as we change the coupling parame- +ters of the second TLS. +The rest of our paper is organized as follows: In Sec. II, +we present the main theory, which includes a description +of the GME, our excitation scheme, as well the various +system Hamiltonians, bath interactions, and observables, +including the cavity-emitted spectra. +In Sec. III, we present the main calculations and re- +sults for the sensing atom approach, and show how the +sensing atom coupling can be used to model the detec- +tion of light. We also show how these results compare to +calculations with the quantum regression theorem and +explore the more general case of different bath couplings +(for the atoms as well as the cavity). Next, in Sec. IV, we +consider the case of two atoms in the USC regime, where +we change the parameters of the second atom, and study +the effect that this has on both the system eigenenergies +as well as the cavity observables. We first show explicitly +how our GME produces gauge-independent results when +using the correct gauge-fixed approaches as described in +the main text. Subsequently, we then present a series of +investigations using the dipole gauge. Finally, we con- +clude in Sec. V. +II. +THEORY +In this section, we present the GME, as well as the +different bath models and system Hamiltonians that we +will use. We also show how these can be used to compute +the cavity spectra, using either the quantum regression +theorem, or a sensing atom approach. +A. +Generalized master equation +We first introduce the main GME that we use to com- +pute the key observables of interest: +∂ +∂tρ = −i +� +ˆH, ρ +� ++ +� +Λ +LΛρ + Lpumpρ, +(5) +where ρ is the composite system (composed of the cavity +and the atom, or atoms) density matrix, and ˆH ≡ ˆHD/C +is the system Hamiltonian in either gauge (dipole, ‘D’, or +Coulomb, ‘C’). +The Lindbladian for each dissipation channel is of the +same form, so we write it generally as [44] +LΛρ = 1 +2 +� +ω,ω′>0 +ΓΛ(ω) +� +X+ +Λ (ω) ρ X− +Λ (ω′) +−X− +Λ (ω′)X+ +Λ (ω) ρ +� ++ ΓΛ(ω′) +� +X+ +Λ (ω) ρ X− +Λ (ω′) +−ρ X− +Λ (ω′)X+ +Λ (ω) +� +. +(6) +Since we now have several possible dissipation channels +for the cavity, and the atoms, Λ indexes the cavity and +the atom, or atoms. +The dressed operators are defined from +X+ +cav(ω) = ⟨j|ˆΠ|k⟩ |j⟩ ⟨k| , +X+ +atom(ω) = ⟨j|σx|k⟩ |j⟩ ⟨k| , +(7) +with ω = ωk − ωj > 0, X− +Λ (ω) = [X+ +Λ (ω)]†, and we +assume that ˆΠ has electric field coupling, such that +ˆΠC = i(a†−a) in the Coulomb gauge, and ˆΠD = i(a′†−a′) +in the dipole gauge [34]. We note that the dressed eigen- +states {|j⟩} are required to construct the correct dressed +operators utilized in the GME; these are the eigenstates +of the full light-matter system Hamiltonian including the +interaction term [34, 44, 76]. The dressed states are nat- +urally gauge-dependent, but the observables are not. +Modeled by a (continuous) superposition of damped +bosonic harmonic oscillators, baths are generally de- +scribed by their correlation functions and, in turn, their + +4 +Figure 1. Cavity-QED schemes with two atoms. Schematics of the cavity-QED model with a second atom, including: +(a) the sensor atom approach and (b) the generalized Dicke model in the USC regime. In the sensor atom approach (a), the +addition of a second TLS shown as a sensor atom is weekly coupled to the cavity (hence shown outside of the cavity). In the +generalized Dicke model (b), the second atom is also considered to be ultrastrongly coupled to the cavity (similar to the first +atom, but it can have different coupling parameters). +spectral densities of states which contain information on +the frequencies of the baths’ modes and their coupling +to the system [44]. +For our purpose, the frequency- +dependence of the baths is modeled as either a flat bath, +Γcav(ω) = κ, +Γatom(ω) = γ +(8) +or an Ohmic bath, +Γcav(ω) = κω +ωc +, +Γatom +a,b (ω) = γa,b ω +ωa,b +. +(9) +However, in the case of a sensor atom, we use +Γsen(ω) = γsω +ωc +, +(10) +since in reality the sensor will also have a center fre- +quency at the main detection frequency of interest, while +we assume is at the cavity resonance frequency. +If the open system also includes a sensing element, spe- +cial considerations for the sensing atom’s bath must be +taken into account. Essentially, we must add the dissipa- +tion channel for this sensor atom in an analogous way to +the primary atom. However, in principle, we require that +the inclusion of the sensor should act as a noninvasive +measurement. Therefore, we must ensure that γs ≪ κ +in either the flat or Ohmic shape of Γsen(ω), where γs +is the sensor atom decay rate, or else the sensor atom +introduces additional broadening to the existing peaks in +the spectra. Careful attention is also needed as the dis- +sipation rate of the sensor puts a limit on the coupling +strength between itself and the cavity. We will cautiously +take into account these considerations in our results. +For a cavity-QED system in the USC regime, the +γ ≪ κ process is usually negligible; however, γ plays +an important role in the sensing atom approach (for its +light detection), so we keep the bath functions general +for such a study. However, in the case of two atoms in +the USC regime, we will use Ohmic baths throughout, +where only Γcav(ω) is generally important. +For the excitation process, we also include the incoher- +ent driving through the pump Lindbladian, with +Lpump = 1 +2Pinc D[X− +cav] ρ, +(11) +where D[ ˆO]ρ = 1 +2(2 ˆOρ ˆO† − ρ ˆO† ˆO − ˆO† ˆOρ), and Pinc is +the incoherent driving strength. +B. +Observables +Now that our main master equation model is estab- +lished, we next present the key observables with which +to explore the dynamics of the system. These can also be +used to ensure we have properly enforced gauge invari- +ance. We will focus on the cavity-emitted spectrum. +The cavity spectrum is typically computed from the +Fourier transform of the two-time cavity correlation func- +tion, which exploits the quantum regression theorem. +In such an approach, the cavity spectrum is defined +from [77] +Scav(ω) +(12) +∝ Re +�� ∞ +0 +dτeiωτ +� ∞ +0 +� +X− +cav(t) X+ +cav(t + τ) +� +dt +� +, +where ω is the emission frequency. +With incoherent +steady-state driving, this simplifies to a single time in- +tegration, +Scav(ω) ∝ Re +�� ∞ +0 +dτeiωτ � +X− +cav(0) X+ +cav(τ) +�� +, +(13) +carried out after the system dynamics has reached steady +state. +An alternative method for computing the spectra is +to include a sensing atom, and compute its excitation +flux. +Reference [78] showed how normal-order correla- +tion functions, used to compute the spectrum and other + +(a) + 0 for every a ∈ A almost surely, (c) unconfoundedness: A ⊥⊥ {T(1), T(0)} | X, (d) +conditionally independent censoring: C ⊥⊥ {T(1), T(0)} | {X, A}, we can nonparametrically +identify E[y(T(a))] by the outcome regression (OR) formula or the inverse probability +weighting (IPW) formula (Van der Laan & Robins 2003). +2.2 +ITR and value function +Without loss of generality, we assume that larger values of T are more desirable. Typically +we aim to identify and estimate an ITR d(x) : X → A, which is a mapping from the +covariate space X to the treatment space A = {0, 1}, that maximizes the expected outcome +in a counterfactual world had this ITR been implemented. +Suppose D is the class of +candidate ITRs of interest, then define the potential outcome T(d) under any d ∈ D by +T(d) = d(X)T(1) + (1 − d(X))T(0), and the value function (Manski 2004) of d is defined +by V (d) = E[y(T(d))]. Then by maximizing V (d) over D, the optimal ITR is defined by +dopt = arg maxd∈D V (d). See Qian & Murphy (2011) for more details. +To estimate the value function, we can use the OR or IPW formulas, and also a doubly +6 + +robust method (Bai et al. 2017): +VDR(d) =E +� +I{A = d(X)}∆ y(U) +Pr(A = d(X) | X)SC(U | A, X) ++ +� +1 − +I{A = d(X)} +Pr(A = d(X) | X) +� +E[y(T) | A = d(X), X] ++ +I{A = d(X)} +Pr(A = d(X) | X) +� ∞ +0 +dMC(u | A, X) +SC(u | A, X) E[y(T) | T ≥ u, A, X] +� +, +(1) +where SC(t | a, x) = Pr(C > t | A = a, X = x) is the conditional survival function for the +censoring process, dMC(u | A = a, X) = dNC(u)−Y (u)dΛC(u | A = a, X) is the martingale +increment for the censoring process, NC(u) = I{U ≤ u, ∆ = 0} and ΛC(u | A = a, X) = +− log(SC(u | A = a, X)). The first term in (1) is the IPW formula, and the augmentation +terms capture additional information from the subjects who do not receive treatment d, +and who receive treatment d but are censored. +In (clinical) practice, it is usually desirable to consider a class of ITRs indexed by a +Euclidean parameter η = (η1, . . . , ηp+1)T ∈ Rp+1 for feasibility and interpretability. Let +V (η) = V (dη). Throughout, we focus on such a class of linear ITRs: +Dη = {dη : dη(X) = I{ηT ˜X ≥ 0}, |ηp+1| = 1}, +where ˜X = (1, XT)T, and for identifiability we assume there exists a continuous covariate +whose coefficient has absolute value one (Zhou et al. 2022); without loss of generality, we +assume |ηp+1| = 1. Therefore, the population parameter η∗ indexing the optimal ITR is +η∗ = arg maxη∈{η∈Rp+1:|ηp+1|=1} V (η), and the optimal value function is V (η∗). +2.3 +Transfer learning +The performance of such a learned ITR may suffer from a covariate shift in which the +population distributions differ (Sugiyama & Kawanabe 2012). Instead of minimizing the +worst-case risk, here we consider a super population framework. Suppose that a source +7 + +sample of size n and a target sample of size m are sampled independently from the target +super population with different mechanisms. Let IS and IT denote the indicator of sampling +from source and target populations, respectively. A covariate shift means that Pr(IS = +1 | X) ̸= Pr(IT = 1 | X). In the source sample, independent and identically distributed +(i.i.d.) data Os = {Xi, Ai, Ui, ∆i, IS,i = 1, IT,i = 0}n +i=1 are observed from n subjects; in the +target sample, it is common that only the covariates information is available, so i.i.d. data +Ot = {Xi, IS,i = 0, IT,i = 1}n+m +i=n+1 are observed from m subjects. The sampling mechanism +and data structure are illustrated in Figure 1. +Figure 1: Schematic of the data structure of the source and target samples within the +target super population framework. +Target super population +Finite population {T(1), T(0), X} +Finite population {T(1), T(0), X} +Source sampling IS +Target sampling IT +Complete source sample +{Ti(1), Ti(0), Xi, IS,i = 1, IT,i = 0}n +i=1 +Complete target sample +{Ti(1), Ti(0), Xi, IS,i = 0, IT,i = 1}n+m +i=n+1 +Treatment assignment A +Censoring C +Only observe covariates X +Observed source sample +{Xi, Ai, Ui, ∆i, IS,i = 1, IT,i = 0}n +i=1 +Observed target sample +{Xi, IS,i = 0, IT,i = 1}n+m +i=n+1 +In this framework, we assume that the source and target sampling mechanisms are +independent, which holds if two separate studies are conducted independently by different +8 + +research projects in different locations or in two separate time periods, and the target +population is sufficiently large. In the context of combining the RCT and observational +study, this framework corresponds to the non-nested study design (Dahabreh et al. 2021). +Remark 1. In the framework illustrated in Figure 1, we also assume the existence of the +finite population of size N, which helps us clarify the sampling mechanism and identification +strategy. The two separate finite populations exemplify the independence of the source and +target sampling processes. We present the identification formulas in Section 3; however, +we do not require N to be fixed and known. Equivalently, it is also possible to assume a +pooled population consisting of a source population and a target population, and similar +identification formulas can be proposed based on the density ratio of the two populations. +3 +Methodology +3.1 +Identification and semiparametric efficiency +To identify the causal effects from the observed data, we make the following assumptions. +Assumption 1. (a) T = T(A) almost surely. (b) Pr(A = a | X, IS = 1) > 0 for every a +almost surely. (c) A ⊥⊥ {T(1), T(0)} | {X, IS = 1}. (d) C ⊥⊥ {T(1), T(0)} | {X, A, IS = 1}. +Assumption 1 includes the standard assumptions as we have introduced in Section 2.1. +Here we only assume them in the source population. Assumption 1(a) implies that the +observed outcome is the potential outcome under the actual assigned treatment. Assump- +tion 1(b) states that each subject has a positive probability of receiving both treatments. +Assumption +1(c) requires that all confounding factors are measured so that treatment +assignment is as good as random conditionally on X. Assumption 1(d) essentially states +9 + +that the censoring process is non-informative conditionally on X. Furthermore, we require +additional assumptions for the source and target populations. +Assumption 2 (Survival mean exchangeability). E[y(T(a)) | X, IS = 1] = E[y(T(a)) | X] +for every a ∈ A. +Assumption 3 (Positivity of Source Inclusion). 0 < Pr(IS = 1 | X) < 1 almost surely. +Assumption 4 (Known target design). The target sample design weight e(x) = π−1 +T (x) = +1/Pr(IT = 1 | X = x) is known by design. +Assumption 2 is similar to the mean exchangeability over trial participation (Dahabreh +et al. 2019), and is weaker than the ignorablility assumption (Stuart et al. 2011), i.e., +IS ⊥⊥ {T(1), T(0)} | X. Assumption 3 states that each subject has a positive probability to +be included in the source sample, and implies adequate overlap of covariate distributions +between the source and target populations. Assumption 4 is commonly assumed in the +survey sampling literature; thus the design-weighted target sample is representative of +the target population. In an observational study with simple random sampling, we have +e(x) = N/m, where N is the target population size. +Under this framework, we have the following key identity that for any g(X) +E +� +IS +πS(X)g(X) +� += E[IT e(X)g(X)] = E[g(X)], +(2) +where πS(X) = Pr(IS = 1 | X) is the sampling score. +Proposition 1 (Identification formulas). Under Assumptions 1 - 4, the value function +V (d) can be identified by the outcome regression formula: +V (d) = E[IT e(X)E[y(T) | A = d(X), X, IS = 1]], +(3) +10 + +and the IPW formula: +V (d) = E +� +IS +πS(X) +I{A = d(X)} +πd(X) +∆ y(U) +SC(U | A, X) +� +, +(4) +where πd(X) = d(X)πA(X) + (1 − d(X))(1 − πA(X)) with the propensity score πA(X) = +Pr(A = 1 | X, IS = 1), and SC(t | a, x) = Pr(C > t | A = a, X = x, IS = 1). +Based on the identification formulas (3) and (4), we can construct plug-in estimators for +V (d), using the sampling score πS(X) or design weights e(X) to account for the sampling +bias. By the identity (2), the design weights IT e(X) in the OR formula (3) with the target +sample can also be replaced by the inverse of sampling score IS/πS(X) using the source +sample. However, these estimators are biased if the posited models are misspecified, and +extreme weights from πS, πA and SC usually lead to large variability. Therefore, we consider +a more efficient and robust approach, motivated by the efficient influence function for V (d). +Proposition 2. Under Assumptions 1 - 4, the efficient influence function of V (d) is +φd = +IS +πS(X) +I{A = d(X)} +πd(X) +∆ y(U) +SC(U | A, X) − V (d) ++ +� +IT e(X) − +IS +πS(X) +I{A = d(X)} +πd(X) +� +µ(d(X), X) ++ +IS +πS(X) +I{A = d(X)} +πd(X) +� ∞ +0 +dMC(u | A, X) +SC(u | A, X) Q(u, A, X). +(5) +where µ(a, x) = E[y(T) | A = a, X = x, IS = 1] and Q(u, a, x) = E[y(T) | T ≥ u, A = +a, X = x, IS = 1] 1. +The semiparametric EIF guides us in constructing efficient estimators combining the +source and target samples. Compared to (1), this EIF captures additional covariates infor- +mation from the target population via the outcome model and thus removes the sampling +1Note that E[y(T) | T ≥ u, A, X] = − +� ∞ +u y(s) dS(s | A, X)/S(u | A, X). +For instance, when y(T) = +I{T ≥ t}, we have E[y(T) | T ≥ u, A, X] = S(t | A, X)/S(u | A, X) for u ≤ t. +11 + +bias. An efficient estimation procedure is proposed in the next section, and we show that +it enjoys the double robustness property, i.e., it is consistent if either the survival outcome +models µ(a, x), Q(u, a, x) or the models of propensity score πA(x), sampling score πS(x) +and censoring process SC(t | a, x) are correct. Moreover, this EIF is Neyman orthogonal +in the sense discussed in Chernozhukov et al. (2018). Therefore, a cross-fitting procedure +is also proposed, allowing flexible machine learning methods for the nuisance parameters +estimation, and +√ +N rate of convergence can be achieved. +3.2 +An efficient and robust estimation procedure +In this section, we focus on estimating the survival function Sd(t) = Pr(T(d) > t) as the +value function under ITR d. Following the asymptotic linear characterization of survival +estimands in Yang et al. (2021), our results are readily extended to a broad class of func- +tionals of survival distributions. For instance, the value function of the RMST under ITR +d is simply +� L +0 Sd(t)dt. +Based on the EIF (5), we propose an estimator for the survival function +ˆSd(t) = 1 +N +N +� +i=1 +� +IS,i +ˆπS(Xi) +I{Ai = d(Xi)} +ˆπd(Xi) +∆i Yi(t) +ˆSC(t | Ai, Xi) ++ +� +IT,i e(Xi) − +IS,i +ˆπS(Xi) +I{Ai = d(Xi)} +ˆπd(Xi) +� +ˆS(t | A = d(Xi), Xi) ++ +IS,i +ˆπS(Xi) +I{Ai = d(Xi)} +ˆπd(Xi) +� ∞ +0 +ˆS(t | Ai, Xi)d ˆ +MC(u | Ai, Xi) +ˆS(u | Ai, Xi) ˆSC(u | Ai, Xi) +� +, +(6) +where S(t | a, x) = Pr(T > t | A = a, X = x, IS = 1) is the treatment-specific con- +ditional survival function. +We posit (semi)parametric models for the nuisance param- +eters. +Let πA(X; θ) be the posited propensity score model, for example, using logis- +tic regression logit{πA(X; θ)} = θT ˜X, where logit(x) = log{x/(1 − x)}. +We use the +Cox proportional hazard model Λ(t | A = a, X = x) = Λ0,a(t) exp(βT +a x) to estimate +12 + +the survival functions S(t | a, x) = exp{−Λ(t | a, x)} and the cumulative baseline haz- +ard function Λ0,a(t) = +� t +0 λ0,a(u)du can be estimated by the Breslow estimator (Breslow +1972). +Similarly, we posit a Cox proportional hazard model for the censoring process +ΛC(t | A = a, X = x) = ΛC0,a(t) exp(αT +a x), and the cumulative baseline hazard function +ΛC0,a(t) is estimated by the Breslow estimator. The sampling score estimation is discussed +in the next section. +Let ˆS(t; η) = ˆSdη(t) be the estimated value function for the ITR class Dη, then the +optimal ITR is given by dˆη(x), where ˆη = arg maxη ˆS(t; η). +3.3 +Calibration weighting +To correct the bias due to the covariate shift between populations, most existing methods +directly model the sampling score (Cole & Stuart 2010), i.e., inverse probability of sampling +weighting (IPSW). However, the IPSW method requires the sampling score model to be +correctly specified, and it could also be numerically unstable. Alternatively, we introduce +the calibration weighting (CW) approach motivated by the identity (2), which is similar to +the entropy balancing method (Hainmueller 2012). +Let g(X) be a vector of functions of X to be calibrated, such as the moments, interac- +tions, and non-linear transformations of X. Each subject i in the source sample is assigned +a weight qi by solving the following optimization task: +min +q1,...,qn +n +� +i=1 +qi log qi, +(7) +subject to qi ≥ 0, +n +� +i=1 +qi = 1, +n +� +i=1 +qig(Xi) = ˜g, +(8) +where ˜g = �n+m +i=n+1 e(Xi)g(Xi)/ �n+m +i=n+1 e(Xi) is a design-weighted estimate of E[g(X)]. +The objective function (7) is the negative entropy of the calibration weights, which ensures +that the empirical distribution of the weights is not too far away from the uniform, such that +13 + +it minimizes the variability due to heterogeneous weights. The final balancing constraint +in (8) calibrates the covariate distribution of the weighted source sample to the target +population in terms of g(X). By introducing the Lagrange multiplier λ, the minimizer of the +optimization task is qi = exp{ˆλTg(Xi)}/ �n +i=1 exp{ˆλTg(Xi)}, where ˆλ solves the estimating +equation �n +i=1 exp{λTg(Xi)}{g(Xi) − ˜g} = 0. +Since we only require specifying g(X), +calibration weighting avoids explicitly modeling the sampling score and evades extreme +weights. +Moreover, suppose that the sampling score follows a loglinear model πS(X; λ) = exp{λT ˜X}, +Lee et al. (2021, 2022) show that there is a direct correspondence between the calibration +weights and the estimated sampling score, i.e., qi = {NπS(Xi; ˆλ)}−1 + op(N −1). We also +note that if the fraction n/N is small, the loglinear model is close to the widely used logistic +regression model; our simulation studies show the robustness of calibration weights. +Remark 2. Other objective functions can also be used for calibration weights estima- +tion. Chu et al. (2022) considers a generic convex distance function h(q) from the Cressie +and Read family of discrepancies (Cressie & Read 1984). Thus the optimization task is +minq1,...,qn +�n +i=1 h(qi) under the constraints (8), and the correspondence between the sam- +pling score model πS and the objective function h has also been established. +3.4 +Cross-fitting +Utilizing the Neyman orthogonality of EIF (5), we consider flexible machine learning meth- +ods for estimating the nuisance parameters, where we want to remain agnostic on model- +ing assumptions for the complex treatment assignment, survival, and censoring processes. +There is extensive recent literature on nonparametric methods for heterogeneous treatment +effect estimation with survival outcomes. Cui et al. (2020) extends the generalized random +14 + +forests (Athey et al. 2019) to estimate heterogeneous treatment effects in a survival and +observational setting. See Xu et al. (2022) for details and practical considerations. A de- +scription of the proposed cross-fitting procedure is given below (Schick 1986, Chernozhukov +et al. 2018). Throughout, we use the subscript CF to denote the cross-fitted version. +Algorithm 1 Pseudo algorithm for the cross-fitting procedure +Step 1 Randomly split the datasets Os and Ot respectively into K-folds with equal size +such that Os = ∪K +k=1Os,k, Ot = ∪K +k=1Ot,k. +For each k ∈ {1, . . . , K}, let Oc +s,k = +Os\Os,k, Oc +t,k = Os\Ot,k. +Step 2 For each k ∈ {1, . . . , K}, estimate the nuisance parameters only using data Oc +s,k +and Oc +t,k; then obtain an estimate of the value function ˆVCF,k(η) using data Os,k. +Step 3 Aggregate the estimates from K folds: ˆVCF(η) = 1 +K +�K +k=1 ˆVCF,k(η). +Step 4 The estimated optimal ITR is indexed by ˆη = arg maxη ˆVCF(η). +4 +Asymptotic properties +In this section, we present the asymptotic properties of the proposed methods. To establish +the asymptotic properties, we require the following assumptions. +Assumption 5. (i) The value function V (η) is twice continuously differentiable in a neigh- +borhood of η∗. (ii) There exists some constant δ0 > 0 such that Pr(0 < | ˜XTη| < δ) = O(δ), +where the big-O term is uniform in 0 < δ < δ0. +Condition (i) is a standard regularity condition to establish uniform convergence. Sim- +ilar margin conditions as (ii), which state that Pr(0 < |γ(X)| < δ) = O(δα) 2, are often +2Let γ(X) = E[T | A = 1, X] − E[T | A = 0, X] denote the conditional average treatment effect, then the +15 + +assumed in the literature of classification (Tsybakov 2004, Audibert & Tsybakov 2007), +reinforcement learning (Farahmand 2011, Hu et al. 2021) and optimal treatment regimes +(Luedtke & van der Laan 2016a, Luedtke & Chambaz 2020), to guarantee a fast conver- +gence rate. Note that α = 0 imposes no restriction, which allows γ(X) = 0 almost surely, +i.e., the challenging setting of exceptional laws where the optimal ITR is not uniquely de- +fined (Robins 2004, Robins & Rotnitzky 2014), while the case α = 1 is of particular interest +and would hold if γ(X) is absolutely continuous with bounded density. +Theorem 1. Under Assumptions 1 - 5 and standard regularity conditions provided in +the Supplementary Material, if either the survival outcome model, or the models of the +propensity score, the sampling score and the censoring process are correct, we have that as +N → ∞, (i) ˆS(t; η) → S(t; η) for any η and 0 < t ≤ L; (ii) +√ +N +� +ˆS(t; η) − S(t; η) +� +con- +verges weakly to a mean zero Gaussian process for any η; (iii) N 1/3 ∥ˆη − η∗∥2 = Op(1); (iv) +√ +N +� +ˆS(t; ˆη) − S(t; η∗) +� +→ N(0, σ2 +t,1), where σt,1 is given in the Supplementary Material. +Next, to characterize the asymptotic behavior of the estimator with the nonparametric +estimation of nuisance parameters, we assume the following consistency and convergence +rate conditions of the nonparametric plug-in nuisance estimators. +Assumption 6. Assume the following convergences in probability: supx∈X |ˆπA(x)−πA(x)| → +0, supx∈X |ˆπS(x) − πS(x)| → 0, and for a = 0, 1, +sup +x∈X,u≤h +| ˆSC(u | a, x) − SC(u | a, x)| → 0, +sup +x∈X,u≤h +����� +ˆλC(u | a, x) +ˆSC(u | a, x) +− λC(u | a, x) +SC(u | a, x) +����� → 0, +sup +x∈X +|ˆµ(a, x) − µ(a, x)| → 0, +sup +x∈X,u≤h +| ˆQ(u, a, x) − Q(u, a, x)| → 0; +and the following rates of convergence: E [supx∈X |ˆπA(x) − πA(x)|] = op(n−1/4), +optimal ITR in an unrestricted class is given by d(X) = I{γ(X) > 0}. +16 + +E [supx∈X |ˆπS(x) − πS(x)|] = op(n−1/4), and for a = 0, 1, +sup +u≤h +E +� +sup +x∈X +��� ˆSC(u | a, x) − SC(u | a, x) +��� +� += op(n−1/4), +sup +u≤h +E +� +sup +x∈X +����� +ˆλC(u | a, x) +ˆSC(u | a, x) +− λC(u | a, x) +SC(u | a, x) +����� +� += op(n−1/4), +E +� +sup +x∈X +|ˆµ(a, x) − µ(a, x)| +� += o(n−1/4), sup +u≤h +E +� +sup +x∈X +| ˆQ(u, a, x) − Q(u, a, x)| +� += o(n−1/4). +The rate conditions in Assumption 6 are generally assumed in the literature (Kennedy +2022). This rate can be achieved by many existing methods under certain structural as- +sumptions on the nuisance parameters. Note that the nuisance parameters do not necessar- +ily need to be estimated at the same rates n−1/4 for our theorems to hold; it would suffice +that the product of rates of any combination of two nuisance parameters is n−1/2. +Theorem 2. Under Assumptions 1 - 6, we have that as N → ∞, (i) ˆSCF(t; η) → S(t; η) +for any η and 0 < t ≤ L; (ii) +√ +N +� +ˆSCF(t; η) − S(t; η) +� +converges weakly to a mean zero +Gaussian process for any η; (iii) N 1/3∥ˆη−η∗∥2 = Op(1); (iv) +√ +N +� +ˆSCF(t; ˆη) − S(t; η∗) +� +→ +N(0, σ2 +t,2), where σt,2 is given in the Supplementary Material. +Besides the survival functions, another common measure of particular interest in sur- +vival analysis is the RMST. Let VRMST(η) = E[min(T(dη), L)]. We present two corollaries. +Corollary 1. Under Assumptions 1 - 5 and standard regularity conditions provided in the +Supplementary material, if either the survival outcome model or the models of the propen- +sity score, the censoring and sampling processes are correct, we have that as N → ∞, (i) +ˆVRMST(η) → VRMST(η) for any η; (ii) N 1/3∥ˆη−η∗∥2 = Op(1); (iii) +√ +N +� +ˆVRMST(ˆη) − VRMST(η∗) +� +→ +N(0, σ2 +3), where σ3 is given in the Supplementary Material. +Corollary 2. Under Assumptions 1 - 6, we have that as N → ∞, (i) ˆVRMST,CF(η) → +VRMST(η) for any η; (ii) N 1/3∥ˆη − η∗∥2 = Op(1); (iii) +√ +N +� +ˆVRMST,CF(ˆη) − VRMST(η∗) +� +→ +N(0, σ2 +4), where σ4 is given in the Supplementary Material.. +17 + +Finally, we show that when the covariate distributions of the source and target pop- +ulations are the same, the semiparametric efficiency bounds of ˆVDR(η) and ˆVCF(η) are +equal. +Theorem 3. Under Assumptions 1 - 6, when the covariate distributions of the source and +target populations are the same, both +√ +N{ˆVDR(η) − V (η)} and +√ +N{ˆVCF(η) − V (η)} are +asymptotically normal with mean zero and same variance. +Theorem 3 implies that when there is no covariate shift, our proposed estimator does not +lose efficiency in comparison to the original double robust estimator since the augmentation +term in EIF (5) from the target population, IT e(X)µ(d(X), X), is asymptotically equal to +this term evaluated on the source population in this case. +Moreover, when the covariate shift exists, we consider the optimal ITR dopt without +restriction on the ITR class. +Theorem 4. Under Assumptions 1 - 6, If dopt ∈ Dη, i.e., dopt = dη∗, both the maximizers +of ˆVDR(η) and ˆVCF(η) converge to η∗. However, ˆVDR(η) is a biased estimator of V (η). +Theorem 4 implies if the true optimal ITR belongs to the restricted ITR class Dη, +standard methods, without accounting for the covariate shift, are still able to recover the +optimal ITR but fail to be consistent for the value function, due to the covariate shift. And +we can only rely on the proposed method to draw valid inferences. +5 +Simulation +In this section, we investigate the finite-sample properties of our method through extensive +numerical simulations 3. +3The +R +code +to +replicate +all +results +is +available +at +https://github.com/panzhaooo/ +transfer-learning-survival-ITR. +18 + +Consider a target population of sample size N = 2 × 105. The covariates (X1, X2, X3)T +are generated from a multivariate normal distribution with mean 0, unit variance with +corr(X1, X3) = 0.2 and all other pairwise correlations equal to 0, and further truncated +below −4 and above 4 to satisfy regularity conditions. The target sample is a random +sample of size m = 8000 from the target population. The sampling score follows πS(X) = +expit(−4.5−0.5X1−0.5X2−0.4X3); thus the source sampling rate is around 1.6%, and the +source sample size around n = 3000. The treatment assignment mechanism in the source +sample follows πA(X) = expit(0.5 + 0.8X1 − 0.5X2). +The counterfactual survival times T(a) are generated according to the hazard functions +λ(t | A = 0, X) = exp(t) · exp(−2.5 − 1.5X1 − X2 − 0.7X3) and λ(t | A = 1, X) = exp(t) · +exp(−1 − X1 − 0.9X2 − X3 − 2X2 +2 + X1X3). The censoring time C is generated according +to the hazard functions λC(t | A = 0, X) = 0.04 exp(t) · exp(−1.6 + 0.8X1 − 1.1X2 − 0.7X3) +and λC(t | A = 1, X) = 0.04 exp(t) · exp(−1.8 − 0.8X1 − 1.7X2 − 1.4X3). The resultant +censoring rate is approximately 20%. +We consider the RMST with the maximal time horizon L = 4 as the value function. +To evaluate the performance of different estimators for optimal ITRs, we compute the +corresponding true value functions and percentages of correct decisions (PCD) for the +target population. Specifically, we generate a large sample with size ˜N = 1 × 105 from +the target population. The true value function of any ITR d(· ; η) is computed by V (η) = +˜N −1 � ˜ +N +i=1 min{d(Xi ; η)Ti(1)+(1−d(Xi ; η))Ti(0), L} and its associated PCD is computed +by 1 − ˜N −1 � ˜ +N +i=1 |d(Xi ; η∗) − d(Xi ; η)|, where η∗ = arg maxη V (η). +We compare the following estimators for the RMST ˆV (η) = +� L +0 ˆS(t; η)dt: +• Naive: ˆSNaive(t; η) = 1 +n +�n +i=1 +I{Ai=d(Xi)} +ˆπd(Xi) +∆iYi(t) +ˆSC(U | A,X); IPW formula (4) without using the +sampling score; +19 + +• IPSW: ˆSIPSW(t; η) = 1 +n +�n +i=1 +IS,i +ˆπS(Xi) +I{Ai=d(Xi)} +ˆπd(Xi) +∆iYi(t) +ˆSC(U | A,X); IPW formula (4) where the +sampling score is estimated via logistic regression; +• CW-IPW: ˆSCW-IPW(t; η) = �n +i=1 qi +I{Ai=d(Xi)} +ˆπd(Xi) +∆iYi(t) +ˆSC(U | A,X) IPW formula (4) where the +sampling score is estimated by calibration weighting; +• CW-OR: ˆSCW-OR(t; η) = �n +i=1 qi ˆS(t | A = d(Xi), Xi); OR formula (3) in combination +with calibration weights by the identity (2); +• ORt: ˆSORt(t; η) = +1 +m +�n+m +i=n+1 ˆS(t | A = d(Xi), Xi); OR formula (3) evaluated on the +target sample; +• ACW: augmented estimator (6), where the sampling score is estimated by calibration +weighting. +Remark 3. Since the estimated value functions are non-convex and non-smooth, multiple +local optimal may exist in the optimization task, and many derivatives-based algorithms do +not work for this challenging setting. Here we utilize the genetic algorithm implemented in +the R package rgenoud (Mebane Jr & Sekhon 2011), which performs well in our numerical +experiments. We refer to Mitchell (1998) for algorithmic details. +5.1 +(Semi)parametric models +We first consider the setting where the nuisance parameters are estimated by posited +(semi)parametric working models as introduced in Section 3.2. To assess the performance +of these estimators under model misspecification, we consider four scenarios: (1) all models +are correct, (2) only the survival outcome model is correct, (3) only the survival outcome +model is wrong, (4) all models are wrong. For the wrong sampling model, the weights are +20 + +estimated using calibration on eX1. The wrong propensity score model is fitted on eX3. +The wrong Cox models for survival and censoring times are fitted on (eX1, eX2, eX3)T. +Figure 2 and Table 1 report the simulation results from 350 Monte Carlo replications. +Variance is estimated by a bootstrap procedure with B = 200 bootstrap replicates. The +proposed ACW estimator is unbiased in scenarios (1) - (3), and the 95% coverage probabil- +ities approximately achieve the nominal level, which shows the double robustness property. +5.2 +Flexible machine learning methods +When utilizing flexible ML methods, we construct the cross-fitted ACW estimator as in- +troduced in Section 3.4. The data generation process is the same as above, except that +the censoring time C is generated according to the hazard functions λC(t | A = 0, X) = +0.2 exp(t)·exp(−1.6+0.8X1−1.1X2−0.7X3) and λC(t | A = 1, X) = 0.2 exp(t)·exp(−1.8− +0.8X1 − 1.7X2 − 1.4X3) which leads to an increased censoring rate of approximately 33%, +so there are enough observations to get an accurate estimate of the censoring process. +The propensity score is estimated by the generalized random forest. The conditional sur- +vival and censoring functions are estimated by the random survival forest. The calibration +weighting uses calibration on the first- and second-order moments of X. +First, we study the impact of sample sizes on the performance of the ML methods, and +simulation results are given in the Supplementary Material. With a small sample size, the +ACW estimator is largely biased, and the bias diminishes as the sample size increases. +Next, we compare the performance of different estimators with target population size +N = 6 × 105 and target sample size m = 24000. Figure 3 shows the simulation results +from 200 Monte Carlo replications. The two IPW-based estimators are biased and perform +poorly due to the large variability of weights. The two OR-based estimators have compa- +21 + +Figure 2: Boxplot of the estimated value, true value and PCD results of estimators under +four model specification scenarios. O: survival outcome, S: sampling score, A: propensity +score, C: censoring; T: True (correctly specified) model, W: Wrong (misspecified) model. +O:T / S:T, A:T, C:T +O:T / S:W, A:W, C:W +O:W / S:T, A:T, C:T +O:W / S:W, A:W, C:W +Estimated Value +PCD +True Value +2 +3 +4 +5 +0.4 +0.6 +0.8 +1.0 +2.0 +2.1 +2.2 +2.3 +2.4 +Estimators +Naive +IPSW +CW−IPW +CW−OR +ORt +ACW +22 + +Table 1: Numerical results under four different model specification scenarios. Bias is the +empirical bias of point estimates; SD is the empirical standard deviation of point estimates; +SE is the average of bootstrap standard error estimates; CP is the empirical coverage +probability of the 95% confidence intervals. +Bias +SD +SE +CP(%) +Bias +SD +SE +CP(%) +O:T / S:T, A:T, C:T +O:T / S:W, A:W, C:W +Naive +−0.8801 +0.4595 +0.2189 +7.43 +−0.3528 +0.5024 +0.4598 +37.43 +IPSW +0.0185 +0.3685 +0.2562 +87.14 +0.3377 +0.7144 +0.6958 +98.29 +CW-IPW +0.0378 +0.3701 +0.2498 +88.29 +0.3406 +0.7144 +0.6957 +97.71 +CW-OR +0.0047 +0.0273 +0.0286 +96.29 +−0.1312 +0.0269 +0.0279 +0.57 +ORt +0.0041 +0.0258 +0.0262 +95.14 +0.0035 +0.0258 +0.0262 +95.71 +ACW +0.0070 +0.0380 +0.0369 +94.29 +0.0055 +0.0316 +0.0334 +95.43 +O:W / S:T, A:T, C:T +O:W / S:W, A:W, C:W +Naive +−0.8801 +0.4595 +0.2207 +6.86 +−0.3528 +0.5024 +0.5018 +38.57 +IPSW +0.0185 +0.3685 +0.2486 +87.71 +0.3377 +0.7144 +0.7586 +99.14 +CW-IPW +0.0378 +0.3701 +0.2418 +88.86 +0.3406 +0.7144 +0.7570 +98.57 +CW-OR +0.0103 +0.0370 +0.0362 +92.29 +−0.2551 +0.0366 +0.0391 +0.00 +ORt +0.0094 +0.0365 +0.0355 +94.00 +0.0115 +0.0328 +0.0355 +95.71 +ACW +−0.0010 +0.0426 +0.0419 +93.14 +0.2644 +0.0422 +0.0475 +0.57 +23 + +rable performance as the ACW estimator in terms of PCD and true value function but still +suffer from the overfitting bias. Only the ACW estimator is consistent and provides valid +inferences. +Figure 3: Boxplots of the estimated value, true value, and PCD of different estimators +using flexible ML methods. +2 +3 +4 +Estimated Value +0.2 +0.4 +0.6 +0.8 +1.0 +PCD +2.0 +2.1 +2.2 +2.3 +2.4 +True Value +Estimators +Naive +IPSW +CW−IPW +CW−OR +ORt +ACW +6 +Real Data Analysis +In this section, to illustrate the proposed method, we study the sodium bicarbonate therapy +for patients with severe metabolic acidaemia in the intensive care unit by leveraging the +RCT data BICAR-ICU (Jaber et al. 2018) and the observational study (OS) data from +Jung et al. (2011). Specifically, we consider the BICAR-ICU data as the source sample and +24 + +the observational study data as the target sample. The BICAR-ICU is a multi-center, open- +label, randomized controlled, phase 3 trial between May 5, 2015, and May 7, 2017, which +includes 387 adult patients admitted within 48 hours to the ICU with severe acidaemia. +The prospective, multiple-center observational study was conducted over thirteen months +in five ICUs, consisting of 193 consecutive patients who presented with severe acidemia +within the first 24 hours of their ICU admission. Some heterogeneity exists between the +two populations. +Both the RCT and OS datasets contain detailed measurements of ICU patients with +severe acidaemia. Motivated by the clinical practice and existing work in the medical liter- +ature, we consider ITRs that depend on the following five variables: SEPSIS, AKIN, SOFA, +SEX, and AGE. A detailed description of the data preprocessing and variable selection is +given in the Supplementary Material. Table 2 summarizes the baseline characteristics of +the two datasets. The baseline covariates distribution of the patients in the BICAR-ICU +differs from the distribution in the observational study; specifically, the BICAR-ICU pa- +tients have higher SOFA scores and the more frequent presence of acute kidney injury and +sepsis. +Table 2: Summary of baseline characteristics of the BICAR-ICU trial sample and the OS +sample. Mean (standard deviation) for continuous and number (proportion) for the binary +covariate. +SEPSIS +AKIN +SOFA +SEX +AGE +BICAR-ICU (n = 387) +236 (60.98%) +181 (46.77%) +10.12 (3.72) +237 (61.24%) +63.95 (14.41) +OS (m = 193) +99(51.30%) +75 (38.86%) +9.10 (4.54) +122 (63.21%) +62.73 (17.49) +We apply our proposed ACW estimator to learn the optimal ITR for the target popu- +lation. The calibration weights are estimated based on the means of continuous covariates +25 + +and the proportions of the binary covariates. The propensity score is estimated using a +logistic regression model, and the Cox proportional hazard model is fitted for the survival +outcome with all covariates. The censoring only occurred on the 28th day when the follow- +up in ICU ends. We consider the class of linear ITRs that depend on all five variables: +D = {I{η1+η2SEPSIS+η3AKIN+η4SOFA+η5SEX+η6AGE > 0} : η1, . . . , η6 ∈ R, |η6| = 1}, +with the aim to maximize the RMST within 28 days in ICU stay. The estimated parame- +ter indexing the optimal ITR is ˆηACW = (22.9, −36.1, 87.4, −9.8, 33.7, 1.0)T, which leads to +an estimated value function ˆV (ˆηACW) = 19.52 days, with confidence interval [17.74, 21.30] +given by 200 bootstraps. In contrast, we also use the standard double robust method to esti- +mate the optimal ITR for the RCT, indexed by ˆηDR.RCT which maximize the value function +ˆVDR(η) in (1) with y(T) = min(T, 28). The estimated value function is ˆV (ˆηDR.RCT) = 15.37 +days for the target population. +7 +Discussion +In this paper, we present an efficient and robust transfer learning framework for estimating +optimal ITR with right-censored survival data that generalizes well to the target population. +The proposed method can be improved or extended in several directions for future work. +Construction and estimation of optimal ITRs for multiple decision points with censored +survival data are challenging, taking into account the timing of censoring, events and +decision points (Jiang et al. 2017, Hager et al. 2018), e.g., using a reinforcement learning +method (Cho et al. 2020). Furthermore, besides the class of ITRs indexed by a Euclidean +parameter, it may be possible to consider other classes of ITRs, such as tree or list-based +ITRs. The current work focus on value functions in the form V (d) = E[y(T(d))] and can also +26 + +be modified in case of optimizing certain easy-to-interpret quantile criteria, which does not +require specifying an outcome regression model and is robust for heavy-tailed distributions +(Zhou et al. 2022). And relaxing the restrictive assumptions such as positivity (Yang & +Ding 2018, Jin et al. 2022) and unconfoundedness (Cui & Tchetgen Tchetgen 2021, Qi +et al. 2021) for learning optimal ITRs is also a fruitful direction. +Acknowledgments +Josse and Zhao gratefully acknowledge the French National Research Agency ANR-16- +IDEX-0006. Yang is partially supported by the USA National Institutes of Health NIA +grant 1R01AG066883 and NIEHS grant 1R01ES031651. +The authors thank Maxime Fosset and Boris Jung for their help and support interpreting +the BICAR-ICU trial and observational study data. +References +Andersen, P. K. & Gill, R. D. (1982), ‘Cox’s regression model for counting processes: a +large sample study’, The annals of statistics pp. 1100–1120. +Athey, S., Tibshirani, J. & Wager, S. (2019), ‘Generalized random forests’, The Annals of +Statistics 47(2), 1148–1178. +Athey, S. & Wager, S. (2021), ‘Policy learning with observational data’, Econometrica +89(1), 133–161. +Audibert, J.-Y. & Tsybakov, A. B. (2007), ‘Fast learning rates for plug-in classifiers’, The +Annals of statistics 35(2), 608–633. +27 + +Bai, X., Tsiatis, A. A., Lu, W. & Song, R. (2017), ‘Optimal treatment regimes for survival +endpoints using a locally-efficient doubly-robust estimator from a classification perspec- +tive’, Lifetime data analysis 23(4), 585–604. +Breslow, N. E. (1972), ‘Contribution to discussion of paper by dr cox’, J. Roy. Statist. Soc., +Ser. B 34, 216–217. +Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. & +Robins, J. (2018), ‘Double/debiased machine learning for treatment and structural pa- +rameters’. +Cho, H., Holloway, S. T., Couper, D. J. & Kosorok, M. R. (2020), ‘Multi-stage optimal dy- +namic treatment regimes for survival outcomes with dependent censoring’, arXiv preprint +arXiv:2012.03294 . +Chu, J., Lu, W. & Yang, S. (2022), ‘Targeted optimal treatment regime learning using +summary statistics’, arXiv preprint arXiv:2201.06229 . +Cole, S. R. & Stuart, E. A. (2010), ‘Generalizing evidence from randomized clinical trials +to target populations: the actg 320 trial’, American journal of epidemiology 172(1), 107– +115. +Colnet, B., Mayer, I., Chen, G., Dieng, A., Li, R., Varoquaux, G., Vert, J.-P., Josse, +J. & Yang, S. (2020), ‘Causal inference methods for combining randomized trials and +observational studies: a review’, arXiv preprint arXiv:2011.08047 . +Cressie, N. & Read, T. R. (1984), ‘Multinomial goodness-of-fit tests’, Journal of the Royal +Statistical Society: Series B (Methodological) 46(3), 440–464. +28 + +Cui, Y., Kosorok, M. R., Sverdrup, E., Wager, S. & Zhu, R. (2020), ‘Estimating hetero- +geneous treatment effects with right-censored data via causal survival forests’, arXiv +preprint arXiv:2001.09887 . +Cui, Y. & Tchetgen Tchetgen, E. (2021), ‘A semiparametric instrumental variable approach +to optimal treatment regimes under endogeneity’, Journal of the American Statistical +Association 116(533), 162–173. +Cui, Y., Zhu, R. & Kosorok, M. (2017), ‘Tree based weighted learning for estimating individ- +ualized treatment rules with censored data’, Electronic journal of statistics 11(2), 3927. +Dahabreh, I. J., Haneuse, S. J. A., Robins, J. M., Robertson, S. E., Buchanan, A. L., Stuart, +E. A. & Hern´an, M. A. (2021), ‘Study designs for extending causal inferences from a +randomized trial to a target population’, American journal of epidemiology 190(8), 1632– +1642. +Dahabreh, I. J. & Hern´an, M. A. (2019), ‘Extending inferences from a randomized trial to +a target population’, European Journal of Epidemiology 34(8), 719–722. +Dahabreh, I. J., Robertson, S. E., Tchetgen, E. J., Stuart, E. A. & Hern´an, M. A. (2019), +‘Generalizing causal inferences from individuals in randomized trials to all trial-eligible +individuals’, Biometrics 75(2), 685–694. +Degtiar, I. & Rose, S. (2021), ‘A review of generalizability and transportability’, arXiv +preprint arXiv:2102.11904 . +D´ıaz, I., Savenkov, O. & Ballman, K. (2018), ‘Targeted learning ensembles for optimal +individualized treatment rules with time-to-event outcomes’, Biometrika 105(3), 723– +738. +29 + +Farahmand, A.-m. (2011), ‘Action-gap phenomenon in reinforcement learning’, Advances +in Neural Information Processing Systems 24. +Goldberg, Y. & Kosorok, M. R. (2012), ‘Q-learning with censored data’, Annals of statistics +40(1), 529. +Hager, R., Tsiatis, A. A. & Davidian, M. (2018), ‘Optimal two-stage dynamic treat- +ment regimes from a classification perspective with censored survival data’, Biometrics +74(4), 1180–1192. +Hainmueller, J. (2012), ‘Entropy balancing for causal effects: A multivariate reweight- +ing method to produce balanced samples in observational studies’, Political analysis +20(1), 25–46. +Hu, Y., Kallus, N. & Uehara, M. (2021), ‘Fast rates for the regret of offline reinforcement +learning’, arXiv preprint arXiv:2102.00479 . +Jaber, S., Paugam, C., Futier, E., Lefrant, J.-Y., Lasocki, S., Lescot, T., Pottecher, J., De- +moule, A., Ferrandiere, M., Asehnoune, K. et al. (2018), ‘Sodium bicarbonate therapy for +patients with severe metabolic acidaemia in the intensive care unit (bicar-icu): a multi- +centre, open-label, randomised controlled, phase 3 trial’, The Lancet 392(10141), 31–40. +Jiang, R., Lu, W., Song, R. & Davidian, M. (2017), ‘On estimation of optimal treatment +regimes for maximizing t-year survival probability’, Journal of the Royal Statistical So- +ciety: Series B (Statistical Methodology) 79(4), 1165–1185. +Jin, Y., Ren, Z., Yang, Z. & Wang, Z. (2022), ‘Policy learning” without”overlap: Pessimism +and generalized empirical bernstein’s inequality’, arXiv preprint arXiv:2212.09900 . +30 + +Jung, B., Rimmele, T., Le Goff, C., Chanques, G., Corne, P., Jonquet, O., Muller, L., +Lefrant, J.-Y., Guervilly, C., Papazian, L. et al. (2011), ‘Severe metabolic or mixed +acidemia on intensive care unit admission: incidence, prognosis and administration of +buffer therapy. a prospective, multiple-center study’, Critical Care 15(5), 1–9. +Kallus, N. (2021), ‘More efficient policy learning via optimal retargeting’, Journal of the +American Statistical Association 116(534), 646–658. +Kennedy, E. H. (2022), ‘Semiparametric doubly robust targeted double machine learning: +a review’, arXiv preprint arXiv:2203.06469 . +Kennedy, E. H., Balakrishnan, S. & G’Sell, M. (2020), ‘Sharp instruments for classifying +compliers and generalizing causal effects’, The Annals of Statistics 48(4), 2008–2030. +Kosorok, M. R. (2008), Introduction to empirical processes and semiparametric inference., +Springer. +Kosorok, M. R. & Laber, E. B. (2019), ‘Precision medicine’, Annual review of statistics +and its application 6, 263. +K¨unzel, S. R., Sekhon, J. S., Bickel, P. J. & Yu, B. (2019), ‘Metalearners for estimat- +ing heterogeneous treatment effects using machine learning’, Proceedings of the national +academy of sciences 116(10), 4156–4165. +Laber, E. B., Linn, K. A. & Stefanski, L. A. (2014), ‘Interactive model building for q- +learning’, Biometrika 101(4), 831–847. +Laber, E. B. & Zhao, Y.-Q. (2015), ‘Tree-based methods for individualized treatment +regimes’, Biometrika 102(3), 501–514. +31 + +Lee, D., Yang, S., Dong, L., Wang, X., Zeng, D. & Cai, J. (2021), ‘Improving trial gener- +alizability using observational studies’, Biometrics . +Lee, D., Yang, S. & Wang, X. (2022), ‘Doubly robust estimators for generalizing treatment +effects on survival outcomes from randomized controlled trials to a target population’, +Journal of Causal Inference (accepted). +Luedtke, A. & Chambaz, A. (2020), Performance guarantees for policy learning, in ‘Annales +de l’IHP Probabilites et statistiques’, Vol. 56, NIH Public Access, p. 2162. +Luedtke, A. R. & van der Laan, M. J. (2016a), ‘Statistical inference for the mean outcome +under a possibly non-unique optimal treatment strategy’, Annals of statistics 44(2), 713. +Luedtke, A. R. & van der Laan, M. J. (2016b), ‘Super-learning of an optimal dynamic +treatment rule’, The international journal of biostatistics 12(1), 305–332. +Manski, C. F. (2004), ‘Statistical treatment rules for heterogeneous populations’, Econo- +metrica 72(4), 1221–1246. +Mebane Jr, W. R. & Sekhon, J. S. (2011), ‘Genetic optimization using derivatives: the +rgenoud package for r’, Journal of Statistical Software 42, 1–26. +Mitchell, M. (1998), An introduction to genetic algorithms, MIT press. +Mo, W., Qi, Z. & Liu, Y. (2021), ‘Learning optimal distributionally robust individualized +treatment rules’, Journal of the American Statistical Association 116(534), 659–674. +Murphy, S. A. (2003), ‘Optimal dynamic treatment regimes’, Journal of the Royal Statistical +Society: Series B (Statistical Methodology) 65(2), 331–355. +Neyman, J. (1923), ‘Sur les applications de la th´eorie des probabilit´es aux experiences +agricoles: Essai des principes’, Roczniki Nauk Rolniczych 10(1), 1–51. +32 + +Qi, Z., Miao, R. & Zhang, X. (2021), ‘Proximal learning for individualized treatment +regimes under unmeasured confounding’, arXiv preprint arXiv:2105.01187 . +Qian, M. & Murphy, S. A. (2011), ‘Performance guarantees for individualized treatment +rules’, Annals of statistics 39(2), 1180. +Rasmussen, S. A., Khoury, M. J. & Del Rio, C. (2020), ‘Precision public health as a key +tool in the covid-19 response’, JAMA 324(10), 933–934. +Robins, J. M. (2004), Optimal structural nested models for optimal sequential decisions, in +‘Proceedings of the second seattle Symposium in Biostatistics’, Springer, pp. 189–326. +Robins, J. & Rotnitzky, A. G. (2014), ‘Discussion of “dynamic treatment regimes: Technical +challenges and applications”’. +Rubin, D. B. (1974), ‘Estimating causal effects of treatments in randomized and nonran- +domized studies.’, Journal of educational Psychology 66(5), 688. +Rubin, D. B. & van der Laan, M. J. (2012), ‘Statistical issues and limitations in personalized +medicine research with clinical trials’, The international journal of biostatistics 8(1), 18. +Sahoo, R., Lei, L. & Wager, S. (2022), ‘Learning from a biased sample’, arXiv preprint +arXiv:2209.01754 . +Schick, A. (1986), ‘On asymptotically efficient estimation in semiparametric models’, The +Annals of Statistics pp. 1139–1151. +Schulte, P. J., Tsiatis, A. A., Laber, E. B. & Davidian, M. (2014), ‘Q-and a-learning +methods for estimating optimal dynamic treatment regimes’, Statistical science: a review +journal of the Institute of Mathematical Statistics 29(4), 640. +33 + +Shi, C., Fan, A., Song, R. & Lu, W. (2018), ‘High-dimensional a-learning for optimal +dynamic treatment regimes’, Annals of statistics 46(3), 925. +Stuart, E. A., Cole, S. R., Bradshaw, C. P. & Leaf, P. J. (2011), ‘The use of propensity +scores to assess the generalizability of results from randomized trials’, Journal of the +Royal Statistical Society: Series A (Statistics in Society) 174(2), 369–386. +Sugiyama, M. & Kawanabe, M. (2012), Machine learning in non-stationary environments: +Introduction to covariate shift adaptation, MIT press. +Trella, A. L., Zhang, K. W., Nahum-Shani, I., Shetty, V., Doshi-Velez, F. & Murphy, +S. A. (2022), ‘Designing reinforcement learning algorithms for digital interventions: Pre- +implementation guidelines’, arXiv preprint arXiv:2206.03944 . +Tsiatis, A. A. (2006), ‘Semiparametric theory and missing data’. +Tsiatis, A. A., Davidian, M., Holloway, S. T. & Laber, E. B. (2019), Dynamic treatment +regimes: Statistical methods for precision medicine, Chapman and Hall/CRC. +Tsybakov, A. B. (2004), ‘Optimal aggregation of classifiers in statistical learning’, The +Annals of Statistics 32(1), 135–166. +Van Buuren, S. & Groothuis-Oudshoorn, K. (2011), ‘mice: Multivariate imputation by +chained equations in r’, Journal of statistical software 45, 1–67. +Van der Laan, M. J. & Robins, J. M. (2003), Unified methods for censored longitudinal +data and causality, Vol. 5, Springer. +Van der Vaart, A. W. & Wellner, J. A. (1996), Weak Convergence and Empirical Processes: +With Applications to Statistics, Springer New York, NY. +34 + +Wu, L. & Yang, S. (2022), ‘Transfer learning of individualized treatment rules from +experimental to real-world data’, Journal of Computation and Graphical Statistics +p. doi.org/10.1080/10618600.2022.2141752. +Xu, Y., Ignatiadis, N., Sverdrup, E., Fleming, S., Wager, S. & Shah, N. (2022), ‘Treatment +heterogeneity with survival outcomes’, arXiv preprint arXiv:2207.07758 . +Yang, S. & Ding, P. (2018), ‘Asymptotic inference of causal effects with observational +studies trimmed by the estimated propensity scores’, Biometrika 105(2), 1–7. +Yang, S., Zhang, Y., Liu, G. F. & Guan, Q. (2021), ‘SMIM: A unified framework of +survival sensitivity analysis using multiple imputation and martingale’, Biometrics p. doi: +10.1111/biom.13555. +Z´avada, J., Hoste, E., Cartin-Ceba, R., Calzavacca, P., Gajic, O., Clermont, G., Bellomo, +R., Kellum, J. A. & investigators, A. (2010), ‘A comparison of three methods to es- +timate baseline creatinine for rifle classification’, Nephrology Dialysis Transplantation +25(12), 3911–3918. +Zhang, B., Tsiatis, A. A., Davidian, M., Zhang, M. & Laber, E. (2012), ‘Estimating optimal +treatment regimes from a classification perspective’, Stat 1(1), 103–114. +Zhang, B., Tsiatis, A. A., Laber, E. B. & Davidian, M. (2012), ‘A robust method for +estimating optimal treatment regimes’, Biometrics 68(4), 1010–1018. +Zhang, Y., Laber, E. B., Davidian, M. & Tsiatis, A. A. (2018), ‘Interpretable dynamic +treatment regimes’, Journal of the American Statistical Association 113(524), 1541– +1549. +35 + +Zhang, Y., Laber, E. B., Tsiatis, A. & Davidian, M. (2015), ‘Using decision lists to construct +interpretable and parsimonious treatment regimes’, Biometrics 71(4), 895–904. +Zhang, Z., Zhu, C., Mo, L. & Hong, Y. (2018), ‘Effectiveness of sodium bicarbonate in- +fusion on mortality in septic patients with metabolic acidosis’, Intensive care medicine +44(11), 1888–1895. +Zhao, Y.-Q., Zeng, D., Tangen, C. M. & Leblanc, M. L. (2019), ‘Robustifying trial- +derived optimal treatment rules for a target population’, Electronic journal of statistics +13(1), 1717. +Zhao, Y., Zeng, D., Rush, A. J. & Kosorok, M. R. (2012), ‘Estimating individualized +treatment rules using outcome weighted learning’, Journal of the American Statistical +Association 107(499), 1106–1118. +Zhou, Y., Wang, L., Song, R. & Zhao, T. (2022), ‘Transformation-invariant learning of opti- +mal individualized decision rules with time-to-event outcomes’, Journal of the American +Statistical Association (just-accepted), 1–35. +36 + +SUPPLEMENTARY MATERIAL +A +Preliminaries +A.1 +Counting processes for Cox model +We use the counting process theory of Andersen & Gill (1982) in our theoretical framework +to study the large sample properties of Cox model. We state the existing results that are +used in our proof. +Let X⊗l denote 1 for l = 0, X for l = 1, and XXT for l = 2. Define +U (l) +a (βa, t) = 1 +na +n +� +i=1 +I{Ai = a}X⊗l +i +exp(βT +a Xi)Yi(t) and u(l) +a (βa, t) = E +� +X⊗l exp(βT +a X)Y (t) +� +, +where na = �n +i=1 I{Ai = a}, and define +Ea(βa, t) = U (1) +a (βa, t) +U (0) +a (βa, t) +and ea(βa, t) = u(1) +a (βa, t) +u(0) +a (βa, t) +. +The maximum partial likelihood estimator ˆβa for the Cox proportional hazards model +solves the estimating equation +Sa,n(βa) = 1 +na +n +� +i=1 +I{Ai = a} +� � +Xi − U (1) +1 (βa, u) +U (0) +1 (βa, u) +� +dNi(u) = 0, +and the cumulative baseline hazard function ˆΛ0,a is estimted by the Breslow estimator: +ˆΛ0,a(t) = +� t +0 +�n +i=1 I{Ai = a}dNi(u) +�n +i=1 I{Ai = a} exp(ˆβT +a Xi)Yi(u) +, a = 0, 1. +Under certain regularity conditions (Andersen & Gill 1982, Conditions A – D), ˆβa and +ˆΛ0,a converge in probability to the limits β∗ +a and Λ∗ +0,a, respectively; and we have +√na(ˆβa − β∗ +a) = Γ−1 +a +1 +√na +n +� +i=1 +I{Ai = a}Ha,i + op(1), +37 + +where Γa = E[−∂Sa,n(β∗ +a)/∂β∗T +a ] is the Fisher information matrix of β∗ +a, Ha,i = +� +I{Ai = +a}{Xi − ea(β∗ +a, u)}dMa,i(u) and dMa,i(u) = dNi(u) − exp(β∗T +a Xi)Yi(u)dΛ∗ +0,a(u). Moreover, +let S∗(t | a, X) = exp{−Λ∗ +0,a(t) exp(β∗T +a X)}; it is shown that √na{ ˆS(t | a, Xi)−S∗(t | a, Xi)} +converges uniformly to a mean-zero Gaussian process for all Xi. +Specifically, we consider the following expansion that we use in our proof of Theorem 1 +and Corollary 1, +ˆS(t | a, Xi) − S∗(t | a, Xi) = − S∗(t | a, Xi)Λ∗ +0,a(t) exp(β∗T +a Xi)XT +i (ˆβa − β∗ +a) +− S∗(t | a, Xi) exp(β∗T +a Xi)(ˆΛ0,a(t) − Λ∗ +0,a(t)), +and furthermore +ˆΛ0,a(t) − Λ∗ +0,a(t) = +� t +0 +� +n−1 +a +�n +i=1 I{Ai = a}dNi(u) +U (0) +a (ˆβa, u) +− n−1 +a +�n +i=1 I{Ai = a}dNi(u) +U (0) +a (β∗ +a, u) +� ++ +� t +0 +� +n−1 +a +�n +i=1 I{Ai = a}dNi(u) +U (0) +a (β∗ +a, u) +− dΛ∗ +0,a(t) +� += − +� +�� +� t +0 +U (1) +a (β∗ +a, u) +� +U (0) +a (β∗ +a, u) +�2 +� +n−1 +a +n +� +i=1 +I{Ai = a}dNi(u) +�� +�� +T +� +ˆβa − β∗ +a +� ++ +� t +0 +n−1 +a +�n +i=1 I{Ai = a}dMa,i(u) +U (0) +a (β∗ +a, u) ++ op(1) += − +�� t +0 +ea(β∗ +a, u)dΛ∗ +0,a(u) +�T � +ˆβa − β∗ +a +� ++ +� t +0 +n−1 +a +�n +i=1 I{Ai = a}dMa,i(u) +U (0) +a (β∗ +a, u) ++ op(1). +Combining the above two equations, we obtain +ˆS(t | a, Xi) − S∗(t | a, Xi) += +� +−S∗(t | a, Xi)Λ∗ +0,a(t) exp(β∗T +a Xi)XT +i − +�� t +0 +ea(β∗ +a, u)dΛ∗ +0,a(u) +�T� � +ˆβa − β∗ +a +� ++ +� t +0 +n−1 +a +�n +i=1 I{Ai = a}dMa,i(u) +U (0) +a (β∗ +a, u) ++ op(1). +38 + +A.2 +Cross-fitting +To show the high-level idea of cross-fitting, we state the lemma from Kennedy et al. (2020), +which is useful in our proof of Theorem 2 and Corollary 2. +Lemma 1. Consider two independent samples O1 = (O1, . . . , On) and O2 = (On+1, . . . , O˜n), +let ˆf(o) be a function estimated from O2 and Pn the empirical measure over O1, then we +have +(Pn − P)( ˆf − f) = OP +� +∥ ˆf − f∥ +√n +� +Proof. First note that by conditioning on O2 we obtain +E +� +Pn( ˆf − f) +�� O2 +� += E( ˆf − f | O2) = P( ˆf − f) +and the conditional variance is +var{(Pn − P)( ˆf − f) | O2} = var{Pn( ˆf − f) | O2} = 1 +nvar( ˆf − f | O2) ≤ ∥ ˆf − f∥2/n +therefore by Chebyshev’s inequality we have +P +� +|(Pn − P)( ˆf − f)| +∥ ˆf − f∥2/n +≥ t +� += E +� +P +� +|(Pn − P)( ˆf − f)| +∥ ˆf − f∥2/n +≥ t +���� O2 +�� +≤ 1 +t2 +thus for any ϵ > 0 we can pick t = 1/√ϵ so that the probability above is no more than ϵ, +which yields the result. +39 + +B +Proof of Proposition 1 +We first show the identification by the outcome regression formula. +V (d) = E[E[y(T(d)) | X]] += E[d(X)E[y(T(1)) | X] + (1 − d(X))E[y(T(0)) | X]] += E[d(X)E[y(T(1)) | X, IS = 1] + (1 − d(X))E[y(T(0)) | X, IS = 1]] += E[d(X)E[y(T(1)) | A = 1, X, IS = 1] ++ (1 − d(X))E[y(T(0)) | A = 0, X, IS = 1]] += E[d(X)E[y(T) | A = 1, X, IS = 1] + (1 − d(X))E[y(T) | A = 0, X, IS = 1]] += E[E[y(T) | A = d(X), X, IS = 1]] += E[IT e(X)E[y(T) | A = d(X), X, IS = 1]]. +Similarly, we show the identification by the IPW formula. +V (d) = E[E[y(T) | A = d(X), X, IS = 1]] += E +� +IS +πS(X)E[y(T) | A = d(X), X, IS = 1] +� += E +� +IS +πS(X) +I{A = d(X)} +πd(X) +∆ y(U) +SC(U | A, X) +� +, +where the last equation follows from the standard IPTW-IPCW formula (Van der Laan & +Robins 2003). +C +Proof of Proposition 2 +While Lee et al. (2022) derived the efficient influence function for the treatment specific sur- +vival function, here we derive the EIF for the value function V (d) = E[IT e(X)µ(d(X), X)]. +40 + +First consider the full data Z = (X, A, T, IS, IT), and we have the factorization as +p(Z) = {p(X)πS(X)p(A|X, IS = 1)p(T|A, X, IS = 1)}IS{p(X)}IT . +Since ISIT = 0, the score function is S(Z) = S(X, A, T, IS) + ITS(X). Let Vϵ(d) = +Eϵ[IT e(X)µϵ(d(X), X)] denote the parameter of interest evaluated under the law pϵ(Z), +where ϵ indexes a regular parametric submodel such that p0(Z) is the true data generating +law. To establish that V (d) is pathwise differentiable with EIF φF +d , we need to show that +∂ +∂ϵVϵ(d) +���� +ϵ=0 += E[φF +d S(Z)]. +First, we compute +∂ +∂ϵVϵ(d) +���� +ϵ=0 += E[IT e(X)µ(d(X), X)S(X)] + E +� ∂ +∂ϵµϵ(d(X), X) +���� +ϵ=0 +� +, +and further write the first term on the right hand side as +E[IT e(X)µ(d(X), X)S(X)] = E[(IT e(X)µ(d(X), X) − V (d))S(X)] += E[(IT e(X)µ(d(X), X) − V (d))S(Z)], +41 + +and the second term as +E +� ∂ +∂ϵµϵ(d(X), X) +���� +ϵ=0 +� += E [d(X)E[y(T)S(T | A, X, IS) | A = 1, X, IS = 1] ++(1 − d(X))E[y(T)S(T | A, X, IS) | A = 0, X, IS = 1]] += E [d(X)E[(y(T) − µ(1, X))S(T | A, X, IS) | A = 1, X, IS = 1] ++(1 − d(X))E[(y(T) − µ(0, X))S(T | A, X, IS) | A = 0, X, IS = 1]] += E +� +d(X)E +� +IS A +πS(X)πA(X)(y(T) − µ(1, X))S(T | A, X, IS) +����X +� ++(1 − d(X))E +� +IS (1 − A) +πS(X)(1 − πA(X))(y(T) − µ(0, X))S(T | A, X, IS) +����X +�� += E +� +IS +πS(X) +� +d(X) +A +πA(X)(y(T) − µ(1, X)) ++(1 − d(X)) +1 − A +1 − πA(X)(y(T) − µ(0, X)) +� +S(T | A, X, IS) +� += E +� +IS +πS(X) +I{A = d(X)} +πd(X) +(y(T) − µ(A, X))S(Z) +� +. +Therefore, the efficient influence function for the full data is +φF +d = IT e(X)µ(d(X), X) + +IS +πS(X) +I{A = d(X)} +πd(X) +(y(T) − µ(A, X)) − V (d). +Next, we consider the observed data O = (X, A, U, ∆, IS, IT) due to right censoring. +According to Tsiatis (2006, Section 10.4), the EIF based on the observed data is given by +φd = +∆ φF +d +SC(U | A, X) + +� ∞ +0 +L(u, A, X) +SC(u | A, X)dMC(u | A, X), +where +L(u, A, X) = E[φF +d | T ≥ u, A, X] += IT e(X)µ(d(X), X) + +IS +πS(X) +I{A = d(X)} +πd(X) +(Q(u, A, X) − µ(A, X)) − V (d). +Since we have +� ∞ +0 +dMC(u | A, X) +SC(u | A, X) += +� ∞ +0 +dNC(u) +SC(u | A, X) − +� U +0 +dΛC(u | A, X) +exp{ΛC(u | A, X)} += 1 − +∆ +SC(U | A, X), +(9) +42 + +we conclude that +φd = +IS +πS(X) +I{A = d(X)} +πd(X) +∆ y(U) +SC(U | A, X) − V (d) ++ +� +IT e(X) − +IS +πS(X) +I{A = d(X)} +πd(X) +� +µ(d(X), X) ++ +IS +πS(X) +I{A = d(X)} +πd(X) +� ∞ +0 +dMC(u | A, X) +SC(u | A, X) Q(u, A, X). +D +Proof of Theorem 1 and Corollary 1 +D.1 +Double robustness +We start with the proof of the double robustness property. +We show that EIF-based +estimator is consistent when either the survival outcome model or the models for the +sampling score, the propensity score and the censoring process are correctly specified. +Under some regularity conditions, the nuisance estimators ˆµ(a, x), ˆQ(u, a, x), ˆπS(x), ˆπA(x) +and ˆSC(t | a, x) converge in probability to µ∗(a, x), Q∗(u, a, x), π∗ +S(x), π∗ +A(x) and S∗ +C(t | a, x), +respectively. It suffices to show that E[V ∗(d)] = V (d), where +V ∗(d) =IT e(X)µ∗(A = d(X), X) ++ +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� +∆ y(U) +S∗ +C(U | A, X) − µ∗(A, X) ++ +� ∞ +0 +dM ∗ +C(u | A, X) +S∗ +C(u | A, X) Q∗(u, A, X) +� +=(I) + (II) + (III). +First, consider the case when the survival outcome model is correct, thus we have +(I) = E[IT e(X)µ∗(A = d(X), X)] = V (d) +43 + +By Equation 9, we obtain +(II) + (III) += +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� +y(T) − µ∗(A, X) − +� ∞ +0 +dM ∗ +C(u | A, X) +S∗ +C(u | A, X) (y(T) − Q∗(u, A, X)) +� +. +In this case, we have +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +(y(T) − µ∗(A, X)) +� += E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +(y(T) − µ∗(A, X)) +���� X +�� += E +� +E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +(y(T) − µ∗(A, X)) +���� A, X, IS = 1 +� ���� X +�� += E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +E[(y(T) − µ∗(A, X)) | A, X, IS = 1] +���� X +�� += E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +(E[y(T) | A, X, IS = 1] − µ∗(A, X)) +���� X +�� += 0. +Also define d ˜ +MC(u | A, X) = d ˜NC(u) − I{C ≥ u}dΛC(u | A, X) where ˜NC(u) = I{C ≤ +u}, so we have +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� ∞ +0 +dM ∗ +C(u | A, X) +S∗ +C(u | A, X) (y(T) − Q∗(u, A, X)) +� += E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� ∞ +0 +d ˜ +MC(u | A, X) +S∗ +C(u | A, X) I{T ≥ u}(y(T) − Q∗(u, A, X)) +� += E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� ∞ +0 +d ˜ +M ( +Cu | A, X) +S∗ +C(u | A, X) I{T ≥ u}(y(T) − Q∗(u, A, X)) +���� X +�� += E +� +E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� ∞ +0 +d ˜ +MC(u | A, X) +S∗ +C(u | A, X) I{T ≥ u} +(y(T) − Q∗(u, A, X)) +���� A, X, C, IS = 1 +� ���� X +�� += E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� ∞ +0 +d ˜ +MC(u | A, X) +S∗ +C(u | A, X) E [I{T ≥ u} +(y(T) − Q∗(u, A, X)) +���� A, X, C, IS = 1 +� ���� X +�� += E +� +E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� ∞ +0 +d ˜ +MC(u | A, X) +S∗ +C(u | A, X) (E[I{T ≥ u}y(T) | A, X, IS = 1] +−E[I{T ≥ u} | A, X, IS = 1]Q∗(u, A, X)) +���� X +�� += 0. +44 + +Next, consider the case when the models for the sampling score, the propensity score +and the censoring process are correctly specified. Rearranging the terms of V ∗(d), we obtain +V ∗(d) = +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +∆ y(U) +S∗ +C(U | A, X) ++ +� +IT e(X) − +IS +π∗ +S(X) +� +µ∗(A = d(X), X) ++ +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +� ∞ +0 +dM ∗ +C(u | A, X) +S∗ +C(u | A, X) Q∗(u, A, X) +=(I) + (II) + (III). +In this case, we have +(I) = E +� +IS +π∗ +S(X) +I{A = d(X)} +π∗ +d(X) +∆ y(U) +S∗ +C(U | A, X) +� += V (d), +(II) = E +�� +IT e(X) − +IS +π∗ +S(X) +� +µ∗(A = d(X), X) +� += E +� +E +� +IT e(X) − +IS +π∗ +S(X) +����X +� +µ∗(A = d(X), X) +� += 0, +and (III) is a stochastic integral with respect to the martingale M ∗ +C(u | A, X), thus equals +0 as well, which completes the double robustness property. +D.2 +Asymptotic properties +To establish the asymptotic results, we need some regularity conditions such that the nui- +sance estimators µ(a, x; ˆβa, ˆΛ0,a), Q(u, a, x; ˆβa, ˆΛ0,a), πS(x; ˆλ), πA(x; ˆθ) and SC(u | a, x; ˆαa, ˆΛC0,a) +converge in probability to µ(a, x; β∗ +a, Λ∗ +0,a), Q(u, a, x; β∗ +a, Λ∗ +0,a), πS(x; λ∗), πA(x; θ∗) and +SC(t | a, x; α∗ +a, Λ∗ +C0,a), respectively. +Condition 1. We assume the following conditions hold: +(C1) X is bounded almost surely. +(C2) The equation E +�� +A − +exp(θT X) +1+exp(θT X) +� +X +� += 0 has a unique solution θ∗. +45 + +(C3) For a = 0, 1, the equation +E +�� L +0 +� +Xi − E[Yi(u) exp(βT +a X)X] +E[Yi(u) exp(βT +a X)] +� +× dNi(u) +� += 0, +has a unique solution β∗ +a, where L > u is a pre-specified time point such that Pr(Ui > L) > +0. Moreover, let +Λ∗ +0,a(u) = E +�� u +0 +dNi(u) +E[Yi(u) exp(β∗T +a Xi)] +� +, +and assume Λ∗ +0,a(L) < ∞. +(C4) For a = 0, 1, the equation +E +�� L +0 +� +Xi − E[Yi(u) exp(αT +a X)X] +E[Yi(u) exp(αT +a X)] +� +× dNi(u) +� += 0, +has a unique solution α∗ +a. Moreover, let +Λ∗ +C0,a(u) = E +�� u +0 +dNi(u) +E[Yi(u) exp(α∗T +a Xi)] +� +, +and assume Λ∗ +C0,a(L) < ∞. +(C5) The estimating equation for the sampling score model πS(X; λ) has a unique solution +λ∗, and achieves root-n rate of convergence. +Under Condition 1, we have the following asymptotic representations: +√n(ˆθ − θ∗) = +1 +√n +n +� +i=1 +φθi + op(1), +√n(ˆλ − λ∗) = +1 +√n +n +� +i=1 +φλi + op(1), +√n(ˆβa − β∗ +a) = +1 +√n +n +� +i=1 +φβai + op(1), +√n(ˆαa − α∗ +a) = +1 +√n +n +� +i=1 +φαai + op(1), +for a = 0, 1. +We focus on the estimation of survival functions by our proposed method: +ˆS(t; η) = 1 +N +N +� +i=1 +� +IT,i e(Xi) ˆS(t | A = dη(Xi), Xi) ++ IS,iI{Ai = dη(Xi)} +ˆπS(Xi)ˆπd(Xi) +� +∆i Yi(t) +ˆSC(t | Ai, Xi) +− ˆS(t | Ai, Xi) ++ +� ∞ +0 +ˆS(t | Ai, Xi)d ˆ +MC(u | Ai, Xi) +ˆS(u | Ai, Xi) ˆSC(u | Ai, Xi) +� � +, +46 + +and for the ease of notation, define +ˆJ(t, a, x) = +∆i Yi(t) +ˆSC(t | a, x) +− ˆS(t | a, x) + +� ∞ +0 +ˆS(t | a, x)d ˆ +MC(u | a, x) +ˆS(u | a, x) ˆSC(u | a, x) +, +J∗(t, a, x) = +∆i Yi(t) +S∗ +C(t | a, x) − S∗(t | a, x) + +� ∞ +0 +S∗(t | a, x)dM ∗ +C(u | a, x) +S∗(u | a, x)S∗ +C(u | a, x) . +Our proof has three main parts as follows. +PART 1. By the double robustness property shown in Section D.1, we have, by the +strong law of large numbers and uniform consistency, that ˆS(t; η) = S(t; η) + op(1), which +proves (i) of Theorem 1. Moreover, define +S∗ +N(t; η) = 1 +N +N +� +i=1 +� +IT,i e(Xi)S∗(t | A = dη(Xi), Xi) + IS,iI{Ai = dη(Xi)} +π∗ +S(Xi)π∗ +d(Xi) +J∗(t, Ai, Xi) +� +, +and by applying the Taylor expansion and the counting processes result in Section A.1, we +obtain +ˆS(t; η) =S∗ +n(t; η) + HT +λ (ˆλ − λ∗) + HT +θ (ˆθ − θ∗) + HT +β0(ˆβ0 − β∗ +0) + HT +β1(ˆβ1 − β∗ +1) ++ HT +α0(ˆα0 − α∗ +0) + HT +α1(ˆα1 − α∗ +1) + RS + op(N −1/2), +where +Hλ = lim +N→∞ +1 +N +N +� +i=1 +∂ ˆS(t; η) +∂λ∗ +, Hθ = lim +N→∞ +1 +N +N +� +i=1 +∂ ˆS(t; η) +∂θ∗ +, +Hβa = lim +N→∞ +1 +N +N +� +i=1 +� +IT,i e(Xi)(−1)a+1G(t, a, Xi) + IS,iI{Ai = a} +π∗ +S(Xi)π∗ +d(Xi) +�� ∞ +0 +G(t, a, Xi)dM ∗ +C(u | a, Xi) +S∗(u | a, Xi)S∗ +C(u | a, Xi) +−G(t, a, Xi) − +� ∞ +0 +G(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗2(u | a, Xi)S∗ +C(u | a, Xi) +�� +, +Hαa = lim +N→∞ +1 +N +N +� +i=1 +IS,iI{Ai = a} +π∗ +S(Xi)π∗ +d(Xi) +� −∆iYi(t) +S∗ +C(t | a, Xi)GC(t, a, Xi) +− +� ∞ +0 +GC(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗2 +C (u | a, Xi)S∗(u | a, Xi) ++ ˜GC(t, a, Xi) +� +, +47 + +RS = 1 +N +N +� +i=1 +� +a=0,1 +� +IT,i e(Xi)(−1)a+1H(t, a, Xi) ++ IS,iI{Ai = a} +π∗ +S(Xi)π∗ +d(Xi) +� � ∞ +0 +H(t, a, Xi)dM ∗ +C(u | a, Xi) +S∗ +C(u | a, Xi)S∗(u | a, Xi) − H(t, a, Xi) +− +� ∞ +0 +H(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗ +C(u | a, Xi)S∗2(u | a, Xi) +− +∆iYi(t) +S∗ +C(t|a, Xi)HC(t, a, Xi) +− +� ∞ +0 +HC(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗2 +C (u | a, Xi)S∗(u | a, Xi) +− ˜HC(t, a, Xi) +�� += 1 +N +N +� +i=1 +φRs,i, +with +G(t, a, x) = −S∗(t | a, x)Λ∗ +0,a(t)xT + S∗(t | a, x) exp(β∗T +a x) +�� t +0 +ea(β∗ +a, u)dΛ∗ +0,a(u) +�T +, +H(t, a, x) = −S∗(t | a, x) exp(β∗T +a x) +� t +0 +n−1 +a +�n +i=1 I{Ai = a}dMa,i(u) +U (0) +a (β∗ +a, u) +, +GC(t, a, x) = −S∗(t | a, x)Λ∗ +0,a(t)xT + S∗(t | a, x) exp(β∗T +a x) +�� t +0 +ea(β∗ +a, u)dΛ∗ +0,a(u) +�T +, +HC(t, a, x) = −S∗(t | a, x) exp(β∗T +a x) +� t +0 +n−1 +a +�n +i=1 I{Ai = a}dMa,i(u) +U (0) +a (β∗ +a, u) +, +˜GC(t, a, x) = +� Ui +0 +S∗(t | a, x)dΛ∗ +C(u | a, x) +S∗ +C(u | a, x)S∗(u | a, x) xT + +�� t +0 +S∗(t | a, x)ea(β∗ +a, u)dΛ∗ +0,a(u) +S∗ +C(u | a, x)S∗(u | a, x) +�T +, +˜HC(t, a, x) = +� t +0 +S∗(t | a, x)n−1 +a +�n +i=1 I{Ai = a}dMa,i(u) +S∗ +C(u | a, x)S∗(u | a, x)U (0) +a (β∗ +a, u) +. +Thus, we have +√ +N +� +ˆS(t; η) − S(t; η) +� += +1 +√ +N +N +� +i=1 +(ξ1,i(t; η) + ξ2,i(t; η)) + op(1), +(10) +where +ξ1,i(t; η) = S∗ +n(t; η) − S(t; η), +ξ2,i(t; η) = HT +λ φλ∗,i + HT +θ φθ∗,i + +� +a=0,1 +HT +βaφβ∗ +0,i + +� +a=0,1 +HT +αaφα∗a,i + HT +α1 + φRs,i, +and ξ1,i(t; η), ξ2,i(t; η) are independent mean-zero processes. +Therefore, we obtain that +√ +N +� +ˆS(t; η) − S(t; η) +� +converges weakly to a mean-zero Gaussian process, which proves +(ii) of Theorem 1. +48 + +PART 2. We show that N 1/3∥ˆη − η∗∥2 = Op(1). Recall that +ˆη = arg max +η +ˆS(t; η) and η∗ = arg max +η +S(t; η). +By Assumption 5 (i), S(t; η) is twice continuously differentiable at a neighborhood of +η∗; in Step 1, we show that ˆS(t; η) = S(t; η) + op(1), ∀η; since ˆη maximizes ˆS(t; η), we have +that ˆS(t; ˆη) ≥ supη ˆS(t; η), thus by the Argmax theorem, we have ˆη +p→ η∗ as N → ∞. +In order to establish the N −1/3 rate of convergence of ˆη, we apply Theorem 14.4 (Rate +of convergence) of Kosorok (2008), and need to find the suitable rate that satisfies three +conditions below. +Condition 1 For every η in a neighborhood of η∗ such that ∥η − η∗∥2 < δ, by Assump- +tion 5 (i), we apply the second-order Taylor expansion, +S(t; η) − S(t; η∗) = S′(η∗)∥η − η∗∥2 + 1 +2S′′(η∗)∥η − η∗∥2 +2 + o(∥η − η∗∥2 +2) += 1 +2S′′(η∗)∥η − η∗∥2 +2 + o(∥η − η∗∥2 +2), +and as S′′(η∗) < 0, there exists c0 = − 1 +2S′′(η∗) > 0 such that S(t; η) − S(t; η∗) ≤ −c0∥η − +η∗∥2 +2. +Condition 2 For all N large enough and sufficiently small δ, we consider the centered +process ˆS − S, and have that +E +� +√ +N +sup +∥η−η∗∥2<δ +��� ˆS(t; η) − S(t; η) − +� +ˆS(t; η∗) − S(t; η∗) +���� +� += E +� +√ +N +sup +∥η−η∗∥2<δ +��� ˆS(t; η) − S∗ +n(t; η) + S∗ +n(t; η) − S(t; η) +− +� +ˆS(t; η∗) − S∗ +n(t; η∗) + S∗ +n(t; η∗) − S(t; η∗) +���� +� +≤ E +� +√ +N +sup +∥η−η∗∥2<δ +��� ˆS(t; η) − S∗ +n(t; η) − +� +ˆS(t; η∗) − S∗ +n(t; η∗) +���� +� +(I) ++ E +� +√ +N +sup +∥η−η∗∥2<δ +|S∗ +n(t; η) − S(t; η) − {S∗ +n(t; η∗) − S(t; η∗)}| +� +, +(II) +49 + +and we bound (I) and (II) respectively as follows. +Condition 2.1 To bound (II), we need the useful facts that +I{A = dη(X)} − I{A = dη∗(X)} = (2A − 1)(dη(X) − dη∗(X)), +S∗(t | dη(Xi), Xi) − S∗(t | dη∗(Xi), Xi) = (S∗(t | 1, Xi) − S∗(t | 0, Xi))(dη(Xi) − dη∗(Xi)), +and obtain +S∗ +n(t; η) − S∗ +n(t; η∗) = 1 +N +N +� +i=1 +(dη(Xi) − dη∗(Xi)) +× +� +IT,i e(Xi)(S∗(t | 1, Xi) − S∗(t | 0, Xi)) + (2Ai − 1)IS,i +π∗ +S(Xi)π∗ +d(Xi)J∗(t, Ai, Xi) +� +. +Define a class of functions +F1 +η = +� +(dη(x) − dη∗(x)) +� +IT e(x)(S∗(t | 1, x) − S∗(t | 0, x)) + (2a − 1)IS +π∗ +a(x)π∗ +S(x)J∗(t, a, x) +� +: +∥η − η∗∥2 < δ +� +, +and let M1 = sup +���IT e(x)(S∗(t | 1, x) − S∗(t | 0, x)) + +(2a−1)IS +π∗a(x)π∗ +S(x)J∗(t, a, x) +���. +By Assump- +tion 1, 3 and Condition 1, we have that M1 < ∞. +When ∥η − η∗∥2 < δ, by Condition 1 (C1), there exists a constant 0 < k0 < ∞ such +that |(1, xT)(η − η∗)| < k0δ; furthermore, we show that |dη(x) − dη∗(x)| = |I{(1, xT)η > +0} − I{(1, xT)η∗ > 0}| ≤ I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}, by considering the three cases: +• when −k0δ ≤ (1, xT)η∗ ≤ k0δ, we have |dη(x) − dη∗(x)| ≤ 1 = I{−k0δ ≤ (1, xT)η∗ ≤ +k0δ}; +• when (1, xT)η∗ > k0δ > 0, we have (1, xT)η = (1, xT)(η − η∗) + (1, xT)η∗ > 0, so +|dη(x) − dη∗(x)| = 0 = I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}; +• when (1, xT)η∗ < −k0δ < 0, we have (1, xT)η = (1, xT)(η − η∗) + (1, xT)η∗ < 0, so +|dη(x) − dη∗(x)| = 0 = I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}. +50 + +Thus we can define the envelope of F1 +η as F1 = M1I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}. By +Assumption 5 (ii), there exists a constant 0 < k1 < ∞ such that +∥F1∥P,2 ≤ M1 +� +Pr(−k0δ ≤ (1, xT)η∗ ≤ k0δ) ≤ M1 +� +2k0k1δ1/2. +By Lemma 9.6 and Lemma 9.9 of Kosorok (2008), we have that F1 +η, a class of indicator +functions, is a Vapnik-Cervonenkis (VC) class with bounded bracketing entropy J∗ +[](1, F1 +η) < +∞. +Since we have the fact that +GNF1 +η = N −1/2 +N +� +i=1 +� +F1 +η − E[F1 +η] +� += +√ +N (S∗ +n(t; η) − S∗ +n(t; η∗) − {S(t; η) − S(t; η∗)}) , +By Theorem 11.2 of Kosorok (2008), we obtain that there exists a constant 0 < c1 < ∞, +(II) = E +� +sup +∥η−η∗∥2<δ +|GNF1 +η| +� +≤ c1J∗ +[](1, F1 +η)∥F1∥P,2 ≤ c1J∗ +[](1, F1 +η)M1 +� +2k0k1δ1/2 = ˜c1δ1/2, +so we conclude that (II) ≤ ˜c1δ1/2 where ˜c1 > 0 is a finite constant. +Condition 2.2 To bound (I), first we have +ˆS(t; η) − S∗ +n(t; η) − { ˆS(t; η∗) − S∗ +n(t; η∗)} = ˆS(t; η) − ˆS(t; η∗) − {S∗ +n(t; η) − S∗ +n(t; η∗)} += 1 +N +N +� +i=1 +(dη(Xi) − dη∗(Xi)) +� +IT,i e(Xi){ ˆS(t|1, Xi) − ˆS(t|0, Xi) − (S∗(t|1, Xi) − S∗(t|0, Xi))} ++ (2Ai − 1)IS,i +ˆπAi(Xi)ˆπS(Xi) +ˆJ(t, Ai, Xi) − (2Ai − 1)IS,i +π∗ +Ai(Xi)π∗ +S(Xi)J∗(t, Ai, Xi) +� +, +and then apply the Taylor expansion and counting processes result in Section A.1, +ˆS(t; η) − S∗ +n(t; η) − { ˆS(t; η∗) − S∗ +n(t; η∗)} += 1 +N +N +� +i=1 +(dη(Xi) − dη∗(Xi)) × +� +Dλ(ˆλ − λ∗) + Dθ(ˆθ − θ∗) + Dβ0(ˆβ0 − β∗ +0) ++Dβ1(ˆβ1 − β∗ +1) + Dα0(ˆα0 − α∗ +0) + Dα1(ˆα1 − α∗ +1) + RS,i +� ++ op(N −1/2), +(11) +51 + +where +Dλ = − (2Ai − 1)IS,i +π∗ +Ai(Xi)π∗2 +S (Xi)J∗(t, Ai, Xi) +�∂π∗ +S(Xi) +∂λ +�T +, +Dθ = − +IS,i +π∗2 +Ai(Xi)π∗ +S(Xi)J∗(t, Ai, Xi) +�∂π∗ +A(Xi) +∂θ +�T +, +Dβa =IT,i e(Xi)(−1)a+1G(t, a, Xi) + (2Ai − 1)I{Ai = a}IS,i +π∗ +Ai(Xi)π∗ +S(Xi) +�� ∞ +0 +G(t, a, Xi)dM ∗ +C(u | a, Xi) +S∗ +C(u | a, Xi)S∗(u | a, Xi) +−G(t, a, Xi) − +� ∞ +0 +G(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗ +C(u | a, Xi)S∗2(u | a, Xi) +� +, +Dαa =(2Ai − 1)I{Ai = a}IS,i +π∗ +Ai(Xi)π∗ +S(Xi) +� +− +∆i Yi(t) +S∗ +C(t | a, Xi)GC(t, a, Xi) +− +� ∞ +0 +GC(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗2 +C (u | a, Xi)S∗(u | a, Xi) ++ ˜GC(t, a, Xi) +� +, +RS,i = +� +a=0,1 +� +IT,i e(Xi)(−1)a+1H(t, a, Xi) + (2Ai − 1)I{Ai = a}IS,i +π∗ +Ai(Xi)π∗ +S(Xi) +� � ∞ +0 +H(t, a, Xi)dM ∗ +C(u | a, Xi) +S∗ +C(u | a, Xi)S∗(u | a, Xi) +− H(t, a, Xi) − +� ∞ +0 +H(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗ +C(u | a, Xi)S∗2(u | a, Xi) +− +∆i Yi(t) +S∗ +C(t | a, Xi)HC(t, a, Xi) − +� ∞ +0 +HC(u, a, Xi)S∗(t | a, Xi)dM ∗ +C(u | a, Xi) +S∗2 +C (u | a, Xi)S∗(u | a, Xi) +− ˜HC(t, a, Xi) +�� +. +Similarly, we define the following classes of functions: +F2 +η = +� +(dη(x) − dη∗(x)) (2a − 1)IS,i +π∗ +a(x)π∗2 +S (x)J∗(t, a, x) +�∂π∗ +S(x) +∂λ +�T +: ∥η − η∗∥2 < δ +� +, +F3 +η = +� +(dη(x) − dη∗(x)) +−IS,i +π∗2 +a (x)π∗ +S(x)J∗(t, a, x) +�∂π∗ +A(x) +∂θ +�T +: ∥η − η∗∥2 < δ +� +, +F4 +η = +� +(dη(x) − dη∗(x)) +� +IT e(x)(−1)a+1G(t, a, x) + (2a − 1)IS +π∗ +a(x)π∗ +S(x) +× +�� ∞ +0 +G(t, a, x)dM ∗ +C(u | a, x) +S∗ +C(u | a, x)S∗(u | a, x) − G(t, a, x) +− +� ∞ +0 +G(u, a, x)S∗(t | a, x)dM ∗ +C(u | a, x) +S∗ +C(u | a, x)S∗2(u | a, x) +� � +: ∥η − η∗∥2 < δ +� +, +52 + +F5 +η = +� +(dη(x) − dη∗(x)) +� +IT e(x)(−1)a+1G(t, a, x) + (2a − 1)IS +π∗ +a(x)π∗ +S(x) +× +�� ∞ +0 +G(t, a, x)dM ∗ +C(u | a, x) +S∗ +C(u | a, x)S∗(u | a, x) − G(t, a, x) +− +� ∞ +0 +G(u, a, x)S∗(t | a, x)dM ∗ +C(u | a, x) +S∗ +C(u | a, x)S∗2(u | a, x) +� � +: ∥η − η∗∥2 < δ +� +, +F6 +η = +� +(dη(x) − dη∗(x)) +� +(2a − 1)IS +π∗ +a(x)π∗ +S(x) +� +− +∆ Y (t) +S∗ +C(t | a, x)GC(t, a, x) +− +� ∞ +0 +GC(u, a, x)S∗(t | a, x)dM ∗ +C(u | a, x) +S∗2 +C (u | a, x)S∗(u | a, x) ++ ˜GC(t, a, x) +� � +: ∥η − η∗∥2 < δ +� +, +F7 +η = +� +(dη(x) − dη∗(x)) +� +(2a − 1)IS +π∗ +a(x)π∗ +S(x) +� +− +∆ Y (t) +S∗ +C(t | a, x)GC(t, a, x) +− +� ∞ +0 +GC(u, a, x)S∗(t | a, x)dM ∗ +C(u | a, x) +S∗2 +C (u | a, x)S∗(u | a, x) ++ ˜GC(t, a, x) +� � +: ∥η − η∗∥2 < δ +� +, +F8 +η = +� +(dη(x) − dη∗(x)) +� � +a=0,1 +� +IT e(x)a+1H(t, a, x) + (2a − 1)IS +π∗ +a(x)π∗ +S(x) +× +� � ∞ +0 +H(t, a, x)dM ∗ +C(u | a, x) +S∗ +C(u | a, x)S∗(u | a, x) − H(t, a, x) +− +� ∞ +0 +H(u, a, x)S∗(t | a, x)dM ∗ +C(u | a, x) +S∗ +C(u | a, x)S∗2(u | a, x) +− +∆ Y (t) +S∗ +C(t | a, x)HC(t, a, x) +− +� ∞ +0 +HC(u, a, x)S∗(t | a, x)dM ∗ +C(u | a, x) +S∗2 +C (u | a, x)S∗(u | a, x) +− ˜HC(t, a, x) +��� +: ∥η − η∗∥2 < δ +� +. +Let +M2 = sup +����� +(2a − 1) +π∗ +a(x) J∗(t, a, x) +�∂π∗ +S(x) +∂λ +�T����� , +where M2 ∈ R+ and the supremum is taken over all the coordinates; and M3, . . . , M8 are +defined accordingly for F3 +η, . . . , F8 +η. By Assumption 1, 3 and Condition 1, we have that +M2, . . . , M8 < ∞. +Using the same technique as in Condition 2.1, we define the envelop of Fj +η as Fj = +MjI{−k0δ ≤ (1, xT)η∗ ≤ k0δ} for j = 2, . . . , 8, and obtain that +∥Fj∥P,2 ≤ ˜ +Mjδ1/2 < ∞, +j = 2, . . . , 8, +53 + +where +˜ +M2, . . . , ˜ +M8 are some finite constants, and that Fj +η is a VC class with bounded +bracketing entropy J∗ +[](1, Fj +η) < ∞, for j = 2, . . . , 8. By Theorem 11.2 of Kosorok (2008), +we obtain +E +� +sup +∥η−η∗∥2<δ +��GNFj +η +�� +� +≤ cjJ∗ +[](1, Fj +η)∥Fj∥P,2, +j = 2, . . . , 8, +where c2, . . . , c8 are some finite constants. That is, we have +E +� +sup +∥η−η∗∥2<δ +��GNF8 +η +�� +� +≤ ˜c8δ1/2, +and furthermore by Theorem 2.14.5 of Van der Vaart & Wellner (1996), we obtain +� +E +� +sup +∥η−η∗∥2<δ +∥GnFj +η∥2 +2 +��1/2 +≤ lj +� +E +� +sup +∥η−η∗∥2<δ +|GnFj +η| +� ++ ∥Fj∥P,2 +� +≤ lj{cjJ∗ +[](1, Fj +η) + 1}∥Fj∥P,2 +≤ ˜cjδ1/2, +j = 2, . . . , 7, +where l2, . . . , l7 and ˜c2, . . . , ˜c7 are some finite constants. +By Equation (11), we have that +(I) = E +� +N 1/2 +sup +∥η−η∗∥2<δ +��� ˆS(t; η) − S∗ +N(t; η) − { ˆS(t; η∗) − S∗ +N(t; η∗)} +��� +� +≤ E +� +sup +∥η−η∗∥2<δ +� +|GnF2 +η(ˆλ − λ∗)| + |GnF3 +η(ˆθ − θ∗)| + |GnF4 +η(ˆβ0 − β∗ +0)| + |GnF5 +η(ˆβ1 − β∗ +1)| ++ |GnF6 +η(ˆα0 − α∗ +0)| + |GnF7 +η(ˆα1 − α∗ +1)| + |GnF8 +η| +� ++ op(1) +� +≤ N −1/2 +� +E +� +sup +∥η−η∗∥2<δ +|GnF2 +η · N 1/2(ˆλ − λ∗)| +� ++ E +� +sup +∥η−η∗∥2<δ +|GnF3 +η · N 1/2(ˆθ − θ∗)| +� ++ E +� +sup +∥η−η∗∥2<δ +|GnF4 +η · N 1/2(ˆβ0 − β∗ +0)| +� ++ E +� +sup +∥η−η∗∥2<δ +|GnF5 +η · N 1/2(ˆβ1 − β∗ +1)| +� ++ E +� +sup +∥η−η∗∥2<δ +|GnF6 +η · N 1/2(ˆα0 − α∗ +0)| +� ++ E +� +sup +∥η−η∗∥2<δ +|GnF7 +η · N 1/2(ˆα1 − α∗ +1)| +� � ++ E +� +sup +∥η−η∗∥2<δ +��GNF8 +η +�� +� ++ op(1), +54 + +and then by the Cauchy-Schwarz inequality, we obtain +(I) ≤ N −1/2 � +E[N∥ˆλ − λ∗∥2 +2] +�1/2 +� +E +� +sup +∥η−η∗∥2<δ +∥GNF2 +η∥2 +2 +��1/2 ++ N −1/2 � +E[N∥ˆθ − θ∗∥2 +2] +�1/2 +� +E +� +sup +∥η−η∗∥2<δ +∥GNF3 +η∥2 +2 +��1/2 ++ N −1/2 � +E[N∥ˆβ0 − β∗ +0∥2 +2] +�1/2 +� +E +� +sup +∥η−η∗∥2<δ +∥GNF4 +η∥2 +2 +��1/2 ++ N −1/2 � +E[N∥ˆβ1 − β∗ +1∥2 +2] +�1/2 +� +E +� +sup +∥η−η∗∥2<δ +∥GNF5 +η∥2 +2 +��1/2 ++ N −1/2 � +E[N∥ˆα0 − α∗ +0∥2 +2] +�1/2 +� +E +� +sup +∥η−η∗∥2<δ +∥GNF6 +η∥2 +2 +��1/2 ++ N −1/2 � +E[N∥ˆα1 − α∗ +1∥2 +2] +�1/2 +� +E +� +sup +∥η−η∗∥2<δ +∥GNF7 +η∥2 +2 +��1/2 ++ E +� +sup +∥η−η∗∥2<δ +��GNF8 +η +�� +� +. +Let Mλ = +� +E[N∥ˆλ − λ∗∥2 +2] +�1/2 +, and Mθ, Mβ0, Mβ1, Mα0, Mα1 are defined accordingly. +By Condition 1, we have that Mλ, Mθ, Mβ0, Mβ1, Mα0, Mα1 < ∞, and therefore +(I) ≤ N −1/2(Mλ˜c2 + Mθ˜c3 + Mβ0˜c4 + Mβ1˜c5 + Mα0˜c6 + Mα1˜c7)δ1/2 + ˜c8δ1/2. +In summary, we obtain that, let N → ∞, the centered process satisfies +E +� +√ +N +sup +∥η−η∗∥2<δ +��� ˆS(t; η) − S(t; η) − { ˆS(t; η∗) − S(t; η∗)} +��� +� +≤ (I) + (II) ≤ (˜c1 + ˜c8)δ1/2. +(12) +Let φN(δ) = δ1/2 and α = 3 +2 < 2, thus we have φn(δ) +δα += δ−1 is decreasing, and α does +not depend on N. That is, the second condition holds. +Condition 3 By the facts that ˆη +p→ η∗ as N → ∞, and that ˆS(t; ˆη) ≥ supη ˆS(t; η), +we choose rN = N 1/3 such that r2 +NφN(r−1 +N ) = N 2/3φN(N −1/3) = N 1/2. The third condition +holds. +55 + +In the end, the three conditions are satisfied with rN = N 1/3; thus we conclude that +N 1/3∥ˆη − η∗∥2 = Op(1), which completes the proof of (iii) of Theorem 1. +PART 3. We characterize the asymptotic distribution of ˆS(t; ˆη). Since we have +√ +N{ ˆS(t; ˆη) − S(t; η∗)} = +√ +N{ ˆS(t; ˆη) − ˆS(t; η∗)} + +√ +N{ ˆS(t; η∗) − S(t; η∗)}, +we study the two terms in two steps. +Step 3.1 To establish +√ +N{ ˆS(t; ˆη)− ˆS(t; η∗)} = op(1), it suffices to show that +√ +N{S(t; ˆη)− +S(t; η∗)} = op(1) and +√ +N( ˆS(t; ˆη) − ˆS(t; η∗) − {S(t; ˆη) − S(t; η∗)}) = op(1). +First, as N 1/3∥ˆη − η∗∥2 = Op(1), we take the second-order Taylor expansion +√ +N{S(t; ˆη) − S(t; η∗)} = +√ +N +� +S′(η∗)∥ˆη − η∗∥2 + 1 +2S′′(η∗)∥ˆη − η∗∥2 +2 + op(∥ˆη − η∗∥2 +2) +� += +√ +N +�1 +2S′′(η∗)∥ˆη − η∗∥2 +2 + op(∥ˆη − η∗∥2 +2) +� += +√ +N +�1 +2S′′(η∗)Op(N −2/3) + op(N −2/3) +� += op(1). +Next, we follow the result (12) obtained in PART 2. As N 1/3∥ˆη − η∗∥2 = Op(1), there +exists ˜δ = c9N −1/3, where c9 < ∞ is a finite constant, such that ∥ˆη − η∗∥2 ≤ ˜δ. Therefore +we have +√ +N( ˆS(t; ˆη) − ˆS(t; η∗) − {S(t; ˆη) − S(t; η∗)}) +≤ E +� +√ +N +sup +∥ˆη−η∗∥2<˜δ +��� ˆS(t; ˆη) − S(t; ˆη) − { ˆS(t; η∗) − S(t; η∗)} +��� +� +≤ (˜c1 + ˜c8)˜δ1/2 = (˜c1 + ˜c8)√c9N −1/6 = op(1), +which yields the result. +Step 3.2 To derive the asymptotic distribution of √n{ ˆS(t; η∗) − S(t; η∗)}, we follow +the result (10) obtained in PART 1 and have that +√ +N +� +ˆS(t; η∗) − S(t; η∗) +� +D→ N(0, σ2 +t,1), +56 + +where σ2 +t,1 = E[(ξ1,i(t; η∗) + ξ2,i(t; η∗))2]. Therefore we obtain in the end +√ +N{ ˆS(t; ˆη) − S(t; η∗)} = +√ +N{ ˆS(t; ˆη) − ˆS(t; η∗)} + +√ +N{ ˆS(t; η∗) − S(t; η∗)} += op(1) + +√ +N{ ˆS(t; η∗) − S(t; η∗)} +D→ N(0, σ2 +t,1), +which completes the proof. +For Corollary 1 where we consider RMST, the proof can follow the same steps as before, +and is thus omitted here. +E +Proof of Theorem 2 and Corollary 2 +Our proof has three main parts below. +PART 1. +Recall that the cross-fitting technique, at a high level as exemplified in +Lemma 1, uses sample splitting to avoid bias due to over-fitting. For simplicity, consider +that the datasets Os and Ot are randomly split into 2 folds with equal size respectively such +that Os = Os,1 ∪Os,2, Ot = Ot,1 ∪Ot,2. The extension to K-folds as described in Algorithm +1 is straightforward. Here the subscript CF is omitted to simplify the notation. Define +I1 = Os,1 ∪ Ot,1, I2 = Os,2 ∪ Ot,2, and N1 = |I1|, N2 = |I2|. The cross-fitted estimator for +the value function under the ITR dη is +ˆV (η) = N1 +N +ˆV I1(η) + N2 +N +ˆV I2(η), +where +ˆV I1(η) = 1 +N1 +� +I1 +� +IT,i e(Xi)ˆµ(dη(Xi), Xi) + +IS,i +ˆπS(Xi) +I{Ai = dη(Xi)} +ˆπd(Xi) +× +� +∆i y(Ui) +ˆSC(Ui | Ai, Xi) +− ˆµ(Ai, Xi) + +� ∞ +0 +d ˆ +MC(u | Ai, Xi) +ˆSC(u | Ai, Xi) +ˆQ(u, Ai, Xi) +� � +, +57 + +and the nuisance parameters are estimated from I2. ˆV I2(η) is defined accordingly. +In this step, we show that +ˆV (η) − VN(η) = op(N −1/2), +and essentially it suffices to prove that +ˆV I1(η) − V I1 +N (η) = op(N −1/2), +where +VN(η) = 1 +N +N +� +i=1 +� +IT,i e(Xi)µ(dη(Xi), Xi) + +IS,i +πS(Xi) +I{Ai = dη(Xi)} +πd(Xi) +× +� +∆i y(Ui) +SC(Ui | Ai, Xi) − µ(Ai, Xi) + +� ∞ +0 +dMC(u | Ai, Xi) +SC(u | Ai, Xi) Q(u, Ai, Xi) +� � +, +and V I1 +N (η) is defined accordingly. +First, we have the following decomposition +ˆV I1(η) − V I1 +N (η) += 1 +N1 +� +I1 +� +IT,i e(Xi)(ˆµ(dη(Xi), Xi) − µ(dη(Xi), Xi)) ++ IS,i +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� I{Ai = dη(Xi)} +πd(Xi) +K(Ai, Xi) ++ IS,iI{Ai = dη(Xi)} +πS(Xi) +� +1 +πd(Xi) − +1 +ˆπd(Xi) +� +K(Ai, Xi) ++ +IS,i +πS(Xi) +I{Ai = dη(Xi)} +πd(Xi) +( ˆK(Ai, Xi) − K(Ai, Xi)) ++ IS,iI{Ai = dη(Xi)} +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� � +1 +πd(Xi) − +1 +ˆπd(Xi) +� +K(Ai, Xi) ++ IS,iI{Ai = dη(Xi)} +πd(Xi) +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� +( ˆK(Ai, Xi) − K(Ai, Xi)) ++ IS,iI{Ai = dη(Xi)} +πS(Xi) +� +1 +πd(Xi) − +1 +ˆπd(Xi) +� +( ˆK(Ai, Xi) − K(Ai, Xi)) ++ IS,iI{Ai = dη(Xi)} +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� � +1 +πd(Xi) − +1 +ˆπd(Xi) +� +( ˆK(Ai, Xi) − K(Ai, Xi)) +� +, +(13) +58 + +where +ˆK(Ai, Xi) = +∆i y(Ui) +ˆSC(Ui | Ai, Xi) +− ˆµ(Ai, Xi) + +� ∞ +0 +d ˆ +MC(u | Ai, Xi) +ˆSC(u | Ai, Xi) +ˆQ(u, Ai, Xi), +K(Ai, Xi) = +∆i y(Ui) +SC(Ui | Ai, Xi) − µ(Ai, Xi) + +� ∞ +0 +dMC(u | Ai, Xi) +SC(u | Ai, Xi) Q(u, Ai, Xi). +In summary, the decomposition (13) consists of two types of terms: four mean-zero +terms and four product terms. For the mean-zero terms, we utilize the method introduced +in Section A.2; since +E[IT,i e(Xi)(ˆµ(dη(Xi), Xi) − µ(dη(Xi), Xi))] = 0, +by applying Lemma 1, we obtain +1 +N1 +� +I1 +IT,i e(Xi)(ˆµ(dη(Xi), Xi) − µ(dη(Xi), Xi)) = op(N −1/2). +Similarly we have +E +� +IS,i +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� I{Ai = dη(Xi)} +πd(Xi) +K(Ai, Xi) +� += 0, +so we obtain +E +� +� +� +1 +N1 +� +I1 +IS,i +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� I{Ai = dη(Xi)} +πd(Xi) +K(Ai, Xi) +�2� +� += E +� +�E +� +� +� +1 +N1 +� +I1 +IS,i +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� I{Ai = dη(Xi)} +πd(Xi) +K(Ai, Xi) +�2 �����I2 +� +� +� +� += E +� +var +� +1 +N1 +� +I1 +IS,i +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� I{Ai = dη(Xi)} +πd(Xi) +K(Ai, Xi) +�����I2 +�� += 1 +N1 +E +� +var +� +IS,i +� +1 +πS(Xi) − +1 +ˆπS(Xi) +� I{Ai = dη(Xi)} +πd(Xi) +K(Ai, Xi) +����I2 +�� +≤ Op(1) +N1 += op( 1 +N ). +We also have +E +�IS,iI{Ai = dη(Xi)} +πS(Xi) +� +1 +πd(Xi) − +1 +ˆπd(Xi) +� +K(Ai, Xi) +� += 0, +59 + +E +� +IS,i +πS(Xi) +I{Ai = dη(Xi)} +πd(Xi) +( ˆK(Ai, Xi) − K(Ai, Xi)) +� += 0, +and using the same technique, we conclude that these two mean-zero terms are op(N −1/2) +as well. +The product terms can be handled simply by the Cauchy-Schwarz inequality and the +rate of convergence conditions in Assumption 6. Additionally we have the decomposition +as follows +1 +N1 +� +I1 +( ˆK(Ai, Xi) − K(Ai, Xi)) += 1 +N1 +� +I1 +� +− (ˆµ(Ai, Xi) − µ(Ai, Xi)) + +1 − ∆i +SC(Ui | Ai, Xi)( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi)) +− +� Ui +0 +λC(u | Ai, Xi) +SC(u | Ai, Xi)( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi))du ++ (1 − ∆i) +� +1 +ˆSC(Ui | Ai, Xi) +− +1 +SC(Ui | Ai, Xi) +� +Q(Ui | Ai, Xi) ++ +� +1 +ˆSC(Ui | Ai, Xi) +− +1 +SC(Ui | Ai, Xi) +� +∆i y(Ui) +− +� Ui +0 +� ˆλC(u | Ai, Xi) +ˆSC(u | Ai, Xi) +− λC(u | Ai, Xi) +SC(u | Ai, Xi) +� +Q(Ui | Ai, Xi)du ++ (1 − ∆i) +� +1 +ˆSC(Ui | Ai, Xi) +− +1 +SC(Ui | Ai, Xi) +� +( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi)) +− +� Ui +0 +� ˆλC(u | Ai, Xi) +ˆSC(u | Ai, Xi) +− λC(u | Ai, Xi) +SC(u | Ai, Xi) +� +( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi))du, +and similarly we have three mean-zero terms which are op(N −1/2) by the same technique +in Section A.2 and the facts that +E[ˆµ(Ai, Xi) − µ(Ai, Xi)] = 0, +E +� +1 − ∆i +SC(Ui | Ai, Xi)( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi)) +− +� Ui +0 +λC(u | Ai, Xi) +SC(u | Ai, Xi)( ˆQ(u | Ai, Xi) − Q(u | Ai, Xi))du +� += 0, +60 + +E +� +(1 − ∆i) +� +1 +ˆSC(Ui | Ai, Xi) +− +1 +SC(Ui | Ai, Xi) +� +Q(Ui | Ai, Xi) ++ +� +1 +ˆSC(Ui | Ai, Xi) +− +1 +SC(Ui | Ai, Xi) +� +∆i y(Ui) +− +� Ui +0 +� ˆλC(u | Ai, Xi) +ˆSC(u | Ai, Xi) +− λC(u | Ai, Xi) +SC(u | Ai, Xi) +� +Q(Ui | Ai, Xi)du +� += 0, +and we can bound the two product terms as well +1 +N1 +� +I1 +� +(1 − ∆i) +� +1 +ˆSC(Ui | Ai, Xi) +− +1 +SC(Ui | Ai, Xi) +� +( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi)) +− +� Ui +0 +� ˆλC(u | Ai, Xi) +ˆSC(u | Ai, Xi) +− λC(u | Ai, Xi) +SC(u | Ai, Xi) +� +( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi))du +� +≤ +� +� 1 +N1 +� +I1 +(1 − ∆i) +� +1 +ˆSC(Ui | Ai, Xi) +− +1 +SC(Ui | Ai, Xi) +�2� +� +1/2 +× +� +1 +N1 +� +I1 +(1 − ∆i)( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi))2 +�1/2 +− +� Ui +0 +� +� 1 +N1 +� +I1 +� ˆλC(u | Ai, Xi) +ˆSC(u | Ai, Xi) +− λC(u | Ai, Xi) +SC(u | Ai, Xi) +�2� +� +1/2 +× +� +1 +N1 +� +I1 +( ˆQ(Ui | Ai, Xi) − Q(Ui | Ai, Xi))2 +�1/2 +du += op(N −1/2), +which proves that +1 +N1 +� +I1( ˆK(Ai, Xi) − K(Ai, Xi)) = op(N −1/2). +Therefore, we conclude that the four product terms in (13) are op(N −1/2) as well, which +completes the proof of (i) in Theorem 2. +PART 2: We show that N 1/3∥ˆη − η∗∥2 = Op(1). +By Assumption 5 (i), V (η) is twice continuously differentiable at a neighborhood of η∗; +in PART 1, we show that ˆV (η) = V (η) + op(1), ∀η; since ˆη maximizes ˆV (η), we have that +ˆV (ˆη) ≥ supη ˆV (η), thus by the Argmax theorem, we have ˆη +p→ η∗ as N → ∞. +In order to establish the N −1/3 rate of convergence of ˆη, we apply Theorem 14.4 (Rate +61 + +of convergence) of Kosorok (2008), and need to find the suitable rate that satisfies three +conditions below. +Condition 1 For every η in a neighborhood of η∗ such that ∥η − η∗∥2 < δ, by Assump- +tion 5 (i), we apply the second-order Taylor expansion, +V (η) − V (η∗) = V ′(η∗)∥η − η∗∥2 + 1 +2V ′′(η∗)∥η − η∗∥2 +2 + o(∥η − η∗∥2 +2) += 1 +2V ′′(η∗)∥η − η∗∥2 +2 + o(∥η − η∗∥2 +2), +and as V ′′(η∗) < 0, there exists c10 = − 1 +2V ′′(η∗) > 0 such that V (η)−V (η∗) ≤ −c10∥η−η∗∥2 +2. +Condition 2 For all N large enough and sufficiently small δ, we consider the centered +process ˆV − V , and have that +E +� +√ +N +sup +∥η−η∗∥2<δ +���ˆV (η) − V (η) − {ˆV (η∗) − V (η∗)} +��� +� += E +� +√ +N +sup +∥η−η∗∥2<δ +���ˆV (η) − Vn(η) + Vn(η) − V (η) − {ˆV (η∗) − Vn(η∗) + Vn(η∗) − V (η∗)} +��� +� +≤ E +� +√ +N +sup +∥η−η∗∥2<δ +���ˆV (η) − Vn(η) − {ˆV (η∗) − Vn(η∗)} +��� +� +(I) ++ E +� +√ +N +sup +∥η−η∗∥2<δ +|Vn(η) − V (η) − {Vn(η∗) − V (η∗)}| +� +(II) +It follows from the result in PART 1 that (I) = op(1). To bound (II), we have +Vn(η) − Vn(η∗) += 1 +N +N +� +i=1 +(dη(Xi) − dη∗(Xi)) × +� +IT,i e(Xi)(µ(1, Xi) − µ(0, Xi)) + (2Ai − 1)IS,i +πAi(Xi)πS(Xi)K(Ai, Xi) +� +. +Define a class of functions +F9 +η = +� +(dη(x)−dη∗(x))× +� +IT e(x)(µ(1, x)−µ(0, x))+ (2a − 1)IS +πa(x)πS(x)K(a, x) +� +: ∥η−η∗∥2 < δ +� +, +and let M9 = sup +���IT e(x)(µ(1, x) − µ(0, x)) + +(2a−1)IS +πa(x)πS(x)K(a, x) +���. By Assumption 1, 3 and +Condition 1, we have that M9 < ∞. Using the same technique as in Section D.2 Condition +62 + +2.1, we define the envelop of F9 +η as F9 = M9I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}, and obtain that +∥F9∥P,2 ≤ ˜ +M9δ1/2 < ∞, where ˜ +M9 is a finite constant, and that F9 +η is a VC class with +bounded entropy J∗ +[](1, F9 +η) < ∞. By Theorem 11.2 of Kosorok (2008), we obtain +E +� +sup +∥η−η∗∥2<δ +��GNF9 +η +�� +� +≤ ˜c9δ1/2, +where ˜c9 is a finite constant. Therefore, we obtain +(II) = E +� +√ +N +sup +∥η−η∗∥2<δ +|VN(η) − V (η) − {VN(η∗) − V (η∗)}| +� += E +� +sup +∥η−η∗∥2<δ +|GnF9 +η| +� +≤ ˜c9δ1/2. +In summary, we obtain that the centered process satisfies +E +� +√ +N +sup +∥η−η∗∥2<δ +��� ˆS(t; η) − S(t; η) − { ˆS(t; η∗) − S(t; η∗)} +��� +� +≤ (I) + (II) ≤ ˜c9δ1/2. +(14) +Let φN(δ) = δ1/2 and α = 3 +2 < 2, thus we have φn(δ) +δα += δ−1 is decreasing, and α does +not depend on N. That is, the second condition holds. +Condition 3 By the facts that ˆη +p→ η∗ as N → ∞, and that ˆS(t; ˆη) ≥ supη ˆS(t; η), +we choose rN = N 1/3 such that r2 +NφN(r−1 +N ) = N 2/3φN(N −1/3) = N 1/2. The third condition +holds. +In the end, the three conditions are satisfied with rN = N 1/3; thus we conclude that +N 1/3∥ˆη − η∗∥2 = Op(1), which completes the proof of (ii) in Theorem 2. +PART 3: We characterize the asymptotic distribution of ˆV (ˆη). Since we have +√ +N{ˆV (ˆη) − V (η∗)} = +√ +N{ˆV (ˆη) − ˆV (η∗)} + +√ +N{ˆV (η∗) − V (t; η∗)}, +we study the two terms in two steps. +Step 3.1 To establish +√ +N{ˆV (ˆη)− ˆV (η∗)} = op(1), it suffices to show that +√ +N{V (ˆη)− +V (η∗)} = op(1) and +√ +N( ˆV (ˆη) − ˆV (η∗) − {V (ˆη) − V (η∗)}) = op(1). +63 + +First, as N 1/3∥ˆη − η∗∥2 = Op(1), we take the second-order Taylor expansion +√ +N{V (ˆη) − V (η∗)} = +√ +N +� +V ′(η∗)∥ˆη − η∗∥2 + 1 +2V ′′(η∗)∥ˆη − η∗∥2 +2 + op(∥ˆη − η∗∥2 +2) +� += +√ +N +�1 +2V ′′(η∗)∥ˆη − η∗∥2 +2 + op(∥ˆη − η∗∥2 +2) +� += +√ +N +�1 +2V ′′(η∗)Op(N −2/3) + op(N −2/3) +� += op(1). +Next, we follow the result (14) obtained in PART 2. As N 1/3∥ˆη − η∗∥2 = Op(1), there +exists ˜δ2 = c11N −1/3, where c11 < ∞ is a finite constant, such that ∥ˆη−η∗∥2 ≤ ˜δ2. Therefore +we have +√ +N( ˆV (ˆη) − ˆV (η∗) − {V (ˆη) − V (η∗)}) +≤ E +� +√ +N +sup +∥ˆη−η∗∥2<˜δ2 +���ˆV (ˆη) − V (ˆη) − {ˆV (η∗) − V (η∗)} +��� +� +≤ ˜c9˜δ1/2 = ˜c9 +√c11N −1/6 = op(1), +which yields the result. +Step 3.2 To derive the asymptotic distribution of +√ +N{ˆV (η∗) − V (η∗)}, we follow the +result obtained in PART 1 that ˆV (η∗) = VN(η∗) + op(N −1/2), and thus +√ +N +� +ˆV (η∗) − V (η∗) +� +D→ N(0, σ2 +2), +where σ2 +2 = E[φ2 +dη∗] is the semiparametric efficiency bound. +Therefore we obtain in the end +√ +N{ˆV (ˆη) − v(η∗)} = +√ +N{ˆV (ˆη) − ˆV (η∗)} + +√ +N{ˆV (η∗) − V (η∗)} += op(1) + +√ +N{ˆV (η∗) − V (η∗)} +D→ N(0, σ2 +2), +which completes the proof of Theorem 2 and Corollary 2. +64 + +F +Proof of Theorem 3 and Theorem 4 +When the source and target populations have the same distributions, both ˆVDR(η) and +ˆVCF(η) converge to V (η). The asymptotic variance of ˆVDR(η) is +σ2 +DR = E +� +IS +P(IS = 1) +� +µ(d(X), X) + I{A = d(X)} +πd(X) +K(A, X) − V (η) +�2� += E +� +IS +P(IS = 1) +� +µ2(d(X), X) + I{A = d(X)} +π2 +d(X) +K2(A, X) − V 2(η) ++2I{A = d(X)} +πd(X) +K(A, X)µ(d(X), X) − 2µ(d(X), X)V (η) +−2I{A = d(X)} +πd(X) +K(A, X)V (η) +�� +, +while the asymptotic variance of ˆVCF(η) is +σ2 +CF = E +�� +IT e(X)µ(d(X), X) + IS I{A = d(X)} +πS(X)πd(X) +K(A, X) − V (η) +�2� += E +�� +IT e2(X)µ2(d(X), X) + IS I{A = d(X)} +π2 +S(X)π2 +d(X) +K2(A, X) − V 2(η) +−2IT e2(X)µ(d(X), X)V (η) − 2IS I{A = d(X)} +πS(X)πd(X) +K(A, X)V (η) +�� +, +where +K(A, X) = +∆ y(U) +SC(U | A, X) − µ(A, X) + +� ∞ +0 +dMC(u | A, X) +SC(u | A, X) Q(u, A, X). +Since we have that +E +� +IS +P(IS = 1) +2I{A = d(X)} +πd(X) +K(A, X)µ(d(X), X) +� += 0, +and for +B ∈ +� +µ2(d(X), X), I{A = d(X)} +π2 +d(X) +K2(A, X), µ(d(X), X)V (η), I{A = d(X)} +π2 +d(X) +K(A, X)V (η) +� +, +we also have that +E +� +IS +P(IS = 1)B +� += E[IT e(X)B] = E +� +IS +πS(X)B +� +, +65 + +we conclude that σ2 +DR = σ2 +CF. +By the law of iterated expectations, the value function Vd = E[y(T(d))] = EX[E[y(T(d)) | X]]. +When there is no restriction on the class of ITRs, the true optimal ITR is +d∗∗(X) = arg max +d +Vd = arg max +d +EX[E[y(T(d)) | X]] += I{E[y(T(1)) | X] > E[y(T(0)) | X]}. +That is, the optimal ITR does not depend on the covariate distributions, but only the bilp +function which is the same in both the source and target populations by Assumption 2. +Thus both the maximizers of ˆVDR(η) and ˆVCF(η) converge to the true population parameter +η∗∗. However, ˆVDR(η) is biased since the expectation EX is taken with respect to the source +population. +G +Additional simulations +We first investigate the performance of the cross-fitted ACW estimator with different sam- +ple sizes (N, m) = (5 × 104, 2000), (1 × 105, 4000), (2 × 105, 8000), (4 × 105, 16000), (6 × +105, 24000), (8×105, 32000). Figure 4 and Table 3 report the results from 200 Monte Carlo +replications. The variance is computed using the EIF. +H +Details of real data analysis +There are around 0.5% and 1.6% missing values in the RCT and OS data, respectively. We +use the mice function in the R package mice (Van Buuren & Groothuis-Oudshoorn 2011) +to impute the missing values. +Motivated by the clinical practice and existing work in the medical literature, we con- +sider ITRs that depend on the following five variables: +66 + +Figure 4: Boxplot of estimated value by ACW estimator with different sample sizes. +2.4 +2.7 +3.0 +3.3 +0.5 +1 +2 +4 +6 +8 +Target super population size +Estimated Value (ACW) +Table 3: Numeric results of the ACW estimator. Bias is the empirical bias of point es- +timates; SD is the empirical standard deviation of point estimates; SE is the average of +standard error estimates; CP is the empirical coverage probability of the 95% Wald confi- +dence intervals. +n; m(×103) +∼ 780; 2 +∼ 1560; 4 +∼ 3120; 8 +∼ 6240; 16 +∼ 9360; 24 +∼ 12480; 32 +Bias +0.1041 +0.0253 +0.0134 +0.0046 +0.0031 +0.0030 +SD +0.1394 +0.0985 +0.0635 +0.0419 +0.0317 +0.0267 +SE +0.1611 +0.0942 +0.0627 +0.0417 +0.0330 +0.0284 +CP(%) +97.5 +93.5 +96.0 +94.5 +97.5 +97.0 +67 + +• AGE, SEX and Sequential Organ Failure Assessment (SOFA) score: these three base- +line variables are well related to mortality in ICUs, so we consider them as important +risk factors. +• Acute Kidney Injury Network (AKIN) score: Jaber et al. (2018) observed that the +infusion of sodium bicarbonate improved survival outcomes and mortality rate in +critically ill patients with severe metabolic acidemia and acute kidney injury. In the +observational data, the AKIN score was not recorded, so we computed the score using +serum creatinine measurement (Z´avada et al. 2010). +• SEPSIS: we consider the presence of sepsis as a risk factor because it is the main +condition associated with severe acidemia at the arrival in ICU. The effect of sodium +bicarbonate infusion on patients with acidema and acute kidney injury was also ob- +served in septic patients (Zhang, Zhu, Mo & Hong 2018). +68 + diff --git a/y9E5T4oBgHgl3EQfNw4V/content/tmp_files/load_file.txt b/y9E5T4oBgHgl3EQfNw4V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1be2042c7b5947bdca1bdd8bffd09be872c0592 --- /dev/null +++ b/y9E5T4oBgHgl3EQfNw4V/content/tmp_files/load_file.txt @@ -0,0 +1,2275 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf,len=2274 +page_content='Efficient and robust transfer learning of optimal individualized treatment regimes with right-censored survival data Pan Zhao∗, Julie Josse † PreMeDICaL, Inria-Inserm, Montpellier, France and Shu Yang‡ Department of Statistics, North Carolina State University, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' January 16, 2023 Abstract An individualized treatment regime (ITR) is a decision rule that assigns treat- ments based on patients’ characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The value function of an ITR is the expected outcome in a counterfactual world had this ITR been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Recently, there has been increasing interest in combining heterogeneous data sources, such as lever- aging the complementary features of randomized controlled trial (RCT) data and a large observational study (OS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Usually, a covariate shift exists between the source and target population, rendering the source-optimal ITR unnecessarily optimal for the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We present an efficient and robust transfer learning framework for estimating the optimal ITR with right-censored survival data that generalizes well to the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The value function accommodates a broad class of functionals of survival distributions, including survival probabilities and restrictive mean survival times (RMSTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We propose a doubly robust estimator of the value function, and the optimal ITR is learned by maximizing the value function within a pre-specified class of ITRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We establish the N−1/3 rate of convergence for the estimated parameter indexing the optimal ITR, and show that the proposed optimal value estimator is consistent and asymptotically normal even with flexible machine learning methods for nuisance parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We evaluate the empirical performance of the proposed method by simulation studies and a real data application of sodium bicar- bonate therapy for patients with severe metabolic acidaemia in the intensive care unit (ICU), combining a RCT and an observational study with heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Keywords: Policy learning, Semiparametric theory, Covariate shift, Transportability, Data integration ∗Email: pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='zhao@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='fr †Email: julie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='josse@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='fr ‡Email: syang24@ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='05491v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='ME] 13 Jan 2023 1 Introduction Data-driven individualized decision making has recently received increasing interest in many fields, such as precision medicine (Kosorok & Laber 2019, Tsiatis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2019), mobile health (Trella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2022), precision public health (Rasmussen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2020) and economet- rics (Athey & Wager 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The goal of optimal ITR estimation is to learn a decision rule that assigns the best treatment among possible options to each patient based on their in- dividual characteristics in order to optimize some functional of the counterfactual outcome distribution in the population of interest, also known as the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The optimal ITR is the one with the maximal value function, and the value function of the optimal ITR is the optimal value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For completely observed data without censoring, one prevailing line of work in the statis- tical and biomedical literature uses model-based methods to solve the optimal ITR problem, such as Q-learning (Robins 2004, Qian & Murphy 2011, Laber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2014) and A-learning (Murphy 2003, Schulte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2014, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Alternatively, direct model-free or policy search methods have been proposed recently, including the classification perspective (Zhang, Tsiatis, Davidian, Zhang & Laber 2012, Zhang, Tsiatis, Laber & Davidian 2012, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2012, Rubin & van der Laan 2012) and interpretable tree or list-based ITRs (Laber & Zhao 2015, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2015, Zhang, Laber, Davidian & Tsiatis 2018), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In clinical studies, right-censored survival data are frequently observed as primary outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Recent extensions of optimal ITR with survival data have been established in Goldberg & Kosorok (2012), Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2017), Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2017), Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2017), D´ıaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Researchers have investigated using machine learning algorithms to estimate the optimal ITR from large classes, which cannot be indexed by a finite-dimensional parameter (Luedtke 2 & van der Laan 2016a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' One typical instance is that the optimal ITR can be learned from the blip function, which is defined as the additive effect of a blip in treatment on a counterfactual outcome, conditional on baseline covariates (Robins 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and most existing regression or supervised learning methods can be directly applied (K¨unzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' However, the ITRs learned by machine learning methods can be too complex to inform policy-making and clinical practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' to facilitate the integration of data-driven ITRs into practice, it is crucial that estimated ITRs be interpretable and parsimonious (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Recently, there has been increasing interest in combining heterogeneous data sources, such as leveraging the complementary features of RCT data and a large OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For example, in biomedical studies and policy research, RCTs are deemed as the gold standard for treatment effects evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' However, due to inclusion or exclusion criteria, data availability, and study design, the enrolled participants in RCT who form the source sample may have systematically different characteristics from the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, findings from RCTs cannot be directly extended to the target population of interest (Cole & Stuart 2010, Dahabreh & Hern´an 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' See also Colnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020) and Degtiar & Rose (2021) for detailed reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Heterogeneity in the populations is of great relevance, and a covariate shift usually exists where the covariate distributions differ between the source and target populations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' thus, the optimal ITR for the source population is not necessarily optimal for the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019) uses data from a single trial study and proposes a two-stage procedure to derive a robust and parsimonious rule for the target population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021) proposes a distributionally robust framework that maximizes the worst-case value function under a set of distributions that are “close” to the training distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Kallus (2021) tackles the lack of overlap for different actions in policy learning based on 3 retargeting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Wu & Yang (2022) and Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022) develop a calibration weighting framework that tailors a targeted optimal ITR by leveraging the individual covariate data or summary statistics from a target population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022) uses distributionally robust optimization and sensitivity analysis tools to learn a decision rule that minimizes the worst-case risk incurred under a family of test distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' However, these methods focus on continuous or binary outcomes and only consider a single sample for worst-case risk minimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' the extension to right-censored survival outcomes within the data integration context has not been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In this paper, we propose a new transfer learning method of finding an optimal ITR from a restricted ITR class under the super population framework where the source sample is subject to selection bias and the target sample is representative of the target population with a known sampling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Specifically, in our value search method, the value function accommodates a broad class of functionals of survival distributions, including survival probabilities and RMSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We characterize the efficient influence function (EIF) of the value function and propose the augmented estimator, which involves models for the survival outcome, propensity score, censoring and sampling processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The proposed estimator is doubly robust in the sense that it is consistent if either the survival outcome model or the models of the propensity score, censoring, and sampling are correctly specified and is locally efficient when all models are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We also consider flexible data-adaptive machine learning algorithms to estimate the nuisance parameters and use the cross-fitting procedure to draw valid inferences under mild regularity conditions and a certain rate of convergence conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' As we consider a restricted class of ITRs indexed by a Euclidean parameter η, we also establish the N −1/3 convergence rate of ˆη, even though its resultant limiting distribution is not standard, and thus very challenging to characterize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Based on 4 this rate of convergence, we show that the proposed estimator for the target value function is consistent and asymptotically normal, even with flexible machine learning methods for nuisance parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Interestingly, when the covariate distributions of the source and target populations are the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', no covariate shift, the semiparametric efficiency bounds of our method and the standard doubly robust method (Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2017) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, if the true optimal ITR belongs to the restricted class of ITRs, the standard doubly robust method can still learn the optimal ITR despite the covariate shift, but only our method provides valid statistical inference for the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The rest of our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In Section 2, we introduce the statistical framework of causal survival analysis and transfer learning of optimal ITR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Section 3 de- velops the main methodology of learning the value function and associated optimal ITR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Section 4 establishes the asymptotic properties of the proposed value estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Extensive simulations are reported in Section 5 to demonstrate the empirical performance of the pro- posed method, followed by a real data application given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The article concludes in Section 7 with a discussion of some remarks and future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' All proofs and additional results are provided in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2 Statistical Framework 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 Causal survival analysis Let X denote the p-dimensional vector of covariates that belongs to a covariate space X ⊂ Rp, A ∈ A = {0, 1} denote the binary treatment, and T ∈ R+ denote the survival time to the event of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In the presence of right censoring, the outcome T may not be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let C ∈ R+ denote the censoring time and ∆ = I{T ≤ C} where I{·} is the 5 indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let U = min{T, C} be the observed outcome, N(t) = I{U ≤ t, ∆ = 1} the counting process, and Y (t) = I{U ≥ t} the at-risk process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We use the potential outcomes framework (Neyman 1923, Rubin 1974), where for a ∈ A = {0, 1}, T(a) is the survival time had the subject received treatment a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The common goal in causal survival analysis is to identify and estimate the counterfactual quantity E[y(T(a))] for some deterministic transformation function y(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Such transfor- mations include y(T) = min(T, L) for the RMST with some pre-specified maximal time horizon L, and y(T) = I{T ≥ t} for the survival probability at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under the standard assumptions (a) consistency: T = T(A), (b) positivity: Pr(A = a | X) > 0 for every a ∈ A almost surely, (c) unconfoundedness: A ⊥⊥ {T(1), T(0)} | X, (d) conditionally independent censoring: C ⊥⊥ {T(1), T(0)} | {X, A}, we can nonparametrically identify E[y(T(a))] by the outcome regression (OR) formula or the inverse probability weighting (IPW) formula (Van der Laan & Robins 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 ITR and value function Without loss of generality, we assume that larger values of T are more desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Typically we aim to identify and estimate an ITR d(x) : X → A, which is a mapping from the covariate space X to the treatment space A = {0, 1}, that maximizes the expected outcome in a counterfactual world had this ITR been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Suppose D is the class of candidate ITRs of interest, then define the potential outcome T(d) under any d ∈ D by T(d) = d(X)T(1) + (1 − d(X))T(0), and the value function (Manski 2004) of d is defined by V (d) = E[y(T(d))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Then by maximizing V (d) over D, the optimal ITR is defined by dopt = arg maxd∈D V (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' See Qian & Murphy (2011) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' To estimate the value function, we can use the OR or IPW formulas, and also a doubly 6 robust method (Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2017): VDR(d) =E � I{A = d(X)}∆ y(U) Pr(A = d(X) | X)SC(U | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) + � 1 − I{A = d(X)} Pr(A = d(X) | X) � E[y(T) | A = d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X] + I{A = d(X)} Pr(A = d(X) | X) � ∞ 0 dMC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) SC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) E[y(T) | T ≥ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1) where SC(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = Pr(C > t | A = a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X = x) is the conditional survival function for the censoring process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' dMC(u | A = a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) = dNC(u)−Y (u)dΛC(u | A = a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) is the martingale increment for the censoring process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' NC(u) = I{U ≤ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ∆ = 0} and ΛC(u | A = a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) = − log(SC(u | A = a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The first term in (1) is the IPW formula, and the augmentation terms capture additional information from the subjects who do not receive treatment d, and who receive treatment d but are censored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In (clinical) practice, it is usually desirable to consider a class of ITRs indexed by a Euclidean parameter η = (η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , ηp+1)T ∈ Rp+1 for feasibility and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let V (η) = V (dη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Throughout, we focus on such a class of linear ITRs: Dη = {dη : dη(X) = I{ηT ˜X ≥ 0}, |ηp+1| = 1}, where ˜X = (1, XT)T, and for identifiability we assume there exists a continuous covariate whose coefficient has absolute value one (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' without loss of generality, we assume |ηp+1| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, the population parameter η∗ indexing the optimal ITR is η∗ = arg maxη∈{η∈Rp+1:|ηp+1|=1} V (η), and the optimal value function is V (η∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3 Transfer learning The performance of such a learned ITR may suffer from a covariate shift in which the population distributions differ (Sugiyama & Kawanabe 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Instead of minimizing the worst-case risk, here we consider a super population framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Suppose that a source 7 sample of size n and a target sample of size m are sampled independently from the target super population with different mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let IS and IT denote the indicator of sampling from source and target populations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A covariate shift means that Pr(IS = 1 | X) ̸= Pr(IT = 1 | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In the source sample, independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=') data Os = {Xi, Ai, Ui, ∆i, IS,i = 1, IT,i = 0}n i=1 are observed from n subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' in the target sample, it is common that only the covariates information is available, so i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' data Ot = {Xi, IS,i = 0, IT,i = 1}n+m i=n+1 are observed from m subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The sampling mechanism and data structure are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Figure 1: Schematic of the data structure of the source and target samples within the target super population framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Target super population Finite population {T(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' T(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X} Finite population {T(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' T(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X} Source sampling IS Target sampling IT Complete source sample {Ti(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ti(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 0}n i=1 Complete target sample {Ti(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ti(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 1}n+m i=n+1 Treatment assignment A Censoring C Only observe covariates X Observed source sample {Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ∆i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 0}n i=1 Observed target sample {Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = 1}n+m i=n+1 In this framework,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' we assume that the source and target sampling mechanisms are independent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' which holds if two separate studies are conducted independently by different 8 research projects in different locations or in two separate time periods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and the target population is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In the context of combining the RCT and observational study, this framework corresponds to the non-nested study design (Dahabreh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In the framework illustrated in Figure 1, we also assume the existence of the finite population of size N, which helps us clarify the sampling mechanism and identification strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The two separate finite populations exemplify the independence of the source and target sampling processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We present the identification formulas in Section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' however, we do not require N to be fixed and known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Equivalently, it is also possible to assume a pooled population consisting of a source population and a target population, and similar identification formulas can be proposed based on the density ratio of the two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 3 Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 Identification and semiparametric efficiency To identify the causal effects from the observed data, we make the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (a) T = T(A) almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (b) Pr(A = a | X, IS = 1) > 0 for every a almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (c) A ⊥⊥ {T(1), T(0)} | {X, IS = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (d) C ⊥⊥ {T(1), T(0)} | {X, A, IS = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 1 includes the standard assumptions as we have introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Here we only assume them in the source population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 1(a) implies that the observed outcome is the potential outcome under the actual assigned treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assump- tion 1(b) states that each subject has a positive probability of receiving both treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 1(c) requires that all confounding factors are measured so that treatment assignment is as good as random conditionally on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 1(d) essentially states 9 that the censoring process is non-informative conditionally on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Furthermore, we require additional assumptions for the source and target populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 2 (Survival mean exchangeability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' E[y(T(a)) | X, IS = 1] = E[y(T(a)) | X] for every a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 3 (Positivity of Source Inclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 0 < Pr(IS = 1 | X) < 1 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 4 (Known target design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The target sample design weight e(x) = π−1 T (x) = 1/Pr(IT = 1 | X = x) is known by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 2 is similar to the mean exchangeability over trial participation (Dahabreh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2019), and is weaker than the ignorablility assumption (Stuart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2011), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', IS ⊥⊥ {T(1), T(0)} | X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 3 states that each subject has a positive probability to be included in the source sample, and implies adequate overlap of covariate distributions between the source and target populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 4 is commonly assumed in the survey sampling literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' thus the design-weighted target sample is representative of the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In an observational study with simple random sampling, we have e(x) = N/m, where N is the target population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under this framework, we have the following key identity that for any g(X) E � IS πS(X)g(X) � = E[IT e(X)g(X)] = E[g(X)], (2) where πS(X) = Pr(IS = 1 | X) is the sampling score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Proposition 1 (Identification formulas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 4, the value function V (d) can be identified by the outcome regression formula: V (d) = E[IT e(X)E[y(T) | A = d(X), X, IS = 1]], (3) 10 and the IPW formula: V (d) = E � IS πS(X) I{A = d(X)} πd(X) ∆ y(U) SC(U | A, X) � , (4) where πd(X) = d(X)πA(X) + (1 − d(X))(1 − πA(X)) with the propensity score πA(X) = Pr(A = 1 | X, IS = 1), and SC(t | a, x) = Pr(C > t | A = a, X = x, IS = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Based on the identification formulas (3) and (4), we can construct plug-in estimators for V (d), using the sampling score πS(X) or design weights e(X) to account for the sampling bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By the identity (2), the design weights IT e(X) in the OR formula (3) with the target sample can also be replaced by the inverse of sampling score IS/πS(X) using the source sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' However, these estimators are biased if the posited models are misspecified, and extreme weights from πS, πA and SC usually lead to large variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, we consider a more efficient and robust approach, motivated by the efficient influence function for V (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 4, the efficient influence function of V (d) is φd = IS πS(X) I{A = d(X)} πd(X) ∆ y(U) SC(U | A, X) − V (d) + � IT e(X) − IS πS(X) I{A = d(X)} πd(X) � µ(d(X), X) + IS πS(X) I{A = d(X)} πd(X) � ∞ 0 dMC(u | A, X) SC(u | A, X) Q(u, A, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (5) where µ(a, x) = E[y(T) | A = a, X = x, IS = 1] and Q(u, a, x) = E[y(T) | T ≥ u, A = a, X = x, IS = 1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The semiparametric EIF guides us in constructing efficient estimators combining the source and target samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Compared to (1), this EIF captures additional covariates infor- mation from the target population via the outcome model and thus removes the sampling 1Note that E[y(T) | T ≥ u, A, X] = − � ∞ u y(s) dS(s | A, X)/S(u | A, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For instance, when y(T) = I{T ≥ t}, we have E[y(T) | T ≥ u, A, X] = S(t | A, X)/S(u | A, X) for u ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 11 bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' An efficient estimation procedure is proposed in the next section, and we show that it enjoys the double robustness property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', it is consistent if either the survival outcome models µ(a, x), Q(u, a, x) or the models of propensity score πA(x), sampling score πS(x) and censoring process SC(t | a, x) are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, this EIF is Neyman orthogonal in the sense discussed in Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, a cross-fitting procedure is also proposed, allowing flexible machine learning methods for the nuisance parameters estimation, and √ N rate of convergence can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 An efficient and robust estimation procedure In this section, we focus on estimating the survival function Sd(t) = Pr(T(d) > t) as the value function under ITR d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Following the asymptotic linear characterization of survival estimands in Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), our results are readily extended to a broad class of func- tionals of survival distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For instance, the value function of the RMST under ITR d is simply � L 0 Sd(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Based on the EIF (5), we propose an estimator for the survival function ˆSd(t) = 1 N N � i=1 � IS,i ˆπS(Xi) I{Ai = d(Xi)} ˆπd(Xi) ∆i Yi(t) ˆSC(t | Ai, Xi) + � IT,i e(Xi) − IS,i ˆπS(Xi) I{Ai = d(Xi)} ˆπd(Xi) � ˆS(t | A = d(Xi), Xi) + IS,i ˆπS(Xi) I{Ai = d(Xi)} ˆπd(Xi) � ∞ 0 ˆS(t | Ai, Xi)d ˆ MC(u | Ai, Xi) ˆS(u | Ai, Xi) ˆSC(u | Ai, Xi) � , (6) where S(t | a, x) = Pr(T > t | A = a, X = x, IS = 1) is the treatment-specific con- ditional survival function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We posit (semi)parametric models for the nuisance param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let πA(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' θ) be the posited propensity score model, for example, using logis- tic regression logit{πA(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' θ)} = θT ˜X, where logit(x) = log{x/(1 − x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We use the Cox proportional hazard model Λ(t | A = a, X = x) = Λ0,a(t) exp(βT a x) to estimate 12 the survival functions S(t | a, x) = exp{−Λ(t | a, x)} and the cumulative baseline haz- ard function Λ0,a(t) = � t 0 λ0,a(u)du can be estimated by the Breslow estimator (Breslow 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Similarly, we posit a Cox proportional hazard model for the censoring process ΛC(t | A = a, X = x) = ΛC0,a(t) exp(αT a x), and the cumulative baseline hazard function ΛC0,a(t) is estimated by the Breslow estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The sampling score estimation is discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = ˆSdη(t) be the estimated value function for the ITR class Dη, then the optimal ITR is given by dˆη(x), where ˆη = arg maxη ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3 Calibration weighting To correct the bias due to the covariate shift between populations, most existing methods directly model the sampling score (Cole & Stuart 2010), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', inverse probability of sampling weighting (IPSW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' However, the IPSW method requires the sampling score model to be correctly specified, and it could also be numerically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Alternatively, we introduce the calibration weighting (CW) approach motivated by the identity (2), which is similar to the entropy balancing method (Hainmueller 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let g(X) be a vector of functions of X to be calibrated, such as the moments, interac- tions, and non-linear transformations of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Each subject i in the source sample is assigned a weight qi by solving the following optimization task: min q1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=',qn n � i=1 qi log qi, (7) subject to qi ≥ 0, n � i=1 qi = 1, n � i=1 qig(Xi) = ˜g, (8) where ˜g = �n+m i=n+1 e(Xi)g(Xi)/ �n+m i=n+1 e(Xi) is a design-weighted estimate of E[g(X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The objective function (7) is the negative entropy of the calibration weights, which ensures that the empirical distribution of the weights is not too far away from the uniform, such that 13 it minimizes the variability due to heterogeneous weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The final balancing constraint in (8) calibrates the covariate distribution of the weighted source sample to the target population in terms of g(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By introducing the Lagrange multiplier λ, the minimizer of the optimization task is qi = exp{ˆλTg(Xi)}/ �n i=1 exp{ˆλTg(Xi)}, where ˆλ solves the estimating equation �n i=1 exp{λTg(Xi)}{g(Xi) − ˜g} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since we only require specifying g(X), calibration weighting avoids explicitly modeling the sampling score and evades extreme weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, suppose that the sampling score follows a loglinear model πS(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' λ) = exp{λT ˜X}, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021, 2022) show that there is a direct correspondence between the calibration weights and the estimated sampling score, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', qi = {NπS(Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆλ)}−1 + op(N −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We also note that if the fraction n/N is small, the loglinear model is close to the widely used logistic regression model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' our simulation studies show the robustness of calibration weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Other objective functions can also be used for calibration weights estima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022) considers a generic convex distance function h(q) from the Cressie and Read family of discrepancies (Cressie & Read 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Thus the optimization task is minq1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=',qn �n i=1 h(qi) under the constraints (8), and the correspondence between the sam- pling score model πS and the objective function h has also been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 Cross-fitting Utilizing the Neyman orthogonality of EIF (5), we consider flexible machine learning meth- ods for estimating the nuisance parameters, where we want to remain agnostic on model- ing assumptions for the complex treatment assignment, survival, and censoring processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' There is extensive recent literature on nonparametric methods for heterogeneous treatment effect estimation with survival outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020) extends the generalized random 14 forests (Athey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2019) to estimate heterogeneous treatment effects in a survival and observational setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' See Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022) for details and practical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A de- scription of the proposed cross-fitting procedure is given below (Schick 1986, Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Throughout, we use the subscript CF to denote the cross-fitted version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Algorithm 1 Pseudo algorithm for the cross-fitting procedure Step 1 Randomly split the datasets Os and Ot respectively into K-folds with equal size such that Os = ∪K k=1Os,k, Ot = ∪K k=1Ot,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , K}, let Oc s,k = Os\\Os,k, Oc t,k = Os\\Ot,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Step 2 For each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , K}, estimate the nuisance parameters only using data Oc s,k and Oc t,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' then obtain an estimate of the value function ˆVCF,k(η) using data Os,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Step 3 Aggregate the estimates from K folds: ˆVCF(η) = 1 K �K k=1 ˆVCF,k(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Step 4 The estimated optimal ITR is indexed by ˆη = arg maxη ˆVCF(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 4 Asymptotic properties In this section, we present the asymptotic properties of the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' To establish the asymptotic properties, we require the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (i) The value function V (η) is twice continuously differentiable in a neigh- borhood of η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (ii) There exists some constant δ0 > 0 such that Pr(0 < | ˜XTη| < δ) = O(δ), where the big-O term is uniform in 0 < δ < δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition (i) is a standard regularity condition to establish uniform convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Sim- ilar margin conditions as (ii), which state that Pr(0 < |γ(X)| < δ) = O(δα) 2, are often 2Let γ(X) = E[T | A = 1, X] − E[T | A = 0, X] denote the conditional average treatment effect, then the 15 assumed in the literature of classification (Tsybakov 2004, Audibert & Tsybakov 2007), reinforcement learning (Farahmand 2011, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2021) and optimal treatment regimes (Luedtke & van der Laan 2016a, Luedtke & Chambaz 2020), to guarantee a fast conver- gence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Note that α = 0 imposes no restriction, which allows γ(X) = 0 almost surely, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', the challenging setting of exceptional laws where the optimal ITR is not uniquely de- fined (Robins 2004, Robins & Rotnitzky 2014), while the case α = 1 is of particular interest and would hold if γ(X) is absolutely continuous with bounded density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 5 and standard regularity conditions provided in the Supplementary Material, if either the survival outcome model, or the models of the propensity score, the sampling score and the censoring process are correct, we have that as N → ∞, (i) ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) → S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) for any η and 0 < t ≤ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (ii) √ N � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) � con- verges weakly to a mean zero Gaussian process for any η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (iii) N 1/3 ∥ˆη − η∗∥2 = Op(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (iv) √ N � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) � → N(0, σ2 t,1), where σt,1 is given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Next, to characterize the asymptotic behavior of the estimator with the nonparametric estimation of nuisance parameters, we assume the following consistency and convergence rate conditions of the nonparametric plug-in nuisance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Assume the following convergences in probability: supx∈X |ˆπA(x)−πA(x)| → 0, supx∈X |ˆπS(x) − πS(x)| → 0, and for a = 0, 1, sup x∈X,u≤h | ˆSC(u | a, x) − SC(u | a, x)| → 0, sup x∈X,u≤h ����� ˆλC(u | a, x) ˆSC(u | a, x) − λC(u | a, x) SC(u | a, x) ����� → 0, sup x∈X |ˆµ(a, x) − µ(a, x)| → 0, sup x∈X,u≤h | ˆQ(u, a, x) − Q(u, a, x)| → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and the following rates of convergence: E [supx∈X |ˆπA(x) − πA(x)|] = op(n−1/4), optimal ITR in an unrestricted class is given by d(X) = I{γ(X) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 16 E [supx∈X |ˆπS(x) − πS(x)|] = op(n−1/4), and for a = 0, 1, sup u≤h E � sup x∈X ��� ˆSC(u | a, x) − SC(u | a, x) ��� � = op(n−1/4), sup u≤h E � sup x∈X ����� ˆλC(u | a, x) ˆSC(u | a, x) − λC(u | a, x) SC(u | a, x) ����� � = op(n−1/4), E � sup x∈X |ˆµ(a, x) − µ(a, x)| � = o(n−1/4), sup u≤h E � sup x∈X | ˆQ(u, a, x) − Q(u, a, x)| � = o(n−1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The rate conditions in Assumption 6 are generally assumed in the literature (Kennedy 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' This rate can be achieved by many existing methods under certain structural as- sumptions on the nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Note that the nuisance parameters do not necessar- ily need to be estimated at the same rates n−1/4 for our theorems to hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' it would suffice that the product of rates of any combination of two nuisance parameters is n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 6, we have that as N → ∞, (i) ˆSCF(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) → S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) for any η and 0 < t ≤ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (ii) √ N � ˆSCF(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) � converges weakly to a mean zero Gaussian process for any η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (iii) N 1/3∥ˆη−η∗∥2 = Op(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (iv) √ N � ˆSCF(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) � → N(0, σ2 t,2), where σt,2 is given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Besides the survival functions, another common measure of particular interest in sur- vival analysis is the RMST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let VRMST(η) = E[min(T(dη), L)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We present two corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 5 and standard regularity conditions provided in the Supplementary material, if either the survival outcome model or the models of the propen- sity score, the censoring and sampling processes are correct, we have that as N → ∞, (i) ˆVRMST(η) → VRMST(η) for any η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (ii) N 1/3∥ˆη−η∗∥2 = Op(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (iii) √ N � ˆVRMST(ˆη) − VRMST(η∗) � → N(0, σ2 3), where σ3 is given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 6, we have that as N → ∞, (i) ˆVRMST,CF(η) → VRMST(η) for any η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (ii) N 1/3∥ˆη − η∗∥2 = Op(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (iii) √ N � ˆVRMST,CF(ˆη) − VRMST(η∗) � → N(0, σ2 4), where σ4 is given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='. 17 Finally, we show that when the covariate distributions of the source and target pop- ulations are the same, the semiparametric efficiency bounds of ˆVDR(η) and ˆVCF(η) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 6, when the covariate distributions of the source and target populations are the same, both √ N{ˆVDR(η) − V (η)} and √ N{ˆVCF(η) − V (η)} are asymptotically normal with mean zero and same variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Theorem 3 implies that when there is no covariate shift, our proposed estimator does not lose efficiency in comparison to the original double robust estimator since the augmentation term in EIF (5) from the target population, IT e(X)µ(d(X), X), is asymptotically equal to this term evaluated on the source population in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, when the covariate shift exists, we consider the optimal ITR dopt without restriction on the ITR class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Assumptions 1 - 6, If dopt ∈ Dη, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', dopt = dη∗, both the maximizers of ˆVDR(η) and ˆVCF(η) converge to η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' However, ˆVDR(η) is a biased estimator of V (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Theorem 4 implies if the true optimal ITR belongs to the restricted ITR class Dη, standard methods, without accounting for the covariate shift, are still able to recover the optimal ITR but fail to be consistent for the value function, due to the covariate shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' And we can only rely on the proposed method to draw valid inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 5 Simulation In this section, we investigate the finite-sample properties of our method through extensive numerical simulations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 3The R code to replicate all results is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='com/panzhaooo/ transfer-learning-survival-ITR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 18 Consider a target population of sample size N = 2 × 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The covariates (X1, X2, X3)T are generated from a multivariate normal distribution with mean 0, unit variance with corr(X1, X3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 and all other pairwise correlations equal to 0, and further truncated below −4 and above 4 to satisfy regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The target sample is a random sample of size m = 8000 from the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The sampling score follows πS(X) = expit(−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5X1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5X2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4X3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' thus the source sampling rate is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6%, and the source sample size around n = 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The treatment assignment mechanism in the source sample follows πA(X) = expit(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8X1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The counterfactual survival times T(a) are generated according to the hazard functions λ(t | A = 0, X) = exp(t) · exp(−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5X1 − X2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7X3) and λ(t | A = 1, X) = exp(t) · exp(−1 − X1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='9X2 − X3 − 2X2 2 + X1X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The censoring time C is generated according to the hazard functions λC(t | A = 0, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='04 exp(t) · exp(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8X1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1X2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7X3) and λC(t | A = 1, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='04 exp(t) · exp(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8X1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7X2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The resultant censoring rate is approximately 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We consider the RMST with the maximal time horizon L = 4 as the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' To evaluate the performance of different estimators for optimal ITRs, we compute the corresponding true value functions and percentages of correct decisions (PCD) for the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Specifically, we generate a large sample with size ˜N = 1 × 105 from the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The true value function of any ITR d(· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) is computed by V (η) = ˜N −1 � ˜ N i=1 min{d(Xi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η)Ti(1)+(1−d(Xi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η))Ti(0), L} and its associated PCD is computed by 1 − ˜N −1 � ˜ N i=1 |d(Xi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − d(Xi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η)|, where η∗ = arg maxη V (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We compare the following estimators for the RMST ˆV (η) = � L 0 ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η)dt: Naive: ˆSNaive(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = 1 n �n i=1 I{Ai=d(Xi)} ˆπd(Xi) ∆iYi(t) ˆSC(U | A,X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IPW formula (4) without using the sampling score;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 19 IPSW: ˆSIPSW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = 1 n �n i=1 IS,i ˆπS(Xi) I{Ai=d(Xi)} ˆπd(Xi) ∆iYi(t) ˆSC(U | A,X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IPW formula (4) where the sampling score is estimated via logistic regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' CW-IPW: ˆSCW-IPW(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = �n i=1 qi I{Ai=d(Xi)} ˆπd(Xi) ∆iYi(t) ˆSC(U | A,X) IPW formula (4) where the sampling score is estimated by calibration weighting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' CW-OR: ˆSCW-OR(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = �n i=1 qi ˆS(t | A = d(Xi), Xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' OR formula (3) in combination with calibration weights by the identity (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ORt: ˆSORt(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = 1 m �n+m i=n+1 ˆS(t | A = d(Xi), Xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' OR formula (3) evaluated on the target sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ACW: augmented estimator (6), where the sampling score is estimated by calibration weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since the estimated value functions are non-convex and non-smooth, multiple local optimal may exist in the optimization task, and many derivatives-based algorithms do not work for this challenging setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Here we utilize the genetic algorithm implemented in the R package rgenoud (Mebane Jr & Sekhon 2011), which performs well in our numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We refer to Mitchell (1998) for algorithmic details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 (Semi)parametric models We first consider the setting where the nuisance parameters are estimated by posited (semi)parametric working models as introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' To assess the performance of these estimators under model misspecification, we consider four scenarios: (1) all models are correct, (2) only the survival outcome model is correct, (3) only the survival outcome model is wrong, (4) all models are wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For the wrong sampling model, the weights are 20 estimated using calibration on eX1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The wrong propensity score model is fitted on eX3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The wrong Cox models for survival and censoring times are fitted on (eX1, eX2, eX3)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Figure 2 and Table 1 report the simulation results from 350 Monte Carlo replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Variance is estimated by a bootstrap procedure with B = 200 bootstrap replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The proposed ACW estimator is unbiased in scenarios (1) - (3), and the 95% coverage probabil- ities approximately achieve the nominal level, which shows the double robustness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 Flexible machine learning methods When utilizing flexible ML methods, we construct the cross-fitted ACW estimator as in- troduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The data generation process is the same as above, except that the censoring time C is generated according to the hazard functions λC(t | A = 0, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 exp(t)·exp(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8X1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1X2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7X3) and λC(t | A = 1, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 exp(t)·exp(−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8X1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7X2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4X3) which leads to an increased censoring rate of approximately 33%, so there are enough observations to get an accurate estimate of the censoring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The propensity score is estimated by the generalized random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The conditional sur- vival and censoring functions are estimated by the random survival forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The calibration weighting uses calibration on the first- and second-order moments of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' First, we study the impact of sample sizes on the performance of the ML methods, and simulation results are given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' With a small sample size, the ACW estimator is largely biased, and the bias diminishes as the sample size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Next, we compare the performance of different estimators with target population size N = 6 × 105 and target sample size m = 24000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Figure 3 shows the simulation results from 200 Monte Carlo replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The two IPW-based estimators are biased and perform poorly due to the large variability of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The two OR-based estimators have compa- 21 Figure 2: Boxplot of the estimated value, true value and PCD results of estimators under four model specification scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' O: survival outcome, S: sampling score, A: propensity score, C: censoring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' T: True (correctly specified) model, W: Wrong (misspecified) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' O:T / S:T, A:T, C:T O:T / S:W, A:W, C:W O:W / S:T, A:T, C:T O:W / S:W, A:W, C:W Estimated Value PCD True Value 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 Estimators Naive IPSW CW−IPW CW−OR ORt ACW 22 Table 1: Numerical results under four different model specification scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Bias is the empirical bias of point estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' SD is the empirical standard deviation of point estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' SE is the average of bootstrap standard error estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' CP is the empirical coverage probability of the 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Bias SD SE CP(%) Bias SD SE CP(%) O:T / S:T, A:T, C:T O:T / S:W, A:W, C:W Naive −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2189 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4598 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='43 IPSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2562 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6958 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='29 CW-IPW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2498 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6957 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='71 CW-OR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0286 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='57 ORt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0262 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0262 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='71 ACW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0369 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0334 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='43 O:W / S:T, A:T, C:T O:W / S:W, A:W, C:W Naive −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2207 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5018 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='57 IPSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2486 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7586 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='14 CW-IPW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2418 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3406 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='00 ORt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0355 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0355 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='71 ACW −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0419 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='57 23 rable performance as the ACW estimator in terms of PCD and true value function but still suffer from the overfitting bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Only the ACW estimator is consistent and provides valid inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Figure 3: Boxplots of the estimated value, true value, and PCD of different estimators using flexible ML methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2 3 4 Estimated Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0 PCD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 True Value Estimators Naive IPSW CW−IPW CW−OR ORt ACW 6 Real Data Analysis In this section, to illustrate the proposed method, we study the sodium bicarbonate therapy for patients with severe metabolic acidaemia in the intensive care unit by leveraging the RCT data BICAR-ICU (Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2018) and the observational study (OS) data from Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Specifically, we consider the BICAR-ICU data as the source sample and 24 the observational study data as the target sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The BICAR-ICU is a multi-center, open- label, randomized controlled, phase 3 trial between May 5, 2015, and May 7, 2017, which includes 387 adult patients admitted within 48 hours to the ICU with severe acidaemia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The prospective, multiple-center observational study was conducted over thirteen months in five ICUs, consisting of 193 consecutive patients who presented with severe acidemia within the first 24 hours of their ICU admission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Some heterogeneity exists between the two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Both the RCT and OS datasets contain detailed measurements of ICU patients with severe acidaemia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Motivated by the clinical practice and existing work in the medical liter- ature, we consider ITRs that depend on the following five variables: SEPSIS, AKIN, SOFA, SEX, and AGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A detailed description of the data preprocessing and variable selection is given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Table 2 summarizes the baseline characteristics of the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The baseline covariates distribution of the patients in the BICAR-ICU differs from the distribution in the observational study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' specifically, the BICAR-ICU pa- tients have higher SOFA scores and the more frequent presence of acute kidney injury and sepsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Table 2: Summary of baseline characteristics of the BICAR-ICU trial sample and the OS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Mean (standard deviation) for continuous and number (proportion) for the binary covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' SEPSIS AKIN SOFA SEX AGE BICAR-ICU (n = 387) 236 (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='98%) 181 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='77%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='12 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='72) 237 (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='24%) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='95 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='41) OS (m = 193) 99(51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='30%) 75 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='86%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='10 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='54) 122 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='21%) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='73 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='49) We apply our proposed ACW estimator to learn the optimal ITR for the target popu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The calibration weights are estimated based on the means of continuous covariates 25 and the proportions of the binary covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The propensity score is estimated using a logistic regression model, and the Cox proportional hazard model is fitted for the survival outcome with all covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The censoring only occurred on the 28th day when the follow- up in ICU ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We consider the class of linear ITRs that depend on all five variables: D = {I{η1+η2SEPSIS+η3AKIN+η4SOFA+η5SEX+η6AGE > 0} : η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , η6 ∈ R, |η6| = 1}, with the aim to maximize the RMST within 28 days in ICU stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The estimated parame- ter indexing the optimal ITR is ˆηACW = (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='9, −36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4, −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='8, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0)T, which leads to an estimated value function ˆV (ˆηACW) = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='52 days, with confidence interval [17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='74, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='30] given by 200 bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In contrast, we also use the standard double robust method to esti- mate the optimal ITR for the RCT, indexed by ˆηDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='RCT which maximize the value function ˆVDR(η) in (1) with y(T) = min(T, 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The estimated value function is ˆV (ˆηDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='RCT) = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='37 days for the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 7 Discussion In this paper, we present an efficient and robust transfer learning framework for estimating optimal ITR with right-censored survival data that generalizes well to the target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The proposed method can be improved or extended in several directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Construction and estimation of optimal ITRs for multiple decision points with censored survival data are challenging, taking into account the timing of censoring, events and decision points (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2017, Hager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2018), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', using a reinforcement learning method (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Furthermore, besides the class of ITRs indexed by a Euclidean parameter, it may be possible to consider other classes of ITRs, such as tree or list-based ITRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The current work focus on value functions in the form V (d) = E[y(T(d))] and can also 26 be modified in case of optimizing certain easy-to-interpret quantile criteria, which does not require specifying an outcome regression model and is robust for heavy-tailed distributions (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' And relaxing the restrictive assumptions such as positivity (Yang & Ding 2018, Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2022) and unconfoundedness (Cui & Tchetgen Tchetgen 2021, Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2021) for learning optimal ITRs is also a fruitful direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Acknowledgments Josse and Zhao gratefully acknowledge the French National Research Agency ANR-16- IDEX-0006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Yang is partially supported by the USA National Institutes of Health NIA grant 1R01AG066883 and NIEHS grant 1R01ES031651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The authors thank Maxime Fosset and Boris Jung for their help and support interpreting the BICAR-ICU trial and observational study data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' References Andersen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Gill, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1982), ‘Cox’s regression model for counting processes: a large sample study’, The annals of statistics pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 1100–1120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Athey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tibshirani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019), ‘Generalized random forests’, The Annals of Statistics 47(2), 1148–1178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Athey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘Policy learning with observational data’, Econometrica 89(1), 133–161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Audibert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Tsybakov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2007), ‘Fast learning rates for plug-in classifiers’, The Annals of statistics 35(2), 608–633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 27 Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Song, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2017), ‘Optimal treatment regimes for survival endpoints using a locally-efficient doubly-robust estimator from a classification perspec- tive’, Lifetime data analysis 23(4), 585–604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Breslow, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1972), ‘Contribution to discussion of paper by dr cox’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B 34, 216–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Demirer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Duflo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Hansen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Newey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Robins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘Double/debiased machine learning for treatment and structural pa- rameters’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Cho, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Holloway, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Couper, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Kosorok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020), ‘Multi-stage optimal dy- namic treatment regimes for survival outcomes with dependent censoring’, arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='03294 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Chu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Targeted optimal treatment regime learning using summary statistics’, arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='06229 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Cole, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Stuart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2010), ‘Generalizing evidence from randomized clinical trials to target populations: the actg 320 trial’, American journal of epidemiology 172(1), 107– 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Colnet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Mayer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Dieng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Varoquaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Vert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Josse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020), ‘Causal inference methods for combining randomized trials and observational studies: a review’, arXiv preprint arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='08047 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Cressie, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Read, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1984), ‘Multinomial goodness-of-fit tests’, Journal of the Royal Statistical Society: Series B (Methodological) 46(3), 440–464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 28 Cui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Kosorok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Sverdrup, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020), ‘Estimating hetero- geneous treatment effects with right-censored data via causal survival forests’, arXiv preprint arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='09887 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Cui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Tchetgen Tchetgen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘A semiparametric instrumental variable approach to optimal treatment regimes under endogeneity’, Journal of the American Statistical Association 116(533), 162–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Cui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zhu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Kosorok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2017), ‘Tree based weighted learning for estimating individ- ualized treatment rules with censored data’, Electronic journal of statistics 11(2), 3927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Dahabreh, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Haneuse, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Robins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Robertson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Buchanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Stuart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Hern´an, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘Study designs for extending causal inferences from a randomized trial to a target population’, American journal of epidemiology 190(8), 1632– 1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Dahabreh, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Hern´an, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019), ‘Extending inferences from a randomized trial to a target population’, European Journal of Epidemiology 34(8), 719–722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Dahabreh, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Robertson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tchetgen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Stuart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Hern´an, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019), ‘Generalizing causal inferences from individuals in randomized trials to all trial-eligible individuals’, Biometrics 75(2), 685–694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Degtiar, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Rose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘A review of generalizability and transportability’, arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='11904 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' D´ıaz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Savenkov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Ballman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘Targeted learning ensembles for optimal individualized treatment rules with time-to-event outcomes’, Biometrika 105(3), 723– 738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 29 Farahmand, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='-m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2011), ‘Action-gap phenomenon in reinforcement learning’, Advances in Neural Information Processing Systems 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Goldberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Kosorok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2012), ‘Q-learning with censored data’, Annals of statistics 40(1), 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Hager, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘Optimal two-stage dynamic treat- ment regimes from a classification perspective with censored survival data’, Biometrics 74(4), 1180–1192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Hainmueller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2012), ‘Entropy balancing for causal effects: A multivariate reweight- ing method to produce balanced samples in observational studies’, Political analysis 20(1), 25–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Kallus, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Uehara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘Fast rates for the regret of offline reinforcement learning’, arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='00479 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Jaber, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Paugam, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Futier, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lefrant, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lasocki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lescot, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Pottecher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', De- moule, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Ferrandiere, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Asehnoune, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘Sodium bicarbonate therapy for patients with severe metabolic acidaemia in the intensive care unit (bicar-icu): a multi- centre, open-label, randomised controlled, phase 3 trial’, The Lancet 392(10141), 31–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Jiang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Song, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2017), ‘On estimation of optimal treatment regimes for maximizing t-year survival probability’, Journal of the Royal Statistical So- ciety: Series B (Statistical Methodology) 79(4), 1165–1185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Jin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Ren, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Policy learning” without”overlap: Pessimism and generalized empirical bernstein’s inequality’, arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='09900 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 30 Jung, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Rimmele, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Le Goff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Chanques, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Corne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Jonquet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Muller, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lefrant, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Guervilly, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Papazian, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2011), ‘Severe metabolic or mixed acidemia on intensive care unit admission: incidence, prognosis and administration of buffer therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a prospective, multiple-center study’, Critical Care 15(5), 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Kallus, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘More efficient policy learning via optimal retargeting’, Journal of the American Statistical Association 116(534), 646–658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Kennedy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Semiparametric doubly robust targeted double machine learning: a review’, arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='06469 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Kennedy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Balakrishnan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & G’Sell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020), ‘Sharp instruments for classifying compliers and generalizing causal effects’, The Annals of Statistics 48(4), 2008–2030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Kosorok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2008), Introduction to empirical processes and semiparametric inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Kosorok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019), ‘Precision medicine’, Annual review of statistics and its application 6, 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' K¨unzel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Sekhon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Bickel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Yu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019), ‘Metalearners for estimat- ing heterogeneous treatment effects using machine learning’, Proceedings of the national academy of sciences 116(10), 4156–4165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Linn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Stefanski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2014), ‘Interactive model building for q- learning’, Biometrika 101(4), 831–847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2015), ‘Tree-based methods for individualized treatment regimes’, Biometrika 102(3), 501–514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 31 Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Dong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zeng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘Improving trial gener- alizability using observational studies’, Biometrics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Doubly robust estimators for generalizing treatment effects on survival outcomes from randomized controlled trials to a target population’, Journal of Causal Inference (accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Luedtke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Chambaz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020), Performance guarantees for policy learning, in ‘Annales de l’IHP Probabilites et statistiques’, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 56, NIH Public Access, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Luedtke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & van der Laan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2016a), ‘Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy’, Annals of statistics 44(2), 713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Luedtke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & van der Laan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2016b), ‘Super-learning of an optimal dynamic treatment rule’, The international journal of biostatistics 12(1), 305–332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Manski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2004), ‘Statistical treatment rules for heterogeneous populations’, Econo- metrica 72(4), 1221–1246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Mebane Jr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Sekhon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2011), ‘Genetic optimization using derivatives: the rgenoud package for r’, Journal of Statistical Software 42, 1–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Mitchell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1998), An introduction to genetic algorithms, MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Mo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Qi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘Learning optimal distributionally robust individualized treatment rules’, Journal of the American Statistical Association 116(534), 659–674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Murphy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2003), ‘Optimal dynamic treatment regimes’, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65(2), 331–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Neyman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1923), ‘Sur les applications de la th´eorie des probabilit´es aux experiences agricoles: Essai des principes’, Roczniki Nauk Rolniczych 10(1), 1–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 32 Qi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Miao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘Proximal learning for individualized treatment regimes under unmeasured confounding’, arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='01187 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Qian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Murphy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2011), ‘Performance guarantees for individualized treatment rules’, Annals of statistics 39(2), 1180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Rasmussen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Khoury, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Del Rio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020), ‘Precision public health as a key tool in the covid-19 response’, JAMA 324(10), 933–934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Robins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2004), Optimal structural nested models for optimal sequential decisions, in ‘Proceedings of the second seattle Symposium in Biostatistics’, Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 189–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Robins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Rotnitzky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2014), ‘Discussion of “dynamic treatment regimes: Technical challenges and applications”’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1974), ‘Estimating causal effects of treatments in randomized and nonran- domized studies.’, Journal of educational Psychology 66(5), 688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Rubin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & van der Laan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2012), ‘Statistical issues and limitations in personalized medicine research with clinical trials’, The international journal of biostatistics 8(1), 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Sahoo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Lei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Learning from a biased sample’, arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='01754 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Schick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1986), ‘On asymptotically efficient estimation in semiparametric models’, The Annals of Statistics pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 1139–1151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Schulte, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2014), ‘Q-and a-learning methods for estimating optimal dynamic treatment regimes’, Statistical science: a review journal of the Institute of Mathematical Statistics 29(4), 640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 33 Shi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Fan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Song, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘High-dimensional a-learning for optimal dynamic treatment regimes’, Annals of statistics 46(3), 925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Stuart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Cole, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Bradshaw, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Leaf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2011), ‘The use of propensity scores to assess the generalizability of results from randomized trials’, Journal of the Royal Statistical Society: Series A (Statistics in Society) 174(2), 369–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Sugiyama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Kawanabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2012), Machine learning in non-stationary environments: Introduction to covariate shift adaptation, MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Trella, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Nahum-Shani, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Shetty, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Doshi-Velez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Murphy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Designing reinforcement learning algorithms for digital interventions: Pre- implementation guidelines’, arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='03944 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2006), ‘Semiparametric theory and missing data’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Holloway, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019), Dynamic treatment regimes: Statistical methods for precision medicine, Chapman and Hall/CRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Tsybakov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2004), ‘Optimal aggregation of classifiers in statistical learning’, The Annals of Statistics 32(1), 135–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Van Buuren, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Groothuis-Oudshoorn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2011), ‘mice: Multivariate imputation by chained equations in r’, Journal of statistical software 45, 1–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Van der Laan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Robins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2003), Unified methods for censored longitudinal data and causality, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 5, Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Van der Vaart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Wellner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (1996), Weak Convergence and Empirical Processes: With Applications to Statistics, Springer New York, NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 34 Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Transfer learning of individualized treatment rules from experimental to real-world data’, Journal of Computation and Graphical Statistics p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1080/10618600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2141752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Ignatiadis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Sverdrup, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Fleming, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Wager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Shah, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Treatment heterogeneity with survival outcomes’, arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='07758 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Ding, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘Asymptotic inference of causal effects with observational studies trimmed by the estimated propensity scores’, Biometrika 105(2), 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Guan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2021), ‘SMIM: A unified framework of survival sensitivity analysis using multiple imputation and martingale’, Biometrics p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1111/biom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='13555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Z´avada, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Hoste, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Cartin-Ceba, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Calzavacca, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Gajic, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Clermont, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Bellomo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Kellum, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & investigators, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2010), ‘A comparison of three methods to es- timate baseline creatinine for rifle classification’, Nephrology Dialysis Transplantation 25(12), 3911–3918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2012), ‘Estimating optimal treatment regimes from a classification perspective’, Stat 1(1), 103–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2012), ‘A robust method for estimating optimal treatment regimes’, Biometrics 68(4), 1010–1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘Interpretable dynamic treatment regimes’, Journal of the American Statistical Association 113(524), 1541– 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 35 Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Laber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tsiatis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Davidian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2015), ‘Using decision lists to construct interpretable and parsimonious treatment regimes’, Biometrics 71(4), 895–904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Mo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Hong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018), ‘Effectiveness of sodium bicarbonate in- fusion on mortality in septic patients with metabolic acidosis’, Intensive care medicine 44(11), 1888–1895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zeng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Tangen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Leblanc, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2019), ‘Robustifying trial- derived optimal treatment rules for a target population’, Electronic journal of statistics 13(1), 1717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Zeng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Rush, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Kosorok, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2012), ‘Estimating individualized treatment rules using outcome weighted learning’, Journal of the American Statistical Association 107(499), 1106–1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=', Song, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' & Zhao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022), ‘Transformation-invariant learning of opti- mal individualized decision rules with time-to-event outcomes’, Journal of the American Statistical Association (just-accepted), 1–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 36 SUPPLEMENTARY MATERIAL A Preliminaries A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 Counting processes for Cox model We use the counting process theory of Andersen & Gill (1982) in our theoretical framework to study the large sample properties of Cox model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We state the existing results that are used in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let X⊗l denote 1 for l = 0, X for l = 1, and XXT for l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Define U (l) a (βa, t) = 1 na n � i=1 I{Ai = a}X⊗l i exp(βT a Xi)Yi(t) and u(l) a (βa, t) = E � X⊗l exp(βT a X)Y (t) � , where na = �n i=1 I{Ai = a}, and define Ea(βa, t) = U (1) a (βa, t) U (0) a (βa, t) and ea(βa, t) = u(1) a (βa, t) u(0) a (βa, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The maximum partial likelihood estimator ˆβa for the Cox proportional hazards model solves the estimating equation Sa,n(βa) = 1 na n � i=1 I{Ai = a} � � Xi − U (1) 1 (βa, u) U (0) 1 (βa, u) � dNi(u) = 0, and the cumulative baseline hazard function ˆΛ0,a is estimted by the Breslow estimator: ˆΛ0,a(t) = � t 0 �n i=1 I{Ai = a}dNi(u) �n i=1 I{Ai = a} exp(ˆβT a Xi)Yi(u) , a = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under certain regularity conditions (Andersen & Gill 1982, Conditions A – D), ˆβa and ˆΛ0,a converge in probability to the limits β∗ a and Λ∗ 0,a, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and we have √na(ˆβa − β∗ a) = Γ−1 a 1 √na n � i=1 I{Ai = a}Ha,i + op(1), 37 where Γa = E[−∂Sa,n(β∗ a)/∂β∗T a ] is the Fisher information matrix of β∗ a, Ha,i = � I{Ai = a}{Xi − ea(β∗ a, u)}dMa,i(u) and dMa,i(u) = dNi(u) − exp(β∗T a Xi)Yi(u)dΛ∗ 0,a(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, let S∗(t | a, X) = exp{−Λ∗ 0,a(t) exp(β∗T a X)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' it is shown that √na{ ˆS(t | a, Xi)−S∗(t | a, Xi)} converges uniformly to a mean-zero Gaussian process for all Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' we consider the following expansion that we use in our proof of Theorem 1 and Corollary 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆS(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) = − S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)Λ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t) exp(β∗T a Xi)XT i (ˆβa − β∗ a) − S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) exp(β∗T a Xi)(ˆΛ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t) − Λ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and furthermore ˆΛ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t) − Λ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t) = � t 0 � n−1 a �n i=1 I{Ai = a}dNi(u) U (0) a (ˆβa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) − n−1 a �n i=1 I{Ai = a}dNi(u) U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) � + � t 0 � n−1 a �n i=1 I{Ai = a}dNi(u) U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) − dΛ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t) � = − � �� � t 0 U (1) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) � U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) �2 � n−1 a n � i=1 I{Ai = a}dNi(u) �� �� T � ˆβa − β∗ a � + � t 0 n−1 a �n i=1 I{Ai = a}dMa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i(u) U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) + op(1) = − �� t 0 ea(β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u)dΛ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(u) �T � ˆβa − β∗ a � + � t 0 n−1 a �n i=1 I{Ai = a}dMa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i(u) U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Combining the above two equations, we obtain ˆS(t | a, Xi) − S∗(t | a, Xi) = � −S∗(t | a, Xi)Λ∗ 0,a(t) exp(β∗T a Xi)XT i − �� t 0 ea(β∗ a, u)dΛ∗ 0,a(u) �T� � ˆβa − β∗ a � + � t 0 n−1 a �n i=1 I{Ai = a}dMa,i(u) U (0) a (β∗ a, u) + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 38 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 Cross-fitting To show the high-level idea of cross-fitting, we state the lemma from Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2020), which is useful in our proof of Theorem 2 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Consider two independent samples O1 = (O1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , On) and O2 = (On+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , O˜n), let ˆf(o) be a function estimated from O2 and Pn the empirical measure over O1, then we have (Pn − P)( ˆf − f) = OP � ∥ ˆf − f∥ √n � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' First note that by conditioning on O2 we obtain E � Pn( ˆf − f) �� O2 � = E( ˆf − f | O2) = P( ˆf − f) and the conditional variance is var{(Pn − P)( ˆf − f) | O2} = var{Pn( ˆf − f) | O2} = 1 nvar( ˆf − f | O2) ≤ ∥ ˆf − f∥2/n therefore by Chebyshev’s inequality we have P � |(Pn − P)( ˆf − f)| ∥ ˆf − f∥2/n ≥ t � = E � P � |(Pn − P)( ˆf − f)| ∥ ˆf − f∥2/n ≥ t ���� O2 �� ≤ 1 t2 thus for any ϵ > 0 we can pick t = 1/√ϵ so that the probability above is no more than ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' which yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 39 B Proof of Proposition 1 We first show the identification by the outcome regression formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' V (d) = E[E[y(T(d)) | X]] = E[d(X)E[y(T(1)) | X] + (1 − d(X))E[y(T(0)) | X]] = E[d(X)E[y(T(1)) | X, IS = 1] + (1 − d(X))E[y(T(0)) | X, IS = 1]] = E[d(X)E[y(T(1)) | A = 1, X, IS = 1] + (1 − d(X))E[y(T(0)) | A = 0, X, IS = 1]] = E[d(X)E[y(T) | A = 1, X, IS = 1] + (1 − d(X))E[y(T) | A = 0, X, IS = 1]] = E[E[y(T) | A = d(X), X, IS = 1]] = E[IT e(X)E[y(T) | A = d(X), X, IS = 1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Similarly, we show the identification by the IPW formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' V (d) = E[E[y(T) | A = d(X), X, IS = 1]] = E � IS πS(X)E[y(T) | A = d(X), X, IS = 1] � = E � IS πS(X) I{A = d(X)} πd(X) ∆ y(U) SC(U | A, X) � , where the last equation follows from the standard IPTW-IPCW formula (Van der Laan & Robins 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' C Proof of Proposition 2 While Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2022) derived the efficient influence function for the treatment specific sur- vival function, here we derive the EIF for the value function V (d) = E[IT e(X)µ(d(X), X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 40 First consider the full data Z = (X, A, T, IS, IT), and we have the factorization as p(Z) = {p(X)πS(X)p(A|X, IS = 1)p(T|A, X, IS = 1)}IS{p(X)}IT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since ISIT = 0, the score function is S(Z) = S(X, A, T, IS) + ITS(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let Vϵ(d) = Eϵ[IT e(X)µϵ(d(X), X)] denote the parameter of interest evaluated under the law pϵ(Z), where ϵ indexes a regular parametric submodel such that p0(Z) is the true data generating law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' To establish that V (d) is pathwise differentiable with EIF φF d , we need to show that ∂ ∂ϵVϵ(d) ���� ϵ=0 = E[φF d S(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' we compute ∂ ∂ϵVϵ(d) ���� ϵ=0 = E[IT e(X)µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)S(X)] + E � ∂ ∂ϵµϵ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) ���� ϵ=0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and further write the first term on the right hand side as E[IT e(X)µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)S(X)] = E[(IT e(X)µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − V (d))S(X)] = E[(IT e(X)µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − V (d))S(Z)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 41 and the second term as E � ∂ ∂ϵµϵ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) ���� ϵ=0 � = E [d(X)E[y(T)S(T | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS) | A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1] +(1 − d(X))E[y(T)S(T | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS) | A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1]] = E [d(X)E[(y(T) − µ(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X))S(T | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS) | A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1] +(1 − d(X))E[(y(T) − µ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X))S(T | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS) | A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1]] = E � d(X)E � IS A πS(X)πA(X)(y(T) − µ(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X))S(T | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS) ����X � +(1 − d(X))E � IS (1 − A) πS(X)(1 − πA(X))(y(T) − µ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X))S(T | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS) ����X �� = E � IS πS(X) � d(X) A πA(X)(y(T) − µ(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) +(1 − d(X)) 1 − A 1 − πA(X)(y(T) − µ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) � S(T | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS) � = E � IS πS(X) I{A = d(X)} πd(X) (y(T) − µ(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X))S(Z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, the efficient influence function for the full data is φF d = IT e(X)µ(d(X), X) + IS πS(X) I{A = d(X)} πd(X) (y(T) − µ(A, X)) − V (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Next, we consider the observed data O = (X, A, U, ∆, IS, IT) due to right censoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' According to Tsiatis (2006, Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4), the EIF based on the observed data is given by φd = ∆ φF d SC(U | A, X) + � ∞ 0 L(u, A, X) SC(u | A, X)dMC(u | A, X), where L(u, A, X) = E[φF d | T ≥ u, A, X] = IT e(X)µ(d(X), X) + IS πS(X) I{A = d(X)} πd(X) (Q(u, A, X) − µ(A, X)) − V (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since we have � ∞ 0 dMC(u | A, X) SC(u | A, X) = � ∞ 0 dNC(u) SC(u | A, X) − � U 0 dΛC(u | A, X) exp{ΛC(u | A, X)} = 1 − ∆ SC(U | A, X), (9) 42 we conclude that φd = IS πS(X) I{A = d(X)} πd(X) ∆ y(U) SC(U | A, X) − V (d) + � IT e(X) − IS πS(X) I{A = d(X)} πd(X) � µ(d(X), X) + IS πS(X) I{A = d(X)} πd(X) � ∞ 0 dMC(u | A, X) SC(u | A, X) Q(u, A, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' D Proof of Theorem 1 and Corollary 1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 Double robustness We start with the proof of the double robustness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We show that EIF-based estimator is consistent when either the survival outcome model or the models for the sampling score, the propensity score and the censoring process are correctly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under some regularity conditions, the nuisance estimators ˆµ(a, x), ˆQ(u, a, x), ˆπS(x), ˆπA(x) and ˆSC(t | a, x) converge in probability to µ∗(a, x), Q∗(u, a, x), π∗ S(x), π∗ A(x) and S∗ C(t | a, x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' It suffices to show that E[V ∗(d)] = V (d), where V ∗(d) =IT e(X)µ∗(A = d(X), X) + IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∆ y(U) S∗ C(U | A, X) − µ∗(A, X) + � ∞ 0 dM ∗ C(u | A, X) S∗ C(u | A, X) Q∗(u, A, X) � =(I) + (II) + (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' First, consider the case when the survival outcome model is correct, thus we have (I) = E[IT e(X)µ∗(A = d(X), X)] = V (d) 43 By Equation 9, we obtain (II) + (III) = IS π∗ S(X) I{A = d(X)} π∗ d(X) � y(T) − µ∗(A, X) − � ∞ 0 dM ∗ C(u | A, X) S∗ C(u | A, X) (y(T) − Q∗(u, A, X)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In this case, we have E � IS π∗ S(X) I{A = d(X)} π∗ d(X) (y(T) − µ∗(A, X)) � = E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) (y(T) − µ∗(A, X)) ���� X �� = E � E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) (y(T) − µ∗(A, X)) ���� A, X, IS = 1 � ���� X �� = E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) E[(y(T) − µ∗(A, X)) | A, X, IS = 1] ���� X �� = E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) (E[y(T) | A, X, IS = 1] − µ∗(A, X)) ���� X �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Also define d ˜ MC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) = d ˜NC(u) − I{C ≥ u}dΛC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) where ˜NC(u) = I{C ≤ u},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' so we have E � IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∞ 0 dM ∗ C(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) S∗ C(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) (y(T) − Q∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) � = E � IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∞ 0 d ˜ MC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) S∗ C(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) I{T ≥ u}(y(T) − Q∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) � = E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∞ 0 d ˜ M ( Cu | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) S∗ C(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) I{T ≥ u}(y(T) − Q∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) ���� X �� = E � E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∞ 0 d ˜ MC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) S∗ C(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) I{T ≥ u} (y(T) − Q∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) ���� A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1 � ���� X �� = E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∞ 0 d ˜ MC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) S∗ C(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) E [I{T ≥ u} (y(T) − Q∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) ���� A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1 � ���� X �� = E � E � IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∞ 0 d ˜ MC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) S∗ C(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) (E[I{T ≥ u}y(T) | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1] −E[I{T ≥ u} | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' IS = 1]Q∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)) ���� X �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 44 Next, consider the case when the models for the sampling score, the propensity score and the censoring process are correctly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Rearranging the terms of V ∗(d), we obtain V ∗(d) = IS π∗ S(X) I{A = d(X)} π∗ d(X) ∆ y(U) S∗ C(U | A, X) + � IT e(X) − IS π∗ S(X) � µ∗(A = d(X), X) + IS π∗ S(X) I{A = d(X)} π∗ d(X) � ∞ 0 dM ∗ C(u | A, X) S∗ C(u | A, X) Q∗(u, A, X) =(I) + (II) + (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In this case, we have (I) = E � IS π∗ S(X) I{A = d(X)} π∗ d(X) ∆ y(U) S∗ C(U | A, X) � = V (d), (II) = E �� IT e(X) − IS π∗ S(X) � µ∗(A = d(X), X) � = E � E � IT e(X) − IS π∗ S(X) ����X � µ∗(A = d(X), X) � = 0, and (III) is a stochastic integral with respect to the martingale M ∗ C(u | A, X), thus equals 0 as well, which completes the double robustness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 Asymptotic properties To establish the asymptotic results, we need some regularity conditions such that the nui- sance estimators µ(a, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆβa, ˆΛ0,a), Q(u, a, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆβa, ˆΛ0,a), πS(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆλ), πA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆθ) and SC(u | a, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆαa, ˆΛC0,a) converge in probability to µ(a, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' β∗ a, Λ∗ 0,a), Q(u, a, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' β∗ a, Λ∗ 0,a), πS(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' λ∗), πA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' θ∗) and SC(t | a, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' α∗ a, Λ∗ C0,a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We assume the following conditions hold: (C1) X is bounded almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (C2) The equation E �� A − exp(θT X) 1+exp(θT X) � X � = 0 has a unique solution θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 45 (C3) For a = 0, 1, the equation E �� L 0 � Xi − E[Yi(u) exp(βT a X)X] E[Yi(u) exp(βT a X)] � × dNi(u) � = 0, has a unique solution β∗ a, where L > u is a pre-specified time point such that Pr(Ui > L) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, let Λ∗ 0,a(u) = E �� u 0 dNi(u) E[Yi(u) exp(β∗T a Xi)] � , and assume Λ∗ 0,a(L) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (C4) For a = 0, 1, the equation E �� L 0 � Xi − E[Yi(u) exp(αT a X)X] E[Yi(u) exp(αT a X)] � × dNi(u) � = 0, has a unique solution α∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, let Λ∗ C0,a(u) = E �� u 0 dNi(u) E[Yi(u) exp(α∗T a Xi)] � , and assume Λ∗ C0,a(L) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (C5) The estimating equation for the sampling score model πS(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' λ) has a unique solution λ∗, and achieves root-n rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Under Condition 1, we have the following asymptotic representations: √n(ˆθ − θ∗) = 1 √n n � i=1 φθi + op(1), √n(ˆλ − λ∗) = 1 √n n � i=1 φλi + op(1), √n(ˆβa − β∗ a) = 1 √n n � i=1 φβai + op(1), √n(ˆαa − α∗ a) = 1 √n n � i=1 φαai + op(1), for a = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We focus on the estimation of survival functions by our proposed method: ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = 1 N N � i=1 � IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i e(Xi) ˆS(t | A = dη(Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = dη(Xi)} ˆπS(Xi)ˆπd(Xi) � ∆i Yi(t) ˆSC(t | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − ˆS(t | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + � ∞ 0 ˆS(t | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)d ˆ MC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆS(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆSC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 46 and for the ease of notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' define ˆJ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = ∆i Yi(t) ˆSC(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − ˆS(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) + � ∞ 0 ˆS(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)d ˆ MC(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) ˆS(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) ˆSC(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' J∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = ∆i Yi(t) S∗ C(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) + � ∞ 0 S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Our proof has three main parts as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' PART 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By the double robustness property shown in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1, we have, by the strong law of large numbers and uniform consistency, that ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) + op(1), which proves (i) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Moreover, define S∗ N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = 1 N N � i=1 � IT,i e(Xi)S∗(t | A = dη(Xi), Xi) + IS,iI{Ai = dη(Xi)} π∗ S(Xi)π∗ d(Xi) J∗(t, Ai, Xi) � , and by applying the Taylor expansion and the counting processes result in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1, we obtain ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) =S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) + HT λ (ˆλ − λ∗) + HT θ (ˆθ − θ∗) + HT β0(ˆβ0 − β∗ 0) + HT β1(ˆβ1 − β∗ 1) + HT α0(ˆα0 − α∗ 0) + HT α1(ˆα1 − α∗ 1) + RS + op(N −1/2), where Hλ = lim N→∞ 1 N N � i=1 ∂ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) ∂λ∗ , Hθ = lim N→∞ 1 N N � i=1 ∂ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) ∂θ∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Hβa = lim N→∞ 1 N N � i=1 � IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i e(Xi)(−1)a+1G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = a} π∗ S(Xi)π∗ d(Xi) �� ∞ 0 G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) −G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 G(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗2(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Hαa = lim N→∞ 1 N N � i=1 IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = a} π∗ S(Xi)π∗ d(Xi) � −∆iYi(t) S∗ C(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 GC(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗2 C (u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + ˜GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 47 RS = 1 N N � i=1 � a=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 � IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i e(Xi)(−1)a+1H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = a} π∗ S(Xi)π∗ d(Xi) � � ∞ 0 H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 H(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗2(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − ∆iYi(t) S∗ C(t|a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 HC(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗2 C (u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − ˜HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �� = 1 N N � i=1 φRs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' with G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = −S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)Λ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t)xT + S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) exp(β∗T a x) �� t 0 ea(β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u)dΛ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(u) �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = −S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) exp(β∗T a x) � t 0 n−1 a �n i=1 I{Ai = a}dMa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i(u) U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = −S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)Λ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(t)xT + S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) exp(β∗T a x) �� t 0 ea(β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u)dΛ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(u) �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = −S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) exp(β∗T a x) � t 0 n−1 a �n i=1 I{Ai = a}dMa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i(u) U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˜GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = � Ui 0 S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dΛ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) xT + �� t 0 S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)ea(β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u)dΛ∗ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='a(u) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˜HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) = � t 0 S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)n−1 a �n i=1 I{Ai = a}dMa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i(u) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)U (0) a (β∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Thus, we have √ N � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) � = 1 √ N N � i=1 (ξ1,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) + ξ2,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η)) + op(1), (10) where ξ1,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η), ξ2,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = HT λ φλ∗,i + HT θ φθ∗,i + � a=0,1 HT βaφβ∗ 0,i + � a=0,1 HT αaφα∗a,i + HT α1 + φRs,i, and ξ1,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η), ξ2,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) are independent mean-zero processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, we obtain that √ N � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) � converges weakly to a mean-zero Gaussian process, which proves (ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 48 PART 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We show that N 1/3∥ˆη − η∗∥2 = Op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Recall that ˆη = arg max η ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) and η∗ = arg max η S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Assumption 5 (i), S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) is twice continuously differentiable at a neighborhood of η∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' in Step 1, we show that ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) = S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) + op(1), ∀η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' since ˆη maximizes ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η), we have that ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) ≥ supη ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η), thus by the Argmax theorem, we have ˆη p→ η∗ as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In order to establish the N −1/3 rate of convergence of ˆη, we apply Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 (Rate of convergence) of Kosorok (2008), and need to find the suitable rate that satisfies three conditions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 1 For every η in a neighborhood of η∗ such that ∥η − η∗∥2 < δ, by Assump- tion 5 (i), we apply the second-order Taylor expansion, S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) = S′(η∗)∥η − η∗∥2 + 1 2S′′(η∗)∥η − η∗∥2 2 + o(∥η − η∗∥2 2) = 1 2S′′(η∗)∥η − η∗∥2 2 + o(∥η − η∗∥2 2), and as S′′(η∗) < 0, there exists c0 = − 1 2S′′(η∗) > 0 such that S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) ≤ −c0∥η − η∗∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 2 For all N large enough and sufficiently small δ, we consider the centered process ˆS − S, and have that E � √ N sup ∥η−η∗∥2<δ ��� ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) ���� � = E � √ N sup ∥η−η∗∥2<δ ��� ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) + S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) + S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) ���� � ≤ E � √ N sup ∥η−η∗∥2<δ ��� ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) ���� � (I) + E � √ N sup ∥η−η∗∥2<δ |S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − {S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)}| � , (II) 49 and we bound (I) and (II) respectively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 To bound (II), we need the useful facts that I{A = dη(X)} − I{A = dη∗(X)} = (2A − 1)(dη(X) − dη∗(X)), S∗(t | dη(Xi), Xi) − S∗(t | dη∗(Xi), Xi) = (S∗(t | 1, Xi) − S∗(t | 0, Xi))(dη(Xi) − dη∗(Xi)), and obtain S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) = 1 N N � i=1 (dη(Xi) − dη∗(Xi)) × � IT,i e(Xi)(S∗(t | 1, Xi) − S∗(t | 0, Xi)) + (2Ai − 1)IS,i π∗ S(Xi)π∗ d(Xi)J∗(t, Ai, Xi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Define a class of functions F1 η = � (dη(x) − dη∗(x)) � IT e(x)(S∗(t | 1, x) − S∗(t | 0, x)) + (2a − 1)IS π∗ a(x)π∗ S(x)J∗(t, a, x) � : ∥η − η∗∥2 < δ � , and let M1 = sup ���IT e(x)(S∗(t | 1, x) − S∗(t | 0, x)) + (2a−1)IS π∗a(x)π∗ S(x)J∗(t, a, x) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Assump- tion 1, 3 and Condition 1, we have that M1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' When ∥η − η∗∥2 < δ, by Condition 1 (C1), there exists a constant 0 < k0 < ∞ such that |(1, xT)(η − η∗)| < k0δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' furthermore, we show that |dη(x) − dη∗(x)| = |I{(1, xT)η > 0} − I{(1, xT)η∗ > 0}| ≤ I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}, by considering the three cases: when −k0δ ≤ (1, xT)η∗ ≤ k0δ, we have |dη(x) − dη∗(x)| ≤ 1 = I{−k0δ ≤ (1, xT)η∗ ≤ k0δ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' when (1, xT)η∗ > k0δ > 0, we have (1, xT)η = (1, xT)(η − η∗) + (1, xT)η∗ > 0, so |dη(x) − dη∗(x)| = 0 = I{−k0δ ≤ (1, xT)η∗ ≤ k0δ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' when (1, xT)η∗ < −k0δ < 0, we have (1, xT)η = (1, xT)(η − η∗) + (1, xT)η∗ < 0, so |dη(x) − dη∗(x)| = 0 = I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 50 Thus we can define the envelope of F1 η as F1 = M1I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Assumption 5 (ii), there exists a constant 0 < k1 < ∞ such that ∥F1∥P,2 ≤ M1 � Pr(−k0δ ≤ (1, xT)η∗ ≤ k0δ) ≤ M1 � 2k0k1δ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6 and Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='9 of Kosorok (2008), we have that F1 η, a class of indicator functions, is a Vapnik-Cervonenkis (VC) class with bounded bracketing entropy J∗ [](1, F1 η) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since we have the fact that GNF1 η = N −1/2 N � i=1 � F1 η − E[F1 η] � = √ N (S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − {S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)}) , By Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 of Kosorok (2008), we obtain that there exists a constant 0 < c1 < ∞, (II) = E � sup ∥η−η∗∥2<δ |GNF1 η| � ≤ c1J∗ [](1, F1 η)∥F1∥P,2 ≤ c1J∗ [](1, F1 η)M1 � 2k0k1δ1/2 = ˜c1δ1/2, so we conclude that (II) ≤ ˜c1δ1/2 where ˜c1 > 0 is a finite constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 To bound (I), first we have ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − { ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − {S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = 1 N N � i=1 (dη(Xi) − dη∗(Xi)) � IT,i e(Xi){ ˆS(t|1, Xi) − ˆS(t|0, Xi) − (S∗(t|1, Xi) − S∗(t|0, Xi))} + (2Ai − 1)IS,i ˆπAi(Xi)ˆπS(Xi) ˆJ(t, Ai, Xi) − (2Ai − 1)IS,i π∗ Ai(Xi)π∗ S(Xi)J∗(t, Ai, Xi) � , and then apply the Taylor expansion and counting processes result in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1, ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − { ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S∗ n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = 1 N N � i=1 (dη(Xi) − dη∗(Xi)) × � Dλ(ˆλ − λ∗) + Dθ(ˆθ − θ∗) + Dβ0(ˆβ0 − β∗ 0) +Dβ1(ˆβ1 − β∗ 1) + Dα0(ˆα0 − α∗ 0) + Dα1(ˆα1 − α∗ 1) + RS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i � + op(N −1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (11) 51 where Dλ = − (2Ai − 1)IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i π∗ Ai(Xi)π∗2 S (Xi)J∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �∂π∗ S(Xi) ∂λ �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Dθ = − IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i π∗2 Ai(Xi)π∗ S(Xi)J∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �∂π∗ A(Xi) ∂θ �T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Dβa =IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i e(Xi)(−1)a+1G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + (2Ai − 1)I{Ai = a}IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i π∗ Ai(Xi)π∗ S(Xi) �� ∞ 0 G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) −G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 G(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗2(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Dαa =(2Ai − 1)I{Ai = a}IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i π∗ Ai(Xi)π∗ S(Xi) � − ∆i Yi(t) S∗ C(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 GC(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗2 C (u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + ˜GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' RS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i = � a=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 � IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i e(Xi)(−1)a+1H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + (2Ai − 1)I{Ai = a}IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i π∗ Ai(Xi)π∗ S(Xi) � � ∞ 0 H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 H(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗2(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − ∆i Yi(t) S∗ C(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − � ∞ 0 HC(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) S∗2 C (u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − ˜HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' we define the following classes of functions: F2 η = � (dη(x) − dη∗(x)) (2a − 1)IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i π∗ a(x)π∗2 S (x)J∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) �∂π∗ S(x) ∂λ �T : ∥η − η∗∥2 < δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' F3 η = � (dη(x) − dη∗(x)) −IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i π∗2 a (x)π∗ S(x)J∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) �∂π∗ A(x) ∂θ �T : ∥η − η∗∥2 < δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' F4 η = � (dη(x) − dη∗(x)) � IT e(x)(−1)a+1G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) + (2a − 1)IS π∗ a(x)π∗ S(x) × �� ∞ 0 G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − � ∞ 0 G(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗2(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) � � : ∥η − η∗∥2 < δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 52 F5 η = � (dη(x) − dη∗(x)) � IT e(x)(−1)a+1G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) + (2a − 1)IS π∗ a(x)π∗ S(x) × �� ∞ 0 G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − G(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − � ∞ 0 G(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗2(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) � � : ∥η − η∗∥2 < δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' F6 η = � (dη(x) − dη∗(x)) � (2a − 1)IS π∗ a(x)π∗ S(x) � − ∆ Y (t) S∗ C(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − � ∞ 0 GC(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗2 C (u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) + ˜GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) � � : ∥η − η∗∥2 < δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' F7 η = � (dη(x) − dη∗(x)) � (2a − 1)IS π∗ a(x)π∗ S(x) � − ∆ Y (t) S∗ C(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − � ∞ 0 GC(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗2 C (u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) + ˜GC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) � � : ∥η − η∗∥2 < δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' F8 η = � (dη(x) − dη∗(x)) � � a=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 � IT e(x)a+1H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) + (2a − 1)IS π∗ a(x)π∗ S(x) × � � ∞ 0 H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − H(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − � ∞ 0 H(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗2(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − ∆ Y (t) S∗ C(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − � ∞ 0 HC(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(t | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)dM ∗ C(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) S∗2 C (u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x)S∗(u | a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) − ˜HC(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' x) ��� : ∥η − η∗∥2 < δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let M2 = sup ����� (2a − 1) π∗ a(x) J∗(t, a, x) �∂π∗ S(x) ∂λ �T����� , where M2 ∈ R+ and the supremum is taken over all the coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and M3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , M8 are defined accordingly for F3 η, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , F8 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Assumption 1, 3 and Condition 1, we have that M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , M8 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Using the same technique as in Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1, we define the envelop of Fj η as Fj = MjI{−k0δ ≤ (1, xT)η∗ ≤ k0δ} for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , 8, and obtain that ∥Fj∥P,2 ≤ ˜ Mjδ1/2 < ∞, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , 8, 53 where ˜ M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , ˜ M8 are some finite constants, and that Fj η is a VC class with bounded bracketing entropy J∗ [](1, Fj η) < ∞, for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 of Kosorok (2008), we obtain E � sup ∥η−η∗∥2<δ ��GNFj η �� � ≤ cjJ∗ [](1, Fj η)∥Fj∥P,2, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , 8, where c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , c8 are some finite constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' That is, we have E � sup ∥η−η∗∥2<δ ��GNF8 η �� � ≤ ˜c8δ1/2, and furthermore by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 of Van der Vaart & Wellner (1996), we obtain � E � sup ∥η−η∗∥2<δ ∥GnFj η∥2 2 ��1/2 ≤ lj � E � sup ∥η−η∗∥2<δ |GnFj η| � + ∥Fj∥P,2 � ≤ lj{cjJ∗ [](1, Fj η) + 1}∥Fj∥P,2 ≤ ˜cjδ1/2, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , 7, where l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , l7 and ˜c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' , ˜c7 are some finite constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Equation (11), we have that (I) = E � N 1/2 sup ∥η−η∗∥2<δ ��� ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S∗ N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − { ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S∗ N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='≤ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='|GnF2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η(ˆλ − λ∗)| + |GnF3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η(ˆθ − θ∗)| + |GnF4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η(ˆβ0 − β∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0)| + |GnF5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η(ˆβ1 − β∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ |GnF6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η(ˆα0 − α∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0)| + |GnF7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η(ˆα1 − α∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1)| + |GnF8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ op(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='≤ N −1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='|GnF2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η · N 1/2(ˆλ − λ∗)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='|GnF3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η · N 1/2(ˆθ − θ∗)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='|GnF4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η · N 1/2(ˆβ0 − β∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='|GnF5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η · N 1/2(ˆβ1 − β∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='|GnF6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η · N 1/2(ˆα0 − α∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='|GnF7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η · N 1/2(ˆα1 − α∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1)| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='∥η−η∗∥2<δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='��GNF8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ op(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 54 and then by the Cauchy-Schwarz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' we obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='(I) ≤ N −1/2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='E[N∥ˆλ − λ∗∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='�1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='��1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='+ N −1/2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='E[N∥ˆθ − θ∗∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='�1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='E ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Let Mλ = � E[N∥ˆλ − λ∗∥2 2] �1/2 , and Mθ, Mβ0, Mβ1, Mα0, Mα1 are defined accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Condition 1, we have that Mλ, Mθ, Mβ0, Mβ1, Mα0, Mα1 < ∞, and therefore (I) ≤ N −1/2(Mλ˜c2 + Mθ˜c3 + Mβ0˜c4 + Mβ1˜c5 + Mα0˜c6 + Mα1˜c7)δ1/2 + ˜c8δ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In summary, we obtain that, let N → ∞, the centered process satisfies E � √ N sup ∥η−η∗∥2<δ ��� ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − { ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} ��� � ≤ (I) + (II) ≤ (˜c1 + ˜c8)δ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (12) Let φN(δ) = δ1/2 and α = 3 2 < 2, thus we have φn(δ) δα = δ−1 is decreasing, and α does not depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' That is, the second condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 3 By the facts that ˆη p→ η∗ as N → ∞, and that ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) ≥ supη ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η), we choose rN = N 1/3 such that r2 NφN(r−1 N ) = N 2/3φN(N −1/3) = N 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The third condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 55 In the end, the three conditions are satisfied with rN = N 1/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' thus we conclude that N 1/3∥ˆη − η∗∥2 = Op(1), which completes the proof of (iii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' PART 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We characterize the asymptotic distribution of ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since we have √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} + √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)}, we study the two terms in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 To establish √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη)− ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = op(1), it suffices to show that √ N{S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη)− S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = op(1) and √ N( ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − {S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)}) = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' First, as N 1/3∥ˆη − η∗∥2 = Op(1), we take the second-order Taylor expansion √ N{S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = √ N � S′(η∗)∥ˆη − η∗∥2 + 1 2S′′(η∗)∥ˆη − η∗∥2 2 + op(∥ˆη − η∗∥2 2) � = √ N �1 2S′′(η∗)∥ˆη − η∗∥2 2 + op(∥ˆη − η∗∥2 2) � = √ N �1 2S′′(η∗)Op(N −2/3) + op(N −2/3) � = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Next, we follow the result (12) obtained in PART 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' As N 1/3∥ˆη − η∗∥2 = Op(1), there exists ˜δ = c9N −1/3, where c9 < ∞ is a finite constant, such that ∥ˆη − η∗∥2 ≤ ˜δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore we have √ N( ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − {S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)}) ≤ E � √ N sup ∥ˆη−η∗∥2<˜δ ��� ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − { ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} ��� � ≤ (˜c1 + ˜c8)˜δ1/2 = (˜c1 + ˜c8)√c9N −1/6 = op(1), which yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 To derive the asymptotic distribution of √n{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)}, we follow the result (10) obtained in PART 1 and have that √ N � ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) � D→ N(0, σ2 t,1), 56 where σ2 t,1 = E[(ξ1,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) + ξ2,i(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗))2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore we obtain in the end √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) − ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} + √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} = op(1) + √ N{ ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} D→ N(0, σ2 t,1), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For Corollary 1 where we consider RMST, the proof can follow the same steps as before, and is thus omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' E Proof of Theorem 2 and Corollary 2 Our proof has three main parts below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' PART 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Recall that the cross-fitting technique, at a high level as exemplified in Lemma 1, uses sample splitting to avoid bias due to over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For simplicity, consider that the datasets Os and Ot are randomly split into 2 folds with equal size respectively such that Os = Os,1 ∪Os,2, Ot = Ot,1 ∪Ot,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The extension to K-folds as described in Algorithm 1 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Here the subscript CF is omitted to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Define I1 = Os,1 ∪ Ot,1, I2 = Os,2 ∪ Ot,2, and N1 = |I1|, N2 = |I2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The cross-fitted estimator for the value function under the ITR dη is ˆV (η) = N1 N ˆV I1(η) + N2 N ˆV I2(η), where ˆV I1(η) = 1 N1 � I1 � IT,i e(Xi)ˆµ(dη(Xi), Xi) + IS,i ˆπS(Xi) I{Ai = dη(Xi)} ˆπd(Xi) × � ∆i y(Ui) ˆSC(Ui | Ai, Xi) − ˆµ(Ai, Xi) + � ∞ 0 d ˆ MC(u | Ai, Xi) ˆSC(u | Ai, Xi) ˆQ(u, Ai, Xi) � � , 57 and the nuisance parameters are estimated from I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆV I2(η) is defined accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In this step, we show that ˆV (η) − VN(η) = op(N −1/2), and essentially it suffices to prove that ˆV I1(η) − V I1 N (η) = op(N −1/2), where VN(η) = 1 N N � i=1 � IT,i e(Xi)µ(dη(Xi), Xi) + IS,i πS(Xi) I{Ai = dη(Xi)} πd(Xi) × � ∆i y(Ui) SC(Ui | Ai, Xi) − µ(Ai, Xi) + � ∞ 0 dMC(u | Ai, Xi) SC(u | Ai, Xi) Q(u, Ai, Xi) � � , and V I1 N (η) is defined accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' we have the following decomposition ˆV I1(η) − V I1 N (η) = 1 N1 � I1 � IT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i e(Xi)(ˆµ(dη(Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − µ(dη(Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i � 1 πS(Xi) − 1 ˆπS(Xi) � I{Ai = dη(Xi)} πd(Xi) K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = dη(Xi)} πS(Xi) � 1 πd(Xi) − 1 ˆπd(Xi) � K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i πS(Xi) I{Ai = dη(Xi)} πd(Xi) ( ˆK(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = dη(Xi)} � 1 πS(Xi) − 1 ˆπS(Xi) � � 1 πd(Xi) − 1 ˆπd(Xi) � K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = dη(Xi)} πd(Xi) � 1 πS(Xi) − 1 ˆπS(Xi) � ( ˆK(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = dη(Xi)} πS(Xi) � 1 πd(Xi) − 1 ˆπd(Xi) � ( ˆK(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) + IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='iI{Ai = dη(Xi)} � 1 πS(Xi) − 1 ˆπS(Xi) � � 1 πd(Xi) − 1 ˆπd(Xi) � ( ˆK(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (13) 58 where ˆK(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) = ∆i y(Ui) ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − ˆµ(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + � ∞ 0 d ˆ MC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆSC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆQ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) = ∆i y(Ui) SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − µ(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + � ∞ 0 dMC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) Q(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In summary, the decomposition (13) consists of two types of terms: four mean-zero terms and four product terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' For the mean-zero terms, we utilize the method introduced in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' since E[IT,i e(Xi)(ˆµ(dη(Xi), Xi) − µ(dη(Xi), Xi))] = 0, by applying Lemma 1, we obtain 1 N1 � I1 IT,i e(Xi)(ˆµ(dη(Xi), Xi) − µ(dη(Xi), Xi)) = op(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Similarly we have E � IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i � 1 πS(Xi) − 1 ˆπS(Xi) � I{Ai = dη(Xi)} πd(Xi) K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' so we obtain E � � � 1 N1 � I1 IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i � 1 πS(Xi) − 1 ˆπS(Xi) � I{Ai = dη(Xi)} πd(Xi) K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �2� � = E � �E � � � 1 N1 � I1 IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i � 1 πS(Xi) − 1 ˆπS(Xi) � I{Ai = dη(Xi)} πd(Xi) K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �2 �����I2 � � � � = E � var � 1 N1 � I1 IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i � 1 πS(Xi) − 1 ˆπS(Xi) � I{Ai = dη(Xi)} πd(Xi) K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �����I2 �� = 1 N1 E � var � IS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='i � 1 πS(Xi) − 1 ˆπS(Xi) � I{Ai = dη(Xi)} πd(Xi) K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ����I2 �� ≤ Op(1) N1 = op( 1 N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We also have E �IS,iI{Ai = dη(Xi)} πS(Xi) � 1 πd(Xi) − 1 ˆπd(Xi) � K(Ai, Xi) � = 0, 59 E � IS,i πS(Xi) I{Ai = dη(Xi)} πd(Xi) ( ˆK(Ai, Xi) − K(Ai, Xi)) � = 0, and using the same technique, we conclude that these two mean-zero terms are op(N −1/2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The product terms can be handled simply by the Cauchy-Schwarz inequality and the rate of convergence conditions in Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Additionally we have the decomposition as follows 1 N1 � I1 ( ˆK(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) = 1 N1 � I1 � − (ˆµ(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − µ(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) + 1 − ∆i SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) − � Ui 0 λC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi))du + (1 − ∆i) � 1 ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − 1 SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + � 1 ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − 1 SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ∆i y(Ui) − � Ui 0 � ˆλC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆSC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − λC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)du + (1 − ∆i) � 1 ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − 1 SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) − � Ui 0 � ˆλC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆSC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − λC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi))du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and similarly we have three mean-zero terms which are op(N −1/2) by the same technique in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 and the facts that E[ˆµ(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − µ(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' E � 1 − ∆i SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) − � Ui 0 λC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)( ˆQ(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi))du � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 60 E � (1 − ∆i) � 1 ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − 1 SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) + � 1 ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − 1 SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ∆i y(Ui) − � Ui 0 � ˆλC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆSC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − λC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)du � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and we can bound the two product terms as well 1 N1 � I1 � (1 − ∆i) � 1 ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − 1 SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) − � Ui 0 � ˆλC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆSC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − λC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) � ( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi))du � ≤ � � 1 N1 � I1 (1 − ∆i) � 1 ˆSC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − 1 SC(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �2� � 1/2 × � 1 N1 � I1 (1 − ∆i)( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi))2 �1/2 − � Ui 0 � � 1 N1 � I1 � ˆλC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) ˆSC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − λC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) SC(u | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) �2� � 1/2 × � 1 N1 � I1 ( ˆQ(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − Q(Ui | Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi))2 �1/2 du = op(N −1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' which proves that 1 N1 � I1( ˆK(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi) − K(Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Xi)) = op(N −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, we conclude that the four product terms in (13) are op(N −1/2) as well, which completes the proof of (i) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' PART 2: We show that N 1/3∥ˆη − η∗∥2 = Op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Assumption 5 (i), V (η) is twice continuously differentiable at a neighborhood of η∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' in PART 1, we show that ˆV (η) = V (η) + op(1), ∀η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' since ˆη maximizes ˆV (η), we have that ˆV (ˆη) ≥ supη ˆV (η), thus by the Argmax theorem, we have ˆη p→ η∗ as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In order to establish the N −1/3 rate of convergence of ˆη, we apply Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 (Rate 61 of convergence) of Kosorok (2008), and need to find the suitable rate that satisfies three conditions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 1 For every η in a neighborhood of η∗ such that ∥η − η∗∥2 < δ, by Assump- tion 5 (i), we apply the second-order Taylor expansion, V (η) − V (η∗) = V ′(η∗)∥η − η∗∥2 + 1 2V ′′(η∗)∥η − η∗∥2 2 + o(∥η − η∗∥2 2) = 1 2V ′′(η∗)∥η − η∗∥2 2 + o(∥η − η∗∥2 2), and as V ′′(η∗) < 0, there exists c10 = − 1 2V ′′(η∗) > 0 such that V (η)−V (η∗) ≤ −c10∥η−η∗∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 2 For all N large enough and sufficiently small δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' we consider the centered process ˆV − V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' and have that E � √ N sup ∥η−η∗∥2<δ ���ˆV (η) − V (η) − {ˆV (η∗) − V (η∗)} ��� � = E � √ N sup ∥η−η∗∥2<δ ���ˆV (η) − Vn(η) + Vn(η) − V (η) − {ˆV (η∗) − Vn(η∗) + Vn(η∗) − V (η∗)} ��� � ≤ E � √ N sup ∥η−η∗∥2<δ ���ˆV (η) − Vn(η) − {ˆV (η∗) − Vn(η∗)} ��� � (I) + E � √ N sup ∥η−η∗∥2<δ |Vn(η) − V (η) − {Vn(η∗) − V (η∗)}| � (II) It follows from the result in PART 1 that (I) = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' To bound (II), we have Vn(η) − Vn(η∗) = 1 N N � i=1 (dη(Xi) − dη∗(Xi)) × � IT,i e(Xi)(µ(1, Xi) − µ(0, Xi)) + (2Ai − 1)IS,i πAi(Xi)πS(Xi)K(Ai, Xi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Define a class of functions F9 η = � (dη(x)−dη∗(x))× � IT e(x)(µ(1, x)−µ(0, x))+ (2a − 1)IS πa(x)πS(x)K(a, x) � : ∥η−η∗∥2 < δ � , and let M9 = sup ���IT e(x)(µ(1, x) − µ(0, x)) + (2a−1)IS πa(x)πS(x)K(a, x) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Assumption 1, 3 and Condition 1, we have that M9 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Using the same technique as in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 Condition 62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1, we define the envelop of F9 η as F9 = M9I{−k0δ ≤ (1, xT)η∗ ≤ k0δ}, and obtain that ∥F9∥P,2 ≤ ˜ M9δ1/2 < ∞, where ˜ M9 is a finite constant, and that F9 η is a VC class with bounded entropy J∗ [](1, F9 η) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 of Kosorok (2008), we obtain E � sup ∥η−η∗∥2<δ ��GNF9 η �� � ≤ ˜c9δ1/2, where ˜c9 is a finite constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore, we obtain (II) = E � √ N sup ∥η−η∗∥2<δ |VN(η) − V (η) − {VN(η∗) − V (η∗)}| � = E � sup ∥η−η∗∥2<δ |GnF9 η| � ≤ ˜c9δ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In summary, we obtain that the centered process satisfies E � √ N sup ∥η−η∗∥2<δ ��� ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η) − { ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗) − S(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)} ��� � ≤ (I) + (II) ≤ ˜c9δ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (14) Let φN(δ) = δ1/2 and α = 3 2 < 2, thus we have φn(δ) δα = δ−1 is decreasing, and α does not depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' That is, the second condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Condition 3 By the facts that ˆη p→ η∗ as N → ∞, and that ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' ˆη) ≥ supη ˆS(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η), we choose rN = N 1/3 such that r2 NφN(r−1 N ) = N 2/3φN(N −1/3) = N 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The third condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In the end, the three conditions are satisfied with rN = N 1/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' thus we conclude that N 1/3∥ˆη − η∗∥2 = Op(1), which completes the proof of (ii) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' PART 3: We characterize the asymptotic distribution of ˆV (ˆη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since we have √ N{ˆV (ˆη) − V (η∗)} = √ N{ˆV (ˆη) − ˆV (η∗)} + √ N{ˆV (η∗) − V (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' η∗)}, we study the two terms in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1 To establish √ N{ˆV (ˆη)− ˆV (η∗)} = op(1), it suffices to show that √ N{V (ˆη)− V (η∗)} = op(1) and √ N( ˆV (ˆη) − ˆV (η∗) − {V (ˆη) − V (η∗)}) = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 63 First, as N 1/3∥ˆη − η∗∥2 = Op(1), we take the second-order Taylor expansion √ N{V (ˆη) − V (η∗)} = √ N � V ′(η∗)∥ˆη − η∗∥2 + 1 2V ′′(η∗)∥ˆη − η∗∥2 2 + op(∥ˆη − η∗∥2 2) � = √ N �1 2V ′′(η∗)∥ˆη − η∗∥2 2 + op(∥ˆη − η∗∥2 2) � = √ N �1 2V ′′(η∗)Op(N −2/3) + op(N −2/3) � = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Next, we follow the result (14) obtained in PART 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' As N 1/3∥ˆη − η∗∥2 = Op(1), there exists ˜δ2 = c11N −1/3, where c11 < ∞ is a finite constant, such that ∥ˆη−η∗∥2 ≤ ˜δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore we have √ N( ˆV (ˆη) − ˆV (η∗) − {V (ˆη) − V (η∗)}) ≤ E � √ N sup ∥ˆη−η∗∥2<˜δ2 ���ˆV (ˆη) − V (ˆη) − {ˆV (η∗) − V (η∗)} ��� � ≤ ˜c9˜δ1/2 = ˜c9 √c11N −1/6 = op(1), which yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='2 To derive the asymptotic distribution of √ N{ˆV (η∗) − V (η∗)}, we follow the result obtained in PART 1 that ˆV (η∗) = VN(η∗) + op(N −1/2), and thus √ N � ˆV (η∗) − V (η∗) � D→ N(0, σ2 2), where σ2 2 = E[φ2 dη∗] is the semiparametric efficiency bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Therefore we obtain in the end √ N{ˆV (ˆη) − v(η∗)} = √ N{ˆV (ˆη) − ˆV (η∗)} + √ N{ˆV (η∗) − V (η∗)} = op(1) + √ N{ˆV (η∗) − V (η∗)} D→ N(0, σ2 2), which completes the proof of Theorem 2 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 64 F Proof of Theorem 3 and Theorem 4 When the source and target populations have the same distributions, both ˆVDR(η) and ˆVCF(η) converge to V (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The asymptotic variance of ˆVDR(η) is σ2 DR = E � IS P(IS = 1) � µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) + I{A = d(X)} πd(X) K(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − V (η) �2� = E � IS P(IS = 1) � µ2(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) + I{A = d(X)} π2 d(X) K2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − V 2(η) +2I{A = d(X)} πd(X) K(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − 2µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)V (η) −2I{A = d(X)} πd(X) K(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)V (η) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' while the asymptotic variance of ˆVCF(η) is σ2 CF = E �� IT e(X)µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) + IS I{A = d(X)} πS(X)πd(X) K(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − V (η) �2� = E �� IT e2(X)µ2(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) + IS I{A = d(X)} π2 S(X)π2 d(X) K2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − V 2(η) −2IT e2(X)µ(d(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)V (η) − 2IS I{A = d(X)} πS(X)πd(X) K(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X)V (η) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' where K(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) = ∆ y(U) SC(U | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) − µ(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) + � ∞ 0 dMC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) SC(u | A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X) Q(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Since we have that E � IS P(IS = 1) 2I{A = d(X)} πd(X) K(A, X)µ(d(X), X) � = 0, and for B ∈ � µ2(d(X), X), I{A = d(X)} π2 d(X) K2(A, X), µ(d(X), X)V (η), I{A = d(X)} π2 d(X) K(A, X)V (η) � , we also have that E � IS P(IS = 1)B � = E[IT e(X)B] = E � IS πS(X)B � , 65 we conclude that σ2 DR = σ2 CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' By the law of iterated expectations, the value function Vd = E[y(T(d))] = EX[E[y(T(d)) | X]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' When there is no restriction on the class of ITRs, the true optimal ITR is d∗∗(X) = arg max d Vd = arg max d EX[E[y(T(d)) | X]] = I{E[y(T(1)) | X] > E[y(T(0)) | X]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' That is, the optimal ITR does not depend on the covariate distributions, but only the bilp function which is the same in both the source and target populations by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Thus both the maximizers of ˆVDR(η) and ˆVCF(η) converge to the true population parameter η∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' However, ˆVDR(η) is biased since the expectation EX is taken with respect to the source population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' G Additional simulations We first investigate the performance of the cross-fitted ACW estimator with different sam- ple sizes (N, m) = (5 × 104, 2000), (1 × 105, 4000), (2 × 105, 8000), (4 × 105, 16000), (6 × 105, 24000), (8×105, 32000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Figure 4 and Table 3 report the results from 200 Monte Carlo replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The variance is computed using the EIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' H Details of real data analysis There are around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='6% missing values in the RCT and OS data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' We use the mice function in the R package mice (Van Buuren & Groothuis-Oudshoorn 2011) to impute the missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Motivated by the clinical practice and existing work in the medical literature, we con- sider ITRs that depend on the following five variables: 66 Figure 4: Boxplot of estimated value by ACW estimator with different sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 1 2 4 6 8 Target super population size Estimated Value (ACW) Table 3: Numeric results of the ACW estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Bias is the empirical bias of point es- timates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' SD is the empirical standard deviation of point estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' SE is the average of standard error estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' CP is the empirical coverage probability of the 95% Wald confi- dence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' m(×103) ∼ 780;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2 ∼ 1560;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 4 ∼ 3120;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 8 ∼ 6240;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 16 ∼ 9360;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 24 ∼ 12480;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 32 Bias 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0030 SD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0267 SE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='1611 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0284 CP(%) 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content='0 67 AGE, SEX and Sequential Organ Failure Assessment (SOFA) score: these three base- line variables are well related to mortality in ICUs, so we consider them as important risk factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' Acute Kidney Injury Network (AKIN) score: Jaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' (2018) observed that the infusion of sodium bicarbonate improved survival outcomes and mortality rate in critically ill patients with severe metabolic acidemia and acute kidney injury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' In the observational data, the AKIN score was not recorded, so we computed the score using serum creatinine measurement (Z´avada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' SEPSIS: we consider the presence of sepsis as a risk factor because it is the main condition associated with severe acidemia at the arrival in ICU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' The effect of sodium bicarbonate infusion on patients with acidema and acute kidney injury was also ob- served in septic patients (Zhang, Zhu, Mo & Hong 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} +page_content=' 68' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf'} diff --git a/yNAyT4oBgHgl3EQfa_ed/content/2301.00254v1.pdf b/yNAyT4oBgHgl3EQfa_ed/content/2301.00254v1.pdf new file mode 100644 index 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A +Missing Event Aware Temporal Graph Neural +Network +Mingyi Liu, Zhiying Tu Member, IEEE, Xiaofei Xu Member, IEEE, Zhongjie Wang Member, IEEE +Abstract—Temporal graph neural network has recently re- +ceived significant attention due to its wide application scenarios, +such as bioinformatics, knowledge graphs, and social networks. +There are some temporal graph neural networks that achieve +remarkable results. However, these works focus on future event +prediction and are performed under the assumption that all +historical events are observable. In real-world applications, events +are not always observable, and estimating event time is as +important as predicting future events. In this paper, we propose +MTGN, a missing event-aware temporal graph neural network, +which uniformly models evolving graph structure and timing +of events to support predicting what will happen in the future +and when it will happen. MTGN models the dynamic of both +observed and missing events as two coupled temporal point +processes, thereby incorporating the effects of missing events into +the network. Experimental results on several real-world temporal +graphs demonstrate that MTGN significantly outperforms exist- +ing methods with up to 89% and 112% more accurate time and +link prediction. Code can be found on https://github.com/HIT- +ICES/TNNLS-MTGN. +Index Terms—Temporal Graph Neural Network, Temporal +Point Process, Missing Events, Temporal Link Prediction, Event +Time Estimation +I. INTRODUCTION +G +Raph structured data has recently received significant +attention due to its wide application scenarios in various +domains such as social networks, knowledge graphs, and +bioinformatics. Graph neural networks (GNNs) are developed +to efficiently learn high-dimensional and non-Euclidean graph +information from graphs. Most existing GNNs [1], [2] are +designed for static graphs. In the real world, Graphs tend to +evolve continuously. For example, new friendships may be +established between people in a social network. Incorporating +dynamics into GNNs is a non-trivial problem. +Recently, a few temporal graph neural networks have been +developed. These methods can be classified as discrete tempo- +ral GNNs and continuous temporal GNNs based on how they +represent the temporal graph [3], [4]. The discrete temporal +GNNs [5], [6], [7] treat temporal graph as a sequence of +graph snapshots to simplify the model, which results in their +inability to capture the fully continuous evolution as fine- +grained temporal information is lost. As a result, the discrete +temporal GNNs can only predict what will happen in the future +Mingyi Liu, Zhiying Tu, Xiaofei Xu, and Zhongjie Wang are with the +Faculty of Computing, Harbin Institute of Technology, Harbin, China (e-mail: +liumy@hit.edu.cn, tzy hit@hit.edu.cn, xiaofei@hit.edu.cn, rainy@hit.edu.cn) +Manuscript received XXXXXX; XXXXXXXXX. Corresponding author: +Zhongjie Wang (email: rainy@hit.edu.cn). +1 +2 +3 +4 +1 +3 +4 +t1 +AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoseiF48V7Qe0oWy2m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6 +bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6xEnC/YgOlQgFo2ilB+x7/XLFrbpzkFXi5aQCORr98ldvELM04gqZpMZ0PTdBP6MaBZN8WuqlhieUjemQdy1VNOLGz+anTsmZVQYkjLUthWSu/p7IaGTMJApsZ0RxZJa9mfif10xvPYzoZIUuWKLRWEqCcZk9jcZCM0ZyoklGlhbyVsRDVlaNMp2RC85ZdXSeui6tWq +l/e1Sv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOk8+K8Ox+L1oKTzxzDHzifPwgwjaQ= +t2 +AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lKRY9FLx4r2lpoQ9lsN+3SzSbsToQS+hO8eFDEq7/Im/GbZuDtj4YeLw3w8y8I +JHCoOt+O4W19Y3NreJ2aWd3b/+gfHjUNnGqGW+xWMa6E1DpVC8hQIl7ySa0yiQ/DEY38z8xyeujYjVA04S7kd0qEQoGEUr3WO/1i9X3Ko7B1klXk4qkKPZL3/1BjFLI6QSWpM13MT9DOqUTDJp6VeanhC2ZgOedSRSNu/Gx+6pScWVAwljbUkjm6u+JjEbGTKLAdkYUR2bZm4n/ed0Uwys/EypJkSu2WBSmkmBMZn+TgdCcoZxYQpkW9lbCRlRThjadkg3BW35lbRrVa9evbirVxrXeRxFOIFTO +AcPLqEBt9CEFjAYwjO8wpsjnRfn3flYtBacfOY/sD5/AEJtI2l +further +AB9XicbVDLTgJBEJzF+IL9ehlIjHxRHYNRo9ELx4xkUcCSGaHXpgwO7uZ6VXJhv/w4kFjvPov3vwbB9iDgpV0UqnqTneXH0th0HW/ndzK6tr6Rn6zsLW9s7tX3D9om +CjRHOo8kpFu+cyAFArqKFBCK9bAQl9C0x9dT/3mA2gjInWH4xi6IRsoEQjO0Er3HYQnTINE4xD0pFcsuWV3BrpMvIyUSIZar/jV6Uc8CUEhl8yYtufG2E2ZRsElTAqdxEDM+IgNoG2pYiGYbjq7ekJPrNKnQaRtKaQz9fdEykJjxqFvO0OGQ7PoTcX/vHaCwWU3FSpOEBSfLwoSTGi0whoX2jgKMeWMK6FvZXyIdOMow2qYEPwFl9eJo2zslcpn9WStWrLI48OSLH5JR45IJUyQ2pkTrhRJNn8krenEfnxXl3PuatOSebOSR/4Hz+AHi/kyc= +Timeline +3 +who? +2 +4 +what time? +who should I +engage with at +what time? +Fig. 1. +A toy example showing the two core tasks and missing events +phenomenon in temporal graph. The solid lines are observed events and the +dash lines are missing events. +but cannot predict when. The continuous temporal GNNs treat +the temporal graph as a stream of events. The continuous +temporal GNNs can be roughly divided into temporal point +process (TPP) based methods [8], [9], [10] and non-TPP based +methods [11], [12], [13]. The non-TPP based methods encode +the event time as an auxiliary feature of the temporal graph +structure using RNNs or Transformers, with the consequence +that they still fail to predict the event time. TPP based methods +model the temporal graph as a temporal point process of events +and parameterize this process using a neural network. TPP can +naturally model the event time so that, theoretically, the TPP +based methods can predict the time of the event. +However, most TPP-based methods still treat event time +prediction as a “second-class citizen” (e.g., not modeled in +the optimization objective) to assist in predicting whether an +event will occur. We believe that predicting when an event will +occur is as important as predicting whether it will occur, e.g., a +task with a 2-hour deadline requires a different response than a +task with a 2-day deadline. In short, we believe that a temporal +GNN should be able to answer “Who should I engage with +at what tim” rather than just “Who should I engage with” (as +shown in Fig. 1). Predicting events and their corresponding +times in one model will broaden the application scenario of +temporal GNN. +Additionally, existing work is often working under the as- +sumption that all events in a temporal graph are fully observed, +but in many real-world scenarios, this assumption is difficult +to hold. We may miss observing events for a variety of reasons +(as the dash lines in Fig. 1). For example, the sensor may fail +at a certain time, and events are lost during that time; in social +networks, we may only observe interactions on Facebook, but +interactions on Twitter or offline are missing. Missing events +are part of temporal graphs and should be modeled in the +temporal GNNs to improve the performance of the temporal +GNNs. +Therefore, in this work, we propose MTGN, a missing event +arXiv:2301.08399v1 [cs.LG] 20 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +aware method that uniformly models evolving graph structure +and event time to support predicting what will happen and +when it will happen. MTGN is a TPP based method that +models the dynamics of both observed and missing events +as two coupled TPPs. Both the observation of events and the +generation of missing events depend on previously observed +events as well as previously generated missing events. +The main contributions are summarized as follows: +• We point out the importance of modeling the timing of +events and the missing events. We present a problem +formulation that unifies modeling missing events, the of +events, and evolving graph structure. +• We propose a novel method called MTGN. MTGN can +effectively learn node’s structure and temporal features +in a uniform embedding. Additionally, MTGN model the +dynamic of observed and missing events as two coupled +TPPs and parameterization with log-norm mixture dis- +tribution, which makes it can effectively estimate event +time. +• We conduct extensive experiments on five real-world +datasets to demonstrate the superiority of the proposed +method MTGN. +The remainder of this paper is organized as follows. Section +II discusses the related works. Section III presents related +preliminaries. Section IV gives a detailed interpretation of +MTGN. Section V shows all experimental results, as well +as detailed analysis and discussion. Section VI offers some +concluding remarks. +II. RELATED WORKS +A. Neural Temporal Point Process +A temporal point process [14] is a stochastic process +whose realizations consist of discrete events in time T += +{t1, t2, ..., tT }, ti ≤ ti+1. There are two common ways to +characterize a temporal point process: intensity-based methods +and intensity-free methods. +Intensity-based methods [15], [16], [17] are defined in terms +of the conditional intensity function λ(t), which specifying the +dependency of next arrival time t on the history Ht = {ti ∈ +T |ti < t}. With the given conditional intensity function λ(t), +we can obtain the conditional probability density function as +follows: +p(t) = λ(t) exp(− +� t +tT +λ(s)ds) +� +�� +� +survival function +(1) +where the survival function of the process [18] denotes that +no events happen during [tT , t). Intensity-based methods are +very flexible as λ(t) can choose different forms (e.g., Poisson, +Renewal, Self-correcting, Hawkes, etc.) to capture different +interests in different scenarios. However, the drawback of the +intensity-based methods are also obvious, as the integral term +involved in the survival function leads to there is no closed +form for Eq. (1), and thus requiring Monte Carlo integration. +To reduce the computationally expensive of intensity-based +methods, intensity-free methods do not directly model the +intensity function. For example, Omi et al. [19] introduce +a fully neural network (FullyNN) to model the cumulative +intensity function Λ(t) = +� t +tT λ(s)ds. Shchur et al. [20] +directly estimate conditional density p(t) by utilizing neural +density estimation [21]. Similar to [20], Gupta et al. [22] also +directly estimate conditional density, and they first introduce +missing events in modeling temporal point process. Inspired +by [22], we introduce the missing events into temporal graph. +For readers who want to learn more about neural temporal +point process, we recommend the survey conducted by Shchur +et al. [23], which provides more details about the background, +models, application scenarios and limitations of using neural +networks to solve temporal point process related problems. +B. Temporal Graph Neural Networks +1) Discrete Temporal Graph Neural Networks: Discrete +temporal graph neural networks treat temporal graphs as +a sequence of graph snapshots[24]. Most discrete dynamic +embedding approaches[25], [26], [27], [28], [29] focus on +the learning representations of entire temporal graphs rather +than node representations. Some approaches[30], [31], [32], +[33], [34] are starting to focus on the dynamic representation +at node level, they encode each graph snapshot using static +embedding approaches[35], [36], [37] to embed each node, +and then combines some time-series models (e.g. LSTM[38], +RNN[39]) for per node to model the discrete dynamic. +The main disadvantages of discrete temporal graph neural +networks are twofold: +• Graph snapshots are generated by aggregating events over +a period of time, and a large amount of fine-grained +temporal information is lost in the aggregation process. +• Discrete temporal GNNs simplify the temporal features. +For example, they treat the temporal graph as a sequence +of graph snapshots consisting of multiple time steps, +while this ignores the fact that the time intervals between +adjacent snapshots may be vary. +These two drawbacks result in discrete temporal GNNs failing +to predict event times and generally performing worse than +continuous temporal graph neural networks. +2) Continuous Temporal Graph Neural Networks: Contin- +uous temporal graph neural networks treat temporal graphs as +a stream of observed events. These methods can be roughly +divided into two main categories: TPP based methods and non- +TPP based methods[3]. +For the non-TPP based methods, the embedding of in- +teracting nodes is updated by an RNN/transformer based +architecture according to the historical information. Represen- +tative works of this type of method are JODIE[40], TGN[41], +APAN[11] and Streaming GNN[12]. JODIE is designed for +user-item networks and uses two RNNs to maintain the embed- +ding of each node. With one RNN for users and another one +for items. Instead of keeping the embedding of nodes directly, +TGN calculates the embedding of nodes at different times by +introducing message and memory mechanisms. These methods +encode the event time as an auxiliary feature of temporal graph +structure, which helps overcome the drawbacks of discrete +temporal GNNs but leads to failing to predict the event time. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +Know-Evolve[42] is the pioneer in bringing the temporal +point processes to dynamic graph representation learning, +which models temporal knowledge graph as multi-relational +timestamped edges by parameterizing a TPP by a deep recur- +rent architecture. DyRep[10] is the successor of Know-Evolve. +DyRep extends Know-Evolve using TPP to model long-term +events and short-term events and introduce aggregation mech- +anisms. LDG[43] argues long-term events are often specific by +humans and can be suboptimal and expensive to obtain. LDG +uses Neural Relational Inference (NRI) model to infer the type +of events on the graph and replaces the self-attention originally +used in DyRep by generating a temporal attention matrix to +better aggregate neighbor information. GHN[44] is another +TPP based approach, which uses an adapted continuous-time +LSTM for Hawkes process. TREND[8] is a Hawkes process +based approach, which captures the individual and collective +characteristics of events by integrating event and node dynam- +ics. Theoretically, these TPP based methods can predict the +time of events. However, most of them still treat event time +prediction as a “second-class citizen” to assist in predicting +events, e.g. event time is not included in optimization objective +and no closed-form event time exception is provided. For a +very recent work, EvoKG[9], which jointly model the evolving +graph structure and timing of events have achieved state-of- +the-art performance. However, EvoKG learns structural and +temporal embeddings separately, which limits its performance +and robustness (detailed discuss in Section V-D2). +Finally, it should be noted that all the temporal GNNs +mentioned above do not model missing events. +III. PRELIMINARIES +A. Notations +A temporal graph G can be represented by a sequence of +|E| discrete events {ei = (ui, vi, ti, ki)}|E| +i=1}, where ui and +vi are two nodes involved in the event ei. ti is the time of the +event, and ti ≤ tj ⇔ i < j. ki ∈ {0, 1} and ki = 1 represents +the event ei is observed and ki = 0 represents the event ei +is missed1. For convenience, we omit k and use O(M) to +denotes observed (missing) temporal graph, o = (u, v, t)(m = +(u, v, t)) to denotes a observed(missing) event. We preserve +boldface lowercase letters (e.g., u) for vectors and boldface +capitals for matrices (e.g., W). Frequently-used notations has +been summarized in Appendix A. +B. Problem Definition +Given a temporal graph G = O ∪M denoted as a sequence +of events, our goal is to model the probability distribution +p(G). We assume that both observed and missing events at +time t depend on the history of previously generated missing +events and observed events. As in practice, only the observed +1In this paper, while we focus on undirected graph without attributes, but +it’s easy to be generalized to directed graphs with attributes +events O can be evaluated for validation, and missing events +are intractable, we can model the probability distribution as: +pθ(OT ) = +T +� +t=1 +� +Mt +pθ(Ot|G∗ +¯t , Mt)pθ(Mt)dMt += Eqφ +T +� +t=1 +pθ(Ot|G∗ +¯t , Mt)pθ(Mt|G∗ +¯t ) +qφ(Mt|Ot, G∗ +¯t ) +(2) +where ¯t is timestamp of last observed event, and we assume +events happen at same time interval (¯t, t] are independent of +each other, then we have: +pθ(Ot|G∗ +¯t , Mt) = +� +(u,v,t)∈Ot +pθ(u, v, t|G∗ +¯t , Mt) +(3) +pθ(Mt|G∗ +¯t ) = +� +(u,v,t)∈Mt +pθ(u, v, t|G∗ +¯t ) +(4) +qφ(Mt|Ot, G∗ +¯t ) = +� +(u,v,t)∈Mt +qφ(u, v, t|Ot, G∗ +¯t ) +(5) +where G∗ +¯t denotes all observed and generated missing events +until time ¯t; Ot and Mt are observed and generated missing +events at time interval (¯t, t], and for convince, we let the +time of the missing event with the same subscript lag slightly +behind that of the observed event; qφ(·) is a variational +approximation posterior distribution, which aims to generate +missing events Mt within the interval (¯t, t), based on the +historical events as well as the further observed events Ot. +Then our goal is to maximize the marginal log-likelihood +of observed events, which can be achieved by maximizing the +following evidence lower bound (ELBO) of the log-likelihood +of observed events: +L(θ, φ; OT ) = Eqφ +T +� +t=1 +� +(u,v,t)∈Ot +log pθ(u, v, t|G∗ +¯t , Mt) +� +�� +� +LO +− +T +� +t=1 +� +(u,v,t)∈Mt +KL(qφ(u, v, t|Ot, G∗ +¯t )||pθ(u, v, t|G∗ +¯t )) +� +�� +� +LM +(6) +where KL(·) is the Kullback-Leibler divergence. Similar +to [9], we further decompose the event joint conditional +probability in Eq. (6) as follows: +pθ(u, v, t|G∗ +¯t , Mt) = pθ(u, v|G∗ +¯t , Mt)pθ(t|u, v, G∗ +¯t , Mt) (7) +pθ(u, v, t|G∗ +¯t ) = pθ(u, v|G∗ +¯t )pθ(t|u, v, G∗ +¯t ) +(8) +qφ(u, v, t|Ot, G∗ +¯t ) = qφ(u, v|Ot, G∗ +¯t )qφ(t|u, v, Ot, G∗ +¯t ) +(9) +Then, by chain rule for relative entropy, the KL divergence +term in Eq. (6) can be denoted as: +KL(qφ(u, v, t|Ot, G∗ +¯t )||pθ(u, v, t|G∗ +¯t )) = +KL(qφ(u, v|Ot, G∗ +¯t )||pθ(u, v|G∗ +¯t )) ++ KL(qφ(t|u, v, Ot, G∗ +¯t )||pθ(t|u, v, G∗ +¯t )) +(10) +Above equations suggest the components of our model from +two perspectives corresponding to the challenges mentioned +in Introduction. From the event perspective, the ELBO in +Eq. (6) suggests our model consist of the three components: + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +one TPP for observed events (pθ(u, v, t|G∗ +¯t , Mt)), one prior +TPP for missing events (pθ(u, v, t|G∗ +¯t )) and one posterior +TPP for missing events (qφ(u, v|Ot, G∗ +¯t )). From the temporal +graph perspective, Eq.(6) - Eq.(8) suggest our model con- +sist of two components: graph structure modeling compo- +nents (pθ(u, v|G∗ +¯t ), pθ(u, v|G∗ +¯t ) and qφ(u, v|Ot, G∗ +¯t )) and event +time modeling components (pθ(t|u, v, G∗ +¯t ), pθ(t|u, v, G∗ +¯t ) and +qφ(t|u, v, Ot, G∗ +¯t )). +IV. MODELING A TEMPORAL GRAPH +Modeling a temporal graph is equivalent to modeling each +observed and missing event in the temporal graph, i.e., Eq.(7)- +Eq. (9). To model these events, we learn the observed (missing) +embeddings of nodes. Unlike EvoKG[9], which learns the +structural embeddings and temporal node embeddings sep- +arately, we learn the time-evolving structural dynamics and +temporal characteristics of nodes in unified embeddings. The +unified embeddings make the model treat events as a whole, +allowing the model to have higher consistency in predicting +link and predicting time, i.e., the model can obtain optimal +link prediction performance and time prediction performance +with the same set of parameters. +We utilize message passing framework to learn node +observed (missing) embeddings. Given concurrent observed +events Ot, we summarise node u’s observed embedding as +follows: +ol+1,t +u += Wl +sol,t +u + +1 +|N Ot +u | +� +(u,v,t)∈Ot +Wl +nol,t +v + Wl +t(t − ¯to +u,v) +(11) +where ol+1,t +u +is the observed embeddings of node u learned +by l-th layer of GNN and o0,t +u +is set to static embedding ou; +N Ot +u +is node u’s neighbors in Ot and ¯to +u,v = max(¯to +u, ¯to +v) is +the last time of node u or node v involved in an observed +event. Wl +• are learnable weights in the l-th layer GNN. +Specially, W• +s models the self-evolution, which indicates a +node evolves with respect to its previous state; W• +n models the +structural message passing from neighborhoods; W• +t models +the event time impact. So that Eq. (11) encodes the structural +and temporal information of the observed events in a uniform +embedding. +For the generated missing events Mt (the generation algo- +rithm is described in Section IV-D), we summarise node u’s +missing embedding as follows: +ml+1,t +u +=Ul +sml,t +u + +1 +|N Mt +u +| +� +(u,v,t′)∈Mt +Ul +nml,t +v +Ul +t(t′−¯tm +u,v) (12) +Eq. (12) is similar to Eq. (11), except that the generated +missing event m1 and m2 may have different time t′ +1 and +t′ +2 (¯t < t′ +1 ≤ t, ¯t < t′ +2 ≤ t ). ¯tm +u,v = max(¯tm +u , ¯tm +v ) is the last +time of node u or node v involved in an generated missing +event. +oL,t +u +(mL,t +u +)only summarizes the interaction patterns in +concurrent events Ot(Mt). To further model the dynamic of +temporal updates, we utilize GRU [45] to capture the temporal +dependency among node embeddings: +o∗,t +u += GRU(oL,t +u , o∗,¯t +u ) +(13) +m∗,t +u += GRU(mL,t +u , m∗,¯t +u ) +(14) +g∗,t +u += [o∗,t +u ; m∗,t +u ] +(15) +where o∗,t +u +and m∗,t +u +are observed evolving-embedding and +missing evolving-embedding of node u. g∗,t +u +is the node u’s +evolving-embedding. [; ] is a concatenation operation. +To enhance the structural information in the node evolve- +embeddings, we concatenate the static embeddings and +evolving-embeddings as follows: +¯ot +u = [ou; o∗,t +u ] +¯mt +u = [mu; m∗,t +u ] +¯gt +u = [¯ot +u; ¯mt +u] +(16) +Graph-level embeddings on observed events, missing events +and all events are obtained by using element-wise max pool- +ing: +¯ot = max({¯ot +u|u ∈ nodes(O∗ +¯t )}) +(17) +¯mt = max({ ¯mt +u|u ∈ nodes(M∗ +¯t )}) +(18) +¯gt = max({¯gt +u|u ∈ nodes(G∗ +¯t )}) +(19) +Based on the above learned embeddings, we give detail +description about parameterization of three temporal point +processes in Section IV-A - Section IV-C. +A. Parameterization of TPP for observed events +Based on Eq. (7), we first parameterize the graph structure +and then parameterize the event time. +For the graph structure part, it is not practical to direct +parameterize the event (u, v), we decompose pθ(u, v|G∗ +¯t , Mt) +as follows: +pθ(u, v|G∗ +¯t , Mt)=pθ(u|G∗ +¯t , Mt)pθ(v|u, G∗ +¯t , Mt) +(20) +and parameterize each component separately: +pθ(u|G∗ +¯t , Mt) = softmax(MLP([¯gt])) +(21) +pθ(v|u, G∗ +¯t , Mt) = softmax(MLP([¯gt +u; ¯gt])) +(22) +where MLP is the abbreviation of multilayer perceptron. +For the event time part, follow [20], [9], we directly +model event time’s probability conditional function (PDF) +p(t) rather than the conditional intensity function λ(t), which +makes MTGN more flex and have a closed-form likelihood +and expectation. We first conduct the context vector for the +observed event as follows: +c = [g∗,t +u ; g∗,t +v ] +(23) +and then obtain the K-dimension (K mixture component) +params of the log-norm mixture distribution as follows: +ω=softmax(MLP(c)) +µ=MLP(c) +σ= exp(MLP(c)) +(24) +With the above-mentioned params, the PDF of the time τ = +t − ¯to +u,v as follows: +pθ(τ|u, v, G∗ +¯t , Mt) = pθ(τ|ω, µ, σ) += +K +� +k=0 +ωk +τσk +√ +2π exp(−(log τ − µk)2 +2σk +) +(25) + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +It is reasonable to model τ rather than directly model t, as t +can be very large thus making the neural network confused. +B. Parameterization of prior TPP for missing events +For the graph structure part, same as observed events, we +decompose pθ(u, v|G∗ +¯t ) as follows: +pθ(u, v|G∗ +¯t ) = pθ(u|G∗ +¯t )pθ(v|u, G∗ +¯t ) +(26) +and parameterize each component separately: +pθ(u|G∗ +¯t ) = softmax(MLP([¯g¯t])) +(27) +pθ(v|u, G∗ +¯t ) = softmax(MLP([¯g¯t +u; ¯g¯t])) +(28) +For the event time part, we also use log-norm mixture +distribution to model missing event time’s PDF. We first +conduct the context vector cθ = [g∗,¯t +u ; g∗,¯t +v ] and obtain the +params of the log-norm mixture params ωθ, µθ, σθ same as +Eq. (24) with cθ as input. Then the PDF of time ∆ = t′ − ¯t +as follows: +pθ(∆|u, v, G∗ +¯t ) = pθ(∆|ωθ, µθ, σθ) += +K +� +k=0 +ωθ +k +∆σθ +k +√ +2π exp(−(log ∆ − µθ +k)2 +2σθ +k +) +(29) +Model time ∆ makes true the missing event time is large than +the latest observed time ¯t. +C. Parameterization of posterior TPP for missing events +For the graph structure part, we decompose qφ(u, v|G∗ +¯t , Ot) +as follows: +qφ(u, v|G∗ +¯t , Ot)=qφ(u|G∗ +¯t , Ot)qφ(v|u, G∗ +¯t , Ot) +(30) +and then parameterize each component separately: +qφ(u|G∗ +¯t , Ot) = softmax(MLP([¯g¯t; ¯ot])) +(31) +qφ(v|u, G∗ +¯t , Mt)=softmax(MLP([¯g¯t +u; ¯g¯t; ¯ot +u; ¯ot])) +(32) +For the event time part, we conduct the context vector cφ = +[g∗,¯t +u ; g∗,¯t +v ; o∗,t +u ; o∗,t +v ] and obtain the params of the log-norm +mixture params ωφ, µφ, σφ same as Eq. (24) with cφ as input. +Then we have the density of the inter-arrival time ∆ as: +qφ(∆|u, v, G∗ +¯t , O∗ +t ) = qφ(∆|ωφ, µφ, σφ) ⊙ I(∆ 10000); +4 +bool flip = false; +5 +if (msg.sender == address(0)) { +6 +{ +while (gasleft() >= 5000) +7 +{ flip = !flip; } } +8 +if (gasleft() < 5000) {test = flip;} } } +Fig. 7. Securify counterexample: gas handling +satisfies the RW property. Still, Securify reports a violation, +since no dependencies between the input to the call and the +call output are modeled. +e) Gas handling: Figure 7 shows a contract that indi- +rectly restricts write access to storage test by consuming the +gas resource in a controlled way. In line 3, the contract ensures +that it is executed with a generous amount of gas; if not enough +gas is available, the execution is aborted and no caller is able +to write to test. The code between lines 5 and 7 essentially +wastes masses of gas if the caller address is equal to 0, and, +otherwise, consumes very little gas. The crux of the contract +is in line 8: From the amount of gas that is left, the contract +can determine if the caller’s address is equal to 0—this is the +case if and only if less than 5000 gas units are left. Hence, +depending on the amount of available gas, either no caller or +only caller 0 can write to test. So, there is always at least +one caller that cannot write to test—the contract satisfies the +RW property. However, Securify reports a violation of this +property. The reason for this wrong analysis result is that +Securify does not track dependencies for the gas resource. +2) Sound Security Properties: Since the dependency predi- +cates do not have a semantic characterization, the soundness of +the security patterns w.r.t. their corresponding property cannot +be proven. Indeed, Schneidewind et al. [24] provide counter +examples for the soundness of 13 out of the 17 security +patterns given in [27]. Above that, the unsoundness of the +RW property undeniably manifests in line 4 of the contract +we constructed in Figure 5. For this example, Securify reports +simultaneously(!) satisfaction of a compliance and a violation +pattern for the RW property w.r.t. a. This refutes the claim that +compliance and violation patterns constitute sufficient criteria +for property compliance and violation, respectively. +V. ANALYSIS FOUNDATIONS +To design a sound static analysis for EVM bytecode based +on program slicing, we instantiate the slicing proof framework +from [30] with a formal bytecode semantics as defined in [14]. +Before discussing the instantiation in Section VI, we shortly +overview both frameworks. +A. EVM bytecode semantics +The EVM semantics was formally defined in [14] in form of +a small-step semantics. We use a linearized representation of +the semantics inspired by Securify, where the use of the stack +is replaced by the usage of local variables in SSA form. We +will call these variables stack variables and, in the following, +always refer to the linearized representation of the semantics. +Formally, the semantics of EVM bytecode is given by a +small-step relation Γ ⊨ S → S′. The relation describes how a +contract, whose execution state is given by a callstack S, can +progress to callstack S′ under a transaction environment Γ. +The transaction environment Γ holds information about the ex- +ternal transaction that initiated execution. We let Γ ⊨ S →∗ S′ +denote the reflexive transitive closure of the small-step relation +and call the pair (Γ, S) a configuration. The details of the +components of the EVM configurations can be found in [14]. +The overall state of an external transaction execution is +captured by a callstack S. The elements of the callstack +model the states of all (pending) internal transactions. In- +ternal transactions can either be pending, as indicated by a +regular execution state (µ, ι, σ), or terminated. The state of +a pending transaction encompasses, the current global state +σ, the execution environment ι and the machine state µ. +The global state σ describes the state of all accounts of the +system and is defined as a partial mapping between account +addresses and account states. The execution environment ι, +among others, contains the code of the currently executing +contract. We model the code of a contract as a function C that +maps program counters to tuples (op(⃗x), pcnext, pre), where +op denotes an opcode from the EVM instruction set, ⃗x is the +vector of input and output (stack) variables to this opcode, and +pcnext denotes the program counter for the next instruction. +Further, we instrument each instruction with a list pre of +precomputed values for the arguments ⃗x. This instrumentation +is only introduced for analysis purposes and does not affect +the execution. +The machine state µ captures the state of the local machine +and holds the amount of gas (g) available for execution, the +program counter (pc), the local memory, and the state of the +(linearized) stack variables (s). +a) Small-step Rules: We illustrate the working of the +EVM bytecode semantics using the example of the ADD +instruction. This instruction takes two values as input and +writes their sum back to its return variable. +ι.code [µ.pc] = (ADD(r, a, b), pcnext, pre) +µ.g ≥ 3 +µ′ = µ[s → µ.s[r → µ.s(a) + µ.s(b)]][pc → pcnext][g −= 3] +Γ ⊨ (µ, ι, σ) :: S +ADD(a,b) +−−−−−→ (µ′, ι, σ) :: S +Given a sufficient amount of gas (here 3 units), an ADD +instruction with result (stack) variable r and operand (stack) +variables a and b writes the sum of the values of a and b to +r and advances the program counter to pcnext. These effects, +as well as the subtraction of the gas cost, are reflected in the +updated machine state µ′. +b) Security properties: Previous work [14] has shown +that there are several generic smart contract security properties, +which are desirable irrespective of the individual contract +logic. The properties formally defined in [14] are integrity +properties that aim at ruling out the influence of attacker +behavior on sensitive contract actions, in particular, the spend- +ing of money. These properties are e.g., the independence +of a contract’s spending behavior from miner-controlled pa- +rameters (as the block timestamp) or mutable contract state. + +Further, [14] introduces the notion of call integrity, which +requires that the spending behavior of a contract is independent +of the code of other smart contracts. Since call integrity is +hard to verify in the presence of reentering exeutions, a proof +strategy is devised that decomposes call integrity into one +reachability property (single-entrancy) that restricts reentering +executions and two local dependency properties. These local +dependency properties ensure that the spending behavior of the +contract does not depend on the return effects of calls to other +(unknown) contracts (effect independence) or immediately on +the code of such contracts (code independence). +Focussing on integrity, the security properties from [14] +are given as non-interference-style notions. We illustrate this +with the example of timestamp independence, a property that +requires that the block timestamp cannot influence a contract’s +spending behavior and hence would rule out vulnerabilities as +those in the roulette example: +Definition 1 (Independence of the block timestamp). A con- +tract C is independent of the block timestamp if for all +reachable configurations (Γ, sC :: S) it holds for all Γ′ that +Γ =/timestamp Γ′ ∧ Γ ⊨ sC :: S +π−→ +∗ s′ +C :: S ∧ final (s′) +∧ Γ′ ⊨ sc :: S +π′ +−→ +∗ +s′′ +C :: S ∧ final (s′′) =⇒ π ↓callsC= π′ ↓callsC +This definition requires that two executions of the contract +C starting in the same execution state sC and in transaction +environments Γ and Γ′ that are equal up to the block timestamp +(denoted by Γ =/timestamp Γ′) exhibit the same calling behavior +(captured by the call traces π ↓callsC). Intuitively, this ensures +that the contract C may not perform different money transfers +based on the block timestamp. The roulette example trivially +violates this property since, based on the block timestamp, the +prize will be paid out to a different user. +B. Program Slicing +Static program slicing is a method for capturing the de- +pendencies between different program points (nodes) and +variables in a program. Intuitively, the program slice of some +program node n in a program P consists of all those nodes n′ +in P that may affect the values of variables written in n. Pro- +gram slices are constructed based on the program dependence +graph (PDG) that models the control and data dependencies +between the nodes of a program. In the following, we will +review the static slicing framework by Wasserraab et al. [30], +which establishes a language-independent correctness result +for slicing based on abstract control flow graphs (CFGs). +a) Abstract control flow graph: An abstract CFG is a +language-agnostic representation of program semantics. Tech- +nically, an abstract CFG is parametrized by a set of program +states Θ and defined by a set of nodes (representing program +points) and a set of directed edges between nodes. Edges may +be of two different types: State-changing edges n− ⇑f −→ n′ +alter the program state θ ∈ Θ by applying the function f to +θ and predicate edges n −(Q)√ −→ n′ guard the transition +between n and n′ with the predicate Q on the program state +θ. We write n +as +−→ +∗ n′ to denote that node n can be reached +n′ using the edges in the list as. Abstract CFG edges can +be related to actual runs of the program by lifting them to a +small-step relation of the form ⟨n, θ⟩ −a−→ ⟨n′, θ′⟩. +b) PDG and backward slices: The PDG for a program +consists of the same nodes as the CFG for this program and +has edges that indicate data and control dependencies. To make +data dependencies inferable, each node n is annotated with a +set of variables that are written (short Def set, written Def(n)) +and a set of variables that are read by the outgoing edges of +the node (short Use set, written Use(n)). A node n′ is data +dependent on node n (written n −→dd n′) if n defines a variable +Y (Y ∈ Def(n)), which is used by n′ (Y ∈ Use(n′)) and n′ +is reachable from n in the CFG without passing another node +that defines Y . A node n′ is (standard) control dependent on +node n (written n −→cd n′) if n′ is reachable from n in the +CFG, but n can as well reach the program’s exit node without +passing through n′ and all other nodes on the path from n +to n′ cannot reach the exit node without passing through n′. +So intuitively, n is the node at which the decision is made +whether n′ will be executed or not. Based on the data and +control flow edges of the PDG, the backward slice of a node +n (written BS(n)) is defined as the set of all nodes n′ that can +reach n within the PDG. +c) Correctness statement: The generic correctness state- +ment for slicing proven in [30] is stated as follows: +Theorem 1. Correctness of Slicing Based on Paths [30] +⟨n, θ⟩ +as +−→ +∗ ⟨n′, θ′⟩ +∃ as′. ⟨n, θ′⟩ +as′ +−→ +∗ +BS(n′) ⟨n′, θ′′⟩ ∧ as ↓BS(n′)= as′ +∧ (∀ V ∈ Use(n′).θ′(V ) = θ′′(V )) +Intuitively, the theorem states that whenever a node n can +reach some node n′ in the PDG (⟨n, θ⟩ +as +−→ +∗ ⟨n′, θ′⟩), then +removing all outgoing edges from nodes not in the backward +slice of n′ (⟨n, θ⟩ +as′ +−→ +∗ +BS(n′) ⟨n′, θ′′⟩) without altering the +path through the PDG in any other way (as ↓BS(n′)= as′) +has no impact on n′. Having no impact on n′ means that +variables used in n′ are assigned to the same values re- +gardless of whether the edges have been removed or not +(∀V ∈ Use(n′). θ′(V ) = θ′′(V )). We call the PDG without +the above-mentioned edges also sliced PDG or sliced graph. +VI. SOUND EVM DEPENDENCY ANALYSIS +In the following, we instantiate the slicing proof frame- +work [30] to accurately capture program dependencies of +EVM smart contracts in terms of program slices. We then +give a logical characterization of such program slices, which +allows for the automatic computation of dependencies between +different program points and variables with the help of a +Datalog solver. The generic correctness statement of the slicing +proof framework guarantees that the slicing-based dependen- +cies soundly over-approximate all real program dependencies. +We show how to use this result to automatically verify relevant +smart contract security properties such as the independence of +the transaction environment and the independence of mutable +account state as defined in [14]. + +A. Instantiation of Slicing Proof Framework +We instantiate the abstract CFG from the slicing framework +with the linearized EVM semantics. +The concrete layout of the instantiation heavily influences +the resulting backward slices and the precision of the analysis. +In the following, we sketch the most interesting aspects of our +instantiation of the CFG components and how they contribute +to the design of a precise dependency analysis. +Preprocessing Information: For a precise analysis, it is +indispensable to preprocess contracts to aggregate as much +statically obtainable information as possible—without com- +promising the soundness of the overall analysis. For example, +knowing the precise destination of jump instructions is crucial +to reconstruct control flow precisely, and, moreover, this +information usually can be easily reconstructed, especially, +when contracts were compiled from a high-level language with +structured control flow. +We require in the following that the preprocessed informa- +tion is correct: +Definition 2 (Sound Preprocessing). A contract C has sound +preprocessing information if for all execution states sC with +an initial machine state running contract C it holds that if +Γ ⊨ sC :: S →∗ s′ +C :: S then +C(s′.µ.pc) = (op(⃗x), pcnext, pre) +⇒ ∀i ∈ [0, |⃗x| − 1]. pre[i] = ⌊µ.s(xi)⌋ ∨ pre[i] = ⊥ +In the remainder, we assume that all existing preprocess- +ing information is correct and sufficient to reconstruct the +contract’s CFG. Recall that, formally, we consider a contract +a function, such that for a program counter pc, C(pc) = +(op(⃗x), pcnext, pre) where pre contains the preprocessing infor- +mation for the instruction op(⃗x): for every ⃗x[i], pre[i] either +holds a precomputed static value, or ⊥ to indicate that no static +value could be inferred. Note that we restrict preprocessing +to stack variables. For our analysis, we are only interested +in precomputed values for memory and storage locations and +jump destinations. +CFG States: The edges of the CFG are labeled with +state-changing functions or predicates on states. For EVM +bytecode programs, the CFG state θ is partitioned into stack +variables (denoted by xls), memory variables (xm), storage +variables (xg) and local (xel) and global (xeg) environmental +variables. Memory and storage variables represent cells in +the local memory, respectively the global storage of the +contract under analysis. Local environment variables contain +the information of the execution environment that is specific +to an internal transaction. Global environmental variables +denote environmental information whose accessibility is not +limited to a single internal transaction, like the state of other +contracts and the block timestamp. Environmental information +that cannot be directly accessed during the execution (such as +the storage of other contracts) is hidden in the dedicated global +environmental variable externaleg. +CFG Nodes, Edges & Def and Use Sets: To transform an +EVM bytecode program into a CFG, we map every program +counter pc to one or more nodes (pc, i) in the CFG (where +C(pc) = (JUMPI(x1 +ls, x2 +ls), pcnext, pre) +f = (λθ.θ ← ge := θ[ge] − 10) +C, cd ⊨ (pc, 0) −⇑f − +→ (pc, 1) +Def = {ge} +Use = {ge} +C(pc) = (JUMPI(x1 +ls, x2 +ls), pcnext, pre) +Q = (λθ.θ[x2 +ls] = 0) +C, cd ⊨ (pc, 1) −(Q)√ − +→ (pcnext, 0) +Def = ∅ +Use = {x2 +ls} +Fig. 8. JUMPI abstract CFG instantiation +i ∈ N is used to distinguish between multiple nodes for +pc). We call a node (pc, 0) initial node (for pc) and nodes +(pc, i) with i > 0 intermediate nodes (for pc). Since the +size of the callstack below the translated callstack element +may influence the contract execution, the rule set defining +the CFG transformation constructs a relation of the form +C, cd ⊨ (pc, i) −a −→ (pc′, i′), where C is the contract for +which the CFG is constructed, cd is the size of the callstack, +and a stands for either a (Q)√ action (for a predicate edge) or +⇑f action (for a state-changing edge). With every rule, we also +provide Def and Use sets. The Use sets contain all variables +whose values are retrieved from the state θ in the definition +of the Q predicate or f function. Similarly, the definition set +contains all variables that are overwritten by the function f +(and is always empty for predicate edges). +Figure 8 shows two exemplary rules for the conditional +jump instruction JUMPI. The first argument to JUMPI is the +jump destination and the second argument is the condition +variable that must be non-zero for the jump to happen. We +only show rules for the case that the condition is false, i.e., the +jump does not happen. The upper rule defines a state-changing +edge that deducts the gas that has to be paid for a JUMPI +instruction. Appropriately, both Def and Use sets contain the +gas variable because the current gas value must be read from +and the reduced value updated in state θ. Note that the edge +goes from the initial node for pc to an intermediate node for +pc, because a second step is necessary to decide whether the +program should jump. The second step, depicted by the lower +rule, continues in the intermediate node for pc and checks if +the condition (in variable x2ls) is false (i.e., if it is zero) via +a predicate edge. In this case, the execution proceeds to the +initial node representing pcnext. x2ls is the only variable used +by Q, hence it is the only variable in the Use set. +It can be shown that the CFG semantics and EVM semantics +coincide via two simulation relations where every (multi-)step +in the CFG semantics between initial nodes is simulated by a +step of the bytecode semantics and vice versa. +B. Core abstractions +We review the most interesting aspects of the CFG seman- +tics and how they lead to a precise dependency analysis. In +this course, we will show how to overcome the challenges +presented in Section IV. +a) Gas abstraction: In the EVM, the execution of in- +structions consumes gas. If the gas is not sufficient to finish + +C(pc) = (ADD(yls, x1 +ls, x2 +ls), pcnext, pre) +f = (λθ.θ ← ge := θ[ge] − 3) +C, cd ⊨ (pc, 0) −⇑f − +→ (pc, 1) +Def = {ge} +Use = {ge} +C(pc) = (ADD(yls, x1 +ls, x2 +ls), pcnext, pre) +f = (λθ.θ ← yls := θ[x1 +ls] + θ[x2 +ls]) +C, cd ⊨ (pc, 1) −⇑f − +→ (pcnext, 0) +Def = {yls} +Use = {x1 +ls, x2 +ls} +Fig. 9. CFG semantics rules for the ADD instruction. +the execution of a contract, it is aborted with an exception. +Modeling this behavior accurately would result in a very +imprecise analysis, since, technically, every instruction would +be control-dependant on all its preceding instructions. This is +as the execution of an instruction depends on whether prior in- +structions led to an out-of-gas exception. However, in practice, +users should only call contracts with a sufficient amount of gas +since, otherwise, the contract execution exceptionally halts. +For this reason, there exist static analysis tools for computing +(sound) gas bounds [4] and even Solidity’s online compiler +provides gas estimates for smart contract execution. +Hence, for our analysis we assume that a contract does not +run out of gas and do not model the corresponding behavior in +the CFG semantics. We remark that Securify also makes this +assumption implicitly; we spell it out explicitly as follows: +Assumption 1 (Absence of local out-of-gas exceptions (infor- +mal)). A contract execution does not exhibit local-out-of-gas +exceptions if each local exception can be attributed to the +execution of an INVALID opcode. +In contrast to Securify, we do not ignore gas entirely, +but model the gas reduction for all instructions. This allows +capturing dependencies such as the one highlighted in Figure 7 +(and missed by Securify). In the CFG, we always model the +gas reduction as a separate edge involving an intermediate +node (e.g., with the upper rule in Figure 8). The Def set of +one node contains only the gas variable, while the Def set +of the other node only contains the (stack) variables involved +in the actual instruction. An example for that is given by the +(simplified) CFG rules of the ADD instruction in Figure 9. +Technically, an ADD instruction performs two types of state +updates: it decreases the gas and performs addition on stack +variables. Since those two state updates are independent, +their execution can be split into two different nodes. As a +consequence, the node (pc, 1) is not data-dependent on nodes +writing the gas variable. +Still, the gas abstraction is sound (under Assumption 1) and +correctly captures the dependencies of the example in +Fig- +ure 7: Figure 10 shows an incomplete and simplified CFG of +the example in Figure 7 with annotated Def and Use sets. The +example illustrates how the CFG captures the dependency of +the storage write (test = flip) on the msg.sender variable. The +storage write in 6 is control dependant on the conditional y2 +in 5 , and 5 depends on node 4 where y2 is defined. 4 +1 +2 +4 +5 +6 +7 +3 +{y1} +{msg.sender} +{y1} +{gas} +{gas} +{y2} +{gas} +{y2} +{test} +{flip} +y1 = msg.sender == address(0); +if (y1){ + while (gasLeft() >= 5000){ +flip = !flip; +} +y2 = gasLeft() >= 5000; +if (y2){ + test = flip; } +} +1 +2 +4 +5 +6 +7 +3 + ⇒  +Fig. 10. Example control flow with gas dependencies. Def sets are given at +the left of each node, Use sets at the right. Data dependencies are indicated +by black arrows, control dependencies by orange ones. +m[x] = msg.sender; +y1 = m[0]; +if (y2){ +y2 = y1 != 0 + test = !test; +1 +2 +3 +4 +5 +6 +} +{⏉m} +{sender,x} ⋃ Xm +{y1} +{0m} +{y2} +{y1} +{y2} +{test} +{test} +Securify (with fix)  +1 +2 +3 +4 +5 +6 +Xm.D +Xm.D ⋃ {sender,x} +{y1} +{0m.S, 0m.D} +{y2} +{y1} +{y2} +{test} +{test} +HoRStify +1 +2 +3 +6 +5 +4 + ⇒  +Xm + ⇒  +Fig. 11. Simplified version of contract in Figure 4 satisfying the RW property +with PDGs depicting the dependencies modeled by Securify and HORSTIFY. +accesses the gas value, so a dependency between 4 and the +gas nodes is established. Node 3 is one of these gas nodes +(there are more not shown in the picture). The execution of +3 depends on condition y1 checked in 2 , so it is control +dependant on 2 . Node 1 defines y1, so 2 depends on 1 . +Thus, there is a transitive dependency between writing to test +in 6 and reading msg.sender in 1 . +b) Memory Abstraction: To precisely model memory and +storage accesses in a CFG, it is important to know statically +as many memory and storage locations as possible. Assume +that such statical information is not available: then memory (or +storage) cannot be separated into regions and all read and write +operations introduce dependencies with the whole memory +(or storage). This would introduce many false dependencies. +During a preprocessing step, such static information can be +inferred. But, as demonstrated in Section IV, using prepro- +cessed data may introduce unsoundness. This requires careful +integration of preprocessing information into the CFG defining +rules. In the following we consider only memory variables; all +ideas equally apply to storage variables. +We propose a, to the best of our knowledge, novel memory +abstraction that is sound and provides high precision. To po- +sition our approach between unsound and imprecise memory +abstractions, we revisit Figure 4 in a simplified version that is +depicted as a CFG in Figure 11. The black and solid line parts +of the left CFG visualize how Securify misses the dependency +between msg.sender ( 1 ) and writing to test ( 5 ). In Securify, +write accesses to unknown memory locations are assumed to +write a special memory variable ⊤m. However, when reading + +{sender, x} +m[1] = msg.sender; +m[x] = 42; +y1 = m[0] +if (y2){ +y2 = y1 != 0 + test = !test; +} +2 +3 +4 +5 +6 +7 +{⏉m} +{sender} +{y1} +{0m} +{y2} +{y1} +{y2} +{test} +{test} +Securify (with fix) +1 +{1m} +{x} ⋃ Xm +2 +3 +4 +5 +6 +7 +Xm.D +{y1} +{0m.D, 0m.S} +{y2} +{y1} +{y2} +{test} +{test} +HoRStify +1 +{1m.S} +{} +1' +{1m.D} +Xm.D +1 +2 +3 +4 +5 +6 +7 +Xm + ⇒  +Fig. 12. +Contract violating the RW property with PDGs depicting the +dependencies as modeled by Securify and HORSTIFY. +from a statically known memory location (as done in 2 ), +Securify does not consider that a value could have been written +to this location when the location was not statically known, +i.e., that the value could have been stored in ⊤m: the Use set +of 2 contains only 0m, but not ⊤m. A hypothetical fix for +this unsoundness is to replace the variable ⊤m by the whole +set Xm of all memory variables. This fix is depicted in violet +in Figure 11. Now, the dependency of the read access in 2 to +the write operation in 1 is naturally established. One should +notice, however, that this interpretation implies that the Use +set of node 1 needs to contain all variables in Xm as well: +a new value is written to one unknown location, but for all +other locations the value is “copied” from the existing memory +cells, and hence, all these cells need to be included in the Use +set. Even though fixing the soundness issue, this modeling +would lead to an imprecise analysis as depicted in Figure 12. +This variant of Figure 11 first writes msg.sender to the known +memory location 1 in node 1 and then writes a value to an +unknown memory location in node 2 . Since the condition +y2 only depends on the value in memory location 0 while +msg.sender was written to location 1, the final write to the +test variable in 6 does not depend on msg.sender. However, +the hypothetical fix of Securify infers a possible dependency +between 6 and msg.sender (shown in violet in the left CFG +in Figure 12). This imprecision is caused by interpreting a +write to an unknown memory location as a write to possibly all +memory locations as this requires the Use set in 2 to contain +Xm. This creates a dependency between the assignment of +location 1 to msg.sender in 1 and the memory access in 2 . +Our memory abstraction is sound but more precise than +the hypothetical fix above. For every memory variable x we +use two sub-variables instead: S-variable xm.S stores values +that are assigned to x when the memory location for x is +statically known, and D-variable xm.D stores values assigned +to x when x’s location is not statically known. During the +execution, every write access to a memory variable x stores +the assigned value in xm.D, unless the memory location for +x is statically known, in which case xm.S stores the value +and xm.D is set to ⊥. Correspondingly, when reading from +a variable (regardless of the memory location being statically +known or not), first, the value of the D-variable is read, and +only if it is ⊥, the value of the S-variable is taken. We model +this read access with the function +load θ x = +� +θ[xm.S] +if θ[xm.D] = ⊥ +θ[xm.D] +otherwise. +This two-layered memory abstraction ensures that the exe- +cution is deterministic and that the read values coincide with +those obtained during an execution without prior preprocess- +ing. load is used in the inference rules in Figure 13 that define +the memory read and write operations for the CFG semantics. +In these rules, we use Xm.S for the set of all S-variables and +Xm.D for the set of all D-variables. The leftmost MSTORE +rule is for the case that a value is written to a memory +location that could not be statically inferred (i.e., pre[0] = ⊥). +There, any of the memory variables from Xm.D might be +redefined, hence the Def set contains all variables in Xm.D. +As discussed for the hypothetical fix of Securify, also the Use +set needs to include Xm.D, because we must not interrupt +potential dependencies for memory cells that are not changed +by this MSTORE instruction. An example for this is node 2 +in Figure 12 (right CFG). The S-variables are not part of the +Use set and hence not part of the value intermingling in 2 . +This removes the imprecision that occurred in the proposed +hypothetical fix above. Still, the MLOAD rules make sure that +no dependencies to S-variables are missed by adding both +D-variables and S-variables to the Use set. This way, the +connection between the memory location and the stored value +is preserved; x1m.D does not inherit any data dependencies +from x2m.S for locations x1 ̸= x2. An example for that is +given in Figure 12, where memory location 0 does not inherit +the dependency from memory location 1 written in 1 . This is +thanks to the node splitting at 1 that breaks the propagation +of dependencies on precomputed locations to dynamic ones. +c) Call Abstraction: Contract calls in Ethereum trigger +a multitude of possible (side) effects. When calling another +account, the control flow is handed over to the code residing +in this account. This code may initiate further internal transac- +tions, e.g., perform money transfers or even reenter the calling +contract before reporting back the result to the callee. +This behavior poses a big challenge to sound static analysis +since all possible effects of interactions with other (potentially +unknown) contracts need to be over-approximated. Securify +avoids this challenge by sacrificing soundness and ignoring +all data dependencies arising from external calls (including +effects of reentrancy) as demonstrated by the examples in Fig- +ure 5 and Figure 6. In contrast, to give a sound and precise +characterization of these dependencies, we first simplify the +problem by restricting our analysis to a set of well-behaved +smart contracts and then model the remaining dependencies in +a fine-grained manner. +The class of smart contracts that we target are such contracts +that cannot write storage variables in reentering executions. +This restriction rules out race conditions on contract variables +and as such is a highly-desirable property that can be easily +achieved (e.g., by a strict local locking discipline). We call +contracts satisfying this restriction store unreachable: + +C(pc) = (MLOAD(yls, xls)), pcnext, pre) +pre[1] = ⊥ +f = (λθ.θ ← yls := load θ (θ[xls])) +C, cd ⊨ (pc, 0) −⇑f − +→ (pc, 1) +Def = {yls} +Use = Xm.D ∪ Xm.S ∪ {xls} +C(pc) = (MSTORE(x1 +ls, x2 +ls), pcnext, pre) +pre[0] = ⊥ +f = (λθ.θ ← θ[x1 +ls].D := θ[x2 +ls]) +C, cd ⊨ (pc, 0) −⇑f − +→ (pc, 1) +Def = Xm.D +Use = Xm.D ∪ {x1 +ls, x2 +ls} +C(pc) = (MLOAD(yls, xls)), pcnext, pre) +pre[1] = ⌊xm⌋ +f = (λθ.θ ← yls := load θ xm) +C, cd ⊨ (pc, 0) −⇑f − +→ (pc, 1) +Def = {yls} +Use = {xm.S, xm.D} +C(pc) = (MSTORE(x1 +ls, x2 +ls), pcnext, pre) +pre[1] = ⌊xm⌋ +f = (λθ.θ ← xm.S := θ[x2 +ls]) +C, cd ⊨ (pc, 0) −⇑f − +→ (pc, 1) +Def = {xm.S} +Use = {x2 +ls} +C(pc) = (MSTORE(x1 +ls, x2 +ls), pcnext, pre) +pre[0] = ⌊xm⌋ +f = (λθ.θ ← xm.D := ⊥) +C, cd ⊨ (pc, 1) −⇑f − +→ (pc, 2) +Def = {xm.D} +Use = ∅ +Fig. 13. MLOAD memory abstraction instantiation +C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) +f1 = λθ.θ ← yls := applyCall(θ, C, pc)[yls] +f2 = λΘ.θ ← externaleg := applyCall(θ, C, pc)[externaleg] +f = λθ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f − +→ (pc, 1) +Def = {yls, externaleg} +Use = {gls, tols, vals, iols, isls, ools, osls, gel, actorel} ∪ Xm ∪ Xeg ∪ Xg +Fig. 14. Simplified CFG rule for the CALL opcode +Assumption 2 (Store unreachability (informal)). A contract +C is store unreachable if all its reentering executions cannot +reach an SSTORE instruction. +The contract in Figure 5 trivially violates store unreachabil- +ity since the field a can be written in a reentering execution. +This could be easily fixed by guarding each function with a +lock that blocks reentering executions. Store unreachability is +a local reachability property of the contract under analysis and +as such falls in the scope of the sound analysis tool eThor [24] +and hence can be automatically verified. +Even when focussing on store unreachable contracts, the +program dependencies induced by external calls are manifold +and often subtle. Figure 14 shows one (slightly simplified) +rule of the CFG semantics for external calls. As seen in +the previous examples, node splitting is used to separate the +dependencies of different variables. The rule displayed in Fig- +ure 14 gives one of the rules for setting a call’s return value +(written to the stack variable yls) and updating the external +environment (represented by variable externaleg) according to +the call effects. +To obtain the updated CFG state after a call, the rule uses +the function applyCall, which executes the internal transaction +initiated by the CALL opcode4. The CFG state resulting from +this execution is then used to describe the state updates (in +the case of the given rule, the updates on the variables yls +and externaleg, as indicated by the Def set). Even though the +whole CFG state θ is taken as an argument by applyCall, +not all variables in θ can influence all aspects of the state +4We define applyCall using the EVM semantics and hence can infer Def +and Use sets from the corresponding EVM semantics rules. +after returning. The variables that indeed may affect yls and +externaleg are given in the Use set. More precisely, the result +of a call may still depend on the global state, so all global +environmental variables (Xeg), as well as the global variables +of the contract under analysis itself (Xg). Additionally, the +execution of the called contract can be influenced by the +parameters given to the call: The argument gls attributes to +the amount of gas given to the call, tols gives the address of +the recipient account and vals the amount of money transferred +with the call. The arguments iols and isls specify the memory +fraction (offset and size) from which input data to the call +is read and ools and osls correspondingly define the memory +fraction where the call’s result data will be written. In the +given simplified rule, we consider that the concrete memory +fragments could not be precomputed and hence all memory +(Xm) could potentially be input data to the call. The Use set +also contains the calling account (as given in actorel), since +this information is made accessible during a call. Finally, the +Use set contains the amount of gas that is available at the +point of calling (given by gel) since this value may influence +the amount of gas given to the call. +We want to highlight two forms of dependencies, which may +erroneously be assumed to be ruled out by the assumption of +store unreachability: First, the Use set explicitly contains the +storage variables (Xg) of the contract under analysis, even +though we assume this contract to be store unreachable and +(by the semantics) its storage variables cannot be accessed +by any other contract. Second, both the Def and the Use set +contain the variable externaleg that represents the external +environment (in particular the state of other contract accounts). +This implies that the rule in Figure 14 explicitly models +information to be stored and retrieved from contract accounts +during an external call. In Figures 15 and 16, we illustrate the +need for these dependencies by two examples. +The example in Figure 15 shows how dependencies on a +storage variable are introduced by reading a contract variable +during a reentering execution. Note that store unreachability +only assures that reentering executions can not write contract +variables, but does not prevent read accesses. The example +gives another version of the Test contract, which performs +the check of msg.sender in an indirect way: First, it writes +msg.sender to the contract variable sender. To read the variable + +1 +contract RetrieveSender { +2 +function getTestSender() public returns (address) { +3 +try Test(msg.sender).getSender() returns (address a) { +4 +return a; } +5 +catch {return address(0); }}} +6 +7 +contract Test { +8 +bool test = false; +9 +address sender; +10 +RetrieveSender rs = RetrieveSender (address(42)); +11 +function getSender () public returns (address) { +12 +return sender;} +13 +function flip () public { +14 +sender = msg.sender; +15 +try rs.getTestSender() returns (address a) +{ +16 +if (a != address(0)){ +17 +test = !test;}} +18 +catch {return; }}} +Fig. 15. Example: Reading storage variables during reentering execution. +1 +contract SaveAddr { +2 +address addr = address(0); +3 +function set(address a) public { +4 +addr = a; } +5 +function get( ) public returns (address) {return addr; }} +6 +7 +contract Test { +8 +bool test = false; +9 +SaveAddr sa = SaveAddr (address(42)); +10 +function flip () public { +11 +try sa.set(msg.sender) { +12 +try c.get() returns (address a) +{ +13 +if (sa != address(0)){ +14 +test = !test; } } +15 +catch {return; }} +16 +catch {return; } } } +Fig. 16. Example: Propagating dependencies via an external contract account. +again, a RetrieveSender contract rs is used as a proxy: 5 The +Test contract calls RetrieveSender’s getTestSender function (in +line 15), which in turn reenters Test via its getSender function +(in line 3) to obtain the value of sender. This value is finally +returned to contract Test. As a consequence, the return variable +a in line 16 contains the value of msg.sender, and so the +assignment of variable test is dependent on msg.sender. This +dependency, however, can only be tracked when considering +that the contract’s own storage variables may influence the +return value of an external call. +The example in Figure 16 shows how dependencies can +be propagated via another contract account. Note that store +unreachability is a contract-specific property that only ensures +that the contract under analysis is not written in reentering +executions. The assumption does not restrict the storage mod- +ification of other contracts. The version of the Test contract +given in Figure 16 uses the contract SaveAddr to propagate +the value of msg.sender. To this end, it first writes the value +of msg.sender into the addr storage variable of the SaveAddr +contract sa using the set function (in line 11). Afterwards, +it retrieves the value back by accessing c’s storage via the +5Note that in Ethereum, a contract is identified by its address. In Solidity, the +syntax RetrieveSender rs = RetrieveSender (address(42)) means +that the contract at address 42 is assumed to be (of the type) RetrieveSender +and accessible via variable rs. +get function (in line 12). Consequently, the return variable +a contains the value of msg.sender in line 13 what makes +the following write to test dependent on that value. This +dependency can only be faithfully modeled when considering +that an external call may change the state of other accounts, +and may also be influenced by this state. This motivates why +the externaleg variable needs to be included in both the Def +and the Use set of the rule in Figure 14. +C. Soundness Reasoning via Dependency Predicates +Inspired by Securify, we define dependency predicates that +can capture the data and control flow dependencies induced +by the PDG (as given through the CFG semantics). They are +inhabited via a set of logical rules (CHCs) R(C) that describe +the data and control flow propagation through the PDG of a +contract C. More formally, the transitive closure of the C’s +PDG is computed as the least fixed point over R(C) (de- +noted by lfp(R(C))). Most prominently, lfp(R(C)) includes +the predicates VarMayDepOn and InstMayDepOn. Intuitively, +VarMayDepOn(y, x) states that the value of variable y may +depend on the value of variable x and InstMayDepOn(n, x) +says that the reachability of node n may depend on the +value of variable x. In the following, let nx and ny denote +nodes that define variables x and y, respectively. The formal +relation between dependency predicates and backward slices +is captured by the following lemma: +Lemma 1 (Fixpoint Characterization of Backward Slices). Let +x and y be variables and C be a contract. The following holds: +1) (∃nx ny. nx ∈ BS(ny)) ⇒ VarMayDepOn(y, x) ∈ lfp(R(C)) +2) (∃ n nif nx. nif −→cd n ∧ nx ∈ BS(nif)) +⇒ InstMayDepOn(n, x) ∈ lfp(R(C)) +Lemma 1 states 1) that whenever there is a node nx +defining x in the backward slice of a node ny defining y, then +VarMayDepOn(y, x) is derivable from the CHCs in R(C) and +2) that whenever there is a node nx defining x in the backward +slice of a node nif on which node n is control dependent then +InstMayDepOn(n, x) is derivable from R(C). The intuition +behind statement 2) is that node n is controlled by nif (by +the definition of standard control dependence), which means +that nif is a branching node. nx ∈ BS(nif) indicates that the +branching condition of nif depends on variable x and, hence, +so does the reachability of n. +Next, we give an explicit semantic characterization of the +dependency predicates, which we prove sound using Theo- +rem 1. This explicit characterization enables us to compose +security patterns as a set of different facts over dependency +predicates and to reason about them in a modular fashion. As a +consequence, we can show in Section VI-D that checking the +inclusion of security patterns in the least fixpoint of the rule set +R(C) is sufficient to prove non-interference-style properties. +Concretely, we can characterize facts from the VarMayDepOn +predicate as follows: +Theorem 2 (Soundness of Dependency Predicates). +∀x y. VarMayDepOn(y, x) ̸∈ lfp(R(C)) ⇒ y ⊥ x + +with y ⊥ x given as: +∀nx i θ1 θ2 θ′ +1. θ1 =/x θ2 ∧ ⟨nx ++, θ1⟩ +Ny +−−→ +i +⟨n, θ′ +1⟩ +⇒ ∃θ′ +2. ⟨nx ++, θ2⟩ +Ny +−−→ +i +⟨n, θ′ +2⟩ ∧ θ′ +1(y) = θ′ +2(y) +where nx ++ denotes the unique successor node of nx, and Ny the +set of all nodes defining y. ⟨nx ++, θ1⟩ +Ny +−−→ +i +⟨n, θ′ +1⟩ describes an +execution from nx to n that passes exactly i nodes defining y. +The theorem states that if VarMayDepOn(y, x) is not in- +cluded in lfp(R(C)) then y is independent of x (y ⊥ x). +A variable y is considered independent of x if for any two +configurations θ1 and θ2 that are equal up to x, and any +execution starting at node nx+, the first node after x is +defined, passing i nodes that define y, and ending in a node +n at state θ′ +1, one can find a matching execution from θ2 +that passes the same number of nodes defining y and ends +at node n in a state θ′ +2 such that θ′ +2 and θ′ +1 agree on y. +This definition ensures loop sensitivity: it captures that during +a looping execution, every individual occurrence of a node +defining y can be matched by the other execution—so that +the values of y agree whenever y gets reassigned. The proof +of Theorem 2 uses Lemma 1 and Theorem 1. For the full +proof and a similar characterization of InstMayDepOn(i, x), +we refer to Appendix A-B. +D. Sound Approximation of Security Properties +With Theorem 2 we are able to formally connect depen- +dency predicates and (independence-based) security proper- +ties. We take trace noninterference as a concrete example, +which comprises a whole class of non-interference-style secu- +rity properties. Concretely, we consider trace noninterference +w.r.t. a set of EVM configuration components Z, which +includes, for example, the block timestamp. A predicate f +defines instructions of interest. If two executions of a contract +C start in configurations that differ only in the components in +Z, then the instructions of interest must coincide in the two +traces that result from these executions. +Definition 3 (Trace noninterference). Let C be an EVM +contract, Z be a set of components of EVM configurations and +f be a predicate on instructions. Then trace noninterference +of contract C w.r.t. Z and f (written TNI(C, Z, f)) is defined +as follows: +TNI(C, Z, f) := ∀ Γ Γ′ s s′ t t′ π. π′ +(Γ, s) =/Z (Γ′, s′) +⇒ Γ ⊨ sC :: S +π−→ +∗ tC :: S ∧ final (t) +⇒ Γ ⊨ s′ +C :: S +π′ +−→ +∗ +t′ +C :: S ∧ final (t′) +⇒ π ↓f= π′ ↓f +where π ↓f denotes the trace filtered by f, so containing only +the instructions satisfying f. +The dependency properties defined in [14] can be expressed +in terms of trace noninterference. E.g., the timestamp indepen- +dence property in Definition 1 is captured as an instance of +trace noninterference as follows: +TNI(C, {Γ.timestamp}, λop.op = CALL) +We show that we can give a sufficient criterion for +trace noninterference in terms of dependency predicates. More +precisely, we give a set PC +Z,f of facts, such that PR(C) +Z,f +∩ +lfp(R(C)) = ∅ implies TNI(C, Z, f). Practically, this means +that we can prove TNI(C, Z, f) by computing the least fix- +point over the CHCs R(C) (e.g., using a datalog engine) +and then check whether it contains any fact from PC +Z,f. For +components in Z, we assume a function toVar that maps +components of the EVM semantic domain to CFG variables. +The dependency predicates constituting a security pattern for +trace noninterference are defined as +PC +Z,f :={InstMayDepOn(pc, toVar(z)) | z ∈ Z +∧ C(pc) = op(⃗x, pcnext, pre) ∧ f(op)} +∪ {VarMayDepOn(xi, toVar(z)) | z ∈ Z ∧ pc ∈ dom(C) +∧ C(pc) = (op(⃗x, pcnext, pre)) ∧ f(op) ∧ xi ∈ ⃗x +}. +The following theorem shows that PC +Z,f is a security pattern +for trace noninterference: +Theorem 3 (Soundness of trace noninterference). Let C be +a contract, Z a set of components, and f an instruction-of- +interest predicate. Then it holds that +(∀p ∈ PC +Z,f. p ̸∈ lfp(R(C))) ⇒ TNI(C, Z, f). +The absence of facts from PC +Z,f in lfp(R(C)) ensures that +the reachability of all instructions satisfying f is independent +of variables representing components in Z and that all argu- +ments xi of such instructions are independent of z as well. +These independences imply trace noninterference since they +ensure that in two executions starting in configurations equal +up to Z, all instructions satisfying f are executed in the same +order (otherwise their reachability would depend on Z) and +with the same arguments (otherwise their argument variables +would depend on Z). Consequently, such executions produce +the same traces, when only considering instructions satisfying +f. A full proof of Theorem 3 can be found in Appendix A-C. +E. Discussion +In this section, we presented a sound analysis pipeline +for checking security properties for linearized EVM bytecode +contracts by means of reasoning about dependencies between +variables or instructions. While our work was inspired by +Securify [27], we developed new formal foundations for +the dependency analysis of EVM bytecode contracts and +in this way revealed several sources of unsoundness in the +analysis of Securify. Further, we provide soundness proofs +for the analysis pipeline end-to-end. The key pillars of the +soundness proof are i) that our EVM CFG semantics satisfies +all conditions to be used with the slicing framework [30], +ii) that the EVM linearized bytecode semantics and the CFG +semantics are equivalent, iii) that our set of CHCs encodes +an over-approximation of dependencies in an EVM contract, +and iv) that the generic security pattern PC +Z,f is a sound +approximation of trace noninterference. The proofs are valid +under assumptions that are clearly stated in this paper. For +Assumptions 1 and 2 we point out the existence of other sound +tools [4], [24] that can check these assumptions. + +We assume that EVM smart contracts are provided in a +(stack-less) linearized form. Transforming into such a rep- +resentation from a stack-based one is a well-studied prob- +lem [19] and a standard step performed by most static analysis +tools [11], [27]. Up to this requirement, our analysis is +parametric with respect to other preprocessing steps. More +precisely, our analysis pipeline is sound for contracts with +sound preprocessing information, and hence, in particular, +for contracts without any preprocessing information but jump +destinations needed for the CFG (cf. Section VI-A). This +gives the flexibility, to enhance the precision of the analysis +through the incorporation of soundly precomputed values +and makes the design of sound preprocessing an orthogonal +problem. There exist already works on soundly precomputing +jump destinations for EVM bytecode [12], which are to be +complemented with other precomputing steps in the future. +VII. EVALUATION +The focus of this paper is on the theoretical foundations +of a sound dependency analysis of smart contracts. However, +we demonstrate the practicality of the presented approach by +developing the prototype analyzer HORSTIFY. We do not +implement the logical rules from Section VI-C directly in +Souffl´e (as done by Securify), but encode them in the HORST +specification language [24]. The HORST language is a high- +level language for the specification of CHCs. By introducing +this additional abstraction layer, we get a close correspondence +between our theoretical rules and their actual implementation +and, hence, anticipate a lower risk of implementation mistakes +that may invalidate soundness claims in the implementation. +HORSTIFY accepts as input a set of dependency facts en- +coding the security patterns specified in the HORST language +and Ethereum smart contracts in the EVM bytecode format. It +first invokes Securify’s decompiler to transform the contract +into a linearized representation and does some lightweight +preprocessing to obtain the precomputable values (cf. Sec- +tion VI-A). Then, HORSTIFY uses our formal specification +of the CFG construction rules and the HORST framework to +create a Souffl´e executable for the analysis and invokes it. +To reduce the risks of implementation mistakes, we pro- +ceeded in two steps. First, we encoded Securify’s RW violation +pattern in the HORST language to execute HoRStify with this +pattern and the contracts in Figures 4, 6 and 76. In contrast +to Securify, HORSTIFY correctly determines that these con- +tracts do not satisfy the RW violation pattern. In addition to +these corner cases, we successfully evaluated HORSTIFY on +Securify’s internal test suite involving 25 contracts. +Next, we conduct a large-scale evaluation of HORSTIFY +and Securify on real-world contracts. To this end, we use +the sanitized dataset from [24] that consists of 720 distinct +smart contracts from the Ethereum blockchain. We compare +the performance of Securify and HORSTIFY on this dataset +for both the RW pattern and for timestamp independence (TS) +6We did not consider the contract in Figure 3 since it concerns the must- +analysis and the contract in Figure 5, which violates Assumption 2. +contracts +errors +timeouts +contracts +∅ time (ms) +\(errors ∪ timeouts) +720 +H 34 +H 46 +634 +H 7055 +S 34 +S 30 +S 3107 +TABLE I +LARGE-SCALE EVALUATION OF HORSTIFY (H) AND SECURIFY (S). +Fig. 17. +Classification of mismatching results of HORSTIFY (above) and +Securify (below) for the RW (left) and TS (right) property8. Ticks indicate +correct matches (tn) and crosses wrong matches (fn) of the respective tool. +tnHor/tnSec, fnHor/fnSec, tpHor/tpSec, fpHor/fpSec denote true negatives, false neg- +atives, true positives, and true negatives of HORSTIFY/Securify, respectively. +as defined in Section VI-D. We manually inspect all contracts +on which Securify and HORSTIFY report a different result. +Table I shows the evaluation results. The average execution +time of HORSTIFY is approximately 2.3 times longer than +for Securify. Consequently, HORSTIFY suffers from more +timeouts than Securify; the execution of both tools is aborted +after one minute. Figure 17 visualizes the manual classification +for those smart contracts where HORSTIFY and Securify dis- +agree. There are only two contracts where HORSTIFY matches +the corresponding pattern, but Securify does not. Recall that +for a sound tool, a pattern match indicates the discovery of +provable independencies that imply either property violation +(RW) or compliance (TS). An erroneous pattern match by +HORSTIFY would present a soundness issue (false negative). +We carefully examined the two examples and could confirm +them not to constitute false negatives of HORSTIFY but +false positives of Securify (fpSec), unveiling an imprecision of +Securify. This seems surprising since our analysis generally +tracks more dependencies than the one of Securify. However, +while HORSTIFY implements standard control dependence to +encode control dependencies (e.g., to compute join points after +loops), Securify implements a less precise custom algorithm. +The contracts where Securify matches a pattern, but HORS- +TIFY does not, can either reveal soundness issues (false nega- +tives) of Securify (fnSec) or a precision loss (false positives) of +HORSTIFY (fpHor). Indeed, in the 29 contracts that are flagged +only by Securify, we find both cases (as shown at the bottom +of Figure 17), as we will illustrate with two examples: +Figure 18 shows a (slightly shortened) version of a contract +classified as safe for TS according to Securify, but that HoRS- +tify (correctly) reports as vulnerable. It is a lottery contract that +pays out a user who manages to guess a random number (func- +tion Guess). The random number is generated from blockchain +and transaction-specific values, including the timestamp (ac- +cessed via now in RandomNumberFromSeed). Hence, the payout in +8For TS we only consider the 165 contracts from the dataset containing a +TIMESTAMP opcode, as Securify labels other contracts as trivially secure. +The manual classification is a conservative best-effort estimate. + +1 +contract RNG { +2 +mapping (address => uint) nonces; +3 +uint public last; +4 +function RandomNumber() returns(uint) { +5 +return RandomNumberFromSeed( +6 +uint(sha3(block.number))ˆuint(sha3(now)) +7 +ˆuint(msg.sender)ˆuint(tx.origin)); } +8 +function RandomNumberFromSeed(uint seed) returns(uint) { +9 +nonces[msg.sender]++; +10 +last = seedˆ(uint(sha3(block.blockhash(block.number), +11 +nonces[msg.sender])) +12 +*0x000b0007000500030001); +13 +return last; } +14 +function Guess(uint _guess) returns (bool) { +15 +if (RandomNumber() == _guess) { +16 +if (!msg.sender.send(this.balance)) throw; +17 +RandomNumberGuessed(_guess, msg.sender); +18 +return true; } +19 +return false; } } +Fig. 18. Lottery Contract 0xaed5a41450b38fc0ea0f6f203a985653fe187d9c +1 +contract lottery{ +2 +address[] public tickets; +3 +function buyTicket(){ +4 +if (msg.value != 1/10) throw; +5 +if (msg.value == 1/10) +6 +tickets.push(msg.sender); +7 +address(0x88a1e54971b31974b2be4d9c67546abbd0a3aa8e) +8 +.send(msg.value/40); +9 +if (tickets.length >= 5) runLottery(); } +10 +function runLottery() internal { +11 +tickets[addmod(now, 0, 5)].send((1/1000)*95); +12 +runJackpot();} +13 +function runJackpot() internal { +14 +if(addmod(now, 0, 150) == 0) +15 +tickets[addmod(now, 0, 5)].send(this.balance); +16 +delete tickets; } } +Fig. 19. Lottery contract 0xe120100349a0b1BF826D2407E519D75C2Fe8f859 +line 16 is not independent of the timestamp. Securify fails to +detect this dependency due to its unsound memory abstraction +(as described in Section IV-B): As Ethreum’s hash function +(sha3) reads input from the local memory, the timestamp is +written to the memory where its dependencies are lost. +Figure 19 shows an example of a false positive for HORS- +TIFY. The contract implements a lottery where users can reg- +ister (via buyTicket) and whenever 5 users were registered, one +of them is selected as a winner. Despite the obvious timestamp +dependency, the contract shows RW violations, which HORS- +TIFY fails to prove.9 E.g., the tickets array is updated without +performing a check on the sender. HORSTIFY does not detect +this vulnerability due to its sound storage abstraction: In line 6, +the caller (msg.sender) is appended to the tickets array. Since +the array position to which msg.sender will be added cannot +be statically known, HORSTIFY needs to assume msg.sender +to be written to any position. When checking the size of +tickets in line 9, the condition is considered dependent on +msg.sender (because in the abstraction, msg.sender is considered +to potentially affect all storage locations, including the one +containing the array size). Thus, the delete operation in line 16 +9Note that this is not a soundness issue since the soundness of HORSTIFY +ensures that independencies can be proven. In the case of violation patterns +as RW the independence constitutes an unwanted effect and hence, we can +only use it to prove the vulnerability of a contract, not its safety. +is considered dependent on msg.sender. One should notice, that +only the unsoundness of Securify’s storage abstraction, enables +Securify to correctly detect the RW violation in this case. +Overall, based on our evaluation results, we can bound the +precision loss of HORSTIFY w.r.t. Securify. More concretely, +when considering that Securify has a specificity10 of SSec on +the full dataset, then one can easily show that it holds for the +specificity SHor of HORSTIFY that SHor ≥ SSec + tnHor−tnSec +|dataset| +where tnHor are the true negatives for HORSTIFY, and tnSec +are the true negatives for Securify found within the man- +ually inspected mismatching contracts. Inserting the results +from Figure 17, we can show that SHor can be at most 0.5 +percentage points less than SSec for RW on the given dataset +and at most 5.4 percent points less for TS. +We refer to horstify.org for more information about +HORSTIFY. +VIII. RELATED WORK +Existing approaches to enforce the correctness of Ethereum +smart contracts can be broadly categorized into analyses at de- +sign time and analyses at runtime. The latter include methods +like runtime monitoring [9], [28] or information flow control +mechanisms [6]. Such dynamic analysis approaches, however, +have limited applicability to the Ethereum blockchain, since +they either require fundamental updates to the workings of +the EVM or impose tremendous costs in terms of gas. Static +analyses, in contrast, verify smart contracts at design time be- +fore they become immutable objects on the blockchain. Most +static analyzers are bug-finding tools (such as Oyente [22], +EthBMC [10], and Maian [23]) that aim to reduce the number +of contracts that are wrongly claimed to be buggy (false +positives). To this end, these tools usually rely on the symbolic +execution of the contract under analysis. The dual objective +of bug-finding is to prove a smart contract secure. Analyzers +following this objective do not only aim at producing a low +number of false negatives in practice but to give provable guar- +antees for their analysis result, e.g., that a contract flagged as +safe is guaranteed to enjoy a corresponding security property. +The only example of a tool, which comes with a provable +soundness claim, so far, is the analyzer eThor [24], whose +analysis relies on abstract interpretation. +Symbolic execution and abstract interpretation have in com- +mon to target properties that can be decided for a finite +prefix of a single (yet arbitrary) execution trace of a smart +contract (so-called reachability properties). However, many +generic security properties for smart contracts (as defined +in [14]) require comparing two execution traces from different +initial configurations and fall into the broader category of +2-safety properties. To check 2-safety properties with tools +whose analysis is limited to reachability properties (such as +eThor) requires an overapproximation of the original property +in terms of reachability. But finding such a meaningful over- +approximation, which does not result in an intolerable preci- +sion loss, is not always possible. In [14], it is, e.g., shown how +10The specificity is a standard precision measure and is calculated as +tn +tn+fp + +to overapproximate the call integrity 2-safety property (char- +acterizing the absence of reentrancy attacks) by a reachability +property (single-entrancy) and two other properties, which are +captured by our notion of trace noninterference. However, +trace noninterference properties still concern two execution +traces and hence cannot be verified using eThor. HORSTIFY +(inspired by the unsound Securify tool [27]) devises a differ- +ent analysis technique, which immediately accommodates the +analysis of trace noninterference. As opposed to the analysis +underlying eThor, this technique does not allow for verifying +general reachability properties, but a special class of 2-safety +properties (including trace noninterference). HORSTIFY and +eThor, hence, can be seen as complementing tools that target +incomparable property classes. The call integrity property +falls neither in the scope of eThor nor HORSTIFY, but its +overapproximation decomposes it into trace noninterference +properties (within the scope of HORSTIFY) and a reachability +property (within the scope of eThor). Other generic security +properties from [14] for characterizing the independence of +miner-controlled parameters (including timestamp indepen- +dence) immediately constitute trace noninterference properties +and as such can be analyzed by HORSTIFY but not by eThor. +More complex properties involving both universal and ex- +istential quantification of execution traces [7], [8] cannot be +checked by either HORSTIFY or eThor. +IX. CONCLUSION +In this work, we present the first provably sound static +dependency analysis for EVM bytecode. Taking up the ap- +proach of the state-of-the-art static analyzer Securify [27], we +uncover conceptual soundness issues of the tool, so we replace +the underlying analysis and spelled out formal soundness +guarantees. The soundness proof of our analysis relies on the +proof framework from [30] for static program slicing, which +we instantiate for EVM bytecode. The slicing framework can +capture the notion of may-dependence, whereas we elucidated +that the must-dependence promoted by Securify raises sound- +ness questions already at the conceptional basis. Although we +removed support for must-dependence, we could show that the +resulting analysis is flexible enough to soundly characterize +relevant smart contract security properties such as timestamp +dependence. Finally, we demonstrate the practicality of the +approach by providing the prototypical analyzer HoRStify. +HoRStify encodes the slicing-based dependency analysis as +logical rules that can be automatically solved by the Datalog +solver Souffl´e; it can verify real-world smart contracts, and +even though being provable sound, shows performance com- +parable to Securify. +REFERENCES +[1] Solidity +programming +language. +https://soliditylang.org/, +https:// +github.com/ethereum/solidity. Accessed: 2022-02-05. +[2] [DEPRECATED] Securify scanner for ethereum smart contracts. https: +//github.com/eth-sri/securify. Accessed: 2022-02-05. +[3] Filip Adamik and Sokol Kosta. Smartexchange: Decentralised trustless +cryptocurrency exchange. +In International Conference on Business +Information Systems, pages 356–367. Springer, 2018. +[4] Elvira Albert, Jes´us Correas, Pablo Gordillo, Guillermo Roman-Diez, +and Albert Rubio. +Don’t run on fumes—parametric gas bounds for +smart contracts. Journal of Systems and Software, 176:110923, 2021. +[5] Moritz Andresen. The biggest smart contract hacks in history or how +to endanger up to us $2.2 billion. https://medium.com/solidified/the- +biggest-smart-contract-hacks-in-history-or-how-to-endanger-up-to-us- +2-2-billion-d5a72961d15d, 2016. +[6] Ethan Cecchetti, Siqiu Yao, Haobin Ni, and Andrew C Myers. Com- +positional security for reentrant applications. In 2021 IEEE Symposium +on Security and Privacy (SP), pages 1249–1267. IEEE, 2021. +[7] Michael R. Clarkson and Fred B. Schneider. +Hyperproperties. +In +CSF’08, pages 51–65, 2008. +[8] Pedro R. D’Argenio, Gilles Barthe, Sebastian Biewer, Bernd Finkbeiner, +and Holger Hermanns. Is your software on dope? - formal analysis of +surreptitiously ”enhanced” programs. pages 83–110. Springer, 2017. +[9] Joshua Ellul and Gordon J Pace. Runtime verification of ethereum smart +contracts. In 2018 14th European Dependable Computing Conference +(EDCC), pages 158–163. IEEE, 2018. +[10] Joel Frank, Cornelius Aschermann, and Thorsten Holz. +Ethbmc: A +bounded model checker for smart contracts. In 29th USENIX Security +Symposium (USENIX Security 20), pages 2757–2774, 2020. +[11] Neville Grech, Lexi Brent, Bernhard Scholz, and Yannis Smaragdakis. +Gigahorse: thorough, declarative decompilation of smart contracts. In +2019 IEEE/ACM 41st International Conference on Software Engineering +(ICSE), pages 1176–1186. IEEE, 2019. +[12] Ilya Grishchenko. +Static Analysis of Low-Level Code. +PhD thesis, +Technische Universit¨at Wien, 2020. +[13] Ilya Grishchenko, Matteo Maffei, and Clara Schneidewind. Foundations +and tools for the static analysis of ethereum smart contracts. +In +Proceedings of the 30th International Conference on Computer-Aided +Verification (CAV), pages 51–78. Springer, 2018. +[14] Ilya Grishchenko, Matteo Maffei, and Clara Schneidewind. A semantic +framework for the security analysis of ethereum smart contracts. +In +International Conference on Principles of Security and Trust, pages 243– +269. Springer, 2018. +[15] Samer Hassan and Primavera De Filippi. +Decentralized autonomous +organization. Internet Policy Review, 10(2):1–10, 2021. +[16] Vanita Jain, Akanshu Raj, Abhishek Tanwar, Mridul Khurana, and Achin +Jain. Coin drop—a decentralised exchange platform. In Cyber Security +and Digital Forensics, pages 391–399. Springer, 2022. +[17] Herbert Jordan, Bernhard Scholz, and Pavle Suboti´c. Souffl´e: On syn- +thesis of program analyzers. In International Conference on Computer +Aided Verification, pages 422–430. Springer, 2016. +[18] Sukrit Kalra, Seep Goel, Mohan Dhawan, and Subodh Sharma. Zeus: +Analyzing safety of smart contracts. NDSS, 2018. +[19] Allen Leung and Lal George. Static single assignment form for machine +code. ACM SIGPLAN Notices, 34(5):204–214, 1999. +[20] Yinsheng Li, Xu Liang, Xiao Zhu, and Bin Wu. A blockchain-based +autonomous credit system. In 2018 IEEE 15th International Conference +on e-Business Engineering (ICEBE), pages 178–186. IEEE, 2018. +[21] Ning Lu, Bin Wang, Yongxin Zhang, Wenbo Shi, and Christian Esposito. +Neucheck: A more practical ethereum smart contract security analysis +tool. Software: Practice and Experience, 2019. +[22] Loi Luu, Duc-Hiep Chu, Hrishi Olickel, Prateek Saxena, and Aquinas +Hobor. Making smart contracts smarter. In Proceedings of the 2016 +ACM SIGSAC Conference on Computer and Communications Security, +pages 254–269. ACM, 2016. +[23] Ivica Nikoli´c, Aashish Kolluri, Ilya Sergey, Prateek Saxena, and Aquinas +Hobor. +Finding the greedy, prodigal, and suicidal contracts at scale. +In Proceedings of the 34th annual computer security applications +conference, pages 653–663, 2018. +[24] Clara Schneidewind, Ilya Grishchenko, Markus Scherer, and Matteo +Maffei. ethor: Practical and provably sound static analysis of ethereum +smart contracts. In Proceedings of the 2020 ACM SIGSAC Conference +on Computer and Communications Security, pages 621–640, 2020. +[25] Clara Schneidewind, Markus Scherer, and Matteo Maffei. The good, the +bad and the ugly: Pitfalls and best practices in automated sound static +analysis of ethereum smart contracts. In International Symposium on +Leveraging Applications of Formal Methods, pages 212–231. Springer, +2020. +[26] Alexandra Sims. Decentralised autonomous organisations: Governance, +dispute resolution and regulation. Dispute Resolution and Regulation +(May 31, 2021), 2021. + +[27] Petar Tsankov, Andrei Dan, Dana Drachsler-Cohen, Arthur Gervais, +Florian Buenzli, and Martin Vechev. Securify: Practical security analysis +of smart contracts. In Proceedings of the 2018 ACM SIGSAC Conference +on Computer and Communications Security, pages 67–82, 2018. +[28] Haijun Wang, Yi Li, Shang-Wei Lin, Lei Ma, and Yang Liu. Vultron: +catching vulnerable smart contracts once and for all. In 2019 IEEE/ACM +41st International Conference on Software Engineering: New Ideas and +Emerging Results (ICSE-NIER), pages 1–4. IEEE, 2019. +[29] Daniel Wasserrab. Towards certified slicing. Archive of Formal Proofs. +http://afp. sf. net/entries/Slicing. shtml, 2008. +[30] Daniel Wasserrab, Denis Lohner, and Gregor Snelting. On pdg-based +noninterference and its modular proof. +In Proceedings of the ACM +SIGPLAN Fourth Workshop on Programming Languages and Analysis +for Security, pages 31–44, 2009. +[31] Gavin Wood et al. +Ethereum: A secure decentralised generalised +transaction ledger. +Ethereum project yellow paper, 151(2014):1–32, +2014. +[32] Lanfranco Zanzi, Antonio Albanese, Vincenzo Sciancalepore, and +Xavier Costa-P´erez. +Nsbchain: a secure blockchain framework for +network slicing brokerage. +In ICC 2020-2020 IEEE International +Conference on Communications (ICC), pages 1–7. IEEE, 2020. +[33] Michal Zima. Coincer: Decentralised trustless platform for exchanging +decentralised cryptocurrencies. In International Conference on Network +and System Security, pages 672–682. Springer, 2017. + +APPENDIX A +SOUND DEPENDENCY ANALYSIS FOR EVM BYTECODE +A. Instantiation of the Slicing Framework +1) State transformation: We formally define the state θ of the EVM as used in the CFG semantics. Afterward, we define +the state transformation functions toCFG and toEVM that convert between the different EVM state representations. +EVM state: We revisit the formal definition of the EVM state as given in [14]. +In the following, we will use B to denote the set {0, 1} of bits and accordingly Bx for sets of bitstrings of size x. We +further let Nx denote the set of non-negative integers representable by x bits and allow for implicit conversion between those +two representations (assuming bitstrings to represent a big-endian encoding of natural numbers). In addition, we will use the +notation [X] (resp. L(X)) for arrays (resp. lists) of elements from the set X. We use standard notations for operations on +arrays and lists. In particular we write a [pos] to access position pos ∈ [1, |a| − 1] of array a ∈ [X] and a[down, up] to access +the subarray of size up − down from position down ∈ [1, |a| − 1] to up ∈ [1, |a| − 1]. In case that down > up this operation +results in the empty array ϵ. In addition, we write a1 · a2 for the concatenation of two arrays a1, a2 ∈ [X]. +In the following formalization, we will make use of bytearrays b ∈ [B8]. To this end, we will assume functions (·)[B8] ∈ +Bx → [B8] and (·)B ∈ [B8] → Bx to chunk bitstrings with size dividable by 8 to bytearrays and vice versa. To denote the zero +byte, we write 08 and, accordingly, for an array of zero bytes of size n, we write 08·n. +For lists, we denote the empty list by ϵ and write x :: xs for placing element x ∈ X on top of list xs ∈ L(X). In addition, +we write xs + +ys for concatenating lists xs, ys ∈ L(X). +We let A denote the set of 160-bit addresses (B160). +In Figure 20 we give a full grammar for call stacks: +Call stacks +S +∋ +S +:= +EXC :: SP | HALT(σ, d, g) :: SP | SP +Plain call stacks +Splain +∋ +SP +:= +(µ, ι, σ) :: SP +Machine states +M +∋ +µ +:= +(g, pc, m, i, s) +Execution environments +I +∋ +ι +:= +(actor, input, sender, value, code) +Global states +Σ +∋ +σ +Account states +A +∋ +acc +:= +(n, b, code, stor) | ⊥ +Transaction environments +Tenv +∋ +Γ +:= +(o, prize, H) +Block headers +H +∋ +H +:= +(parent, beneficiary, difficulty, number, gaslimit, timestamp) +Notations: +d ∈ [B8], +g ∈ N256, +η ∈ N, +o ∈ A, +prize ∈ N256, +H ∈ H +g ∈ N256, +pc ∈ N256, +m ∈ N256 → N256 +i ∈ N256, +s ∈ N8 → N256 +sender ∈ A +input ∈ [B8] +sender ∈ A +value ∈ N256 +code ∈ [B8] +b ∈ N256 +stor ∈ N256 → N256 +L ∈ L(Evlog) +S† ⊆ A +Σ = A → A +parent ∈ N256 +beneficiary ∈ A +difficulty ∈ N256 +numberN256 +gaslimit ∈ N256 +timestamp ∈ N256 +Fig. 20. Grammar for calls stacks and transaction environments +Note that the grammar was slightly adapted to account for the fact that the local stack is assumed to be precomputed to local +variables. Further, the intermediate representation assumes all memory accesses to be aligned, meaning that memory acceses +only occure at addresses that are multiples of 32. +CFG state: We formally define the state for the CFG semantics of EVM bytecode. The formal definition of the CFG +state is given as follows: + +State +θ := (ls, m.S, m.D, g.S, g.D, el, eg) +Stack +ls ∈ N8 → N256 +Local Static Memory +m.S ∈ N256 → N256 +Local Dynamic Memory +m.D ∈ N256 → N256 ∪ {⊥} +Global Static Storage +g.S ∈ N256 → N256 +Global Dynamic Storage +g.D ∈ N256 → N256 ∪ {⊥} +Local Environment +el := (g, i, actor, input, sender, value) +Global Environment +eg := (parent, beneficiary, difficulty, number, gaslimit, timestamp, o, prize, external) +External Global Environment +external := (b, n, σ) +We will treat the CFG state as a heterogeneous mapping and write θ[xls] for θ.ls(x); θ[xm.S] for θ.m.S(x); θ[xm.D] for +θ.m.D(x); θ[xg.S] for θ.g.S(x); θ[xg.D] for θ.g.D(x); θ[xel] for θ.el.x; and θ[xeg] for θ.eg.x. In particular, we will treat (static +and dynamic) memory and storage locations as variables and will write Xm.D for the set of all dynamic memory locations, +Xm.S for the set of all static memory locations, Xg.D for the set of all dynamic storage locations, and Xg.S for the set of +all static storage locations. Further, we use Xm and Xg to denote the (distinct) sets of all memory and, respectively, storage +locations. +The state is partitioned according to the granularity of the analysis. All components whose dependencies are explicitly +tracked occur on the top level. +a) State transformation: In the following we will assume the load function to be defined on both memory locations xm +and storage locations xg. +load θ x = +� +θ[x.D] +if θ[x.S] = ⊥ +θ[x.S] +otherwise. +where x ∈ Xm ∪ Xg +Using this, the translation between the different state types can be defined as follows: +toCFG(Γ, s) := +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(θ, C, pc) +(µ, ι, σ) = s ∧ ls = µ.s ∧ m.S = λ(i, pc).µ.m(i) ∧ m.D = λ(i, pc).⊥ +∧ g.S = λ(i, pc).σ(ι.actor).stor(i) ∧ g.D = λ(i, pc).⊥ +∧ el = (λpc.µ.g, λpc.µ.i, ι.actor, ι.actor, ι.sender, ι.va) +∧ external = λpc.(σ(ι.actor).b, σ(ι.actor).n, σ) ∧ (o, prize, H) = Γ +∧ (parent, beneficiary, difficulty, number, gaslimit, timestamp) = H +∧ eg = (parent, beneficiary, difficulty, number, gaslimit, timestamp, o, prize, external) +∧ θ = (ls, m.S, m.D, g.S, g.D, el, eg) ∧ C = ι.code +∧ pc = µ.pc +Note that we will usually write θ = toCFG(Γ, s) to implicitely drop the reconstructed contract C and program counter pc. +toEVM(θ, C, pc) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(Γ, s) +(ls, m.S, m.D, g.S, g.D, el, eg) = θ +∧ (parent, beneficiary, difficulty, number, gaslimit, timestamp, o, prize, external) = eg +∧ (b, n, σ′) = external +∧ (g, i, actor, input, sender, value) = el +∧ code = λpc.(C(pc).op, C(pc).pcnext) +∧ µ = (g, pc, λx.load θ xm, i, ls) ∧ ι = (actor, input, sender, value, code) +∧ σ = σ′[actor → (b, n, code, λx.load θ xg)] +∧ H = (parent, beneficiary, difficulty, number, gaslimit, timestamp) ∧ Γ = (o, prize, H) +For defining the EVM CFG semantics, we assume Θ = θ⊎θ to represent the heterogeneous mapping that maps two copies of +the variables from θ to their respective values. We will denote the copy of variable x from θ in θ as x. Variables x in θ function +as temporal variables and are initially set to ⊥. We denote with θ⊥ the partial mapping that maps all variables x to ⊥. Temporal +variables are only needed to track individual variable dependencies for opcodes that are initiating internal transactions. In this +case, many variables are updated simultanously and to distinguish the different dependencies in a fine-grained manner, the +updates are first written into temporal variables and copied to their corresponding variables in θ only later. +Note that for state updates only touching variables in θ, by convention we write λθ.⟨exp⟩ while we use λΘ.⟨exp⟩ to denote +state updates that may also touch temporal variables. + +2) CFG semantics: We closely follow the semantic rules given in [14] and group the rules whenever possible. +a) Binary Stack Operations: We first give the rules for binary stack operations We define +Instbin := {ADD, SUB, LT, GT, EQ, AND, OR, XOR, SLT, SGT, MUL, DIV, SDIV, +MOD, SMOD, SIGNEXTEND, BYTE} +and +costbin(ibin) = +� +3 +ibin ∈ {ADD, SUB, LT, GT, SLT, SGT, EQ, AND, OR, XOR, BYTE} +5 +ibin ∈ {MUL, DIV, SDIV, MOD, SMOD, SIGNEXTEND} +and +funbin(ibin) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +λ(a, b). a + b +mod 2256 +ibin = ADD +λ(a, b). a − b +mod 2256 +ibin = SUB +λ(a, b). a < b ? 1 : 0 +ibin = LT +λ(a, b). a > b ? 1 : 0 +ibin = GT +λ(a, b). a− < b− ? 1 : 0 +ibin = SLT +λ(a, b). a− > b− ? 1 : 0 +ibin = SGT +λ(a, b). a = b ? 1 : 0 +ibin = EQ +λ(a, b). a&b +ibin = AND +λ(a, b). a∥b +ibin = OR +λ(a, b). a ⊕ b +ibin = XOR +λ(a, b). a · b +mod 2256 +ibin = MUL +λ(a, b). (b = 0) ? 0 : ⌊a ÷ b⌋ +ibin = DIV +λ(a, b). (b = 0) ? 0 : a +mod b +ibin = MOD +λ(a, b). (b = 0)? 0 : (a = 2255 ∧ b− = −1)? 2256 : +let x = a− ÷ b− in (sign(x) · ⌊|x|⌋)+ +ibin = SDIV +λ(a, b). (b = 0) ? 0 : (sign(a) · |a| +mod |b|)+ +ibin = SMOD +λ(o, b). (o ≥ 32) ? 0 : b[8 · o, 8 · o + 7] · 0248 +ibin = BYTE +λ(a, b). let x = 256 − 8(a + 1) in +let s = b [x] in sx · b[x, 255] +ibin = SIGNEXTEND +where sign(·) : Intx → {−1, 1} is defined as +sign(x) = +� +1 +x ≥ 0 +0 +otherwise +and &, ∥ and ⊕ are bitwise and, or and xor, respectively. +Exceptions to the normal binary operations are the exponentiation as this instruction uses non-constant costs and the +computation of the Keccack-256 hash. +The CFG rules for these operations are given as follows: +C(pc) = (op(yls, x1 +ls, x2 +ls), pc′, pre) +op ∈ Instbin +f = λθ.θ ← yls := funbin(op)(θ[x1 +ls], θ[x2 +ls]) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {x1 +ls, x2 +ls} +C(pc) = (op(yls, x1 +ls, x2 +ls), pc′, pre) +op ∈ Instbin +f = λθ.θ ← ge := θ[ge] − costbin(op) +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +Exceptions to the normal binary operations are the exponentiation as this instruction uses non-constant costs and the +computation of the Keccack-256 hash. We give their CFG semantics rules separately: + +C(pc) = (EXP(yls, x1 +ls, x2 +ls), pc′, pre) +f = λθ.θ ← yls := θ[x1 +ls]θ[x2 +ls] +mod 2256 +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {x1 +ls, x2 +ls} +C(pc) = (EXP(yls, x1 +ls, x2 +ls), pc′, pre) +c = λθ.(θ[x2 +ls] = 0) ? 10 : 10 + 10 ∗ (1 + +� +log256 θ[x2 +ls] +� +) +f = λθ.θ ← ge := θ[ge] − c(θ) +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel, x2 +ls} +C(pc) = (SHA3(yls, x1 +ls, x2 +ls), pc′, pre) +v = λθ.loadm θ[x1 +ls] θ[x2 +ls] +f = λθ.θ ← yls := Keccak(v(θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {x1 +ls, x2 +ls} ∪ Xm.D ∪ Xm.S +C(pc) = (SHA3(yls, x1 +ls, x2 +ls), pc′, pre) +pos = λθ.θ[x1 +ls] +size = λθ.θ[x2 +ls] +aw = λθ.M (θ[iel], pos(θ), size(θ)) +c = λθ.Cmem (θ[iel], aw(θ)) + 30 + 6 · +�size(θ) +32 +� +f = λθ.θ ← ge := θ[ge] − c(θ) +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = {gel, iel, x1 +ls, x2 +ls} +C(pc) = (SHA3(yls, x1 +ls, x2 +ls), pc′, pre) +pos = λθ.θ[x1 +ls] +size = λθ.θ[x2 +ls] +aw = λθ.M (θ[iel], pos(θ), size(θ)) +f = λθ.θ ← ie := aw(θ) +C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) +Def = {iel} +Use = {iel, x1 +ls, x2 +ls} +Where loadm θ o s is defined as +loadm θ o s := +� +0 +s = 0 +(load θ om) ∗ 256(s−1) + load θ (o + 1) (s − 1) +s > 0 +b) Unary Stack Operations: +C(pc) = (ISZERO(yls, xls), pc′, pre) +r = λθ.(θ[xls] = 0) ? 1 : 0 +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {xls} +C(pc) = (ISZERO(yls, xls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 3 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +C(pc) = (NOT(yls, xlsls), pc′, pre) +r = λθ.¬(θ[x]) +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {xls} +C(pc) = (NOT(yls, xls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 3 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +where ¬ is bitwise negation. + +c) Ternary Stack Operations: +C(pc) = (ADDMOD(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +b = λθ.θ[x2 +ls] +c = λθ.θ[x3 +ls] +r = λθ.(c(θ) = 0) ? 0 : (a(θ) + b(θ)) +mod c(θ) +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {x1 +ls, x2 +ls, x3 +ls} +C(pc) = (ADDMOD(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 8 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +C(pc) = (MULMOD(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +b = λθ.θ[x2 +ls] +c = λθ.θ[x3 +ls] +r = λθ.(c(θ) = 0) ? 0 : (a(θ) · b(θ)) +mod c(θ) +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {x1 +ls, x2 +ls, x3 +ls} +C(pc) = (MULMOD(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 8 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +d) Accessing the execution environment: +C(pc) = (ADDRESS(yls), pc′, pre) +r = λθ.θ[actorel] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {actorel} +Most instructions for accessing the execution environment have the same gas cost. For this reason we summarize the rule +for gas substraction for them. +C(pc) = (op(yls), pc′, pre) +op ∈ {ADDRESS, CALLER, CALLVALUE, CODESIZE, CALLDATASIZE} +f = λθ.θ ← ge := θ[ge] − 2 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +C(pc) = (CALLER(yls), pc′, pre) +r = λθ.θ[senderel] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {senderel} +C(pc) = (CALLVALUE(yls), pc′, pre) +r = λθ.θ[valueel] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {valueel} +C(pc) = (CODESIZE(yls), pc′, pre) +r = λθ.|θ[codeel]| +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {codeel} + +C(pc) = (CALLDATASIZE(yls), pc′, pre) +r = λθ.|θ[inputel]| +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {inputel} +We give individual rules for accessing the code and input data. +The CALLDATALOAD instruction accesses a word of the call data at a specified position +C(pc) = (CALLDATALOAD(yls, xls), pc′, pre) +a = λθ.=θ[xls] +d = λθ.θ[inputel] +size = λθ.|d(θ)| +k = λθ.(size(θ) − a(θ) < 0) ? 0 : min (size(θ) − a(θ), 32) +v = λθ.d(θ) [a(θ), a(θ) + k(θ) − 1] +r = λθ.v(θ) · 0256−k(θ)·8 +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {codeel} +C(pc) = (CALLDATALOAD(yls, xls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 3 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +C(pc) = (CALLDATACOPY(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +posm = λθ.θ[x1 +ls] +posd = λθ.θ[x2 +ls] +size = λθ.θ[x3 +ls] +d = λθ.θ[inputel] +k = λθ.(|d(θ)| − posd(θ) < 0 ? 0 : min (|d(θ)| − posd(θ), size(θ)) +d′ = λθ.d(θ) [posd(θ), posd(θ) + k(θ) − 1] +d = λθ.d′(θ) · 08·(size(θ)−k(θ)) +f = λθ.θ ← (im.D := d(θ)[i])i∈[posm(θ),posm(θ)+size(θ)−1] +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = Xm.D +Use = {inputel, x1 +ls, x2 +ls, x3 +ls} +C(pc) = (CALLDATACOPY(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +posm = λθ.θ[x1 +ls] +posd = λθ.θ[x2 +ls] +size = λθ.θ[x3 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +c = λθ.Cmem (θ[iel], aw(θ)) + 3 + 3 · +�size(θ) +32 +� +f = λθ.θ ← ge := θ[ge] − c(θ) +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 2) +Def = {gel} +Use = {gel, iel, x1 +ls, x2 +ls, x3 +ls} +C(pc) = (CALLDATACOPY(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +posm = λθ.θ[x1 +ls] +posd = λθ.θ[x2 +ls] +size = λθ.θ[x3 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +f = λθ.θ ← ie := aw(θ) +C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) +Def = {iel} +Use = {iel, x1 +ls, x3 +ls} +The rules for copying a fraction of the code to memory (CODECOPY) are similar: +C(pc) = (CODECOPY(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +posm = λθ.θ[x1 +ls] +poscode = λθ.θ[x2 +ls] +size = λθ.θ[x3 +ls] +d = λθ.θ[codeel] +k = λθ.(|d(θ)| − poscode(θ) < 0 ? 0 : min (|d(θ)| − posd(θ), size(θ)) +d′ = λθ.d(θ) [poscode(θ), poscode(θ) + k(θ) − 1] +d = λθ.d′(θ) · STOPsize(θ)−k(θ) +f = λθ.θ ← (im.D := d(θ)[i])i∈[posm(θ),posm(θ)+size(θ)−1] +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = Xm.D +Use = {codeel, x1 +ls, x2 +ls, x3 +ls} + +C(pc) = (CODECOPY(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +posm = λθ.θ[x1 +ls] +poscode = λθ.θ[x2 +ls] +size = λθ.θ[x3 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +c = λθ.Cmem (θ[iel], aw(θ)) + 3 + 3 · +�size(θ) +32 +� +f = λθ.θ ← ge := θ[ge] − c(θ) +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 2) +Def = {gel} +Use = {gel, iel, x1 +ls, x2 +ls, x3 +ls} +C(pc) = (CODECOPY(yls, x1 +ls, x2 +ls, x3 +ls), pc′, pre) +posm = λθ.θ[x1 +ls] +poscode = λθ.θ[x2 +ls] +size = λθ.θ[x3 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +f = λθ.θ ← ie := aw(θ) +C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) +Def = {iel} +Use = {iel, x1 +ls, x3 +ls} +Note that the rules for CALLDATACOPY and CODECOPY could easily be refined to account for preprocessing information. +e) Accessing the transaction environment: +C(pc) = (ORIGIN(yls), pc′, pre) +r = λθ.θ[origineg] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {origineg} +Most instructions for accessing the transaction environment have the same gas cost. For this reason we summarize the rule +for gas substraction for them. +C(pc) = (op(yls), pc′, pre) +op ∈ {ORIGIN, GASPRICE, COINBASE, TIMESTAMP, NUMBER, GASLIMIT, DIFFICULTY} +f = λθ.θ ← ge := θ[ge] − 2 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +C(pc) = (GASPRICE(yls), pc′, pre) +r = λθ.θ[prizeeg] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {prizeeg} +C(pc) = (COINBASE(yls), pc′, pre) +r = λθ.θ[beneficiaryeg] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {beneficiaryeg} +C(pc) = (TIMESTAMP(yls), pc′, pre) +r = λθ.θ[timestampeg] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {timestampeg} +C(pc) = (NUMBER(yls), pc′, pre) +r = λθ.θ[numbereg] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {numbereg} + +C(pc) = (GASLIMIT(yls), pc′, pre) +r = λθ.θ[gaslimiteg] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {gaslimiteg} +C(pc) = (DIFFICULTY(yls), pc′, pre) +r = λθ.θ[difficultyeg] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {difficultyeg} +C(pc) = (BLOCKHASH(yls, xls), pc′, pre) +n = λθ.θ[xls] +r = λθ.P (θ[parenteg], n(θ), 0) +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {parenteg} +where the function P (h, n, a) tries to access the block with number n by traversing the block chain starting from h until +the counter a reaches the limit of 256 or the genesis block is reached. +P (h, n, a) := +� +� +� +� +� +0 +n > h.number ∨ a = 256 ∨ h = 0 +h +n = h.number +P (h.parent, n, a + 1) +otherwise +C(pc) = (BLOCKHASH(yls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 20 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +f) Accessing the global state: +C(pc) = (BALANCE(yls, xls), pc′, pre) +a = λθ.θ[xls] +r = λθ.(θ[externaleg].σ(a(θ) +mod 2160) = (nonce, balance, stor, code)) ? balance : 0 +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {externaleg} +C(pc) = (BALANCE(yls, xls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 400 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +C(pc) = (EXTCODESIZE(yls, xls), pc′, pre) +a = λθ.θ[xls] +r = λθ.|( +� +θ[externaleg].σ(a(θ) +mod 2160) +� +.code| +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {externaleg} +C(pc) = (EXTCODESIZE(yls, xls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 700 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} + +C(pc) = (EXTCODECOPY(yls, x1 +ls, x2 +ls, x3 +ls, x4 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +posm = λθ.θ[x2 +ls] +poscode = λθ.θ[x3 +ls] +size = λθ.θ[x4 +ls] +d = λθ. +� +θ[externaleg].σ(a +mod 2160) +� +.code +k = λθ.(|d(θ)| − poscode(θ) < 0 ? 0 : min (|d(θ)| − posd(θ), size(θ)) +d′ = λθ.d(θ) [poscode(θ), poscode(θ) + k(θ) − 1] +d = λθ.d′(θ) · STOPsize(θ)−k(θ) +f = λθ.θ ← (im.D := d(θ)[i])i∈[posm(θ),posm(θ)+size(θ)−1] +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = Xm.D +Use = {externaleg, x1 +ls, x2 +ls, x3 +ls, x4 +ls} +C(pc) = (EXTCODECOPY(yls, x1 +ls, x2 +ls, x3 +ls, x4 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +posm = λθ.θ[x2 +ls] +poscode = λθ.θ[x3 +ls] +size = λθ.θ[x4 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +c = λθ.Cmem (θ[iel], aw(θ)) + 700 + 3 · +�size(θ) +32 +� +f = λθ.θ ← ge := θ[ge] − c(θ) +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 2) +Def = {gel} +Use = {gel, iel, x2 +ls, x3 +ls, x4 +ls} +C(pc) = (EXTCODECOPY(yls, x1 +ls, x2 +ls, x3 +ls, x4 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +posm = λθ.θ[x2 +ls] +poscode = λθ.θ[x3 +ls] +size = λθ.θ[x4 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +f = λθ.θ ← ie := aw(θ) +C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) +Def = {iel} +Use = {iel, x2 +ls, x4 +ls} +g) Stack Operations: Since we assume the code to be in SSA form, all stack operations have been replaced by assignments. +C(pc) = (ASSIGN(yls, xls), pc′, pre) +r = λθ.θ[xls] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {xls} +C(pc) = (ASSIGN(yls, xls), pc′, pre) +f = λθ.θ ← ge := θ[ge] − 3 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +h) Jump Instructions: For the case of jump instructions by assumption, the jump destination has been precomputed. Since +the JUMP instruction has no other effect than updating the program counter, we only need to add a rule for updating the gas +and stepping to the next program counter: +C(pc) = (JUMP(yls, xls), pc′, pre) +pre[0] = ⌊pc′⌋ +f = λθ.θ ← ge := θ[ge] − 8 +C, cd ⊨ (pc, 0) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} +For the conditional jump instruction, in addition to deducing the gas, the next program counter needs to be decided based +on the condition. +We first give the rules for updating the gas value: +C(pc) = (JUMPI(yls, x1 +ls, x2 +ls), pc′, pre) +pre[0] = ⌊pc′′⌋ +f = λθ.θ ← ge := θ[ge] − 10 +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {gel} +Use = {gel} + +Finally, we give the rules for branching: +C(pc) = (JUMPI(yls, x1 +ls, x2 +ls), pc′, pre) +pre[0] = ⌊pc′′⌋ +Q = λθ.θ[x2 +ls] = 0 +C, cd ⊨ (pc, 1) −(Q)√ −→ (pc′, 0) +Def = ∅ +Use = {x2 +ls} +C(pc) = (JUMPI(yls, x1 +ls, x2 +ls), pc′, pre) +pre[0] = ⌊pc′′⌋ +Q = λθ.θ[x2 +ls] ̸= 0 +C, cd ⊨ (pc, 1) −(Q)√ −→ (pc′′, 0) +Def = ∅ +Use = {x2 +ls} +i) Memory Instructions: +C(pc) = (MLOAD(yls, xls)), pc′, pre) +pre[1] = None +f = λθ.θ ← yls := load θ θ[xls] +m +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = Xm.D ∪ Xm.S +C(pc) = (MLOAD(yls, xls)), pc′, pre) +pre[1] = ⌊x⌋ +f = λθ.θ ← yls := load θ xm +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {xm.S, xm.D} +C(pc) = (MLOAD(yls, xls)), pc′, pre) +a = λθ.θ[xls] +aw = λθ.M (θ[iel], a(θ), 32) +c = λθ.Cmem (θ[iel], aw(θ)) + 3 +f = λθ.θ ← gel := θ[g] − c(θ) +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = {gel, iel, xls} +C(pc) = (MLOAD(yls, xls)), pc′, pre) +a = λθ.θ[xls] +aw = λθ.M (θ[iel], a(θ), 32) +f = λθ.θ ← iel := aw(θ) +C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) +Def = {iel} +Use = {iel, xls} +Note that we do not distinguish between the MLOAD and the MLOADbyte instruction, because we assume the preprocessing +to already check for consistent memory accesses. +C(pc) = (MSTORE(x1 +ls, x2 +ls), pc′, pre) +pre[1] = ⊥ +f = λθ.θ ← +� +θ[x1 +ls] +�m.D := θ[x2 +ls] +C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 2) +Def = Xm.D +Use = Xm.D ∪ {x1 +ls, x2 +ls} +C(pc) = (MSTORE(x1 +ls, x2 +ls), pc′, pre) +pre[0] = ⌊x⌋ +f = λθ.θ ← xm.S := θ[x2 +ls] +C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 1) +Def = {xm.S} +Use = {x2 +ls} +C(pc) = (MSTORE(x1 +ls, x2 +ls), pc′, pre) +pre[0] = ⌊x⌋ +f = λθ.θ ← xm.D := ⊥ +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {xm.D} +Use = ∅ +C(pc) = (MSTORE(x1 +ls, x2 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +aw = λθ.M (θ[iel], a(θ), 32) +c = λθ.Cmem (θ[iel], aw(θ)) + 3 +f = λθ.θ ← gel := θ[g] − c(θ) +C, cd ⊨ (pc, 2) −⇑f −→ (pc, start + 5) +Def = {gel} +Use = {gel, iel, x1 +ls} + +C(pc) = (MSTORE(x1 +ls, x2 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +aw = λθ.M (θ[iel], a(θ), 32) +f = λθ.θ ← iel := aw(θ) +C, cd ⊨ (pc, start + 5) −⇑f −→ (pc′, 0) +Def = {iel} +Use = {iel, x1 +ls} +j) Storage Instructions: The storage instructions closely resemble the instructions for memory access: +C(pc) = (SLOAD(yls, xls)), pc′, pre) +pre[1] = None +f = λθ.θ ← yls := load θ θ[xls] +g +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = Xg.D ∪ Xg.S +C(pc) = (SLOAD(yls, xls)), pc′, pre) +pre[1] = ⌊x⌋ +f = λθ.θ ← yls := load θ xg +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {xg.S, xg.D} +C(pc) = (SLOAD(yls, xls)), pc′, pre) +a = λθ.θ[xls] +f = λθ.θ ← gel := θ[g] − 200 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel, } +C(pc) = (SSTORE(x1 +ls, x2 +ls), pc′, pre) +pre[1] = ⊥ +f = λθ.θ ← +� +θ[x1 +ls] +�g.D := θ[x2 +ls] +C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 2) +Def = Xg.D +Use = Xg.D ∪ {x1 +ls, x2 +ls} +C(pc) = (SSTORE(x1 +ls, x2 +ls), pc′, pre) +pre[0] = ⌊x⌋ +f = λθ.θ ← xg.S := θ[x2 +ls] +C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 1) +Def = {xg.S} +Use = {x2 +ls} +C(pc) = (SSTORE(x1 +ls, x2 +ls), pc′, pre) +pre[0] = ⌊x⌋ +f = λθ.θ ← xg.D := ⊥ +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {xg.D} +Use = ∅ +C(pc) = (SSTORE(x1 +ls, x2 +ls), pc′, pre) +a = λθ.θ[x1 +ls] +b = λθ.θ[x2 +ls] +c = λθ.(b(θ) ̸= 0 ∧ (load θ (θ[a(θ)])g = 0) ? 20000 : 5000 +f = λθ.θ ← gel := θ[g] − c(θ) +C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel, x1 +ls} ∪ Xm.D ∪ Xm.S +k) Accessing the machine state: +C(pc) = (GAS(yls), pc′, pre) +r = λθ.θ[gel] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {gel} +Most instructions for accessing the machine state have the same gas cost. For this reason we summarize the rule for gas +substraction for them. +C(pc) = (op(yls), pc′, pre) +op ∈ {GAS, PC, MSIZE} +f = λθ.θ ← ge := θ[ge] − 2 +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {gel} +Use = {gel} + +C(pc) = (MSIZE(yls), pc′, pre) +r = λθ.θ[iel] +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {iel} +C(pc) = (PC(yls), pc′, pre) +r = λθ.pc +f = λθ.θ ← yls := r(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = ∅ +l) Logging: Since we do not model the logged events, for the logging instructions we only need to model the effect on +gas and local memory. +C(pc) = (LOGn(x1 +ls, x2 +ls, . . . , xn+2 +ls), pc′, pre) +posMem = λθ.θ[x1 +ls] +size = λθ.θ[x2 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +c = λθ.Cmem (θ[iel], aw(θ)) + 375 + 8 · size(θ) + n · 375 +f = λθ.θ ← gel := θ[gel] − c(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {gel} +Use = {gel, iel, x1 +ls, x2 +ls} +C(pc) = (LOGn(x1 +ls, x2 +ls, . . . , xn+2 +ls), pc′, pre) +posMem = λθ.θ[x1 +ls] +size = λθ.θ[x2 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +f = λθ.θ ← iel := aw(θ) +C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) +Def = {iel} +Use = {iel, x1 +ls, x2 +ls} +m) Halting instructions: +C(pc) = (RETURN(x1 +ls, x2 +ls, pc′, pre) +posMem = λθ.θ[x1 +ls] +size = λθ.θ[x2 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +c = λθ.Cmem (θ[iel], aw(θ)) +f = λθ.θ ← gel := θ[gel] − c(θ) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {gel} +Use = {gel, iel, x1 +ls, x2 +ls} +C(pc) = (RETURN(x1 +ls, x2 +ls, pc′, pre) +posMem = λθ.θ[x1 +ls] +size = λθ.θ[x2 +ls] +aw = λθ.M (θ[iel], posm(θ), size(θ)) +f = λθ.θ ← iel := aw(θ) +C, cd ⊨ (pc, 1) −⇑f −→ halt +Def = {iel} +Use = {iel, x1 +ls, x2 +ls} +C(pc) = (STOP, pc′, pre) +f = λθ.θ +C, cd ⊨ (pc, 0) −⇑f −→ halt +Def = ∅ +Use = ∅ +C(pc) = (INVALID, pc′, pre) +f = λθ.θ +C, cd ⊨ (pc, 0) −⇑f −→ exception +Def = ∅ +Use = ∅ + +The SELFDESTRUCT instruction allows for the self destruction of the executing account. Since we only model the +execution of a single contract and since the execution halts with the execution of SELFDESTRUCT, we do not model the +deletion of the contract itself. However, the gas cost of the SELFDESTRUCT instruction, as well as the balances of accounts +are affected by the execution of the SELFDESTRUCT instruction. +C(pc) = (SELFDESTRUCT(xls), pc′, pre) +aben = λθ.θ[xls] +a = λθ.aben(θ) +mod 2160 +σ = λθ.θ[externalel].σ +φ = λθ.σ(θ)(a) = ⊥ +f1 = λθ.θ ← externaleg := θ[externaleg][σ → σ(θ) +� +θ[actorel] → σ(θ)[b → 0] +� +] +f2 = λθ.θ ← externaleg := θ[externaleg][σ → σ(θ) +� +a(θ) → σ(θ)[b += σ(θ)(θ[actorel]).b] +� +] +f3 = λθ.θ ← externaleg := θ[externaleg][σ → σ(θ) +� +a(θ) → (0, σ(θ)(θ[actorel]).b, λx. 0, ϵ) +� +] +f = λθ.(φ(θ)) ? f1(f2(θ)) : f1(f3(θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {externaleg} +Use = {externaleg, xls, actorel} +C(pc) = (SELFDESTRUCT(xls), pc′, pre) +aben = λθ.θ[xls] +a = λθ.aben(θ) +mod 2160 +σ = λθ.θ[externalel].σ +φ = λθ.σ(θ)(a) = ⊥ +c = λθ.(φ(θ)) ? 37000 : 5000 +f = λθ.θ ← gel := θ[gel] − c(θ) +C, cd ⊨ (pc, 1) −⇑f −→ halt +Def = {gel} +Use = {externaleg, xls, gel} +n) Rules for transaction initiating instructions.: Since we will exclude the execution of DELEGATECALL and +CALLCODE statements by assumption, we will only give the CFG rules for CALL, STATICCALL, and CREATE. +CALL: The definition of the CALL rule comes with small technical difficulties. In order to give a precise analysis, the +different changes in the state that are triggered by the call need to be done in separate nodes whenever they have different +dependencies. While e.g., the output data fragment can be influenced by the input to the call as well as by the global environment, +the memory outside of the output fragment will simply be propagated. Similarly, the number of active words i only depends +on the arguments specifying the input and output memory fragment, but not on any other inputs or the global environment. +To model these dependencies accurately, the corresponding updates need to happen in different nodes. However, we can +only characterize the overall call effects on a state (using the EVM small-step semantics). +We first define the function applyCall that mimics the effects of a function call on a CFG state θ. For +simplicity, we define the function here as a relation. However, this relation is functional given that C(pc) += +(CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre). +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre)) +(Γ, s) = toEVM(θ, C, pc) +Γ ⊨ s :: S +T−→ s′ :: S +θ′ = toCFG(Γ, s′) +applyCall(θ, C, pc) = θ′ +Note that applyCall operates on a CFG state without temporal variables (since this is what toEVM is expecting and toCFG +is returning). When using applyCall in the following rules in conjunction with full CFG states Θ, we will write Θ ↓D(θ) to +denote the restriction of Θ to the set of non-temporal variables (D(θ)). +The rules update the different state components one after the other, grouping those updates together that have the same +dependencies. Technically, for propagating pc-indexed state components, we need to update all variables in individual notes. +To this end, first, the temporal variables in θ are set to the values of θ′ (the state after applying the call). Finally, the variables +of θ are (one by one) updated to the values of θ and θ is set to θ⊥. +We now define the CFG rules for the CALL instruction. +For the precise treatment of dependencies, the other state updates are treated differently depending on the preprocessing +information. We always need to consider all possible effects on the state. More precisely, the effect on the local return memory +fraction, on the return value y, on the external environment, on the gas, and on the active words and memory. The effects on +the return value and the external environment depend on the same variables (which determine the overall outcome of the call): +• the arguments to the call +• the input memory fragment (as specified by the arguments to the call) +• the current amount of gas available +• the global environment (including the state of all other accounts and all globally accessible values) +• the global variables of the contract (since those may be read during reentrancy) + +The return memory fragment after the call, in addition, may depend on the previous values in this memory fraction. This is, +because in the case that the call was unsuccessful (returned with an exception), the return memory fragment stays unchanged. +The gas value after the call (in addition to the call outcome that influences the amount of gas refunded) depends, also, on +the active words in memory (since this influences the costs for memory access for writing to the return memory fragment). +Finally, the active words in memory only depend on the location and size of the input and return memory fragment and the +previous number of active words in memory. +We give rules for these four different forms of dependencies. To track the dependencies precisely, we first write the updated +values into the corresponding temporal variables and only update the original variables later. This is required so that we can use +the applyCall function on the original state to obtain the updated values separately. Since we are considering purely functional +state updates, we cannot simply save the original result of the applyCall function, but need to recompute it at every node in +order to obtain the needed values. To denote that we are updating the full CFG state, we write the state update function as +λΘ.(⟨exp⟩). To refer to the restriction of the Θ to the non-temporal variables, we write Θ ↓D(θ). +We first define the rules for the updates of y and the external environment. These rules are different because the STATICCALL +does not change the static environment. +C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +f1 = λΘ.Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls, externaleg} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} +C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +f1 = λΘ.Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls, externaleg} +Use = Xm.S ∪ Xm.D ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg +∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} + +C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3 +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +f1 = λΘ.Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} +C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3 +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +f1 = λΘ.Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls} +Use = Xm.S ∪ Xm.D ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg +∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} +Next, we define the rules for the update of the gas value: +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +f = λΘ.Θ ← gel := applyCall(Θ ↓D(θ), C, pc)[gel] +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {gls, tols, vals, gel, iel, actorel} ∪ Xeg ∪ Xg ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +f = λΘ.Θ ← gel := applyCall(Θ ↓D(θ), C, pc)[gel] +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = Xm.S ∪ Xm.D ∪ {gls, tols, vals, gel, iel, actorel} ∪ Xeg ∪ Xg +∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} + +Next, we give the rule for the update of the active words in memory: +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +f = λΘ.Θ ← iel := applyCall(Θ ↓D(θ), C, pc)[iel] +C, cd ⊨ (pc, 2) −⇑f −→ (pc, 3) +Def = {iel} +Use = {iel} ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = +⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} +Finally, we give the rules for the memory update. These rules are the most interesting ones since they differ heavily depending +on the available pre-processing information. +We first consider the case that the input and the result memory fragment are known. In this case, the values of memory +locations im.S are assigned in individual nodes. The node splitting, in this case, allows for precise treatment, since it only +needs to be considered that the value at memory location i may depend on the input to the call or the previous value at +exactly this location i. For assigning the ⊥ value to dynamic memory locations im.D, this distinction is not needed since no +dependencies are propagated in the first place. +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← im.S := load applyCall(Θ ↓D(θ), C, pc) im +i ∈ [xoo, xoo + xos − 1] +C, cd ⊨ (pc, 3 + (i − xoo)) −⇑f −→ (pc, 3 + (i − xoo) + 1) +Def = {im.S} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} +∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {im.S, im.D} +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← (xm.D := ⊥)x∈[xoom,xoom+xosm−1] +C, cd ⊨ (pc, 3 + xos) −⇑f −→ (pc, 4 + xos) +Def = {xm.D | x ∈ [xoo, xoo + xos − 1]} +Use = ∅ +After updating the memory, it is still required to set the values of all updated variables to the corresponding temporal +variables and to set the temporal variables back to ⊥. To determine the right offset for the different cases, we define a +function getNodeOffset that given the precomputed memory fragments outputs the corresponding node offsets. More precisely, +it computes the number of intermediate nodes required for the node splitting in the different cases. +getNodeOffset(io, is, oo, os) = +� +� +� +� +� +� +� +� +� +xos +io = ⌊xio⌋ ∧ is = ⌊xis⌋ ∧ oo = ⌊xoo⌋ ∧ os = ⌊xos⌋ +MAXInt256 +io = ⌊xio⌋ ∧ is = ⌊xis⌋ ∧ (oo = ⊥ ∨ os = ⊥) +1 +(io = ⊥ ∨ is = ⊥) ∧ oo = ⌊xoo⌋ ∧ os = ⌊xos⌋ +0 +otherwise +We give the rules for the stack/external environment, gas and active words in memory only once, since they are the same +for all cases (for different node offsets). + +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +f = λΘ.(Θ ← yls := Θ[yls]) ← externale := Θ[externale] +C, cd ⊨ (pc, reset) −⇑f −→ (pc, reset + 1) +Def = {yls, externale} +Use = {yls, externale} +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +f = λΘ.Θ ← ge := Θ[ge] +C, cd ⊨ (pc, reset + 1) −⇑f −→ (pc, reset + 2) +Def = {ge} +Use = {ge} +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +f = λΘ.Θ ← ie := Θ[ie] +C, cd ⊨ (pc, reset + 2) −⇑f −→ (pc, reset + 3) +Def = {ie} +Use = {ie} +Next, the values of the updated memory locations need to be carried over one by one. The number of nodes needed for that +again depends on the pre-computed values for the input and output memory fraction. +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← im.S := Θ[im.S] +i ∈ [xoo, xoo + xos − 1] +C, cd ⊨ (pc, reset + 3 + (i − xoo)) −⇑f −→ (pc, reset + 3 + (i − xoo) + 1) +Def = {im.S} +Use = {im.S} +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← (xm.D := Θ[xm.D])x∈[xoom,xoom+xosm−1] +C, cd ⊨ (pc, reset + 3 + xos) −⇑f −→ (pc, reset + xos + 4) +Def = {xm.D | x ∈ [xoo, xoo + xos − 1]} +Use = {xm.D | x ∈ [xoo, xoo + xos − 1]} + +Finally, all temporal variables are set to ⊥. This is the same for all cases, irrespective of the pre-computed values for the +input and output memory fraction. +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 2 ∗ (4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3])) +f = λΘ.Θ ← (x := ⊥)x∈D(θ) +C, cd ⊨ (pc, reset) −⇑f −→ (pc′, 0) +Def = D(θ) +Use = ∅ +In the case that only the input memory is known, the dependencies need to be propagated to the whole memory (as potential +output). Note that in this case, we still gain precision by node splitting since, otherwise, we would need to propagate the +dependencies of the whole memory to the whole memory again. By node splitting, we ensure that only the dependencies of +the input memory fragment are propagated to the whole memory. +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +(pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) +f = λΘ.Θ ← im.D := load applyCall(Θ ↓D(θ), C, pc) im +i ∈ N256 +C, cd ⊨ (pc, 3 + i) −⇑f −→ (pc, 3 + i + 1) +Def = {im.D} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {im.S, im.D} ∪{ools | pre[omem +2] = ⌊xoo⌋} ∪{osls | pre[omem +3] = ⌊xos⌋} +Again, the values of the temporary memory variables need to be one-by-one written to the non-temporal variables: +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +(pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) +f = λΘ.Θ ← im.D := Θ[im.D] +i ∈ N256 +C, cd ⊨ (pc, reset + 3 + i) −⇑f −→ (pc, reset + 3 + i + 1) +Def = {im.D} +Use = {im.D} +In the case that only the output memory is known, the dependencies from the whole memory need to be propagated, but +only to a small memory fraction. In this scenario, node splitting does not help since anyway each affected memory node gets +already the dependencies from the whole memory assigned: + +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← (im.S := load applyCall(Θ ↓D(θ), C, pc) im)i∈[xoo,xoo+xos−1] +C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) +Def = {xm.S | x ∈ [xoo, xoo + xos − 1]} +Use = Xm.S ∪ Xm.D ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪{iols | pre[omem] = ⌊xio⌋} ∪{isls | pre[omem+1] = ⌊xis⌋} +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← (xm.D := ⊥)x∈[xoo,xoo+xos−1] +C, cd ⊨ (pc, 4) −⇑f −→ (pc, 5) +Def = {xm.D | x ∈ [xoo, xoo + xos − 1]} +Use = ∅ +Similar to the previous cases, the temporal memory variables are carried over afterward: +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← (im.S := Θ[im.S])i∈[xoo,xoo+xos−1] +C, cd ⊨ (pc, reset + 3) −⇑f −→ (pc, reset + 4) +Def = {xm.S | x ∈ [xoo, xoo + xos − 1]} +Use = {xm.S | x ∈ [xoo, xoo + xos − 1]} +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +pre[omem + 2] = ⌊xoo⌋ +pre[omem + 3] = ⌊xos⌋ +f = λΘ.Θ ← (xm.D := Θ[xm.D])x∈[xoo,xoo+xos−1] +C, cd ⊨ (pc, reset + 4) −⇑f −→ (pc, reset + 5) +Def = {xm.D | x ∈ [xoo, xoo + xos − 1]} +Use = {xm.D | x ∈ [xoo, xoo + xos − 1]} +Finally, if neither input nor result memory fraction can be determined the whole memory needs to be considered the whole +memory. We can characterize this by a single rule as follows: + +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +(pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) +f = λΘ.Θ ← (im.D := load applyCall(Θ ↓D(θ), C, pc) im)i∈[Θ[ools],Θ[ools]+Θ[ools]−1] +C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) +Def = Xm.D +Use = Xm.S ∪ Xm.D ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = +⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} +The rules for carrying over the temporal memory variables are as follows: +(C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 +∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) +reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +(pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) +f = λΘ.Θ ← (im.D := Θ[im.D])i∈[Θ[ools],Θ[ools]+Θ[ools]−1] +C, cd ⊨ (pc, reset + 3) −⇑f −→ (pc, reset + 4) +Def = Xm.D +Use = Xm.D +It is important to note that the function applyCall(θ, C, pc) in all of the different rules returns the same result since only +temporal variables are altered before each call of applyCall (and those are not used by applyCall(θ, C, pc)). +Note that we make here use of the distinction between local and global environment variables. Intuitively, the global +environment variables can be accessed by other contracts as well and hence may influence the outcome of the call. Consequently, +they need to be included in the Use sets of all the rules applying the effects of the call to the state. The local environment +variable gel plays a special role in that the current amount of gas may influence the amount of gas given to the call and hence +also the outcome of the execution. For this reason, gel needs to be included in the Use set. +We give the rules for the CREATE opcode in a similar fashion: +We first devise a rule for the application of a create transaction: +(C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) ∨ C(pc) = (CREATE2(yls, vals, saltlsiols, isls, pc′, pre)) +(Γ, s) = toEVM(θ, C, pc) +Γ ⊨ s :: S +T−→ s′ :: S +θ′ = toCFG(Γ, s′) +applyCreate(θ, C, pc) = θ′ +As opposed to call instructions, create instructions do not expect a return value written to memory, but only the resulting +address of the created account is written to the stack. However, we need to consider that in case an exception occurs, 0 is +written to the stack. An exception can occur if the execution of the initialization code causes an exception. Since the behavior +of the initialization code may again depend on the environment, the value written to the stack can be dependent on the global +environment. +We, hence, can again summarize the rules for the return value yls and the external environment. Similar to the CALL rules, +all updates are first done to temporal variables and only later transferred to the actual ones. + +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +f1 = λθ.Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls, externaleg} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {vals, gel, actorel} ∪ Xeg ∪ Xg +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +f1 = λΘ.Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls, externaleg} +Use = Xm.S ∪ Xm.D ∪ {vals, gel, actorel} ∪ Xeg ∪ Xg +∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} +We give the rules for the gas computation: +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +f = λΘ.Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {vals, gel, iel, actorel} ∪ Xeg ∪ Xg +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +f = λΘ.Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = Xm.S ∪ Xm.D ∪ {vals, gel, iel, actorel} ∪ Xeg ∪ Xg +∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} +Next, we give the rule for the update of the active words in memory: +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +f = λΘ.Θ ← iel := applyCreate(Θ ↓D(θ), C, pc)[iel] +C, cd ⊨ (pc, 2) −⇑f −→ (pc, 3) +Def = {iel} +Use = {iel} ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} +We give the rules for writing the temporal variables into the actual ones one by one: + +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +f1 = λθ.Θ ← yls := Θ[yls] +f2 = λΘ.Θ ← externaleg := Θ[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) +Def = {yls, externaleg} +Use = {yls, externaleg} +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +f = λθ.Θ ← gel := Θ[gel] +C, cd ⊨ (pc, 4) −⇑f −→ (pc, 5) +Def = {gel} +Use = {gel} +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +f = λθ.Θ ← iel := Θ[iel] +C, cd ⊨ (pc, 5) −⇑f −→ (pc, 6) +Def = {iel} +Use = {iel} +Finally, all temporal variables are set to ⊥ again: +C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) +omem = 2 +f = λΘ.Θ ← (x := ⊥)x∈D(θ) +C, cd ⊨ (pc, 6) −⇑f −→ (pc′, 0) +Def = D(θ) +Use = ∅ +Finally, the instruction CREATE2 operates in a similar fashion as CREATE with the main difference being that the newly +created contract is assigned an address that can be predetermined. To this end, CREATE2 takes an additional argument salt, +which together with the creation code determines the address. Correspondingly, the rules for CREATE2 closely follow those +of CREATE. +We can again summarize the rules for the return value yls and the external environment. Similar to the CREATE and CALL +rules, all updates are first done to temporal variables and only later transferred to the actual ones. +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +f1 = λθ.Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls, externaleg} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {vals, saltls, gel, actorel} ∪ Xeg ∪ Xg + +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +f1 = λΘ.Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] +f2 = λΘ.Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) +Def = {yls, externaleg} +Use = Xm.S ∪ Xm.D ∪ {vals, saltlsgel, actorel} ∪ Xeg ∪ Xg +∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} +We give the rules for the gas computation: +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +pre[omem] = ⌊xio⌋ +pre[omem + 1] = ⌊xis⌋ +f = λΘ.Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = {xm.S | x ∈ [xio, xio + xis − 1]} ∪ {xm.D | x ∈ [xio, xio + xis − 1]} +∪ {vals, saltls, gel, iel, actorel} ∪ Xeg ∪ Xg +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +(pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) +f = λΘ.Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] +C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) +Def = {gel} +Use = Xm.S ∪ Xm.D ∪ {vals, saltls, gel, iel, actorel} ∪ Xeg ∪ Xg +∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} +Next, we give the rule for the update of the active words in memory: +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +f = λΘ.Θ ← iel := applyCreate(Θ ↓D(θ), C, pc)[iel] +C, cd ⊨ (pc, 2) −⇑f −→ (pc, 3) +Def = {iel} +Use = {iel} ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} +We give the rules for writing the temporal variables into the actual ones one by one: +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +f1 = λθ.Θ ← yls := Θ[yls] +f2 = λΘ.Θ ← externaleg := Θ[externaleg] +f = λΘ.f2(f1(Θ)) +C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) +Def = {yls, externaleg} +Use = {yls, externaleg} + +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +f = λθ.Θ ← gel := Θ[gel] +C, cd ⊨ (pc, 4) −⇑f −→ (pc, 5) +Def = {gel} +Use = {gel} +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +f = λθ.Θ ← iel := Θ[iel] +C, cd ⊨ (pc, 5) −⇑f −→ (pc, 6) +Def = {iel} +Use = {iel} +Finally, all temporal variables are set to ⊥ again: +C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) +omem = 2 +f = λΘ.Θ ← (x := ⊥)x∈D(θ) +C, cd ⊨ (pc, 6) −⇑f −→ (pc′, 0) +Def = D(θ) +Use = ∅ +o) From CFG semantics to Logical Rules: We illustrate how the logical rules describing the PDG derived from the CFG +semantics are constructed. +The CFG semantics describes the PDG by giving control dependencies (via the CFG) and data dependencies via the Def +and Use sets. Each transition rule introduces data dependencies from all variables in the Def set to all variables in the Use +set. The node splitting allows for enhancing precision since assignments of several variables that do not share the same Use +set can be distinguished in a more fine-grained manner. +For translating the CFG rules into dependency predicates, it is simply required to model the resulting data and control flow +dependencies. However, we need to introduce a further abstraction step to account for the fact that Def and Use sets may be +infinite (or at least unreasonably large, assuming that memory and storage locations can be represented by 256 bits). More +precisely, we will introduce a symbolic variable ⊤, which we will use to summarize memory and storage variables. Intuitively, +⊤ when used for modeling variable access (in the Use set) will represent the union of all dynamic and static memory (or +storage) variables (Xm.D ∪ Xm.S or Xg.D ∪ Xg.S, respectively). When used to model writing variables (in the Def set), ⊤ +will represent all dynamic memory (or storage) variables (Xm.D or Xg.D, respectively). +To model this, we will assume the following types for our predicate domains: +L := N256 ∪ {⊤} +T := I +where I is the set of all instructions. Intuitively, L encodes the type of all (symbolic) storage locations and T encodes the +types of so-called tags. Tags model those variables on which we explicitly want to track dependencies. For the scope of this +work, we will only track dependencies on static environment variables, which we will (for simplicity) all represent by the +opcodes that access these variables. For this reason, we define T to consist of the set of all instructions I. +To capture the dependencies as induced by the Def and Use sets, we define local data dependency predicates that describe +the data dependencies between variables at specific nodes. +As opposed to directly specifying the Def and Use sets, these predicates enumerate all pairs of variables in the Def and the +Use set (the cross-product between them). In this way, we do not need to make the subnodes at a given program counter (as +given in the CFG semantics) explicit. These subnodes result from node splitting and only aim for separating the dependencies +for different nodes in the Def set. Consequently, we can easily mimic this effect by directly modeling dependencies between +variables as they are induced by the Def and Use set at a given subnode. +For efficiency reasons, we consider different variable types and devise predicates that describe the local dependencies between +these types (as induced by the CFG nodes for a specific program counter). This results in improved performance since it enables + +the underlying datalog solver to compute several smaller fixpoints (for each variable type) instead of a big fixpoint (that captures +the dependencies for all variables). +More precisely, we define the following predicates (indexed by the program counter) for the different combinations of variable +types as follows, where their name indicates the corresponding type (Var for stack variables, Mem for memory variables, Store +for storage variables, Gas for local environmental variable gel, Msize for the local environmental variable iel, and External for +the global environmental variables externaleg, and Source the static local and global environmental variables). +VarVarpc ⊆ N256 × N256 +VarMempc ⊆ N256 × L +VarStorpc ⊆ N256 × L +VarExternalpc ⊆ N256 +VarGaspc ⊆ N256 +VarSourcepc ⊆ N256 × T +StoreVarpc ⊆ L × N256 +MemMempc ⊆ L × L × L +MemVarpc ⊆ L × N256 +MemExternalpc ⊆ L +MemGaspc ⊆ L +MemMsizepc ⊆ L +MemStorepc ⊆ L × L +MemSourcepc ⊆ L × T +GasMempc ⊆ L +GasVarpc ⊆ N256 +GasExternalpc ⊆ B +GasMsizepc ⊆ B +GasStorepc ⊆ L +GasSourcepc ⊆ T +MsizeVarpc ⊆ N256 +ExternalMempc ⊆ L +ExternalVarpc ⊆ N256 +ExternalGaspc ⊆ B +ExternalMsizepc ⊆ B +ExternalStorepc ⊆ L +ExternalSourcepc ⊆ T +Where +⟨write⟩⟨read⟩ +indicates +for +write +∈ +{Var, Mem, Store, Gas.Msize, External} +and +read +∈ +{Var, Mem, Store, Gas.Msize, External, Source} that variable kind write is written and variable kind read is read. +E.g., VarMem(x, y) indicates that stack variable xls is written dependent on dynamic and static memory variables +{ym.S, ym.D} and VarMem(x, ⊤) indicates that stack variable xls depends on all static and dynamic memory variables +(Xm.S ∪ Xm.D). Similarly, MemVar(x, y) indicates that the static memory location xm.S depends on stack variable yls, +and MemVar(⊤, y) indicates that all dynamic memory variables Xm.D depend on stack variable yls. Note that the variable ⊤ +is used in the symbolic fashion described above. +A special case of this symbolic treatment is the predicate MemMem, which takes three arguments to give a more fine-grained +symbolic modeling of whole memory intervals: The first position of MemMem specifies a memory location, and the two next +positions specify a memory interval, which is given by its start offset and size. Start offset and size can again be of type L, +such that MemMempc(x, ⊤, ⊤) indicates that the (static) memory variable xm.S depends on all static and dynamic memory +variables and MemMempc(x, i, s) indicates that xm depends on all static and dynamic memory variables starting at memory +position i until i + s − 1. +Note that write can never be Source since static local and global environment variables can never be written. Similarly, other +write-read combinations are omitted for cases that never occur (e.g., for store unreachable contracts, the only way to write +the contract’s storage is the SSTORE opcode, that allows for storing a stack variable. Consequently, within a CFG node, a +storage variable can only depend on the stack variables, so the predicate StoreVar is sufficient to capture all local dependencies +of storage variables). +The local dependency predicates can be simply inhabited by rules that closely follow the CFG semantics: For each program +counter pc, instruction-specific rules are generated that reflect the dependencies induced by the Def and Use sets of the subnodes +of pc. We give the example for the MLOAD instruction: +{⊤ ⇒ VarMem(y, ⊤) | +C(pc) = (MLOAD(yls, xls, pc′, pre) ∧ pre[0] = ⊥ +⊤ ⇒ MsizeVarpc(x) +⊤ ⇒ GasVarpc(x) +⊤ ⇒ GasMsizepc(⊤)} +{⊤ ⇒ VarMem(y, v) | +C(pc) = (MLOAD(yls, xls, pc′, pre) ∧ pre[0] = v +⊤ ⇒ GasMsizepc(⊤)} +The given rules describe the dependencies induced by the corresponding CFG rules: The result variable yls either depends on +all static and dynamic memory locations (if the memory location is unknown) or on the specific static and dynamic memory +locations {vm.S, vm.D}. The gas gel depends on the value of the active words in memory and on the stack variable xls, which +holds the memory position. Similarly, iel depends on xls. Note that we do not explicitly model that gel and iel always depends on +themselves since this is always the case, and hence we account for this by generic propagation rules, which always propagate +gas and active word dependencies to the next program counter. + +In addition to the local dependency predicates there exist special predicates that indicate that a variable is written in the first +place: +MsizeWritepc ⊆ B +ExternalWritepc ⊆ B +These predicates encode that the corresponding variable (here iel or externaleg) is written at a specific program counter pc. +We need these predicates for expressing the interaction between data and control dependence (for building backward slices): +If a variable x is written at a certain node n′, which is control-dependent on another node n, and x is read at another node +n′′ (without being overwritten before), then n′′ is data dependent on n′ and by transitivity, n′′ depends on n (via one data +dependency and one control dependency edge). For this reason, it is important to model when a variable is written. When +making Def and Use sets fully explicit, this is easy to see, however, in our modeling, we immediately consider the cross-product +from the Def and Use sets (at a program counter). Consequently, it can happen that there are no entries for certain variables +in the Def Set (if there is no variable in the Use set). So, we need to cover these cases explicitly. +For all other variables (but iel or externaleg), we can use existing predicates as indicators for writing. E.g., whenever a +variable is written the VarSource predicate is inhabited. Similarly, whenever memory or storage variables are written MemVar +or, respectively StorVar are inhabited for the corresponding variables. Gas is written at any program counter. +To model the transitive dependencies (as induced by the PDG), we use the local data dependencies (as modeled by the +local dependency predicates above) and the control dependencies (pre-computed according to the definition of standard control +dependence) and build their transitive closure. The control dependence is available via a predicate Controls. First, the transitive +closure for control dependence is modeled via the predicate MayControlspc ⊆ N256 ×N256. Intuitively, MayControlspc(pcb, xb) +means that the program counter pc is transitively controled by the program counter pcb where at pcb there is a brach instruction +(JUMPI) with condition stack variable xbls. +Next, we define fixed point rules, which inhabit the following transitive closure predicates for program dependence: +VarMayDependOn ⊆ N × T +MemMayDependOnpc ⊆ L × T +StorMayDependOnpc ⊆ L × T +MsizeDependOnpc ⊆ T +GasDependOnpc ⊆ T +ExternalDependOnpc ⊆ T +Intuitively, VarMayDependOn(x, t) denotes that variable x may depend on tag t, so that a node n′ where t is in the Use set, +is in the backward slice of the (unique) node where x is written. Note that the tag t represents a static environment variable. +We index the predicates (with exception of VarMayDependOn) by the program counter to precisely characterize data +dependence: A node n′ is considered data dependant on another node n if n defines a variable x that is used by n′ and +n′ is reachable from n without passing through another node defining x. Since we aim at staying within a characterization of +dependencies that only uses grounded Horn clauses, we cannot simply express the second requirement (namely that no other +node defining x should be passed). Instead, we explicitly formulate rules propagating dependencies in the case that at a certain +program counter a variable is not (over)written. E.g., we can formulate a generic rule for gas propagation, since gas is updated +at every program counter (and hence is contained in the Def and Use set). +Note that dependencies of stack variables (VarMayDependOn), as opposed to the other dependency predicates, is not indexed +by the concrete node. This is because the contract is assumed to be in SSA form (for stack variables), and those should hence +only appear at only a single program location. +Note that (similar to Securify), we currently only explicitly track transitive dependencies on local and global static +environment variables (of type T ). E.g., we can express that a stack variable transitively depends on, e.g., the block timestamp, +but not that it transitively depends on e.g., a specific memory variable (however, it is, of course, captured that dependencies +on global static environment variables can be introduced through dependencies on other variables). The analysis can easily +be extended to track further dependencies explicitly by adding rules introducing the corresponding dependencies to the +corresponding ⟨write⟩Source predicate. +The rules inhabiting the fixed point predicates are fairly standard. They are slightly complicated by the fact that we consider +different variable types with different predicates so we need to consider data dependencies described by all the different local + +dependence predicates. Considering all possible combinations, introduces a slight overhead in rules (as compared to having a +single predicate for variable types), but results in better performance, since it splits the fixed point computations into several +smaller fixpoints. Further, there are some subtleties to consider for our symbolic treatment of dynamic memory locations and +memory intervals (as discussed above). +We illustrate this by the fixpoint rules for the MemMayDependOn predicate given in Figures 21 and 22. +Figure 21 shows the rules for describing transitive data dependencies for memory locations (captured by the +MemMayDependOn predicate). To this end, there are rules for all local dependency predicates, which indicate that the +local memory is written. Intuitively, the rules model the data dependencies introduced by the nodes at pc′, by propagating +dependencies known for the previous program counter pc′. Rule 1 simply introduces dependencies from the MemSource +predicates (constituting the base case). The rules for propagating dependencies from variables(2), gas (4), active words in +memory (5), and the external environment (6) are fully standard. We need to consider that for all memory locations which are +not overwritten, the dependencies from the previous program counter (pc) are propagated. To this end, we define the predicate +NoReassignMem that contains those opcodes that do not overwrite any memory location. It only contains all instructions but +MSTORE and the copy operations. For all other operations, all previous dependencies are propagated (by rule 14). For the +overwriting operations, we need to consider that they do not write all memory variables. In particular, MSTORE may only +write a single memory location, so for all other memory locations, the dependencies shall be propagated (this is done by +rule 3). 11 Additionally, we have a general rule that always propagates the dependencies of the symbolic memory position ⊤ +(10). This rule accounts for the fact that no opcode overwrites all (dynamic) memory locations. +Finally, the most involved rules are those that involve symbolic reads from memory/storage locations. For those, we need +to consider that reading from a static location, also always implies reading from ⊤. This is shown by rules 7, 8 and 9, which +account for the influences of storage locations on memory locations (as given by the local dependency predicate MemStore): +Here MemStorepc′(ℓ, ⊤) indicates that memory variable ℓ depends on any static or dynamic storage variable. For this reason, +any of their dependencies are propagated. Similarly, MemStorepc′(ℓ, ℓS) (for ℓS ∈ N256) indicates that the storage location +from which is read is statically known. In this case, both the dependencies of the static storage locations within this interval +are propagated, as well as the dependencies from ⊤ (indicating the corresponding dynamic storage locations). 12. +The treatment of the MemMem predicate is similar: Rule 12 considers the case that potentially the whole memory is +read (MemMempc′(ℓ, ⊤, ℓs)), rule 13 considers the case where the specified interval that is read is concretely known and all +dependencies from the corresponding static locations are read, and rule 13 ensures that also in this case all rules from the +dynamic location (⊤) are propagated. +Finally, Figure 22 shows the rule that models the influence of the control flow on the dependencies of memory variables +(captured by MemMayDependOn). The rule states that a memory location ℓ depends on tag t (at pc) if the memory is written +at pc (indicated by MemVarpc(ℓ, v1)) and there is another program counter pc′, which controls pc and at pc′ (which needs to +be a branch instruction), the conditional variable is v2, which again depends (transitively) on t. +3) Equivalence proof: We first introduce preliminary notions for the equivalence theorem. +For reasoning about contract executions that span several internal transactions, we introduce the notion of contract annotation +as used in [24]. For the sake of simplicity, in the main body of the paper, we annotated execution states only with the contract +code C. However, it gives more flexibility to characterize a contract as a pair c = (a, C) where a is the address of the contract +and C is its code. This allows to distinguish executions of different contracts that share the same code. In the following, we +will use the simplified annotation when sufficient and otherwise use the full annotation. +We recall the notion of strong consistency from [24]. +Definition 4 (Annotation consistency). An execution state s is consistent with contract annotation c if the following two +conditions hold +1) isRegular(s) =⇒ s.ι.actor = c.addr +2) isRegular(s) ∨ isHalt(s) =⇒ s.σ(c.addr).code = c.code +where isRegular(·) and isHalt(·) are predicates on execution states indicating whether they are regular execution states or +halting states, respectively. +Definition 5 (Strong annotation consistency). An execution state s is strongly consistent with contract annotation c (written +s-consistent(s, c)) if it is consistent with c and additionally +isRegular(s) =⇒ s.ι.code = c.code +Intuitively, a contract annotation c being strongly consistent with execution state s requires that s executes the contract as +it resides in the global state. +11Note that technically, we would need to have a similar rule for the copy operations, but since we anyway overapproximate them to only write the ⊤ +variable, the generic ⊤-propagation rule (10) captures this case. +12Note that in the implementation we currently omit this last rule, since, anyway, MemStore can only be inhabited with value ⊤ in the CALL-like rules. + +{MemSourcepc′(ℓ, t) ⇒ MemMayDependOnpc′(ℓ, t), +| C(pc) = (op(⃗x), pc′, pre) +(1) +MemVarpc′(ℓ, v) ∧ VarMayDependOn(v, t) +(2) +⇒ MemMayDependOnpc′(ℓ, t), +MemVarpc′(ℓ, v) ∧ ℓ ̸= ℓ′ ∧ MemMayDependOnpc(ℓ′, t) +(3) +⇒ MemMayDependOnpc′(ℓ′, t), +MemGaspc′(ℓ) ∧ GasMayDependOn(t) +(4) +⇒ MemMayDependOnpc′(ℓ, t), +MemMsizepc′(ℓ) ∧ MsizeMayDependOn(t) +(5) +⇒ MemMayDependOnpc′(ℓ, t), +MemExternalpc′(ℓ) ∧ ExternalMayDependOn(t) +(6) +⇒ MemMayDependOnpc′(ℓ, t), +MemStorepc′(ℓ, ⊤) ∧ StoreMayDependOn(ℓS, t) +(7) +⇒ MemMayDependOnpc′(ℓ, t), +MemStorepc′(ℓ, ℓS) ∧ ℓS ∈ N256 ∧ StoreMayDependOn(ℓS, t) +(8) +⇒ MemMayDependOnpc′(ℓ, t), +MemStorepc′(ℓ, ℓS) ∧ ℓS ∈ N256 ∧ StoreMayDependOn(⊤, t) +(9) +⇒ MemMayDependOnpc′(ℓ, t), +MemMayDependOnpc(⊤, t) +(10) +⇒ MemMayDependOnpc′(⊤, t), +MemMempc′(ℓ, ⊤, ℓs) ∧ MemMayDependOnpc(ℓ′, t) +(11) +⇒ MemMayDependOnpc′(ℓ, t), +MemMempc′(ℓ, ℓo, ℓs) ∧ ℓo ∈ N256 ∧ ℓs ∈ N256 +(12) +∧ i ≥ ℓo ∧ i < ℓo + ℓs ∧ MemMayDependOnpc(i, t) +⇒ MemMayDependOnpc′(ℓ, t), +MemMempc′(ℓ, ℓo, ℓs) ∧ ℓo ∈ N256 ∧ ℓs ∈ N256 +(13) +∧ MemMayDependOnpc(⊤, t) +⇒ MemMayDependOnpc′(ℓ, t), +MemMayDependOnpc(ℓ, t) ∧ NoReassignMem(pc) +(14) +⇒ MemMayDependOnpc′(ℓ, t), +} +Fig. 21. Horn Clauses describing the data flow dependencies captured by the MemMayDependOn predicate. + +{MemVarpc(ℓ, v1) ∧ MayControlspc(pc′, v2) ∧ VarMayDependOn(v2, t) +| pc ∈ D(C) +(15) +⇒ MemMayDependOnpc(ℓ, t)} +Fig. 22. Horn Clauses describing the control flow dependencies captured by the MemMayDependOn predicate. +Note that the function toEVM(θ, C, pc) maps states in the CFG semantics θ to execution states that are strongly consistent +with C. +Lemma 2. Let (Γ, s) = toEVM(θ, C, pc). Then s is strongly consistent with (s.ι.actor, λpc.(C(pc).op, C(pc).pre)). +Proof. Trivially follows from the definition of toEVM since s.σ is set to hold code toEVM(θ, C, pc) at address s.ι.actor. +We formally define the notion of a transaction step Γ ⊨ sC :: S +T−→ s′ +C :: S and the the relation Γ ⊨ sC :: S �→ s′ +C :: S. +Definition 6 (Transaction step). Let S be a callstack, s, s′ be execution states and Γ be a transaction environment. Further +let C be a contract. Then +Γ ⊨ sC :: S +T−→ s′ +C :: S := ∃s∗s†C∗. Γ ⊨ sC :: S → s∗ +C∗ :: sC :: S →∗ s† +C∗ :: sC :: S → s′ +C :: S +A transaction step describes the execution of a transaction being initiated in s (since in the execution step thereafter the +element s∗ is added to the call stack) and ends in s′ (since this is the execution state immediately after removing the additional +stack element). Note that this definition excludes that the execution might have returned before and have triggered another +internal transaction since the execution state s on the stack would have otherwise changed (at least due to a decrease in gas). +Definition 7 (Medium step). Let S be a callstack, s, s′ be execution states and Γ be a transaction environment. Further let +C be a contract. Then +Γ ⊨ sC :: S �→ s′ +C :: S := Γ ⊨ sC :: S → s′ +C :: S ∨ Γ ⊨ sC :: S +T−→ s′ +C :: S +Due to the two-layered memory abstraction, multiple states in the CFG semantics represent a single state in the EVM +semantics. Consequently, we define a notion of equivalence on CFG states that takes this into account: +Definition 8 (CFG state equivalence). Two CFG state θ and θ′ are considered equivalent (written θ ≈θ θ′) if the following +holds: +θ.ls = θ′.ls ∧ θ.el = θ′.el ∧ θ.eg = θ′.eg ∧ ∀xm. load θ xm = load θ′ xm ∧ ∀xg. load θ xg = load θ′ xg +We state some basic properties on CFG state equivalences: +Lemma 3 (CFG state equivalence properties). The following hold: +• θ ≈θ θ ← x := load θ x +• θ ≈θ (θ ← x := load θ x) ← x := ⊥ +Proof. Follows immediately from the definition ≈θ and load. +Most importantly, equivalent CFG states will be mapped to the same EVM states: +Lemma 4. For all contracts C, all program counters pc and all CFG state θ θ′ it holds that +θ ≈θ θ′ ⇔ toEVM(θ, C, pc) = toEVM(θ′, C, pc) +Proof. Follows immediately from the definitions of toEVM, ≈θ and load. +We formally define the medium step version of the CFG semantics: +C, cd ⊨ ⟨(pc, 0), θ⟩ =⇒ ⟨n, θ′⟩ := ∃ n (θi)i∈[0,n] (ai)i∈[0,n]. C, cd ⊨ ⟨(pc, 0), θ⟩ (−ai −→ ⟨(pc, i), θi⟩)i∈[0,n−1] −an −→ ⟨n, θ′⟩ +We now state the equivalence statement. In particular, we explicitly state the assumptions on the execution (excluding +exceptions and reentering storage modification). Further, we consider the cases where exception or halting states are entered. +Theorem 4 (Equivalence of EVM and CFG semantics). Let C be a store unreachable contract with sound preprocessing +information. Then the following holds: + +1) Let Γ ⊨ sC :: S �→ s′ +C :: S be an execution of contract C that does not exhibit local out-of-gas exceptions and let s be +strongly consistent with C. Then either +a) s′ = (µ′, ι′, σ′) and C, |S| ⊨ +⟨(µ.pc, 0), toCFG(Γ, s) ⊎ θ⊥⟩ +=⇒ ⟨(µ′.pc, 0), θ′ ⊎ θ⊥⟩ for some θ′ with θ′ ≈θ +toCFG(Γ, s′) +b) s′ = EXC and C, |S| ⊨ ⟨(µ.pc, 0), toCFG(Γ, s) ⊎ θ⊥⟩ =⇒ ⟨exception, toCFG(Γ, s) ⊎ θ⊥⟩ +c) s′ = HALT(σ′, g, d) and C, |S| ⊨ ⟨(µ.pc, 0), toCFG(Γ, s) ⊎ θ⊥⟩ =⇒ ⟨halt, toCFG(Γ, s) ⊎ θ⊥⟩ +2) Let C, cd ⊨ ⟨(pc, 0), θ ⊎ θ⊥⟩ =⇒ ⟨n, Θ⟩. Then either +a) n = (pc′, 0) and Θ = θ′ ⊎ θ⊥ and for (Γ, s) = toEVM(θ, C, pc) and (Γ′, s′) = toEVM(θ′, C, pc′) it holds that +Γ = Γ′ and for all S s.t. |S| = cd it holds that either Γ ⊨ sC :: S �→ s′ +C :: S or Γ ⊨ sC :: S → EXCC :: S and +C[pc] ̸= INVALID. +b) n = exception and for (Γ, s) = toEVM(θ, C, pc) and for all S s.t. |S| = cd it holds that Γ ⊨ sC :: S → EXCC :: S +and C[pc] = INVALID +c) n = halt and for (Γ, s) = toEVM(θ, C, pc) and for all S s.t. |S| = cd it holds that Γ ⊨ sC :: S → HALT(s.σ, g, d)C :: S +for some g and d. +Note that the CFG semantics only aims at modeling a single contract execution. In particular, it does not consider the effects +that a finalized execution may have on the caller (e.g., it does not model that global state variables are reverted in case that +the execution halted exceptionally). +Proof. We prove the two directions of the proof separately: +⇒ Let Γ ⊨ sC :: S �→ s′ +C :: S be an execution of contract C that does not exhibit local out-of-gas exceptions and let s be +strongly consistent with C. We do a case distinction on s′ +1) s′ = (µ′, ι′, σ′). We do a further case distinction on Γ ⊨ sC :: S �→ s′ +C :: S. +– The execution step was a local step and hence Γ ⊨ sC :: S → s′ +C :: S. In this case we know that s.ι = ι′ and +s.σ = σ′. The proof trivially follows by case distinction over the instruction in s.ι.code[s.µ.pc] using the fact that +due to strong consistency s.ι.code = C. +– The execution step was a call step and hence Γ ⊨ sC :: S +T−→ s′ +C :: S. By the definition of the CFG rule for calling +which leverages the notion of a transaction step on the EVM semantics, we only need to argue that the changes on +θ′ (in the corresponding rule sequences) preserve equivalence. This immediately follows from Lemma 3. +2) s′ = EXC. Since the execution does not exhibit local out of gas exceptions we know by Assumption 1 that in this case +C(s.µ.pc) = INVALID. This case, hence, follows immediately from the CFG semantics rule for the INVALID opcode. +3) s′ = HALT(σ, g, d). This case follows immediately from the CFG semantics rules for the halting instructions STOP +and RETURN. +⇐ Let C, cd ⊨ ⟨(pc, 0), θ ⊎ θ⊥⟩ =⇒ ⟨n, Θ⟩. The proof follows by a simple case distinction on C[pc] using the CFG semantics +rules for the individual instructions and taking advantage of Lemma 4 to reason about the equality of the execution states. +Note that we did not need to make use of the assumption that the contract is store unreachable for the equivalence proof. +This is since this requirement is only needed to prove the consistency of the Def and Use sets. +We revisit the assumption of store unreachability using full contract annotations: +Assumption 3 (Store unreachability). A contract c is store unreachable if {DELEGATECALL, CALLCODE} ∩ c.code = ∅ +and for all regular execution states (µ, ι, σ) that are strongly consistent with c, it holds that for all transaction environments +Γ and all callstacks S +¬∃s,S′. Γ ⊨ (µ, ι, σ)c :: S →∗ sc :: S′ + +S +∧ |S′| > 0 ∧ c.code(s.µ.pc) = (op(⃗x), pcnext) ∧ op ∈ InstSSTORE +Where the set InstSSTORE of store instructions is defined as +InstSSTORE = {SSTORE} +The key property following from the store unreachability is the following: +Lemma 5 (Store unreachability implies global storage preservation). Let C be a store unreachable contract and s an execution +state that is strongly consistent with C. Then for all callstacks S and execution states s′ +Γ ⊨ sc :: S +T−→ s′ +c :: S ⇒ s.σ(s.ι.actor).stor = s′.σ(s′.ι.actor).stor + +Proof. Assume towards contradiction that s.σ(s.ι.actor).stor ̸= s′.σ(s′.ι.actor).stor. Then there must have been a callstack S∗ +and an execution state s∗ and a contract C∗ such that Γ ⊨ sC :: S →∗ s∗ +c∗ :: S∗ + +sc :: S → s+ +c + :: S+++sc :: S →∗ s′ +c :: S +such that s∗.σ(s.ι.actor).stor ̸= s+.σ(s.ι.actor).stor. Since storage can only be altered via the SSTORE instruction (which is +a local instruction), we know that c∗ = c+ and S∗ = S+ and s∗.ι = s+.ι and s∗.ι.code(s∗.µ.pc) = (SSTORE(x, y), pcnext). +Further, since one can only write the storage of the active account, we know that s∗.ι.actor = c.a. Due to strong consistency +(which is preserved during execution for contacts without DELEGATECALL and CALLCODE), we hence know that +s∗.ι.code = c.code and so also c∗ = c. From Assumption 3, we can derive a contradition. +B. Correctness of May Analysis +We start by introducing definitions and lemmas needed for our soundness claims. +Definition 9 (Backward Slice of Set). +BS(N) = +� +n∈N +BS(n) +Definition 10 (Execution steps). Execution steps with states are defined as follows +n −(Q)√ −→ n′ +Q(θ) +⟨n, θ⟩ −(Q)√ −→ ⟨n′, θ⟩ +n−⇑f −→ n′ +θ′ = f(θ) +⟨n, θ⟩ −⇑f −→ ⟨n′, θ′⟩ +and −→∗ denotes the transitive closure. +Definition 11 (Deterministic successor function). We define the function “ds” as +ds(n) = +� +n′ +if ∃n′. n−⇑f −→ n′ +n′ +if ∃n′. n −(Q)√ −→ n′ ∧ Q 1 +Definition 12 (Sliced execution steps). +⟨n, θ⟩ +a−→ ⟨n′, θ′⟩ +n ∈ BS(N) +⟨n, θ⟩ +a−→BS(N) ⟨n′, θ′⟩ +⟨n, θ⟩ +a−→ ⟨n′, θ′⟩ +n ̸∈ BS(N) +n′′ = ds(n) +⟨n, θ⟩ +τ−→BS(N) ⟨n′′, θ⟩ +Definition 13 (Sliced execution paths). We define the sliced execution path +as +−→ +∗ +BS(N) as the transitive τ closure of Definition 12 +and require that +as +−→ +∗ +BS(N) must end at a node in BS(N). We write +as +−→ +m +BS(N) if path “as” in +as +−→ +∗ +BS(N) has exactly m observable +steps. +Definition 14 (Up-to equality). Let Var be the set of all variables used in a contract C. States θ1 and θ2 are equal except for +variable X if and only if +∀V ∈ Var/{X}. θ1(V ) = θ2(V ) +We write θ1 =/X θ2 in that case. +Definition 15 (Deterministic successor). Node n+1 is defined as the successor of node n in a contract C if it is the sole +successor of n and undefined otherwise: +n+1 = +� +n′ +if ∀n′, n′′. n → n′ ∧ n → n′′ =⇒ n′ = n′′ +undefined +otherwise +Definition 16 (Step-indexed execution). We define +as +−→ +∗|N +i +by +⟨n, θ⟩ +as +−→ +∗ ⟨n′, θ′⟩ +| as ↓N|= i +⟨n, θ⟩ +as +−→ +∗|N +i +⟨n′, θ′⟩ +where ↓N filters out the actions where the source node is not in N. +Definition 17 (Relevant variables [30]). Relevant variables of backward slice of S at node n are defined as follows +n +as +−→ +∗|NV +0 +n′ +n′ ∈ BS(S) +V ∈ Use(n′) +V ∈ rv S n + +Lemma 6 (Slice usage set property). +(∀nX. nX ̸∈ BS(NY )) =⇒ ⟨n+1 +X , θ⟩ −→m +BS(NY ) ⟨n+, +⟩ +=⇒ (∀k ≤ m. ⟨n+1 +X , θ⟩ −→k +BS(NY ) ⟨n∗, +⟩ =⇒ X ̸∈ Use(n∗)) +The lemma states that for all paths to nodes n+ of the backward slice NY , it holds that each step (n∗) in the backward +slice towards n+ does not use X. +Proof. We assume nX ̸∈ BS(NY ) and ⟨n+1 +X , θ⟩ −→m +BS(NY ) ⟨n+, +⟩. +Towards contradiction we assume there would be a k such that +⟨n+1 +X , θ1⟩ −→k +BS(NY ) ⟨n∗, +⟩ ∧ X ∈ Use(n∗) +Then we define n∗ +X with l ≤ k such that +⟨n+1 +X , θ1⟩ −→l +BS(NY ) ⟨n∗ +X, θ∗ +1⟩ ∧ X ∈ Def(n∗ +X) +where n∗ +X identifies the last node where X was defined on the path to node n∗ or if such an l does not exists, we know +⟨nX, +⟩ −→ ⟨n+1 +X , θ1⟩ −→∗ +BS(NY )|NX +0 +⟨n∗, +⟩ +This means that X is either defined along the path or at the predecessor of the start node since we start at a successor of a +definition of X. This node n∗ +X (nX) is then by definition in the backward slide of NY since X is used at node n∗ in the +backward slice and it was the closest definition. This contradicts the assumption n∗ +X ̸∈ BS(NY ). +The proofs use the Slicing framework’s soundness claim for backward slices of a set of nodes instead of single nodes. +Theorem 5 (Correctness of Slicing Based on Paths and Sets [29]). +⟨n, θ⟩ +as +−→ +∗ ⟨n′, θ′⟩ +n′ ∈ S +∃ as′. ⟨n, θ⟩ +as′ +−→ +∗ +BS(S) ⟨n′, θ′′⟩ ∧ (∀ V ∈ Use(n′).θ′(V ) = θ′′(V )) ∧ as ↓BS(S)= as′ +The following lemma lifts sliced executions while preserving the number of visits at nodes in the backward slice. +Lemma 7. All executions within the backward slice of node-set N can be mapped to real executions with the same number +of visits in any subset of the backward slice and all relevant variables concerning the final node n′ of the execution were +computed correctly. +n′ ∈ N =⇒ N ′ ⊆ N =⇒ ⟨n, θ⟩ +as′ +−→ +∗ +BS(N)|N ′ +i +⟨n′, θ′⟩ +=⇒ ∃as, θ′′.⟨n, θ⟩ +as +−→ +∗|N ′ +i +⟨n′, θ′′⟩ ∧ ∀V ∈ rv N n′. θ′(V ) = θ′′(V ) +Proof. We first show that path as exists outside of the backward slice: +(1) ⟨n, θ⟩ +as +−→ ⟨n′, θ′′⟩ +Path as is constructed by extending the sliced path as′ with nodes outside of the backward slice. By Definition 12 and since +every step in ⟨n, θ⟩ +as +−→ ⟨n′, θ′⟩ would be deterministic by the slicing framework requirements, we can reconstruct an unique +original path as by adding nodes along the path as′ where τ edges were followed. +Therefore, we construct the following path as where all nodes ni with a line on top were not part of the original sliced path. +(2) +as = +n +n1 +n1 +ni +n′ +. . . +as′ +1 +as′ +1 +as′ +i+1 +where the original path was +as′ = +n +n1 +n′ +. . . +as′ +1 +τ∗ +as′ +i+1 +The construction described above is feasible by the definition of +as +−→ +∗ +BS(N) and satisfies (1). We remark that path as′ is an +ordered sublist of the original sliced-path as as nodes were only inserted while construction in step (2) but not removed: +(3) as′ ⊆od as + +It is left to be shown that path as has the same number of visits in the set N ′ as the path as′ in the sliced graph: +(4) ⟨n, θ⟩ +as +−→ +∗|N ′ +i +⟨n′, θ′′⟩ +Recall that by assumption, all nodes in N ′ are part of N and therefore elements of the backward slice of BS(N). By construction +of (2) and (3), we know that the number of visits at N ′ is the same in both paths and (4) holds. We now show that for final +state θ′′ from (4) and the final state θ′ from the sliced-path it holds that +(5) ∀V ∈ rv N n′. θ′(V ) = θ′′(V ) +All nodes where any variable V in rv N n′ is defined are part of the backward slice by Definition 17. Therefore, no node of +this kind was added or deleted in the construction of as and they were already present in as′. +With that, we can use the correctness statement of slicing to show that all relevant variables reassigned along as′ have the +same value in as. Non-reassigned variables are not changed and stay the same in both states. Therefore, all relevant variables +have the same value in θ′ and θ′′. +The statement follows from (4) and (5). +a) Variable Independence: +Lemma 8 (Variable Dependency Predicate). Let C be a contract and X and Y be variables thereof. Then it holds that +VarMayDepOn(Y, X) ̸∈ lfp(R(C)) ⇒ ∀nY , nX. nX ̸∈ BS(nY ) +Definition 18 (Variable Independence). A variable Y is independent of a variable X if and only if for all states θ1, θ2 and +θ′ +1 it holds that +∀nX, nY , i. θ1 =/X θ2 ∧ ⟨n+1 +X , θ1⟩ −→∗|NY +i +⟨n+1 +Y , θ′ +1⟩ +=⇒ ∃θ′ +2. ⟨n+1 +X , θ2⟩ −→∗|NY +i +⟨n+1 +Y , θ′ +2⟩ ∧ θ′ +1(Y ) = θ′ +2(Y ) +We will refer to this definition by VarIndOf(n, X). +Theorem 6 (Soundness of Variable Independence). Let X and Y be variables and nX ̸∈ BS(nY ) for all nodes nX, nY . Then +variable Y is independent of variable X. +Proof. Let C be a contract and C be a contract with sound preprocessing information that is consistent with C. We assume +that (1) nX ̸∈ BS(nY ) holds for all nodes nX and nY and an executions from node n+1 +X to node n+1 +Y +that starts in states θ1. +Let θ2 be a state such that (2) θ1 =/X θ2 and +(3) ⟨n+1 +X , θ1⟩ +as +−→ +∗|NY +i +⟨n+1 +Y , θ′ +1⟩ +identity the previous execution with path as. +From (3) we know from correctness of slicing (Theorem 5) that +(5) ∃as′.⟨n+1 +X , θ1⟩ +as′ +−−→ +m +BS(NY )|NY +j +⟨n+1 +Y , θ∗ +1⟩, +(6) θ′ +1(Y ) = θ∗ +1(Y ) +and (7) as ↓NY += as′ +Equations (5) and (6) say that there exists a corresponding path in the backward slice and it has the correct value for Y . +Equation (7) implies that all nodes in the backward slice of NY are visited in the same order in as and as′ [30]. Since all +nodes where Y is defined are by defintion of NY in the backward slice, we know from (7) that i = j. +We now show that path as′ exists in the sliced-graph when starting from state θ2: +(9) ⟨n+1 +X , θ2⟩ +as′ +−−→ +m +BS(nY )|NY +j +⟨n+1 +Y , θ∗ +2⟩ +and +θ∗ +1(Y ) = θ∗ +2(Y ) +With Lemma 6 we get from (1) and (5) for all k ≤ m that +(8) ⟨n+1 +X , θ1⟩ −→k +BS(NY ) ⟨n+, θ+ +1 ⟩ =⇒ X ̸∈ Use(n+) +We can use (8) to show (9) since at all nodes n+, that would have effected the trace, X is not used. From (2) we know by the +slicing framework’s well-formedness properties that X is only source of different effects. Therefore, we know that (9) holds. +From (9) we can use Lemma 7 to conclude that +⟨n+1 +X , θ2⟩ +as′′ +−−→ +∗ +|NY +i +⟨n+1 +Y , θ′ +2⟩ ∧ ∀V ∈ rv NY nY . θ′ +2(V ) = θ∗ +2(V ) + +and in particular +θ′ +2(Y ) = θ∗ +2(Y ) +since Y ∈ rv NY nY . +b) Instruction Independence: +Lemma 9 (Instruction Dependency Predicate). Let C be a contract, n be a node and X be variable thereof. Then it holds +that +InstMayDepOn(n, X) ̸∈ lfp(R(C)) ⇒ ∀nif, nX. nif −→cd n ⇒ nX ̸∈ BS(nif) +Definition 19 (Instruction Independence). A node n is independent of a variable X if and only if for all states θ1 and θ1 it +holds that +∀nX, n′, i. θ1 =/X θ2 ∧ ⟨n+1 +X , θ1⟩ +as +−→ +∗|n′ +i +⟨n′, +⟩ ∧ ⟨n+1 +X , θ2⟩ +as′ +−→ +∗ +|n′ +i +⟨n′, +⟩ +=⇒ | as ↓n| += +| as′ ↓n| +We will refer to this definition by InstIndOf(n, X). +Theorem 7 (Soundness of Instruction Independence). Let n be a node, X be a variable and nX ̸∈ BS(nif) hold for all nodes +nX and nif where nif −→cd n. Then node n is independent of variable X. +Proof. Let C be a contract and C be a contract with sound preprocessing information that is consistent with C. We assume +that (1) nX ̸∈ BS(nif) holds for all nodes nX and nif such that nif −→cd n and two executions from node n+1 +X to node n′ that +starts in states θ1 and θ2 with (4) θ1 =/X θ2 and visit n′ on their paths i times. Let +(2) ⟨n+1 +X , θ1⟩ +as +−→ +∗|n′ +i +⟨n′, +⟩ +and +(3) ⟨n+1 +X , θ2⟩ +as′ +−→ +∗ +|n′ +i +⟨n′, +⟩ +be these two executions with paths as and as′. +The interesting case is node n′ ̸= n since the conclusion for node n′ = n follows with assumption (3) and (4) by definition +of −→∗|n′ +i +. Therefore, we show the statement for n′ ̸= n. +We show that paths as and as′ visit node n the same number of times by contradiction. W.l.o.g we assume that path as +visits node n at least once more than path as′: +(∗) +| as ↓n| +> +| as′ ↓n| +From (∗) we know that there exists a prefix as of path as such that it can visit node n once more than path as′: +(5.1) ⟨n+1 +X , θ1⟩ +as +−→ +∗ +|n +k ⟨n, +⟩ −→∗|n′ +0 +⟨n′, +⟩ ∧ as ⊆od as +with k =| as′ ↓n| such that as ends at the k +1 occurrence of n. Node n′ is reachable from n because the prefix as can allows +be completed to the full path from (3). +Next, we define a corresponding prefix as′ of as′ that vists n the maximal number of k times and ends at the same appearence +of n′. +(5.2) ⟨n+1 +X , θ2⟩ +as′ +−−−−−−−−−−−−→ +∗ +|n′ +j +⟨n′, +⟩ ∧ as′ ⊆od as′ +with j =| as ↓n′|. That path exists based on (4) and (∗). We know that path as′ visits n less often than as; and as′ consists +of all visits of n in as′. +We know by construction of (5.1) and (5.2) that as and as′ split up after the k-th visit at n because otherwise both would +visit node n at least k + 1 number of times. We call this split node nif and make the split explicit: +(6) ∃nif, w, z. ⟨n+1 +X , θ1⟩ +as1 +−−→ ∗ |nif +w +⟨nif, θif +1 ⟩ +as2 +−−→ ∗ ⟨n, +⟩ −→ ∗ ⟨n′, +⟩ +∧ ⟨n+1 +X , θ2⟩ +as′ +1 +−−→ ∗ |nif +z +⟨nif, θif +2 ⟩ +as′ +2 +−−−−−−−−−−−→ ∗ ⟨n′, +⟩ +where as = as1 @ as2 such that as2 and as′ +2 do not share a single node, i.e. nif is the last node where the paths could split +up. From (3) – (6), we can conclude with standard control dependence that +(7) nif −→∗ +cd n +By instantiating (1) with (7) we know +(8) ∀n0 +X. n0 +X ̸∈ BS(nif) + +From (7) we know that states θif +1 +and θif +2 +have at least one different value for some input variable at nif. Only a different +value in the usage set can lead to splitting control flow. Therefore, we get: +(9) ¬∀ V ∈ Use(nif). θif +1 (V ) = θif +2 (V ) +We now map paths as1 and as′ +1 from (5) into the sliced graph with Theorem 5 of correctness of slicing. Those paths compute +correct input values at nif. We get +(10) ∃ as1. ⟨n+1 +X , θ1⟩ +as1 +−−→ +g +BS(nif)|nif +w +⟨nif, θ∗ +1⟩ ∧ ∀ V ∈ Use(nif). θif +1 (V ) = θ∗ +1(V ) +(11) ∃ as′ +1. ⟨n+1 +X , θ2⟩ +as′ +1 +−−→ +h +BS(nif)|nif +z +⟨nif, θ∗ +2⟩ ∧ ∀ V ∈ Use(nif). θif +2 (V ) = θ∗ +2(V ) +We apply paths as1 and as′ +1 to Lemma 6 with (8) and get +(12) ∀ q′ ≤ q. ⟨n+1 +X , θp⟩ −→q′ +BS(Lif) ⟨n+, θ∗ +p⟩ =⇒ X ̸∈ Use(n+) +with (p, q) ∈ {(1, g), (2, h)}. +Assumption (2) states that only X can propagate changes, but we know from (12) that no node in the backward slice on either +path uses X. Therefore, executions ⟨n+1 +X , θ1⟩ −→∗ +BS(nif) +and ⟨n+1 +X , θ2⟩ −→∗ +BS(nif) +have the same states (up to X) after the +same number of steps: +(13) ⟨n+1 +X , θ1⟩ +xs +−→ +t +BS(nif) ⟨ +, ˆθ1⟩ ∧ ⟨n+1 +X , θ2⟩ +ys +−→ +t +BS(nif) ⟨ +, ˆθ2⟩ +=⇒ ˆθ1 =/X ˆθ2 +Since nif ∈ BS(nif), we know that paths in (10) and (11) reach the split node nif after the same number of steps in the backward +slice: +(14) w = z +and therefore choose +t = +| as1 | += +| as1 | +Finally, by (13) and (14) on paths (10) and (11) we get θ∗ +1 =/X θ∗ +2 and conclude by (12): +(15) ∀ V ∈ Use(nif). θ∗ +1(V ) = θ∗ +2(V ) +The state equivalence reasoning is summarized in the following for all V ∈ Use(nif): +θif +1 (V ) +θ∗ +1(V ) +θ∗ +2(V ) +θif +2 (V ) +(10) += +(15) += +(11) += +Thereby, we conclude ∀ V ∈ Use(nif). θif +1 (V ) = θif +2 (V ) which contradicts (9). +Definition 20 (Environmental Instruction Independence). A node n is independent of a constant environmental variable X if +and only if for all states θ1 and θ2 it holds that +∀n0, n′, i. θ1 =/X θ2 ∧ ⟨n0, θ1⟩ +as +−→ +∗|n′ +i +⟨n′, +⟩ ∧ ⟨n0, θ2⟩ +as′ +−→ +∗ +|n′ +i +⟨n′, +⟩ +=⇒ | as ↓n| += +| as′ ↓n| +Lemma 10 (Soundness of Environmental Instruction Independence). Let n be a node, X be a constant environmental variable +and nX ̸∈ BS(nif) for all nodes nX and nif where nif −→cd n. Then node n is independent of variable X. +Proof. Definite assignment was ensured by n+1 +X in Theorem 7. Since constant environmental variables are definitely assigned +and final by design, we can drop this requirement in the soundness claim and reuse the proof of Theorem 7. + +C. Correctness of Trace Noninterference Pattern +To prove Theorem 3 +Lemma 11. Let V be a set of CFG state variables and let +• Nf(C) := {n | ∃ i. n = (pc, i) land ∃ op ⃗x pcnext pre. C(pc) = (op(⃗x), pcnext, pre) ∧ f(op)} +• Args(C, n) := {x | ∃ i. ∧ n = (pc, i) ∧ ∃ op ⃗x pcnext pre. C(pc) = (op(⃗x), pcnext, pre) ∧ x ∈ ⃗x} +• Varf(C) := {x | ∃ nf. nf ∈ Nf(C) ∧ x ∈ Args(C, nf)} +Further, let m ∈ N, C, n, n1 +f, n1 +f ++, · · · , nm +f , nm +f ++, as1, · · · , asm, a1, · · · , am, and θ, θ′, θ1, θ+ +1 , · · · , θm, θ+ +m be arbitrary and +assume that +1) ∀θ =/V θ′ +2) ∀xf ∈ Varf(C). ∀v ∈ V. VarIndOf(xf, v) +3) ∀nf ∈ Nf(C). ∀v ∈ V. InstIndOf(nf, v) +4) ⟨n, θ⟩ +� asi +−−→ +∗|Nf (C) +0 +⟨ni +f, θi⟩ +ai +−→ ⟨ni +f ++, θ+ +i ⟩ +�m +i=1 +Then there exist as′ +1, · · · , as′ +m, a′ +1, · · · , a′ +m, and θ′ +1, θ′+ +1 · · · , θ′ +m, θ′+ +m such that +⟨n, θ⟩ +� +as′ +i +−−→ +∗ +|Nf (C) +0 +⟨ni +f, θ′ +i⟩ +a′ +i +−→ ⟨ni +f ++, θ′+ +i ⟩ +�m +i=1 +∧ ∀i ∈ [1, m]. ∀x ∈ Args(C, ni +f). θi(x) = θ′ +i(x) +Proof. By induction on m ∈ N. +1) Let m = 0. The claim trivially holds. +2) Let m > 0. Then +� asi +−−→ +∗|Nf (C) +0 +⟨ni +f, θi⟩ +ai +−→ ⟨ni +f ++, θ+ +i ⟩ +�m−1 +i=1 +asm +−−→ +∗|Nf (C) +0 +⟨nm +f , θm⟩ +am +−−→ +⟨nm +f ++, θ+ +m⟩ and by the +inductive hypothesis also ⟨n, θ⟩ +� +as′ +i +−−→ +∗ +|Nf (C) +0 +⟨ni +f, θ′ +i⟩ +a′ +i +−→ ⟨ni +f ++, θ′+ +i ⟩ +�m−1 +i=1 +for some as′ +1, · · · , as′ +m−1, a′ +1, · · · , am−1, +and θ′ +1, θ′+ +1 , · · · , θ′ +m−1, θ′+ +m−1. such that ∀i ∈ [1, m − 1]. ∀x ∈ Args(C, ni +f). θi(x) = θ′ +i(x). We are hence left to show +that there exists some as′ +m, a′ +m, θ′ +m, θ′+ +m such that ⟨nm−1 +f ++, θ′+ +m−1⟩ +as′ +m +−−→ +∗ +|Nf (C) +0 +⟨nm +f , θ′ +m⟩ +a′ +m +−−→ +⟨nm +f ++, θ′+ +m ⟩, and +∀x ∈ Args(C, nm +f ). θm(x) = θ′ +m(x). Assume towards contradiction that there is no as′ +m, a′ +m and θ′ +m, θ′+ +m such that +⟨nm−1 +f ++, θ′+ +m−1⟩ +as′ +m +−−→ +∗ +|Nf (C) +0 +⟨nm +f , θ′ +m⟩ +a′ +m +−−→ ⟨nm +f ++, θ′+ +m ⟩, meaning that nm +f is not reachable from ⟨nm−1 +f +, θ′ +m−1⟩ without +stepping through another node in Nf(C). We consider two cases +a) nm +f +is not reachable from ⟨nm−1 +f ++, θ′+ +m−1⟩. Since every execution ends in the node exit, we know that +⟨nm−1 +f ++, θ′+ +m−1⟩ +as′ +m +−−→ +∗ +| +{nm +f } +0 +⟨exit, θ′ +exit⟩ and ⟨nm +f , θm⟩ +asm+1 +−−−−→ +∗ +⟨exit, θexit⟩ for some θexit and θ′ +exit, and asm+1. +Consequently, we know that must be executions ⟨n, θ⟩ +as +−→ +∗ +⟨exit, θexit⟩ and ⟨n, θ′⟩ +as′ +−→ +∗ +⟨exit, θ′ +exit⟩ such that +|as ↓nm +f | > |as′ ↓nm +f | (with as = � +i∈[1,m+1] asi · ai and as′ = � +i∈[0,m] as′ +i · a′ +i). This immediately contradicts +InstIndOf(nm +f , v) (assumption 3). +b) A node n∗ +f from Nf(C) different from nm +f , which is reached from ⟨nm−1 +f +, θ′ +m−1⟩ before reaching nm +f . So, there +are as′ +m, a′∗ +m, as′∗ +m, θ′∗, θ′∗+ such that ⟨nm−1 +f ++, θ′+ +m−1⟩ +as′ +m +−−→ +∗ +|Nf (C) +0 +⟨n∗ +f, θ′∗⟩ +a′∗ +m +−−→ +⟨n∗ +f ++, θ′∗+⟩ +as′∗ +m +−−→ +∗ +⟨nm +f , θ′ +m⟩. +But then there must be executions ⟨n, θ⟩ +as +−→ +∗ +⟨nm +f , θm⟩ and ⟨n, θ′⟩ +as′ +−→ +∗ +⟨nm +f , θ′ +m⟩ such that |as ↓n∗ +f | < |as′ ↓n∗ +f | +(with as = � +i∈[1,m+1] asi · ai and as′ = (� +i∈[0,m−1] as′ +i · a′ +i) · as′ +m · a′ +m ∗ ·as′∗ +m). This again immediately contradicts +InstIndOf(n∗ +f, v) (assumption 3). +Hence, we know that θ′ +m, θ′+ +m such that ⟨nm−1 +f ++, θ′+ +m−1⟩ +as′ +m +−−→ +∗ +|Nf (C) +0 +⟨nm +f , θ′ +m⟩ +a′ +m +−−→ +⟨nm +f ++, θ′+ +m ⟩. Finally, we need to +show that ∀x ∈ Args(C, nm +f ). θm(x) = θ′ +m(x). This immediately follows from assumption 2 and concludes the proof. +We define the set of components of EVM configurations. +Definition 21 (EVM configuration components). The set of EVM configuration components CompEVM is defined as follows +CompEVM := Comps +EVM ∪ CompΓ +EVM ∪ CompS +EVM +with +Comps +EVM := {µ.gas, µ.msize, ι.input, ι.sender, ι.value} +∪ {µ.stack(x) | x ∈ N8} ∪ {µ.m(x) | x ∈ N256} + +and +CompΓ +EVM := {H.parent, H.beneficiary, H.difficulty, H.number, H.gaslimit, H.timestamp, origin, gasprice} +and +CompS +EVM := {σ(other).stor, σ(other).bal, σ(other).code, σ(this).bal, σ(this).nonce} +∪ {σ(this).stor(x) | x ∈ N256} +Note that we explicitly exclude the program counter, the active account and the active code. The reason for this is that we +will only compare executions of the same contract starting from the same instruction (program counter). +Definition 22. Let Z ⊆ CompEVM be a set of EVM components. We define two EVM configurations (Γ, s), (Γ′, s′) equal up +to Z (written (Γ, s) =/Z (Γ′, s′)) if the following holds +∀z.z ̸∈ Z ⇒ (z ∈ Comps +EVM ⇒ s.z = s′.z) +∧ (z ∈ CompΓ +EVM ⇒ Γ.z = Γ′.z) +∧ (z = σ(other).stor ⇒ ∀x ∈ N256 a ∈ N160. a ̸= s.ι.actor ⇒ s.σ(a).stor(x) = s′.σ(a).stor(x)) +∧ (z = σ(other).bal ⇒ ∀a ∈ N160. a ̸= s.ι.actor ⇒ s.σ(a).bal = s′.σ(a).bal) +∧ (z = σ(other).nonce ⇒ ∀a ∈ N160. a ̸= s.ι.actor ⇒ s.σ(a).nonce = s′.σ(a).nonce) +∧ (z = σ(other).code ⇒ ∀a ∈ N160. a ̸= s.ι.actor ⇒ s.σ(a).code = s′.σ(a).code) +∧ (∀x. z = σ(this).stor(x) ⇒ ∀a ∈ N160. a = s.ι.actor ⇒ s.σ(a).stor(x) = s′.σ(a).stor(x)) +∧ (z = σ(this).bal ⇒ ∀a ∈ N160. a = s.ι.actor ⇒ s.σ(a).bal = s′.σ(a).bal) +∧ (z = σ(this).nonce ⇒ ∀a ∈ N160. a = s.ι.actor ⇒ s.σ(a).nonce = s′.σ(a).nonce) +We formally define the function toVar, which maps components of EVM configurations to CFG variables. Note, that as +opposed to the slightly simplified version in the main body of the paper, toVar maps to a set of variables. +Definition 23. The function toVar ∈ CompEVM → P(V ) is defined as follows: +toVar(z) := +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +{xls} +z = µ.stack(x) +{xm.D, xm.S} +z = µ.m(x) +{gel} +z = µ.gas +{iel} +z = µ.msize +{inputel} +z = ι.input +{senderel} +z = ι.sender +{vael} +z = ι.value +{parenteg} +z = H.parent +{beneficiaryeg} +z = H.beneficiary +{difficultyeg} +z = H.difficulty +{numbereg} +z = H.number +{gaslimiteg} +z = headerc.gaslimit +{timestampeg} +z = H.timestamp +{origineg} +z = origin +{prizeeg} +z = gasprice +{externaleg} +z ∈ {σ(other).stor, σ(other).bal, σ(this).bal, σ(other).nonce, σ(this).nonce, σ(other).code} +{xg.D, xg.S} +z = σ(this).stor(x) +Next, we establish the relation between the notions of equivalence up to components: +Lemma 12. Let Γ, Γ′ be transaction environments, s, s′ execution states and Z a set if EVM state components such that +(Γ, s) =/Z (Γ′, s′) and (θ, C, pc) = toCFG(Γ, s) and (θ′, C, pc) = toCFG(Γ, s) for some θ, θ′, C. Further, let Z = {x | ∃z ∈ +Z. x ∈ toVar(z)}. Then θ =/Z θ′. +Proof. Trivially follows from the definition of toCFG that maps components into variables in the same way as toVar. + +Finally, by combining the previous lemmas, we can prove Theorem 3. We restate the theorem for completeness with all +assumptions here. To this end, we restate the definition of trace noninterferenceto explicitly state the consistency assumption +the involved executions: +Definition 24 (Trace noninterference). Let C be an EVM contract, Z ∈ CompEVM be a set of components of EVM configurations +and f ∈ I → B be a predicate on instructions. Then trace noninterference of contract C w.r.t. Z and f (written TNI(C, Z, f)) +is defined as follows: +TNI(C, Z, f) := ∀ Γ Γ′ s s′ t t′ π. π′ +s-consistent(s, C) +⇒ s-consistent(s′, C) +⇒ (Γ, s) =/Z (Γ′, s′) +⇒ Γ ⊨ sC :: S +π−→ +∗ tC :: S ∧ final (t) +⇒ Γ ⊨ s′ +C :: S +π′ +−→ +∗ +t′ +C :: S ∧ final (t′) +⇒ π ↓f= π′ ↓f +where π ↓f denotes the trace filtered by f, so containing only the instructions satisfying f. and s and s′ are strongly consistent +with contract C. +Next, we formally define PC +Z,f considering the proper definition of toVar: +Definition 25 (Trace non-interference pattern). +PC +Z,f :={InstMayDepOn(pc, z) | ∃ z z op ⃗x pcnext pre. z ∈ Z ∧ z ∈ toVar(z) ∧ C(pc) = op(⃗x, pcnext, pre) ∧ f(op)} +∪ {VarMayDepOn(xi, z) | ∃ z z op ⃗x pcnext pre. z ∈ Z ∧ z ∈ toVar(z) ∧ C(pc) = (op(⃗x, pcnext, pre)) ∧ f(op) ∧ xi ∈ ⃗x}. +Theorem 8 (Soundness of trace noninterference). Let Z ⊆ CompEVM be a set of EVM components, and f ∈ I → B an +instruction-of-interest predicate. Further, let C be a store unreachable contract that does not exhibit local out-of-gas exceptions. +Then it holds that +(∀p ∈ PC +Z,f. p ̸∈ lfp(R(C))) ⇒ TNI(C, Z, f). +Note that due to the definition of EVM components, we can rely on the fact that +(Γ, s) =/Z (Γ′, s′) ⇒ s.µ.pc = s′.µ.pc +(16) +Proof. Assume that ∀p ∈ PC +Z,f. p ̸∈ lfp(R(C)). From the definition of PC +Z,f ( Definition 25), we know that hence +∀ z ∈ Z. ∀ z ∈ toVar(z). ∀ pc i pcnext pre op ⃗x. C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ InstMayDepOn((pc, i), z) ̸∈ lfp(R(C)) +and +∀ z ∈ Z. ∀ z ∈ toVar(z). ∀ x pc pcnext pre op ⃗x. C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ x ∈ ⃗x ⇒ VarMayDepOn(xi, z) ̸∈ lfp(R(C)) +From Theorem 6,Lemma 8, Theorem 7,Lemma 9, we get that +∀ z ∈ Z. ∀ z ∈ toVar(z). ∀ pc i pcnext pre op ⃗x. C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ InstIndOf((pc, i), , z) +and +∀ z ∈ Z. ∀ z ∈ toVar(z). ∀ x pc pcnext pre op ⃗x. C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ x ∈ ⃗x ⇒ VarIndOf(xi, z) +Which is equivalent to +∀ v ∈ {v | ∃z ∈ Z. v ∈ toVar(z)}. ∀nf ∈ Nf(C). InstIndOf(nf, v) +(17) +and +∀ v ∈ {v | ∃z ∈ Z. v ∈ toVar(z)}. ∀x ∈ Varf(C). VarIndOf(x, v) +(18) +. +Now let Γ, Γ′ be transaction environments and s, s′, t, t′ be execution states and π, and π′ be traces. Assume the following: +1) (Γ, s) =/Z (Γ′, s′) +2) Γ ⊨ sC :: S +π−→ +∗ tC :: S +3) final (t) + +4) Γ ⊨ s′ +C :: S +π′ +−→ +∗ +t′ +C :: S +5) final (t′) +We need to show that π ↓f= π′ ↓f. +We define the following relation to reason about steps that do not produce any actions satisfying f: +Γ ⊨ s :: S �−→∗|f s′ :: S := Γ ⊨ s :: S +π +�−→ +∗ +s′ :: S ∧ π ↓f= ϵ +Using this definition, we can decompose every run Γ +⊨ +sC :: S +π−→ +∗ +tC :: S +into a run Γ +⊨ +sC +:: +S +� +�−→∗|f sf,i +C :: S +opi(⃗vi) +�−−−−→ sf +,i +C +:: S +�m +i=1 +�−→∗|f +tC :: S. Such that π ↓f= �m +i=1 opi(⃗vi). Similarly, we can deconstruct +every run Γ ⊨ s′ +C :: S +π′ +−→ +∗ +t′ +C :: S into a run Γ ⊨ s′ +C :: S +� +�−→∗|f s′f,i +C +:: S +op′ +i(⃗v′i) +�−−−−→ s′f +,i +C +:: S +�n +i=1 +�−→∗|f +t′ +C :: S. Such +that π′ ↓f= �n +i=1 op′ +i(⃗v′i). Towards contradiction, we assume that π ↓f̸= π′ ↓f. So, either there is a position i such that +opi(⃗vi) ̸= op′ +i(⃗v′i) and for all j < i it holds that opj(⃗vj) = op′ +j(⃗v′j) or |π ↓f| < |π′ ↓f| and π ↓f is a prefix of π′ ↓f or +|π′ ↓f| < |π ↓f| and π′ ↓f is a prefix of π ↓f. We proceed by case distinction based on these cases: +1) Assume that there is a position i such that opi(⃗vi) ̸= op′ +i(⃗v′i) and for all j < i it holds that opj(⃗vj) = op′ +j(⃗v′j). Then +we know that Γ ⊨ sC :: S +� +�−→∗|f sf,j +C :: S +opj(⃗vj) +�−−−−→ sf +,j +C +:: S +�i−1 +j=1 +�−→∗|f +sf,i +C :: S +opi(⃗vi) +�−−−−→ sf +,i +C +:: S and hence for +µi = sf,i.µ we have that C(µi.pc) = (opi(⃗xi), pci +next, prei) and for all vi +k ∈ ⃗vi it holds that vi +k = µi.s(xi +k). Similarly, +we know that Γ ⊨ s′ +C :: S +� +�−→∗|f s′f,j +C +:: S +op′ +j(⃗v′j) +�−−−−−→ s′f +,j +C +:: S +�i−1 +j=1 +�−→∗|f +s′f,i +C +:: S +op′ +i(⃗v′i) +�−−−−→ s′f +,i +C +:: S and hence +for µ′ +i = s′f,i.µ we have that C(µ′ +i.pc) = (op′ +i(⃗x′i), pc′i +next, pre′ +i) and for all v′i +k ∈ ⃗v′i it holds that v′i +k = µ′ +i.s(x′i +k). From +Theorem 4 we get that +a) C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨nsf,j, toCFG(Γ, sf,j) ⊎ θ⊥⟩ ⇒ ⟨nsf+,j, toCFG(Γ, sf +,j) ⊎ θ⊥⟩ +�i +j=1 +such that all nsf,j ∈ Nf(C) +b) C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨ns′f,j, toCFG(Γ, s′f,j) ⊎ θ⊥⟩ ⇒ ⟨ns′f+,j, toCFG(Γ, s′f +,j) ⊎ θ⊥⟩ +�i +j=1 +such that all ns′f,j ∈ Nf(C). +Note that from assumption 16 we know that ns = (s.µ.pc, 0) = (s′.µ.pc, 0) = ns′. Further, we know that +C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨n∗ +j, θj⟩ −→ ⟨n∗ +j, θj+⟩ +�l +j=1 +for some l ≥ i such that there is some g ∈ N → N for which it holds that 1) ∀ n m. n < m ⇒ g(n) < g(m) and 2) +∀ j ∈ [1, i]. nsf,j = n∗ +g(j) and 3) ∀ j ∈ [1, i]. toCFG(Γ, sf,j) ⊎ θ⊥ = θg(j) and 4) ∀ j ∈ [1, i]. ∀k. k > g(i) ⇒ k < +g(i + 1) ⇒ ∀pc. n∗ +g(j) = (pc, 0) ⇒ ∃q. n∗ +k = (pc, q). This is as all ⇒-steps from nodes in Nf(C) can be expanded into +the individual steps between the subnodes of the same pc. +Consequently, we can apply Lemma 11 (using Lemma 12) to obtain +C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨n∗ +j, θ′ +j⟩ −→ ⟨n∗ +j, θ′ +j+⟩ +�l +j=1 +and ∀p ∈ [1, l]. ∀x ∈ Args(C, n∗ +p). θp(x) = θ′ +p(x). Consequently, we can also conclude that +C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨nsf,j, θ† +j⟩ ⇒ ⟨nsf+,j, θ† +j⟩ +�i +j=1 +such that ∀ j ∈ [1, i].θ† +j = θ′ +g(j). In particular, this means that ∀ j ∈ [1, i − 1]. ∀x ∈ Args(C, nsf,j). θ† +j(x) = +toCFG(Γ, sf,j) ⊎ θ⊥(x). +Since opi(⃗vi) ̸= op′ +i(⃗v′i), it must either holds that opi(⃗xi) ̸= op′ +i(⃗x′i) or there is some position k such that vi +k ̸= v′i +k . We +do another case distinction: +a) Assume that opi(⃗xi) ̸= op′ +i(⃗x′i). Then it must hold that µi.pc ̸= µ′ +i.pc (since C deterministically maps program +counters triples (op(⃗x), pcnext, pre) ) and consequently ns′f,i ̸= nsf,i. However, since execution is deterministic, and +we know that we can reach nsf,i as the ith node from Nf(C) (with a different pc) when starting the execution in +⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩, this leads to a contradiction. +b) Assume that there is some position k such that vi +k ̸= v′i +k . This means that there exists xi +k ∈ ⃗xi (with C(µi.pc) = +(op(⃗xi), pci +next, prei)) such that µ′ +i.s(xi +k) ̸= µi.s(xi +k). However, since execution is deterministic, and we know that we can + +reach the configuration ⟨nsf,i, θ† +i ⟩ as the ith node from Nf(C) when starting the execution in ⟨ns′, toCFG(Γ, s′)⊎θ⊥⟩, +we know that θ† +i = toCFG(Γ, s′f,j) ⊎ θ⊥. Further, we know that θ† +i (xi +k) = toCFG(Γ, sf,i) ⊎ θ⊥(xi +k) and consequently +also toCFG(Γ, s′f,j) ⊎ θ⊥(xi +k) = toCFG(Γ, sf,i) ⊎ θ⊥(xi +k). This contradicts µ′ +i.s(xi +k) ̸= µi.s(xi +k). +2) Assume that |π ↓f| < |π′ ↓f| and π ↓f is a prefix of π′ ↓f. So, m < n, which means that +Γ ⊨ s′ +C :: S +� +�−→∗|f s′f,i +C +:: S +op′ +i(⃗v′i) +�−−−−→ s′f +,i +C +:: S +�m +i=1 +� +�−→∗|f s′f,i +C +:: S +op′ +i(⃗v′i) +�−−−−→ s′f +,i +C +:: S +�m+k +i=m +�−→∗|f t′ +C :: S +for some k > 1. From Theorem 4 we get that +C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨ns′f,i, toCFG(Γ, s′f,i) ⊎ θ⊥⟩ ⇒ ⟨ns′f+,i, toCFG(Γ, s′f +,i) ⊎ θ⊥⟩ +�n +i=1 +such that all ns′f,j ∈ Nf(C). From this, we can conclude that +C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨n∗ +j, θj⟩ −→ ⟨n∗ +j, θj+⟩ +�l +j=1 +for some l ≥ n such that there is some g ∈ N → N for which it holds that 1) ∀ n m. n < m ⇒ g(n) < g(m) and 2) +∀ j ∈ [1, n]. ns′f,j = n∗ +g(j) and 3) ∀ j ∈ [1, n].toCFG(Γ, s′f,j) ⊎ θ⊥ = θg(j) and 4) ∀ j ∈ [1, i]. ∀k. k > g(i) ⇒ k < +g(i + 1) ⇒ ∀pc. n∗ +g(j) = (pc, 0) ⇒ ∃q. n∗ +k = (pc, q). This is as all ⇒-steps from nodes in Nf(C) can be expanded into +the individual steps between the subnodes of the same pc. +Note that from assumption 16 we know that ns = (s.µ.pc, 0) = (s′.µ.pc, 0) = ns′. Consequently, we can apply Lemma 11 +(using Lemma 12) to obtain +C, |S| ⊨ ⟨ns, toCFG(Γ, s′) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨n∗ +j, θ′ +j⟩ −→ ⟨n∗ +j, θ′ +j+⟩ +�l +j=1 +and ∀p ∈ [1, l]. ∀x ∈ Args(C, n∗ +p). θp(x) = θ′ +p(x). Consequently, we can also conclude that +C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨ns′f,j, θ† +j⟩ ⇒ ⟨ns′f+,j, θ† +j⟩ +�n +j=1 +such that ∀ j ∈ [1, n].θ† +j = θ′ +g(j). Consequently, we have that +C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨ns′f,j, θ† +j⟩ ⇒ ⟨ns′f+,j, θ† +j⟩ +�m +j=1 +� +−→∗|Nf (C) +0 +⟨ns′f,j, θ† +j⟩ ⇒ ⟨ns′f+,j, θ† +j⟩ +�m+k +j=m +However, from Theorem 4 we know that +C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ +� +−→∗|Nf (C) +0 +⟨nsf,j, toCFG(Γ, sf,j) ⊎ θ⊥⟩ ⇒ ⟨nsf+,j, toCFG(Γ, sf +,j) ⊎ θ⊥⟩ +�m +j=1 +−→∗|Nf (C) +0 +⟨nt, toCFG(Γ, t) ⊎ θ⊥⟩ +such that all nsf,j ∈ Nf(C). This leads to a contradiction, since execution is deterministic and like this we obtain two +executions starting in ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩, which step through a different number of nodes (with different pcs) in +Nf(C). +3) Assume that |π′ ↓f| < |π ↓f| and π′ ↓f is a prefix of π ↓f. The proof is fully analogous to the previous case. +APPENDIX B +SECURIFY +A. Analysis specification +The following rules are extracted from the Securify fixed-point calculation [2] [27]. We split the appendix in input facts in +Section Section B-A1, May-semantic rules in Section B-A2, and Must-semantic rules in Section B-A3. Must-analysis rules that +are identical to the May-analysis semantic rules are left out. Identical rules are denoted with ⇐ instead of ⇐May in Appendix +B-A2. Additionally, analogous rules for storage are omitted. + +1) Input facts: +Source(L, Y0, inst) ← inst(L, Y0, . . .) +(19) +AssignVar(L, Yi, Xj) ← inst(L, . . . , Yi, . . . +� +�� +� +Outputs +, . . . , Xj, . . . +� +�� +� +Inputs +) +(20) +inst ̸∈ {mload, sload, sha3} (no propagation for known accesses) +mstore(L, MO, X) ← mstore(L, O, X), hasConstantV alue(O) +(21) +mstore(L, ⊤, X) ← mstore(L, O, X), ¬hasConstantV alue(O) +(22) +mload(L, Y, MO) ← mload(L, Y, O), hasConstantV alue(O) +(23) +mload(L, Y, ⊤) ← mload(L, Y, O), ¬hasConstantV alue(O) +(24) +AssignVar(L, Y, O) ←May mload(L, Y, O), ¬hasConstantV alue(O) +(25) +a) Optional Source Rules: +Source(L, Y, MO) ← mload(L, Y, O), hasConstantV alue(O) +(26) +Source(L, Y, M⊤) ← mload(L, Y, O), ¬hasConstantV alue(O) +(27) +Source(L, Y, Y ) ← call(L, Y, . . .) or staticcall(L, Y, . . .) +(28) +b) May Control Flow and Dependency Propagation: +Follow(L1, L2) ← instpc(L1, . . .), instpc+1(L2, . . .), hasLinearSuc(L1) +(29) +Follow(L1, L3) ← jumpI(L1, , L3) +(30) +Follow(L1, L2) ← jump(L1, L2) +(31) +Taint(L1, L3, X) ←May jumpI(L1, X, L3), L3 ̸= MergeInstr(L1) +(32) +Taint(L1, L2, X) ←May ”jumpI(L1, X, L3); inst(L2, . . .)”, L2 ̸= MergeInstr(L1) +(33) +Join(L1, L2) ←May jumpI(L1, X, L3), L2 = MergeInstr(L1) +(34) +c) Must Control Flow and Dependency Propagation: +OneBranchTag(L) ←Must jumpDest(L), one incoming branch, no prev inst in BB +(35) +Tag(L) ←Must jumpDest(L) +(36) +Jump(L1, L2, L4), Follows(L1, L2) ←Must ”jumpI(L1, . . .); inst(L2, . . .)”, L4 = MergeInstr +(37) +Jump(L1, L3, L4) ←Must jumpI(L1, , L3), L4 = MergeInstr +(38) +Jump(L1, L2, L2) ←Must jump(L1, L2) +(39) +Follow(L1, L2) ←Must ”inst(L1, . . .); inst(L2, . . .)” (same basic block) +(40) +JoinIncBranches (JIB) connects all incoming branches such that the Must-analysis can check if a predicate holds on all +preceeding nodes: +JIB(L1, L2, Linc +2 ), . . . , JIB(Linc +n , Ln, L′) ←Must ∀i, j, k. +� +� +� +� +� +� +� +”instpc(Li, . . .); jumpDestpc+1(L′)” +jump(Lj, L′) +jumpI(Lk, , L′) +(41) + +2) May-Semantic Rules: +VarMayDepOn(Y, X) ⇐May AssignVar( , Y, Y ′), VarMayDepOn(Y ′, X) +(42) +VarMayDepOn(Y, X) ⇐May AssignVar(L, Y, ), Taint( , L, Y ′), VarMayDepOn(Y ′, X) +(43) +VarMayDepOn(Y, X) ⇐May Source(L, Y, ), Taint( , L, Y ′), VarMayDepOn(Y ′, X) +(44) +VarMayDepOn(Y, X) ⇐May Source( , Y, X) +(45) +InstMayDepOn(L, X) ⇐May Taint( , L, X) +(46) +InstMayDepOn(L, X) ⇐May Taint( , L, Y ), VarMayDepOn(Y, X) +(47) +a) Memory Dependency Propagation: +MemMayDepOn(L, O, T) ⇐May mstore(L, O, X), VarMayDepOn(X, T) +(48) +MemMayDepOn(L, O, T) ⇐May Follows(L1, L), MemMayDepOn(L1, O, T), +(49) +¬ReassignMem(L, O) +ReassignMem(L, O) ⇐ mstore(L, O, ), isConst(O) +(50) +Source(L, Y, T) ⇐ mload(L, Y, O), MemMayDepOn(L, O, T), isConst(O) +(51) +Source(L, Y, T) ⇐May mload(L, Y, O), MemMayDepOn(L, , T), ¬isConst(O) +(52) +b) Control Dependence Propagation: +Taint(L1, L2, X) ⇐May Follow(L3, L2), Taint(L1, L3, X), ¬Join(L1, L2) +(53) +3) Must-Semantic Rules: +MustFollow(L1, L2) ⇐Must MustPrecedeStep(L1, L2) +(54) +MustFollow(L1, L3) ⇐Must MustFollow(L1, L2), MustFollow(L2, L3) +(55) +MustPrecedeStep(L1, L2) ⇐Must Follow(L1, L2), ¬Tag(L2) +(56) +MustPrecedeStep(L1, L3) ⇐Must Jump(L1, L2, ), oneBranchTag(L2) +(57) +MustPrecedeStep(L1, L2) ⇐Must Jump(L1, , L2) +(58) +DetBy(L, Y, X) ⇐Must Source(L, Y, X) +(59) +DetBy(L, Y, X) ⇐Must AssignVar(L, Y, Y ′), DetBy(L, Y ′, X) +(60) +DetBy(L2, Y, X) ⇐Must MustFollow(L1, L2), DetBy(L1, Y, X) +(61) +DetBy(L, Y, X) ⇐Must JoinIncBr(L1, L2, L), DetBy(L1, Y, X), DetBy(L2, Y, X) +(62) +MemDetBy(L, O, T) ⇐Must mstore(L, O, X), DetBy(L, X, T), isConst(O) +(63) +MemDetBy(L2, O, T) ⇐Must MustPrecedeStep(L1, L2), MemDetBy(L1, O, T), +(64) +¬ReassignMem(L2, O) +MemDetBy(L, O, T) ⇐Must JoinIncBr(L1, L2, L), MemDetBy(L1, O, T), +(65) +MemDetBy(L2, O, T), (isConst(O)) + diff --git a/ydFST4oBgHgl3EQfTTiJ/content/tmp_files/load_file.txt b/ydFST4oBgHgl3EQfTTiJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..843ac6f47e78bfa3089d20dc3798354dfcb39e71 --- /dev/null +++ b/ydFST4oBgHgl3EQfTTiJ/content/tmp_files/load_file.txt @@ -0,0 +1,2935 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf,len=2934 +page_content='HORSTIFY: Sound Security Analysis of Smart Contracts Sebastian Holler‡∗, Sebastian Biewer†, Clara Schneidewind∗ ∗Max-Planck-Institute for Security & Privacy Universit¨atsstraße, Bochum, Germany †Saarland University, ‡Saarbr¨ucken Graduate School of Computer Science Saarland Informatics Campus, Saarbr¨ucken, Germany Abstract—The cryptocurrency Ethereum is the most widely used execution platform for smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Smart contracts are distributed applications, which govern financial assets and, hence, can implement advanced financial instruments, such as decentralized exchanges or autonomous organizations (DAOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Their financial nature makes smart contracts an attractive attack target, as demonstrated by numerous exploits on popular contracts resulting in financial damage of millions of dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This omnipresent attack hazard motivates the need for sound static analysis tools, which assist smart contract developers in eliminating contract vulnerabilities a priori to deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Vulnerability assessment that is sound and insightful for EVM contracts is a formidable challenge because contracts execute low-level bytecode in a largely unknown and potentially hostile execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So far, there exists no provably sound automated analyzer that allows for the verification of security properties based on program dependencies, even though prevalent attack classes fall into this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this work, we present HORSTIFY, the first automated analyzer for dependency properties of Ethereum smart contracts based on sound static analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' HORSTIFY grounds its soundness proof on a formal proof framework for static program slicing that we instantiate to the semantics of EVM bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We demonstrate that HORSTIFY is flexible enough to soundly verify the absence of famous attack classes such as timestamp dependency and, at the same time, performant enough to analyze real-world smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Index Terms—Ethereum, Smart Contract, Blockchain, Depen- dency Analysis, Security, Tool I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' INTRODUCTION Modern cryptocurrencies enable mutually mistrusting users to conduct financial operations without relying on a cen- tral trusted authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Foremost, the cryptocurrency Ethereum supports the trustless execution of arbitrary quasi Turing- complete programs, so-called smart contracts [31], which manage money in the virtual currency Ether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The expressiveness of smart contracts gives rise to a whole distributed financial ecosystem known as Decentralized Fi- nance (DeFi), which encompasses a multitude of (financial) applications such as brokerages [20], [32], decentralized ex- changes [3], [16], [33] or decentralized autonomous orga- nizations [15], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, smart contracts have shown This work has been supported by the Heinz Nixdorf Foundation through a Heinz Nixdorf Research Group (HN-RG) and funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Ger- many’s Excellence Strategy—EXC 2092 CASA—390781972, and through grant 389792660 as part of TRR 248—CPEC, see https://perspicuous- computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' to be particularly prone to programming errors that lead to devastating financial losses [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These severe incidents can be attributed to different factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' First, smart contracts are agents that interact with a widely unpredictable and potentially hostile environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Accounting for all possible environment behav- iors adds a layer of complexity to smart contract development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Second, smart contracts manage real money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This financial nature makes them an extraordinarily lucrative attack target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Third, transactions in blockchain-based cryptocurrencies, like Ethereum, are inherently immutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As a consequence, not only the effects of exploits are persistent, but also vulnerable smart contracts cannot be patched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Given this state of affairs, it is of utmost importance to preempt contract vulnerabilities a priori to contract deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Sound static analysis tools allow for reasoning about all possible runtime behaviors without deploying a contract on the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this way, smart contract developers and users can reliably identify and eliminate harmful behavior before publishing or interacting with Ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' How- ever, as shown in recent works [24], [25], most automatic static analyzers for Ethereum smart contracts that promise soundness guarantees cannot live up to their soundness claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To the best of our knowledge, the only tools targeting sound and automated static analyses of smart contract security properties are Securify [27], ZEUS [18], EtherTrust [13], NeuCheck [21], and eThor [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The soundness claims of ZEUS, Securify, EtherTrust, and NeuCheck are systematically confuted in [25] and [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The analysis tool eThor [24] comes with a rigorous sound- ness proof but only supports the verification of reachability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' While this is sufficient to characterize the absence of interesting attack classes, many other smart contract security properties do not fall within this property fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Grishenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [14] give a semantic characterization of security prop- erties that characterize the absence of prominent classes of smart contract bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Most of these properties fall into the class of non-interference-style two-safety properties that we will refer to as dependency properties and fall out of the scope of eThor’s analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The only tool that, up to now, targeted the (sound) verification of dependency properties was the tool Securify [27]—which was empirically shown unsound in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Our Contributions: In this work, we revisit Securify’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this course, we analyze the peculiar challenges arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='13769v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='CR] 31 Jan 2023 in designing a sound static dependency analysis tool for Ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We show how to overcome these obstacles with a principled approach based on rigorous formal foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Leveraging a formal proof framework for static program slicing [30], we design a provably sound dependency analysis for Ethereum smart contracts on the level of Ethereum Virtual Machine (EVM) bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, we give an im- plementation of the analyzer HORSTIFY that performs the static dependency analysis via a logical encoding, which can be automatically solved by Datalog solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We demonstrate how to use HORSTIFY to automatically verify dependency properties on smart contracts, such as the ones defined in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Concretely, we make the following contributions: We study the root causes of the soundness issues of the state-of-the-art Ethereum smart contract analysis tool Securify [27] that so far had only been reported through empirical evidence in [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this course, we uncover new soundness problems in Securify’s analysis, which we can show to affect real-world smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We devise a new dependency analysis for EVM bytecode based on program slicing following the static program framework presented in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We prove this dependency analysis to be sound with respect to a formal semantics of EVM bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We show how to approach relevant smart contract security properties presented in [14] with the dependency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We present HORSTIFY, an automated prototype static analysis tool that implements the dependency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We demonstrate that HORSTIFY overcomes the sound- ness issues of Securify while showing comparable per- formance and small precision loss on real-world smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The remainder of the paper is organized as follows: Sec- tion II overviews our approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Section III introduces the necessary background on Ethereum smart contract execution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Section IV discusses the challenges in designing sound static analysis tools for smart contract dependency analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Sec- tion V introduces the slicing proof framework from [30] that our analysis builds on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Section VI presents our static analysis based on program slicing and its soundness proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Section VII reports on our prototype implementation HORSTIFY and its practical evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' and Section IX concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' OVERVIEW In this paper, we develop a dependency analysis tool for EVM bytecode that is designed in accordance with formal correctness statements providing overall soundness guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The correctness proof is modularized as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The core module is a generic proof framework [30] for backward slicing using abstract control flow graphs (CFGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In these CFGs, each node is annotated with all variables it reads and all variables it writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The backward slice of a node is a set containing all nodes that possibly influence the variables written in the respective node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The framework extends the abstract CFG to a program dependence graph (PDG) by explicitly defining the data and control dependencies Abstract CFG Program Dependence Graph (PDG) Backward Slices Generic correctness statement EVM Semantics Security Properties Horn Clauses Security Patterns Solver Correctness Statement Correctness Statement secure/ insecure EVM CFG Semantics A B C HoRStify EVM Backward Slices Slicing Framework D E Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Overview on the formal guarantees of HORSTIFY between the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this PDG, the framework establishes a generic correctness statement for slicing: whenever a node influences another, the influencing node appears in the back- ward slice of the influenced node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To obtain the correctness result for a concrete programming language the abstract CFG representation is instantiated for a concrete program semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We instantiate the framework for EVM bytecode by devising a new EVM CFG semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We show ( A ) that the EVM CFG semantics satisfies all requirements for instantiation and ( B ) that it is equivalent to a formalisation of the EVM bytecode semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From this, we obtain backward slicing for EVM contracts with a corresponding correctness statement ( C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For the actual analysis, we express dependencies in EVM contracts by means of dependency predicates which we char- acterize by (fixpoints over) a set of logical rules, given in the form of Constrained Horn Clauses (CHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Most importantly, we show that if the backward slice of some program point contains some other program point, then the (potential) de- pendency between these two program points is also captured by the predicate encoding ( D ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From the EVM bytecode analyzer Securify [27] we adopt the idea of defining so-called security patterns to soundly approximate the satisfaction (or violation) of a security prop- erty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A security pattern is a set of facts over dependency predicates, which characterize the form of dependencies that are ruled out by the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In contrast to Securify, our formal characterization of dependency predicates enables a correctness statement for the approximating behavior of the security patterns w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' their corresponding property ( E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, we present the prototype tool HORSTIFY that implements our dependency analysis and uses the Datalog engine Souffl´e to perform the fixpoint computation and to check whether a security pattern is matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A pattern match guarantees (in)security w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' the respective security property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Challenges: The main challenge of designing a practical and sound dependency analysis for EVM bytecode is finding precise and performant abstractions that tame the complexity of EVM bytecode while maintaining soundness guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As we will show in Section IV, EVM bytecode’s language design makes this task particularly hard: Non-standard language fea- tures introduce corner cases that are easily overlooked or make it necessary to enhance the analysis with custom optimizations that can lead to unsoundness when done in an ad-hoc manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As a consequence, it is of paramount importance to construct a sound analysis tool with formal foundations that are flexible enough to cover those subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The slicing framework [30] enables a modular soundness proof that separates the standard argument for the correctness of slicing from the characterization of program dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, even though this reduces the proof effort, a naive instantiation of the framework would introduce a multitude of superfluous dependencies and hence lead to a highly imprecise analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason, the key challenge lies in the design of the EVM CFG semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We will show how to approach these challenges with a solid theoretical foundation and by circumventing the bothersome technical hurdles without com- promising the soundness of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' BACKGROUND ON ETHEREUM SMART CONTRACTS Ethereum smart contracts are distributed applications that are jointly executed by the users of the Ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, we shortly overview the workings of Ethereum and the resulting particularities of the Ethereum smart contract execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) Ethereum: The cryptocurrency Ethereum supports smart contracts via an account-based execution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The global state of the system is given by accounts whose states are modified through the execution of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' All accounts have in common that they hold a balance in the currency Ether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An account can be either an external account that is owned by a user of the system and that solely supports user-authorized money transfers, or a contract account that manages its spending behavior autonomously by means of a program associated with the contract that may use its own persistent storage to provide advanced stateful functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Users interact with accounts via transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Transactions either call existing accounts or create new contract accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A call transaction transfers an amount of money (that could be 0) to the target account and triggers the execution of the account’s code if the target is a contract account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A contract execution can modify the contract’s persistent storage and potentially initiates further transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this case, we speak of internal transactions, as opposed to external transactions, which are initiated by users on behalf of external accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) Smart Contract Languages: Smart contracts are spec- ified in EVM bytecode and executed by the Ethereum Virtual Machine (EVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' EVM bytecode is a stack-based low-level language that supports standard instructions for stack ma- nipulation, arithmetics, jumps, and memory access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' On top, EVM’s instruction set includes blockchain-specific opcodes, for example, to access transaction information and to initiate internal transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' While the EVM bytecode is technically Turing-complete, the execution of smart contracts is bounded by a transaction-specific resource limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' With each transaction, the originator sets this limit in the unit gas and pays for it upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' During the execution, instructions consume gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The execution halts with an exception if running out of gas and reverts all effects of the prior execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In practice, Ethereum smart contracts are written in high- level languages—foremost, Solidity [1]—and compiled to EVM bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Solidity is an imperative language that mimics features of object-oriented languages like Java but supports additional primitives for accessing blockchain information and performing transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For better readability, we will give examples using the Solidity syntax even though our analysis operates on EVM bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We will introduce relevant Solid- ity language features throughout the paper when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' c) Adversarial Execution Environment: The blockchain environment poses novel challenges to the programmers of smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As opposed to programs that run locally, smart contracts are executed in an untrusted environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This means, in particular, that certain system parameters cannot be fully trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A prominent example of this issue is Ethereum’s block timestamp: In Ethereum’s blockchain-based consensus mechanism, the system is advanced by appending a bulk of transactions grouped into a block to the blockchain, a distributed tamper-resistant data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These blocks are created by special system users, so-called miners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' While all system users check that blocks only contain valid transactions, the correctness of a block’s metadata cannot easily be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So is each block required to carry a timestamp, but due to the lack of synchronicity in the system, this timestamp can only be checked to lie within a plausible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This enables a miner to choose the value of the block timestamp freely within this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The following example illustrates how this peculiarity can be exploited in a smart contract: 1 function spinWheel() private (uint) { 2 return block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='timestamp % 37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } The function spinWheel() implements a spinning wheel that determines a random number between 0 and 36 based on the block timestamp (accessed via block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='timestamp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Based on such a function, a contract could implement a roulette game where players bet money on the outcome of the spinning wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' While the system timestamp may serve as a decent source of randomness for programs that run locally, this is not the case for smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A miner could easily tweak the timestamp of a block containing an invocation of the spinWheel function and thereby influence its outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this way, a miner could ensure to win the roulette game themself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' CHALLENGES IN SOUND DEPENDENCY ANALYSIS As recently demonstrated in the literature [25], the sound analysis of Ethereum smart contracts is a challenging problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' most analysis tools aiming at provable soundness guarantees fall short of their goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This can be mainly attributed to the non-standard language features of the EVM bytecode language and the unusual execution model of the EVM: Smart contracts are executed in a (potentially) hostile environment, which can interact with, and even, schedule contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The smart contract execution is dependent on the gas resource and the low-level compliance: all jump(L1, Y, ), sstore(L2, , ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MustFollow(L1, L2) ∧ DetBy(L1, Y, caller) violation: some sstore(L1, X, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬MayDepOn(X, caller) ∧ ¬MayDepOn(L1, caller) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Restricted Write compliance and violation pattern [27] EVM bytecode language features little static information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As a consequence, execution heavily depends on unknown runtime parameters, which makes it hard to reason statically about contract behaviors in a sound and reasonably precise and efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This incentives the incorporation of ad-hoc optimizations, which increase the complexity of the analysis even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, it is crucial to establish rigorous formal foundations for EVM bytecode analysis and to align the implementation with these foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, we demonstrate how the lack of formal foundations affects the guarantees of the state-of-the-art analysis tool Securify [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify The automated analyzer Securify is the only analysis tool up to now that aims at giving provable guarantees for dependency analyses of EVM bytecode contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' It decompiles the byte- code into a stackless intermediate representation (IR), where values are stored in variables in static single assignment (SSA) form rather than on a stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, it determines the CFG of the contract and encodes the transitive control and data flow dependencies between variables and program locations as a set of dependency predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' While it is not possible to specify arbitrary (security) properties in Securify, the tool allows for defining compliance patterns and violation patterns that serve as “approximations” for the satisfaction and, respectively, the violation of the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These patterns are defined over the dependency predicates and can be checked automatically using the Datalog solver Souffl´e [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A compliance pattern is sound w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a property, if satisfying the pattern implies satisfaction of the property, and, analogously, a violation pattern is sound w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a property if satisfying the pattern implies violation of the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' If neither of the patterns is satisfied, the satisfaction of the property is inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Obviously, it cannot be that for the same contract and for the same property a sound compliance and violation pattern hold simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An example of a security property is the restricted write (RW) property for storage locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, a contract satisfies RW, if for all storage locations, there is at least one caller address that cannot write to this location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Figure 2 shows a compliance1 and violation pattern for RW [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The compliance pattern for RW states that for all conditional jump instructions at program location L1 that branch on con- dition Y (jump(L1, Y, )) and for all storage write instructions at location L2 (sstore(L2, , )) that are necessarily preceded by such jump instructions (MustFollow(L1, L2)), it must hold that at location L1 the condition Y must be determined 1The Securify implementation contains two compliance patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' one is shown in [27], the other one is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 contract Start { bool test = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2 function flipper() public { 3 if (uint(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender) * 0 == 0) 4 { test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify counterexample: must-analysis by the caller of the transaction (DetBy(L1, Y, caller)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The violation pattern for RW states that there is some storage write instruction at location L1 writing to storage address X (sstore(L1, X, )) such that that neither the address X nor the execution of the storage instruction at L1 may depend on the caller of the transaction (¬MayDepOn(X, caller) ∧ ¬MayDepOn(L1, caller)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Soundness Issues Even though Securify characterizes security properties and their corresponding compliance and violation patterns, no formal connection between patterns and properties is estab- lished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular, they do not prove the soundness of the patterns they propose in [27] w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' the properties they are supposed to approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Doing so would require 1) to prove that the dependency predicates imply semantic notions of independence (sound core analysis) and 2) to prove that the semantic notions implied by the security patterns indeed imply the security properties (sound security patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, we use the example of the RW property to show how the absence of formal soundness arguments causes Securify to miss corner cases that undermine its soundness guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1) Sound Core Analysis: Securify does not draw a con- nection between the dependency predicates and the EVM bytecode semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This leads to mismatches between the intuitions for the predicates and their definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) Must-analysis: Securify’s dependency analysis and predicates can be attributed to one of two categories: a may-analysis aims at over-approximating possible control and dataflow dependencies, encoded by may-predicates, and a must-analysis aims at capturing dependencies and deducing must-predicates that show a definite effect on the actual execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' According to their usage in the security patterns, negated may-predicates imply a notion of independence, while must-predicates should imply a form of determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, it is stated that the must-predicate DetBy(L, Y, T) “indicates that a different value of T guarantees that the value of Y changes.” [27] This guarantee, however, is violated in the contract shown in Figure 3, where Securify inferred that test is determined by the caller although every caller can change the value of test: The check of the conditional evaluates to true for any value of msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender, hence allowing every caller to write the test field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Still, Securify reports this contract to match the compliance pattern, indicating that the condition in line 3 would be determined by the caller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The underlying reason for this problem is of substantial nature: The must-analysis under- approximates control flows but over-approximates data flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, a variable X is considered to be determined by a variable Y if Y occurs in the expression assigned to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since 1 contract Start { bool test = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2 function storeTest(uint c) public { 3 address[] memory a = new address[](7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 4 for (uint i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' i < 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' i++) { 5 a[i] = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='} 6 if (a[0] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='= address(0)) {test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='} } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify counterexample: storage abstraction msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender appears in the condition expression in line 3, the condition is considered to be determined by msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender even though it actually is independent of msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This treatment makes the must-analysis inherently unsound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Due to this substantial mismatch between the intuition for the DetBy predicate and its implementation, it is unclear whether adjusting the implementation of the must-analysis such that it is sound, could result in a performant and precise analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So, in this work, we will focus on the may-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) Memory Abstraction: For establishing a sound may- analysis, it is crucial to overapproximate dependencies for all relevant system components that can interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular, this includes stack, memory and storage variables, because values are written from the stack to the local memory and persistent storage, and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, the addresses of memory and storage accesses are not statically known but specified on the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', the EVM instruction MSTORE(x, y) denotes that the value in stack variable y should be written to the address as given in stack variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, the concrete memory address at which the value in y will be stored may only be known at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This poses a big challenge to static analysis since for precisely modeling the dependencies on different memory and storage cells, their accesses need to be known statically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Otherwise, the dependen- cies on all memory and storage cells would need to be merged, resulting in a substantial precision loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In practice, memory and storage addresses can in most cases be precomputed by partial evaluation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Hence, this preprocessing information can be used to enhance the analysis precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify implements this optimization in an unsound way, as illustrated by the example in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Here, function storeTest locally defines a new address array a of size 7 and initializes all its elements with the contract caller msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The write access to the test variable is restricted by the condition that the first array element a[0] (which obviously contains the caller address) is not 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, the contract satisfies the RW property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Still, Securify certifies a violation of the RW pattern w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' test2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The example illustrates that the analysis does not consider that a memory address may be statically unknown at the point of writing but known at the point of reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since writing to the array is done in a loop, for the assignment a[i] = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender the memory address cannot be statically determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For the condition in line 6, in contrast, the memory address for a[0] can be precomputed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, Securify fails to account for the fact that dependencies of 2This is indeed unsoundness and not imprecision: Securify guarantees that a property does not hold if the violation pattern matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Only inconclusive cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', no compliance and no violation pattern matches) cause imprecision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 contract Start { bool test = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2 address a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3 function setAddress(address addr) public 4 { a = addr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 5 function flipper () public { 6 try Start(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='setAddress(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender) { 7 if (a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='= address(0)) { test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 8 } catch { revert();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify counterexample: reentrancy handling 1 contract Check { 2 function testZero (address a) public { 3 assert (a == address(0));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } 4 contract Start { 5 bool test = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 6 address check = address(42);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 7 function flip() public { 8 try Check(check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='testZero(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender){ 9 test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 10 } catch {return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='} } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify counter example: external call handling an unknown memory access should propagate to all concrete memory addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' c) Reentrancy handling: Smart contracts are reactive programs in the sense that they can transfer control to other contracts and are subject to reentrancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', while awaiting the return of the other contract, this contract may call the waiting contract again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Figure 5 shows a simple case of reentrancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this variant of Figure 3, function flipper calls the contract’s function setAddress within a new internal transaction3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' flipper uses setAddress to store the caller msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender in the storage lo- cation a (defined in line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then, flipper modifies the critical storage location test if and only if the address stored in a is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Ethereum contracts are executed non-concurrently, so the value of a remains unchanged after line 6 and before the evaluation of the condition in line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, a caller with address 0 can never write to the test field and the contract satisfies the RW property w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Still, Securify reports a match of the violation pattern for test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Inspection of the Securify code reveals that it does not model potential dependencies between arguments of external calls and storage locations accessible via reentrancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' d) External call handling: Aside from reentrancy, exter- nal calls may affect the local execution state in multiple ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The success of an external call is indicated by placing a corresponding flag on the stack and the return value of the call (if existent) is written to a memory fragment that is specified as an argument to the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These effects may depend on the recipient and the arguments of the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The example in Figure 6 illustrates how ignoring those dependencies causes an unsoundness in Securify: In this example, the sender check is outsourced to the method testZero of another contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The assignment of variable test depends on whether testZero returns without the assert throwing an exception, which in turn depends on (the input data) msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Hence, this contract 3A reasonable contract would call a function of the same contract directly so that such a call would be translated to a JUMP by the compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The chosen syntax enforces that the function call will be translated to a CALL instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 contract Start { bool test = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2 function flipper () public { 3 require(gasleft() > 10000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 4 bool flip = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 5 if (msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender == address(0)) { 6 { while (gasleft() >= 5000) 7 { flip = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } 8 if (gasleft() < 5000) {test = flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='} } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify counterexample: gas handling satisfies the RW property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Still, Securify reports a violation, since no dependencies between the input to the call and the call output are modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' e) Gas handling: Figure 7 shows a contract that indi- rectly restricts write access to storage test by consuming the gas resource in a controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In line 3, the contract ensures that it is executed with a generous amount of gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' if not enough gas is available, the execution is aborted and no caller is able to write to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The code between lines 5 and 7 essentially wastes masses of gas if the caller address is equal to 0, and, otherwise, consumes very little gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The crux of the contract is in line 8: From the amount of gas that is left, the contract can determine if the caller’s address is equal to 0—this is the case if and only if less than 5000 gas units are left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Hence, depending on the amount of available gas, either no caller or only caller 0 can write to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So, there is always at least one caller that cannot write to test—the contract satisfies the RW property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, Securify reports a violation of this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The reason for this wrong analysis result is that Securify does not track dependencies for the gas resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2) Sound Security Properties: Since the dependency predi- cates do not have a semantic characterization, the soundness of the security patterns w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' their corresponding property cannot be proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Indeed, Schneidewind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [24] provide counter examples for the soundness of 13 out of the 17 security patterns given in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Above that, the unsoundness of the RW property undeniably manifests in line 4 of the contract we constructed in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this example, Securify reports simultaneously(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=') satisfaction of a compliance and a violation pattern for the RW property w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This refutes the claim that compliance and violation patterns constitute sufficient criteria for property compliance and violation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ANALYSIS FOUNDATIONS To design a sound static analysis for EVM bytecode based on program slicing, we instantiate the slicing proof framework from [30] with a formal bytecode semantics as defined in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Before discussing the instantiation in Section VI, we shortly overview both frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' EVM bytecode semantics The EVM semantics was formally defined in [14] in form of a small-step semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We use a linearized representation of the semantics inspired by Securify, where the use of the stack is replaced by the usage of local variables in SSA form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We will call these variables stack variables and, in the following, always refer to the linearized representation of the semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Formally, the semantics of EVM bytecode is given by a small-step relation Γ ⊨ S → S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The relation describes how a contract, whose execution state is given by a callstack S, can progress to callstack S′ under a transaction environment Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The transaction environment Γ holds information about the ex- ternal transaction that initiated execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We let Γ ⊨ S →∗ S′ denote the reflexive transitive closure of the small-step relation and call the pair (Γ, S) a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The details of the components of the EVM configurations can be found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The overall state of an external transaction execution is captured by a callstack S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The elements of the callstack model the states of all (pending) internal transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In- ternal transactions can either be pending, as indicated by a regular execution state (µ, ι, σ), or terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The state of a pending transaction encompasses, the current global state σ, the execution environment ι and the machine state µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The global state σ describes the state of all accounts of the system and is defined as a partial mapping between account addresses and account states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The execution environment ι, among others, contains the code of the currently executing contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We model the code of a contract as a function C that maps program counters to tuples (op(⃗x), pcnext, pre), where op denotes an opcode from the EVM instruction set, ⃗x is the vector of input and output (stack) variables to this opcode, and pcnext denotes the program counter for the next instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, we instrument each instruction with a list pre of precomputed values for the arguments ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This instrumentation is only introduced for analysis purposes and does not affect the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The machine state µ captures the state of the local machine and holds the amount of gas (g) available for execution, the program counter (pc), the local memory, and the state of the (linearized) stack variables (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) Small-step Rules: We illustrate the working of the EVM bytecode semantics using the example of the ADD instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This instruction takes two values as input and writes their sum back to its return variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code [µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc] = (ADD(r, a, b), pcnext, pre) µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g ≥ 3 µ′ = µ[s → µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s[r → µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(a) + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(b)]][pc → pcnext][g −= 3] Γ ⊨ (µ, ι, σ) :: S ADD(a,b) −−−−−→ (µ′, ι, σ) :: S Given a sufficient amount of gas (here 3 units), an ADD instruction with result (stack) variable r and operand (stack) variables a and b writes the sum of the values of a and b to r and advances the program counter to pcnext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These effects, as well as the subtraction of the gas cost, are reflected in the updated machine state µ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) Security properties: Previous work [14] has shown that there are several generic smart contract security properties, which are desirable irrespective of the individual contract logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The properties formally defined in [14] are integrity properties that aim at ruling out the influence of attacker behavior on sensitive contract actions, in particular, the spend- ing of money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These properties are e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', the independence of a contract’s spending behavior from miner-controlled pa- rameters (as the block timestamp) or mutable contract state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, [14] introduces the notion of call integrity, which requires that the spending behavior of a contract is independent of the code of other smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since call integrity is hard to verify in the presence of reentering exeutions, a proof strategy is devised that decomposes call integrity into one reachability property (single-entrancy) that restricts reentering executions and two local dependency properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These local dependency properties ensure that the spending behavior of the contract does not depend on the return effects of calls to other (unknown) contracts (effect independence) or immediately on the code of such contracts (code independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Focussing on integrity, the security properties from [14] are given as non-interference-style notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We illustrate this with the example of timestamp independence, a property that requires that the block timestamp cannot influence a contract’s spending behavior and hence would rule out vulnerabilities as those in the roulette example: Definition 1 (Independence of the block timestamp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A con- tract C is independent of the block timestamp if for all reachable configurations (Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' sC :: S) it holds for all Γ′ that Γ =/timestamp Γ′ ∧ Γ ⊨ sC :: S π−→ ∗ s′ C :: S ∧ final (s′) ∧ Γ′ ⊨ sc :: S π′ −→ ∗ s′′ C :: S ∧ final (s′′) =⇒ π ↓callsC= π′ ↓callsC This definition requires that two executions of the contract C starting in the same execution state sC and in transaction environments Γ and Γ′ that are equal up to the block timestamp (denoted by Γ =/timestamp Γ′) exhibit the same calling behavior (captured by the call traces π ↓callsC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, this ensures that the contract C may not perform different money transfers based on the block timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The roulette example trivially violates this property since, based on the block timestamp, the prize will be paid out to a different user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Program Slicing Static program slicing is a method for capturing the de- pendencies between different program points (nodes) and variables in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, the program slice of some program node n in a program P consists of all those nodes n′ in P that may affect the values of variables written in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Pro- gram slices are constructed based on the program dependence graph (PDG) that models the control and data dependencies between the nodes of a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, we will review the static slicing framework by Wasserraab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [30], which establishes a language-independent correctness result for slicing based on abstract control flow graphs (CFGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) Abstract control flow graph: An abstract CFG is a language-agnostic representation of program semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Tech- nically, an abstract CFG is parametrized by a set of program states Θ and defined by a set of nodes (representing program points) and a set of directed edges between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Edges may be of two different types: State-changing edges n− ⇑f −→ n′ alter the program state θ ∈ Θ by applying the function f to θ and predicate edges n −(Q)√ −→ n′ guard the transition between n and n′ with the predicate Q on the program state θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We write n as −→ ∗ n′ to denote that node n can be reached n′ using the edges in the list as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Abstract CFG edges can be related to actual runs of the program by lifting them to a small-step relation of the form ⟨n, θ⟩ −a−→ ⟨n′, θ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) PDG and backward slices: The PDG for a program consists of the same nodes as the CFG for this program and has edges that indicate data and control dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To make data dependencies inferable, each node n is annotated with a set of variables that are written (short Def set, written Def(n)) and a set of variables that are read by the outgoing edges of the node (short Use set, written Use(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A node n′ is data dependent on node n (written n −→dd n′) if n defines a variable Y (Y ∈ Def(n)), which is used by n′ (Y ∈ Use(n′)) and n′ is reachable from n in the CFG without passing another node that defines Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A node n′ is (standard) control dependent on node n (written n −→cd n′) if n′ is reachable from n in the CFG, but n can as well reach the program’s exit node without passing through n′ and all other nodes on the path from n to n′ cannot reach the exit node without passing through n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So intuitively, n is the node at which the decision is made whether n′ will be executed or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Based on the data and control flow edges of the PDG, the backward slice of a node n (written BS(n)) is defined as the set of all nodes n′ that can reach n within the PDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' c) Correctness statement: The generic correctness state- ment for slicing proven in [30] is stated as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Correctness of Slicing Based on Paths [30] ⟨n, θ⟩ as −→ ∗ ⟨n′, θ′⟩ ∃ as′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n, θ′⟩ as′ −→ ∗ BS(n′) ⟨n′, θ′′⟩ ∧ as ↓BS(n′)= as′ ∧ (∀ V ∈ Use(n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ′(V ) = θ′′(V )) Intuitively, the theorem states that whenever a node n can reach some node n′ in the PDG (⟨n, θ⟩ as −→ ∗ ⟨n′, θ′⟩), then removing all outgoing edges from nodes not in the backward slice of n′ (⟨n, θ⟩ as′ −→ ∗ BS(n′) ⟨n′, θ′′⟩) without altering the path through the PDG in any other way (as ↓BS(n′)= as′) has no impact on n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Having no impact on n′ means that variables used in n′ are assigned to the same values re- gardless of whether the edges have been removed or not (∀V ∈ Use(n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ′(V ) = θ′′(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We call the PDG without the above-mentioned edges also sliced PDG or sliced graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SOUND EVM DEPENDENCY ANALYSIS In the following, we instantiate the slicing proof frame- work [30] to accurately capture program dependencies of EVM smart contracts in terms of program slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We then give a logical characterization of such program slices, which allows for the automatic computation of dependencies between different program points and variables with the help of a Datalog solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The generic correctness statement of the slicing proof framework guarantees that the slicing-based dependen- cies soundly over-approximate all real program dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We show how to use this result to automatically verify relevant smart contract security properties such as the independence of the transaction environment and the independence of mutable account state as defined in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Instantiation of Slicing Proof Framework We instantiate the abstract CFG from the slicing framework with the linearized EVM semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The concrete layout of the instantiation heavily influences the resulting backward slices and the precision of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, we sketch the most interesting aspects of our instantiation of the CFG components and how they contribute to the design of a precise dependency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Preprocessing Information: For a precise analysis, it is indispensable to preprocess contracts to aggregate as much statically obtainable information as possible—without com- promising the soundness of the overall analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For example, knowing the precise destination of jump instructions is crucial to reconstruct control flow precisely, and, moreover, this information usually can be easily reconstructed, especially, when contracts were compiled from a high-level language with structured control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We require in the following that the preprocessed informa- tion is correct: Definition 2 (Sound Preprocessing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A contract C has sound preprocessing information if for all execution states sC with an initial machine state running contract C it holds that if Γ ⊨ sC :: S →∗ s′ C :: S then C(s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc) = (op(⃗x), pcnext, pre) ⇒ ∀i ∈ [0, |⃗x| − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre[i] = ⌊µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(xi)⌋ ∨ pre[i] = ⊥ In the remainder, we assume that all existing preprocess- ing information is correct and sufficient to reconstruct the contract’s CFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Recall that, formally, we consider a contract a function, such that for a program counter pc, C(pc) = (op(⃗x), pcnext, pre) where pre contains the preprocessing infor- mation for the instruction op(⃗x): for every ⃗x[i], pre[i] either holds a precomputed static value, or ⊥ to indicate that no static value could be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that we restrict preprocessing to stack variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For our analysis, we are only interested in precomputed values for memory and storage locations and jump destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' CFG States: The edges of the CFG are labeled with state-changing functions or predicates on states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For EVM bytecode programs, the CFG state θ is partitioned into stack variables (denoted by xls), memory variables (xm), storage variables (xg) and local (xel) and global (xeg) environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Memory and storage variables represent cells in the local memory, respectively the global storage of the contract under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Local environment variables contain the information of the execution environment that is specific to an internal transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Global environmental variables denote environmental information whose accessibility is not limited to a single internal transaction, like the state of other contracts and the block timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Environmental information that cannot be directly accessed during the execution (such as the storage of other contracts) is hidden in the dedicated global environmental variable externaleg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' CFG Nodes, Edges & Def and Use Sets: To transform an EVM bytecode program into a CFG, we map every program counter pc to one or more nodes (pc, i) in the CFG (where C(pc) = (JUMPI(x1 ls, x2 ls), pcnext, pre) f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 10) C, cd ⊨ (pc, 0) −⇑f − → (pc, 1) Def = {ge} Use = {ge} C(pc) = (JUMPI(x1 ls, x2 ls), pcnext, pre) Q = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] = 0) C, cd ⊨ (pc, 1) −(Q)√ − → (pcnext, 0) Def = ∅ Use = {x2 ls} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' JUMPI abstract CFG instantiation i ∈ N is used to distinguish between multiple nodes for pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We call a node (pc, 0) initial node (for pc) and nodes (pc, i) with i > 0 intermediate nodes (for pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since the size of the callstack below the translated callstack element may influence the contract execution, the rule set defining the CFG transformation constructs a relation of the form C, cd ⊨ (pc, i) −a −→ (pc′, i′), where C is the contract for which the CFG is constructed, cd is the size of the callstack, and a stands for either a (Q)√ action (for a predicate edge) or ⇑f action (for a state-changing edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' With every rule, we also provide Def and Use sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The Use sets contain all variables whose values are retrieved from the state θ in the definition of the Q predicate or f function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, the definition set contains all variables that are overwritten by the function f (and is always empty for predicate edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Figure 8 shows two exemplary rules for the conditional jump instruction JUMPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The first argument to JUMPI is the jump destination and the second argument is the condition variable that must be non-zero for the jump to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We only show rules for the case that the condition is false, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', the jump does not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The upper rule defines a state-changing edge that deducts the gas that has to be paid for a JUMPI instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Appropriately, both Def and Use sets contain the gas variable because the current gas value must be read from and the reduced value updated in state θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that the edge goes from the initial node for pc to an intermediate node for pc, because a second step is necessary to decide whether the program should jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The second step, depicted by the lower rule, continues in the intermediate node for pc and checks if the condition (in variable x2ls) is false (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', if it is zero) via a predicate edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this case, the execution proceeds to the initial node representing pcnext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' x2ls is the only variable used by Q, hence it is the only variable in the Use set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' It can be shown that the CFG semantics and EVM semantics coincide via two simulation relations where every (multi-)step in the CFG semantics between initial nodes is simulated by a step of the bytecode semantics and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Core abstractions We review the most interesting aspects of the CFG seman- tics and how they lead to a precise dependency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this course, we will show how to overcome the challenges presented in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) Gas abstraction: In the EVM, the execution of in- structions consumes gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' If the gas is not sufficient to finish C(pc) = (ADD(yls, x1 ls, x2 ls), pcnext, pre) f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 3) C, cd ⊨ (pc, 0) −⇑f − → (pc, 1) Def = {ge} Use = {ge} C(pc) = (ADD(yls, x1 ls, x2 ls), pcnext, pre) f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := θ[x1 ls] + θ[x2 ls]) C, cd ⊨ (pc, 1) −⇑f − → (pcnext, 0) Def = {yls} Use = {x1 ls, x2 ls} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' CFG semantics rules for the ADD instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' the execution of a contract, it is aborted with an exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Modeling this behavior accurately would result in a very imprecise analysis, since, technically, every instruction would be control-dependant on all its preceding instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is as the execution of an instruction depends on whether prior in- structions led to an out-of-gas exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, in practice, users should only call contracts with a sufficient amount of gas since, otherwise, the contract execution exceptionally halts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason, there exist static analysis tools for computing (sound) gas bounds [4] and even Solidity’s online compiler provides gas estimates for smart contract execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Hence, for our analysis we assume that a contract does not run out of gas and do not model the corresponding behavior in the CFG semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We remark that Securify also makes this assumption implicitly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' we spell it out explicitly as follows: Assumption 1 (Absence of local out-of-gas exceptions (infor- mal)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A contract execution does not exhibit local-out-of-gas exceptions if each local exception can be attributed to the execution of an INVALID opcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In contrast to Securify, we do not ignore gas entirely, but model the gas reduction for all instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This allows capturing dependencies such as the one highlighted in Figure 7 (and missed by Securify).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the CFG, we always model the gas reduction as a separate edge involving an intermediate node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', with the upper rule in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The Def set of one node contains only the gas variable, while the Def set of the other node only contains the (stack) variables involved in the actual instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An example for that is given by the (simplified) CFG rules of the ADD instruction in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Technically, an ADD instruction performs two types of state updates: it decreases the gas and performs addition on stack variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since those two state updates are independent, their execution can be split into two different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As a consequence, the node (pc, 1) is not data-dependent on nodes writing the gas variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Still, the gas abstraction is sound (under Assumption 1) and correctly captures the dependencies of the example in Fig- ure 7: Figure 10 shows an incomplete and simplified CFG of the example in Figure 7 with annotated Def and Use sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The example illustrates how the CFG captures the dependency of the storage write (test = flip) on the msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The storage write in 6 is control dependant on the conditional y2 in 5 , and 5 depends on node 4 where y2 is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 4 1 2 4 5 6 7 3 {y1} {msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender} {y1} {gas} {gas} {y2} {gas} {y2} {test} {flip} y1 = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender == address(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' if (y1){ while (gasLeft() >= 5000){ flip = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } y2 = gasLeft() >= 5000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' if (y2){ test = flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } 1 2 4 5 6 7 3 ⇒ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Example control flow with gas dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Def sets are given at the left of each node, Use sets at the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Data dependencies are indicated by black arrows, control dependencies by orange ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' m[x] = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' y1 = m[0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' if (y2){ y2 = y1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='= 0 test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 2 3 4 5 6 } {⏉m} {sender,x} ⋃ Xm {y1} {0m} {y2} {y1} {y2} {test} {test} Securify (with fix) 1 2 3 4 5 6 Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ⋃ {sender,x} {y1} {0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, 0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} {y2} {y1} {y2} {test} {test} HoRStify 1 2 3 6 5 4 ⇒ Xm ⇒ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Simplified version of contract in Figure 4 satisfying the RW property with PDGs depicting the dependencies modeled by Securify and HORSTIFY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' accesses the gas value, so a dependency between 4 and the gas nodes is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Node 3 is one of these gas nodes (there are more not shown in the picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The execution of 3 depends on condition y1 checked in 2 , so it is control dependant on 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Node 1 defines y1, so 2 depends on 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Thus, there is a transitive dependency between writing to test in 6 and reading msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender in 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) Memory Abstraction: To precisely model memory and storage accesses in a CFG, it is important to know statically as many memory and storage locations as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Assume that such statical information is not available: then memory (or storage) cannot be separated into regions and all read and write operations introduce dependencies with the whole memory (or storage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This would introduce many false dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' During a preprocessing step, such static information can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' But, as demonstrated in Section IV, using prepro- cessed data may introduce unsoundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This requires careful integration of preprocessing information into the CFG defining rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following we consider only memory variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' all ideas equally apply to storage variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We propose a, to the best of our knowledge, novel memory abstraction that is sound and provides high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To po- sition our approach between unsound and imprecise memory abstractions, we revisit Figure 4 in a simplified version that is depicted as a CFG in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The black and solid line parts of the left CFG visualize how Securify misses the dependency between msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender ( 1 ) and writing to test ( 5 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Securify, write accesses to unknown memory locations are assumed to write a special memory variable ⊤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, when reading {sender, x} m[1] = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' m[x] = 42;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' y1 = m[0] if (y2){ y2 = y1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='= 0 test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 2 3 4 5 6 7 {⏉m} {sender} {y1} {0m} {y2} {y1} {y2} {test} {test} Securify (with fix) 1 {1m} {x} ⋃ Xm 2 3 4 5 6 7 Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D {y1} {0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, 0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} {y2} {y1} {y2} {test} {test} HoRStify 1 {1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content="S} {} 1' {1m." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D 1 2 3 4 5 6 7 Xm ⇒ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Contract violating the RW property with PDGs depicting the dependencies as modeled by Securify and HORSTIFY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' from a statically known memory location (as done in 2 ), Securify does not consider that a value could have been written to this location when the location was not statically known, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', that the value could have been stored in ⊤m: the Use set of 2 contains only 0m, but not ⊤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A hypothetical fix for this unsoundness is to replace the variable ⊤m by the whole set Xm of all memory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This fix is depicted in violet in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Now, the dependency of the read access in 2 to the write operation in 1 is naturally established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' One should notice, however, that this interpretation implies that the Use set of node 1 needs to contain all variables in Xm as well: a new value is written to one unknown location, but for all other locations the value is “copied” from the existing memory cells, and hence, all these cells need to be included in the Use set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Even though fixing the soundness issue, this modeling would lead to an imprecise analysis as depicted in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This variant of Figure 11 first writes msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender to the known memory location 1 in node 1 and then writes a value to an unknown memory location in node 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since the condition y2 only depends on the value in memory location 0 while msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender was written to location 1, the final write to the test variable in 6 does not depend on msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, the hypothetical fix of Securify infers a possible dependency between 6 and msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender (shown in violet in the left CFG in Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This imprecision is caused by interpreting a write to an unknown memory location as a write to possibly all memory locations as this requires the Use set in 2 to contain Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This creates a dependency between the assignment of location 1 to msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender in 1 and the memory access in 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Our memory abstraction is sound but more precise than the hypothetical fix above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For every memory variable x we use two sub-variables instead: S-variable xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S stores values that are assigned to x when the memory location for x is statically known, and D-variable xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D stores values assigned to x when x’s location is not statically known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' During the execution, every write access to a memory variable x stores the assigned value in xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, unless the memory location for x is statically known, in which case xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S stores the value and xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D is set to ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Correspondingly, when reading from a variable (regardless of the memory location being statically known or not), first, the value of the D-variable is read, and only if it is ⊥, the value of the S-variable is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We model this read access with the function load θ x = � θ[xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S] if θ[xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D] = ⊥ θ[xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D] otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This two-layered memory abstraction ensures that the exe- cution is deterministic and that the read values coincide with those obtained during an execution without prior preprocess- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' load is used in the inference rules in Figure 13 that define the memory read and write operations for the CFG semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In these rules, we use Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S for the set of all S-variables and Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D for the set of all D-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The leftmost MSTORE rule is for the case that a value is written to a memory location that could not be statically inferred (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', pre[0] = ⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' There, any of the memory variables from Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D might be redefined, hence the Def set contains all variables in Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As discussed for the hypothetical fix of Securify, also the Use set needs to include Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, because we must not interrupt potential dependencies for memory cells that are not changed by this MSTORE instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An example for this is node 2 in Figure 12 (right CFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The S-variables are not part of the Use set and hence not part of the value intermingling in 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This removes the imprecision that occurred in the proposed hypothetical fix above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Still, the MLOAD rules make sure that no dependencies to S-variables are missed by adding both D-variables and S-variables to the Use set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This way, the connection between the memory location and the stored value is preserved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' x1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D does not inherit any data dependencies from x2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S for locations x1 ̸= x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An example for that is given in Figure 12, where memory location 0 does not inherit the dependency from memory location 1 written in 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is thanks to the node splitting at 1 that breaks the propagation of dependencies on precomputed locations to dynamic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' c) Call Abstraction: Contract calls in Ethereum trigger a multitude of possible (side) effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' When calling another account, the control flow is handed over to the code residing in this account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This code may initiate further internal transac- tions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', perform money transfers or even reenter the calling contract before reporting back the result to the callee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This behavior poses a big challenge to sound static analysis since all possible effects of interactions with other (potentially unknown) contracts need to be over-approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify avoids this challenge by sacrificing soundness and ignoring all data dependencies arising from external calls (including effects of reentrancy) as demonstrated by the examples in Fig- ure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In contrast, to give a sound and precise characterization of these dependencies, we first simplify the problem by restricting our analysis to a set of well-behaved smart contracts and then model the remaining dependencies in a fine-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The class of smart contracts that we target are such contracts that cannot write storage variables in reentering executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This restriction rules out race conditions on contract variables and as such is a highly-desirable property that can be easily achieved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', by a strict local locking discipline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We call contracts satisfying this restriction store unreachable: C(pc) = (MLOAD(yls, xls)), pcnext, pre) pre[1] = ⊥ f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := load θ (θ[xls])) C, cd ⊨ (pc, 0) −⇑f − → (pc, 1) Def = {yls} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ {xls} C(pc) = (MSTORE(x1 ls, x2 ls), pcnext, pre) pre[0] = ⊥ f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← θ[x1 ls].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := θ[x2 ls]) C, cd ⊨ (pc, 0) −⇑f − → (pc, 1) Def = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {x1 ls, x2 ls} C(pc) = (MLOAD(yls, xls)), pcnext, pre) pre[1] = ⌊xm⌋ f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := load θ xm) C, cd ⊨ (pc, 0) −⇑f − → (pc, 1) Def = {yls} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} C(pc) = (MSTORE(x1 ls, x2 ls), pcnext, pre) pre[1] = ⌊xm⌋ f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S := θ[x2 ls]) C, cd ⊨ (pc, 0) −⇑f − → (pc, 1) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} Use = {x2 ls} C(pc) = (MSTORE(x1 ls, x2 ls), pcnext, pre) pre[0] = ⌊xm⌋ f = (λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := ⊥) C, cd ⊨ (pc, 1) −⇑f − → (pc, 2) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} Use = ∅ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MLOAD memory abstraction instantiation C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) f1 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := applyCall(θ, C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← externaleg := applyCall(θ, C, pc)[externaleg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f − → (pc, 1) Def = {yls, externaleg} Use = {gls, tols, vals, iols, isls, ools, osls, gel, actorel} ∪ Xm ∪ Xeg ∪ Xg Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Simplified CFG rule for the CALL opcode Assumption 2 (Store unreachability (informal)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A contract C is store unreachable if all its reentering executions cannot reach an SSTORE instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The contract in Figure 5 trivially violates store unreachabil- ity since the field a can be written in a reentering execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This could be easily fixed by guarding each function with a lock that blocks reentering executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Store unreachability is a local reachability property of the contract under analysis and as such falls in the scope of the sound analysis tool eThor [24] and hence can be automatically verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Even when focussing on store unreachable contracts, the program dependencies induced by external calls are manifold and often subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Figure 14 shows one (slightly simplified) rule of the CFG semantics for external calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As seen in the previous examples, node splitting is used to separate the dependencies of different variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The rule displayed in Fig- ure 14 gives one of the rules for setting a call’s return value (written to the stack variable yls) and updating the external environment (represented by variable externaleg) according to the call effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To obtain the updated CFG state after a call, the rule uses the function applyCall, which executes the internal transaction initiated by the CALL opcode4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The CFG state resulting from this execution is then used to describe the state updates (in the case of the given rule, the updates on the variables yls and externaleg, as indicated by the Def set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Even though the whole CFG state θ is taken as an argument by applyCall, not all variables in θ can influence all aspects of the state 4We define applyCall using the EVM semantics and hence can infer Def and Use sets from the corresponding EVM semantics rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' after returning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The variables that indeed may affect yls and externaleg are given in the Use set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, the result of a call may still depend on the global state, so all global environmental variables (Xeg), as well as the global variables of the contract under analysis itself (Xg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Additionally, the execution of the called contract can be influenced by the parameters given to the call: The argument gls attributes to the amount of gas given to the call, tols gives the address of the recipient account and vals the amount of money transferred with the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The arguments iols and isls specify the memory fraction (offset and size) from which input data to the call is read and ools and osls correspondingly define the memory fraction where the call’s result data will be written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the given simplified rule, we consider that the concrete memory fragments could not be precomputed and hence all memory (Xm) could potentially be input data to the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The Use set also contains the calling account (as given in actorel), since this information is made accessible during a call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, the Use set contains the amount of gas that is available at the point of calling (given by gel) since this value may influence the amount of gas given to the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We want to highlight two forms of dependencies, which may erroneously be assumed to be ruled out by the assumption of store unreachability: First, the Use set explicitly contains the storage variables (Xg) of the contract under analysis, even though we assume this contract to be store unreachable and (by the semantics) its storage variables cannot be accessed by any other contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Second, both the Def and the Use set contain the variable externaleg that represents the external environment (in particular the state of other contract accounts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This implies that the rule in Figure 14 explicitly models information to be stored and retrieved from contract accounts during an external call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Figures 15 and 16, we illustrate the need for these dependencies by two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The example in Figure 15 shows how dependencies on a storage variable are introduced by reading a contract variable during a reentering execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that store unreachability only assures that reentering executions can not write contract variables, but does not prevent read accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The example gives another version of the Test contract, which performs the check of msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender in an indirect way: First, it writes msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender to the contract variable sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To read the variable 1 contract RetrieveSender { 2 function getTestSender() public returns (address) { 3 try Test(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='getSender() returns (address a) { 4 return a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 5 catch {return address(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' }}} 6 7 contract Test { 8 bool test = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 9 address sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 10 RetrieveSender rs = RetrieveSender (address(42));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 11 function getSender () public returns (address) { 12 return sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='} 13 function flip () public { 14 sender = msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 15 try rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='getTestSender() returns (address a) { 16 if (a !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='= address(0)){ 17 test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='}} 18 catch {return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' }}} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Example: Reading storage variables during reentering execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 contract SaveAddr { 2 address addr = address(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3 function set(address a) public { 4 addr = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 5 function get( ) public returns (address) {return addr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' }} 6 7 contract Test { 8 bool test = false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 9 SaveAddr sa = SaveAddr (address(42));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 10 function flip () public { 11 try sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='set(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender) { 12 try c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='get() returns (address a) { 13 if (sa !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='= address(0)){ 14 test = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } 15 catch {return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' }} 16 catch {return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Example: Propagating dependencies via an external contract account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' again, a RetrieveSender contract rs is used as a proxy: 5 The Test contract calls RetrieveSender’s getTestSender function (in line 15), which in turn reenters Test via its getSender function (in line 3) to obtain the value of sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This value is finally returned to contract Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As a consequence, the return variable a in line 16 contains the value of msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender, and so the assignment of variable test is dependent on msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This dependency, however, can only be tracked when considering that the contract’s own storage variables may influence the return value of an external call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The example in Figure 16 shows how dependencies can be propagated via another contract account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that store unreachability is a contract-specific property that only ensures that the contract under analysis is not written in reentering executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The assumption does not restrict the storage mod- ification of other contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The version of the Test contract given in Figure 16 uses the contract SaveAddr to propagate the value of msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, it first writes the value of msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender into the addr storage variable of the SaveAddr contract sa using the set function (in line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Afterwards, it retrieves the value back by accessing c’s storage via the 5Note that in Ethereum, a contract is identified by its address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Solidity, the syntax RetrieveSender rs = RetrieveSender (address(42)) means that the contract at address 42 is assumed to be (of the type) RetrieveSender and accessible via variable rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' get function (in line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, the return variable a contains the value of msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender in line 13 what makes the following write to test dependent on that value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This dependency can only be faithfully modeled when considering that an external call may change the state of other accounts, and may also be influenced by this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This motivates why the externaleg variable needs to be included in both the Def and the Use set of the rule in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Soundness Reasoning via Dependency Predicates Inspired by Securify, we define dependency predicates that can capture the data and control flow dependencies induced by the PDG (as given through the CFG semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' They are inhabited via a set of logical rules (CHCs) R(C) that describe the data and control flow propagation through the PDG of a contract C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More formally, the transitive closure of the C’s PDG is computed as the least fixed point over R(C) (de- noted by lfp(R(C))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Most prominently, lfp(R(C)) includes the predicates VarMayDepOn and InstMayDepOn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, VarMayDepOn(y, x) states that the value of variable y may depend on the value of variable x and InstMayDepOn(n, x) says that the reachability of node n may depend on the value of variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, let nx and ny denote nodes that define variables x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The formal relation between dependency predicates and backward slices is captured by the following lemma: Lemma 1 (Fixpoint Characterization of Backward Slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let x and y be variables and C be a contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The following holds: 1) (∃nx ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nx ∈ BS(ny)) ⇒ VarMayDepOn(y, x) ∈ lfp(R(C)) 2) (∃ n nif nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nif −→cd n ∧ nx ∈ BS(nif)) ⇒ InstMayDepOn(n, x) ∈ lfp(R(C)) Lemma 1 states 1) that whenever there is a node nx defining x in the backward slice of a node ny defining y, then VarMayDepOn(y, x) is derivable from the CHCs in R(C) and 2) that whenever there is a node nx defining x in the backward slice of a node nif on which node n is control dependent then InstMayDepOn(n, x) is derivable from R(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The intuition behind statement 2) is that node n is controlled by nif (by the definition of standard control dependence), which means that nif is a branching node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nx ∈ BS(nif) indicates that the branching condition of nif depends on variable x and, hence, so does the reachability of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Next, we give an explicit semantic characterization of the dependency predicates, which we prove sound using Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This explicit characterization enables us to compose security patterns as a set of different facts over dependency predicates and to reason about them in a modular fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As a consequence, we can show in Section VI-D that checking the inclusion of security patterns in the least fixpoint of the rule set R(C) is sufficient to prove non-interference-style properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Concretely, we can characterize facts from the VarMayDepOn predicate as follows: Theorem 2 (Soundness of Dependency Predicates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀x y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarMayDepOn(y, x) ̸∈ lfp(R(C)) ⇒ y ⊥ x with y ⊥ x given as: ∀nx i θ1 θ2 θ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1 =/x θ2 ∧ ⟨nx +, θ1⟩ Ny −−→ i ⟨n, θ′ 1⟩ ⇒ ∃θ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨nx +, θ2⟩ Ny −−→ i ⟨n, θ′ 2⟩ ∧ θ′ 1(y) = θ′ 2(y) where nx + denotes the unique successor node of nx, and Ny the set of all nodes defining y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨nx +, θ1⟩ Ny −−→ i ⟨n, θ′ 1⟩ describes an execution from nx to n that passes exactly i nodes defining y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The theorem states that if VarMayDepOn(y, x) is not in- cluded in lfp(R(C)) then y is independent of x (y ⊥ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A variable y is considered independent of x if for any two configurations θ1 and θ2 that are equal up to x, and any execution starting at node nx+, the first node after x is defined, passing i nodes that define y, and ending in a node n at state θ′ 1, one can find a matching execution from θ2 that passes the same number of nodes defining y and ends at node n in a state θ′ 2 such that θ′ 2 and θ′ 1 agree on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This definition ensures loop sensitivity: it captures that during a looping execution, every individual occurrence of a node defining y can be matched by the other execution—so that the values of y agree whenever y gets reassigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The proof of Theorem 2 uses Lemma 1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For the full proof and a similar characterization of InstMayDepOn(i, x), we refer to Appendix A-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Sound Approximation of Security Properties With Theorem 2 we are able to formally connect depen- dency predicates and (independence-based) security proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We take trace noninterference as a concrete example, which comprises a whole class of non-interference-style secu- rity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Concretely, we consider trace noninterference w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a set of EVM configuration components Z, which includes, for example, the block timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A predicate f defines instructions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' If two executions of a contract C start in configurations that differ only in the components in Z, then the instructions of interest must coincide in the two traces that result from these executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 3 (Trace noninterference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be an EVM contract, Z be a set of components of EVM configurations and f be a predicate on instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then trace noninterference of contract C w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Z and f (written TNI(C, Z, f)) is defined as follows: TNI(C, Z, f) := ∀ Γ Γ′ s s′ t t′ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' π′ (Γ, s) =/Z (Γ′, s′) ⇒ Γ ⊨ sC :: S π−→ ∗ tC :: S ∧ final (t) ⇒ Γ ⊨ s′ C :: S π′ −→ ∗ t′ C :: S ∧ final (t′) ⇒ π ↓f= π′ ↓f where π ↓f denotes the trace filtered by f, so containing only the instructions satisfying f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The dependency properties defined in [14] can be expressed in terms of trace noninterference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', the timestamp indepen- dence property in Definition 1 is captured as an instance of trace noninterference as follows: TNI(C, {Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='timestamp}, λop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='op = CALL) We show that we can give a sufficient criterion for trace noninterference in terms of dependency predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, we give a set PC Z,f of facts, such that PR(C) Z,f ∩ lfp(R(C)) = ∅ implies TNI(C, Z, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Practically, this means that we can prove TNI(C, Z, f) by computing the least fix- point over the CHCs R(C) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', using a datalog engine) and then check whether it contains any fact from PC Z,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For components in Z, we assume a function toVar that maps components of the EVM semantic domain to CFG variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The dependency predicates constituting a security pattern for trace noninterference are defined as PC Z,f :={InstMayDepOn(pc, toVar(z)) | z ∈ Z ∧ C(pc) = op(⃗x, pcnext, pre) ∧ f(op)} ∪ {VarMayDepOn(xi, toVar(z)) | z ∈ Z ∧ pc ∈ dom(C) ∧ C(pc) = (op(⃗x, pcnext, pre)) ∧ f(op) ∧ xi ∈ ⃗x }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The following theorem shows that PC Z,f is a security pattern for trace noninterference: Theorem 3 (Soundness of trace noninterference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be a contract, Z a set of components, and f an instruction-of- interest predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then it holds that (∀p ∈ PC Z,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' p ̸∈ lfp(R(C))) ⇒ TNI(C, Z, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The absence of facts from PC Z,f in lfp(R(C)) ensures that the reachability of all instructions satisfying f is independent of variables representing components in Z and that all argu- ments xi of such instructions are independent of z as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These independences imply trace noninterference since they ensure that in two executions starting in configurations equal up to Z, all instructions satisfying f are executed in the same order (otherwise their reachability would depend on Z) and with the same arguments (otherwise their argument variables would depend on Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, such executions produce the same traces, when only considering instructions satisfying f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A full proof of Theorem 3 can be found in Appendix A-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Discussion In this section, we presented a sound analysis pipeline for checking security properties for linearized EVM bytecode contracts by means of reasoning about dependencies between variables or instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' While our work was inspired by Securify [27], we developed new formal foundations for the dependency analysis of EVM bytecode contracts and in this way revealed several sources of unsoundness in the analysis of Securify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, we provide soundness proofs for the analysis pipeline end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The key pillars of the soundness proof are i) that our EVM CFG semantics satisfies all conditions to be used with the slicing framework [30], ii) that the EVM linearized bytecode semantics and the CFG semantics are equivalent, iii) that our set of CHCs encodes an over-approximation of dependencies in an EVM contract, and iv) that the generic security pattern PC Z,f is a sound approximation of trace noninterference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The proofs are valid under assumptions that are clearly stated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For Assumptions 1 and 2 we point out the existence of other sound tools [4], [24] that can check these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We assume that EVM smart contracts are provided in a (stack-less) linearized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Transforming into such a rep- resentation from a stack-based one is a well-studied prob- lem [19] and a standard step performed by most static analysis tools [11], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Up to this requirement, our analysis is parametric with respect to other preprocessing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, our analysis pipeline is sound for contracts with sound preprocessing information, and hence, in particular, for contracts without any preprocessing information but jump destinations needed for the CFG (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Section VI-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This gives the flexibility, to enhance the precision of the analysis through the incorporation of soundly precomputed values and makes the design of sound preprocessing an orthogonal problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' There exist already works on soundly precomputing jump destinations for EVM bytecode [12], which are to be complemented with other precomputing steps in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' EVALUATION The focus of this paper is on the theoretical foundations of a sound dependency analysis of smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, we demonstrate the practicality of the presented approach by developing the prototype analyzer HORSTIFY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We do not implement the logical rules from Section VI-C directly in Souffl´e (as done by Securify), but encode them in the HORST specification language [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The HORST language is a high- level language for the specification of CHCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' By introducing this additional abstraction layer, we get a close correspondence between our theoretical rules and their actual implementation and, hence, anticipate a lower risk of implementation mistakes that may invalidate soundness claims in the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' HORSTIFY accepts as input a set of dependency facts en- coding the security patterns specified in the HORST language and Ethereum smart contracts in the EVM bytecode format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' It first invokes Securify’s decompiler to transform the contract into a linearized representation and does some lightweight preprocessing to obtain the precomputable values (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Sec- tion VI-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then, HORSTIFY uses our formal specification of the CFG construction rules and the HORST framework to create a Souffl´e executable for the analysis and invokes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To reduce the risks of implementation mistakes, we pro- ceeded in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' First, we encoded Securify’s RW violation pattern in the HORST language to execute HoRStify with this pattern and the contracts in Figures 4, 6 and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In contrast to Securify, HORSTIFY correctly determines that these con- tracts do not satisfy the RW violation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In addition to these corner cases, we successfully evaluated HORSTIFY on Securify’s internal test suite involving 25 contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Next, we conduct a large-scale evaluation of HORSTIFY and Securify on real-world contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, we use the sanitized dataset from [24] that consists of 720 distinct smart contracts from the Ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We compare the performance of Securify and HORSTIFY on this dataset for both the RW pattern and for timestamp independence (TS) 6We did not consider the contract in Figure 3 since it concerns the must- analysis and the contract in Figure 5, which violates Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' contracts errors timeouts contracts ∅ time (ms) \\(errors ∪ timeouts) 720 H 34 H 46 634 H 7055 S 34 S 30 S 3107 TABLE I LARGE-SCALE EVALUATION OF HORSTIFY (H) AND SECURIFY (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Classification of mismatching results of HORSTIFY (above) and Securify (below) for the RW (left) and TS (right) property8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Ticks indicate correct matches (tn) and crosses wrong matches (fn) of the respective tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' tnHor/tnSec, fnHor/fnSec, tpHor/tpSec, fpHor/fpSec denote true negatives, false neg- atives, true positives, and true negatives of HORSTIFY/Securify, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' as defined in Section VI-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We manually inspect all contracts on which Securify and HORSTIFY report a different result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Table I shows the evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The average execution time of HORSTIFY is approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='3 times longer than for Securify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, HORSTIFY suffers from more timeouts than Securify;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' the execution of both tools is aborted after one minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Figure 17 visualizes the manual classification for those smart contracts where HORSTIFY and Securify dis- agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' There are only two contracts where HORSTIFY matches the corresponding pattern, but Securify does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Recall that for a sound tool, a pattern match indicates the discovery of provable independencies that imply either property violation (RW) or compliance (TS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An erroneous pattern match by HORSTIFY would present a soundness issue (false negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We carefully examined the two examples and could confirm them not to constitute false negatives of HORSTIFY but false positives of Securify (fpSec), unveiling an imprecision of Securify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This seems surprising since our analysis generally tracks more dependencies than the one of Securify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, while HORSTIFY implements standard control dependence to encode control dependencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', to compute join points after loops), Securify implements a less precise custom algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The contracts where Securify matches a pattern, but HORS- TIFY does not, can either reveal soundness issues (false nega- tives) of Securify (fnSec) or a precision loss (false positives) of HORSTIFY (fpHor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Indeed, in the 29 contracts that are flagged only by Securify, we find both cases (as shown at the bottom of Figure 17), as we will illustrate with two examples: Figure 18 shows a (slightly shortened) version of a contract classified as safe for TS according to Securify, but that HoRS- tify (correctly) reports as vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' It is a lottery contract that pays out a user who manages to guess a random number (func- tion Guess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The random number is generated from blockchain and transaction-specific values, including the timestamp (ac- cessed via now in RandomNumberFromSeed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Hence, the payout in 8For TS we only consider the 165 contracts from the dataset containing a TIMESTAMP opcode, as Securify labels other contracts as trivially secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The manual classification is a conservative best-effort estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 contract RNG { 2 mapping (address => uint) nonces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3 uint public last;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 4 function RandomNumber() returns(uint) { 5 return RandomNumberFromSeed( 6 uint(sha3(block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='number))ˆuint(sha3(now)) 7 ˆuint(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender)ˆuint(tx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='origin));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 8 function RandomNumberFromSeed(uint seed) returns(uint) { 9 nonces[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender]++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 10 last = seedˆ(uint(sha3(block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='blockhash(block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='number), 11 nonces[msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender])) 12 0x000b0007000500030001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 13 return last;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 14 function Guess(uint _guess) returns (bool) { 15 if (RandomNumber() == _guess) { 16 if (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='send(this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='balance)) throw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 17 RandomNumberGuessed(_guess, msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 18 return true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 19 return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Lottery Contract 0xaed5a41450b38fc0ea0f6f203a985653fe187d9c 1 contract lottery{ 2 address[] public tickets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3 function buyTicket(){ 4 if (msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='value !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='= 1/10) throw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 5 if (msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='value == 1/10) 6 tickets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='push(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 7 address(0x88a1e54971b31974b2be4d9c67546abbd0a3aa8e) 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='send(msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='value/40);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 9 if (tickets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='length >= 5) runLottery();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } 10 function runLottery() internal { 11 tickets[addmod(now, 0, 5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='send((1/1000)*95);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 12 runJackpot();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='} 13 function runJackpot() internal { 14 if(addmod(now, 0, 150) == 0) 15 tickets[addmod(now, 0, 5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='send(this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='balance);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 16 delete tickets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Lottery contract 0xe120100349a0b1BF826D2407E519D75C2Fe8f859 line 16 is not independent of the timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify fails to detect this dependency due to its unsound memory abstraction (as described in Section IV-B): As Ethreum’s hash function (sha3) reads input from the local memory, the timestamp is written to the memory where its dependencies are lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Figure 19 shows an example of a false positive for HORS- TIFY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The contract implements a lottery where users can reg- ister (via buyTicket) and whenever 5 users were registered, one of them is selected as a winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Despite the obvious timestamp dependency, the contract shows RW violations, which HORS- TIFY fails to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='9 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', the tickets array is updated without performing a check on the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' HORSTIFY does not detect this vulnerability due to its sound storage abstraction: In line 6, the caller (msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender) is appended to the tickets array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since the array position to which msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender will be added cannot be statically known, HORSTIFY needs to assume msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender to be written to any position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' When checking the size of tickets in line 9, the condition is considered dependent on msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender (because in the abstraction, msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender is considered to potentially affect all storage locations, including the one containing the array size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Thus, the delete operation in line 16 9Note that this is not a soundness issue since the soundness of HORSTIFY ensures that independencies can be proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the case of violation patterns as RW the independence constitutes an unwanted effect and hence, we can only use it to prove the vulnerability of a contract, not its safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' is considered dependent on msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' One should notice, that only the unsoundness of Securify’s storage abstraction, enables Securify to correctly detect the RW violation in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Overall, based on our evaluation results, we can bound the precision loss of HORSTIFY w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More concretely, when considering that Securify has a specificity10 of SSec on the full dataset, then one can easily show that it holds for the specificity SHor of HORSTIFY that SHor ≥ SSec + tnHor−tnSec |dataset| where tnHor are the true negatives for HORSTIFY, and tnSec are the true negatives for Securify found within the man- ually inspected mismatching contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Inserting the results from Figure 17, we can show that SHor can be at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='5 percentage points less than SSec for RW on the given dataset and at most 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='4 percent points less for TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We refer to horstify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='org for more information about HORSTIFY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' RELATED WORK Existing approaches to enforce the correctness of Ethereum smart contracts can be broadly categorized into analyses at de- sign time and analyses at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The latter include methods like runtime monitoring [9], [28] or information flow control mechanisms [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Such dynamic analysis approaches, however, have limited applicability to the Ethereum blockchain, since they either require fundamental updates to the workings of the EVM or impose tremendous costs in terms of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Static analyses, in contrast, verify smart contracts at design time be- fore they become immutable objects on the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Most static analyzers are bug-finding tools (such as Oyente [22], EthBMC [10], and Maian [23]) that aim to reduce the number of contracts that are wrongly claimed to be buggy (false positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, these tools usually rely on the symbolic execution of the contract under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The dual objective of bug-finding is to prove a smart contract secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Analyzers following this objective do not only aim at producing a low number of false negatives in practice but to give provable guar- antees for their analysis result, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', that a contract flagged as safe is guaranteed to enjoy a corresponding security property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The only example of a tool, which comes with a provable soundness claim, so far, is the analyzer eThor [24], whose analysis relies on abstract interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Symbolic execution and abstract interpretation have in com- mon to target properties that can be decided for a finite prefix of a single (yet arbitrary) execution trace of a smart contract (so-called reachability properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, many generic security properties for smart contracts (as defined in [14]) require comparing two execution traces from different initial configurations and fall into the broader category of 2-safety properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To check 2-safety properties with tools whose analysis is limited to reachability properties (such as eThor) requires an overapproximation of the original property in terms of reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' But finding such a meaningful over- approximation, which does not result in an intolerable preci- sion loss, is not always possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In [14], it is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', shown how 10The specificity is a standard precision measure and is calculated as tn tn+fp to overapproximate the call integrity 2-safety property (char- acterizing the absence of reentrancy attacks) by a reachability property (single-entrancy) and two other properties, which are captured by our notion of trace noninterference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, trace noninterference properties still concern two execution traces and hence cannot be verified using eThor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' HORSTIFY (inspired by the unsound Securify tool [27]) devises a differ- ent analysis technique, which immediately accommodates the analysis of trace noninterference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As opposed to the analysis underlying eThor, this technique does not allow for verifying general reachability properties, but a special class of 2-safety properties (including trace noninterference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' HORSTIFY and eThor, hence, can be seen as complementing tools that target incomparable property classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The call integrity property falls neither in the scope of eThor nor HORSTIFY, but its overapproximation decomposes it into trace noninterference properties (within the scope of HORSTIFY) and a reachability property (within the scope of eThor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Other generic security properties from [14] for characterizing the independence of miner-controlled parameters (including timestamp indepen- dence) immediately constitute trace noninterference properties and as such can be analyzed by HORSTIFY but not by eThor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More complex properties involving both universal and ex- istential quantification of execution traces [7], [8] cannot be checked by either HORSTIFY or eThor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' CONCLUSION In this work, we present the first provably sound static dependency analysis for EVM bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Taking up the ap- proach of the state-of-the-art static analyzer Securify [27], we uncover conceptual soundness issues of the tool, so we replace the underlying analysis and spelled out formal soundness guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The soundness proof of our analysis relies on the proof framework from [30] for static program slicing, which we instantiate for EVM bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The slicing framework can capture the notion of may-dependence, whereas we elucidated that the must-dependence promoted by Securify raises sound- ness questions already at the conceptional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Although we removed support for must-dependence, we could show that the resulting analysis is flexible enough to soundly characterize relevant smart contract security properties such as timestamp dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, we demonstrate the practicality of the approach by providing the prototypical analyzer HoRStify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' HoRStify encodes the slicing-based dependency analysis as logical rules that can be automatically solved by the Datalog solver Souffl´e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' it can verify real-world smart contracts, and even though being provable sound, shows performance com- parable to Securify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' REFERENCES [1] Solidity programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' https://soliditylang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='org/, https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='com/ethereum/solidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Accessed: 2022-02-05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [2] [DEPRECATED] Securify scanner for ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='com/eth-sri/securify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Accessed: 2022-02-05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [3] Filip Adamik and Sokol Kosta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Smartexchange: Decentralised trustless cryptocurrency exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In International Conference on Business Information Systems, pages 356–367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [4] Elvira Albert, Jes´us Correas, Pablo Gordillo, Guillermo Roman-Diez, and Albert Rubio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Don’t run on fumes—parametric gas bounds for smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Journal of Systems and Software, 176:110923, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [5] Moritz Andresen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The biggest smart contract hacks in history or how to endanger up to us $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='2 billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' https://medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='com/solidified/the- biggest-smart-contract-hacks-in-history-or-how-to-endanger-up-to-us- 2-2-billion-d5a72961d15d, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [6] Ethan Cecchetti, Siqiu Yao, Haobin Ni, and Andrew C Myers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Com- positional security for reentrant applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In 2021 IEEE Symposium on Security and Privacy (SP), pages 1249–1267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [7] Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Clarkson and Fred B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Schneider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In CSF’08, pages 51–65, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [8] Pedro R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' D’Argenio, Gilles Barthe, Sebastian Biewer, Bernd Finkbeiner, and Holger Hermanns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Is your software on dope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' - formal analysis of surreptitiously ”enhanced” programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pages 83–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [9] Joshua Ellul and Gordon J Pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Runtime verification of ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In 2018 14th European Dependable Computing Conference (EDCC), pages 158–163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [10] Joel Frank, Cornelius Aschermann, and Thorsten Holz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Ethbmc: A bounded model checker for smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In 29th USENIX Security Symposium (USENIX Security 20), pages 2757–2774, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [11] Neville Grech, Lexi Brent, Bernhard Scholz, and Yannis Smaragdakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Gigahorse: thorough, declarative decompilation of smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In 2019 IEEE/ACM 41st International Conference on Software Engineering (ICSE), pages 1176–1186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [12] Ilya Grishchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Static Analysis of Low-Level Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' PhD thesis, Technische Universit¨at Wien, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [13] Ilya Grishchenko, Matteo Maffei, and Clara Schneidewind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Foundations and tools for the static analysis of ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Proceedings of the 30th International Conference on Computer-Aided Verification (CAV), pages 51–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [14] Ilya Grishchenko, Matteo Maffei, and Clara Schneidewind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A semantic framework for the security analysis of ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In International Conference on Principles of Security and Trust, pages 243– 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [15] Samer Hassan and Primavera De Filippi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Decentralized autonomous organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Internet Policy Review, 10(2):1–10, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [16] Vanita Jain, Akanshu Raj, Abhishek Tanwar, Mridul Khurana, and Achin Jain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Coin drop—a decentralised exchange platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Cyber Security and Digital Forensics, pages 391–399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [17] Herbert Jordan, Bernhard Scholz, and Pavle Suboti´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Souffl´e: On syn- thesis of program analyzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In International Conference on Computer Aided Verification, pages 422–430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [18] Sukrit Kalra, Seep Goel, Mohan Dhawan, and Subodh Sharma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Zeus: Analyzing safety of smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' NDSS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [19] Allen Leung and Lal George.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Static single assignment form for machine code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ACM SIGPLAN Notices, 34(5):204–214, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [20] Yinsheng Li, Xu Liang, Xiao Zhu, and Bin Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A blockchain-based autonomous credit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In 2018 IEEE 15th International Conference on e-Business Engineering (ICEBE), pages 178–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [21] Ning Lu, Bin Wang, Yongxin Zhang, Wenbo Shi, and Christian Esposito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Neucheck: A more practical ethereum smart contract security analysis tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Software: Practice and Experience, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [22] Loi Luu, Duc-Hiep Chu, Hrishi Olickel, Prateek Saxena, and Aquinas Hobor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Making smart contracts smarter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pages 254–269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ACM, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [23] Ivica Nikoli´c, Aashish Kolluri, Ilya Sergey, Prateek Saxena, and Aquinas Hobor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finding the greedy, prodigal, and suicidal contracts at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Proceedings of the 34th annual computer security applications conference, pages 653–663, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [24] Clara Schneidewind, Ilya Grishchenko, Markus Scherer, and Matteo Maffei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ethor: Practical and provably sound static analysis of ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, pages 621–640, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [25] Clara Schneidewind, Markus Scherer, and Matteo Maffei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The good, the bad and the ugly: Pitfalls and best practices in automated sound static analysis of ethereum smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In International Symposium on Leveraging Applications of Formal Methods, pages 212–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [26] Alexandra Sims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Decentralised autonomous organisations: Governance, dispute resolution and regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Dispute Resolution and Regulation (May 31, 2021), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [27] Petar Tsankov, Andrei Dan, Dana Drachsler-Cohen, Arthur Gervais, Florian Buenzli, and Martin Vechev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Securify: Practical security analysis of smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, pages 67–82, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [28] Haijun Wang, Yi Li, Shang-Wei Lin, Lei Ma, and Yang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Vultron: catching vulnerable smart contracts once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In 2019 IEEE/ACM 41st International Conference on Software Engineering: New Ideas and Emerging Results (ICSE-NIER), pages 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [29] Daniel Wasserrab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Towards certified slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Archive of Formal Proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' http://afp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' sf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' net/entries/Slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' shtml, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [30] Daniel Wasserrab, Denis Lohner, and Gregor Snelting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' On pdg-based noninterference and its modular proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Proceedings of the ACM SIGPLAN Fourth Workshop on Programming Languages and Analysis for Security, pages 31–44, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [31] Gavin Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Ethereum: A secure decentralised generalised transaction ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Ethereum project yellow paper, 151(2014):1–32, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [32] Lanfranco Zanzi, Antonio Albanese, Vincenzo Sciancalepore, and Xavier Costa-P´erez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Nsbchain: a secure blockchain framework for network slicing brokerage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In ICC 2020-2020 IEEE International Conference on Communications (ICC), pages 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' [33] Michal Zima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Coincer: Decentralised trustless platform for exchanging decentralised cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In International Conference on Network and System Security, pages 672–682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' APPENDIX A SOUND DEPENDENCY ANALYSIS FOR EVM BYTECODE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Instantiation of the Slicing Framework 1) State transformation: We formally define the state θ of the EVM as used in the CFG semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Afterward, we define the state transformation functions toCFG and toEVM that convert between the different EVM state representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' EVM state: We revisit the formal definition of the EVM state as given in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, we will use B to denote the set {0, 1} of bits and accordingly Bx for sets of bitstrings of size x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We further let Nx denote the set of non-negative integers representable by x bits and allow for implicit conversion between those two representations (assuming bitstrings to represent a big-endian encoding of natural numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In addition, we will use the notation [X] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L(X)) for arrays (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' lists) of elements from the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We use standard notations for operations on arrays and lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular we write a [pos] to access position pos ∈ [1, |a| − 1] of array a ∈ [X] and a[down, up] to access the subarray of size up − down from position down ∈ [1, |a| − 1] to up ∈ [1, |a| − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In case that down > up this operation results in the empty array ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In addition, we write a1 · a2 for the concatenation of two arrays a1, a2 ∈ [X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following formalization, we will make use of bytearrays b ∈ [B8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, we will assume functions (·)[B8] ∈ Bx → [B8] and (·)B ∈ [B8] → Bx to chunk bitstrings with size dividable by 8 to bytearrays and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To denote the zero byte, we write 08 and, accordingly, for an array of zero bytes of size n, we write 08·n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For lists, we denote the empty list by ϵ and write x :: xs for placing element x ∈ X on top of list xs ∈ L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In addition, we write xs + +ys for concatenating lists xs, ys ∈ L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We let A denote the set of 160-bit addresses (B160).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In Figure 20 we give a full grammar for call stacks: Call stacks S ∋ S := EXC :: SP | HALT(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' g) :: SP | SP Plain call stacks Splain ∋ SP := (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ι,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' σ) :: SP Machine states M ∋ µ := (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' s) Execution environments I ∋ ι := (actor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' sender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' code) Global states Σ ∋ σ Account states A ∋ acc := (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' code,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' stor) | ⊥ Transaction environments Tenv ∋ Γ := (o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' prize,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' H) Block headers H ∋ H := (parent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' beneficiary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' difficulty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' gaslimit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' timestamp) Notations: d ∈ [B8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' g ∈ N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' η ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' o ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' prize ∈ N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' H ∈ H g ∈ N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc ∈ N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' m ∈ N256 → N256 i ∈ N256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' s ∈ N8 → N256 sender ∈ A input ∈ [B8] sender ∈ A value ∈ N256 code ∈ [B8] b ∈ N256 stor ∈ N256 → N256 L ∈ L(Evlog) S† ⊆ A Σ = A → A parent ∈ N256 beneficiary ∈ A difficulty ∈ N256 numberN256 gaslimit ∈ N256 timestamp ∈ N256 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Grammar for calls stacks and transaction environments Note that the grammar was slightly adapted to account for the fact that the local stack is assumed to be precomputed to local variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, the intermediate representation assumes all memory accesses to be aligned, meaning that memory acceses only occure at addresses that are multiples of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' CFG state: We formally define the state for the CFG semantics of EVM bytecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The formal definition of the CFG state is given as follows: State θ := (ls, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, el, eg) Stack ls ∈ N8 → N256 Local Static Memory m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∈ N256 → N256 Local Dynamic Memory m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∈ N256 → N256 ∪ {⊥} Global Static Storage g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∈ N256 → N256 Global Dynamic Storage g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∈ N256 → N256 ∪ {⊥} Local Environment el := (g, i, actor, input, sender, value) Global Environment eg := (parent, beneficiary, difficulty, number, gaslimit, timestamp, o, prize, external) External Global Environment external := (b, n, σ) We will treat the CFG state as a heterogeneous mapping and write θ[xls] for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ls(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ[xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S] for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ[xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D] for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ[xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S] for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ[xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D] for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ[xel] for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='el.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' and θ[xeg] for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular, we will treat (static and dynamic) memory and storage locations as variables and will write Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D for the set of all dynamic memory locations, Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S for the set of all static memory locations, Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D for the set of all dynamic storage locations, and Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S for the set of all static storage locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, we use Xm and Xg to denote the (distinct) sets of all memory and, respectively, storage locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The state is partitioned according to the granularity of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' All components whose dependencies are explicitly tracked occur on the top level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) State transformation: In the following we will assume the load function to be defined on both memory locations xm and storage locations xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' load θ x = � θ[x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D] if θ[x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S] = ⊥ θ[x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S] otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' where x ∈ Xm ∪ Xg Using this, the translation between the different state types can be defined as follows: toCFG(Γ, s) := � � � � � � � � � � � � � � � � � � � � � � � � � (θ, C, pc) (µ, ι, σ) = s ∧ ls = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s ∧ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S = λ(i, pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='m(i) ∧ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D = λ(i, pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='⊥ ∧ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S = λ(i, pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(i) ∧ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D = λ(i, pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='⊥ ∧ el = (λpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g, λpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='i, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='va) ∧ external = λpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (σ(ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='b, σ(ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='n, σ) ∧ (o, prize, H) = Γ ∧ (parent, beneficiary, difficulty, number, gaslimit, timestamp) = H ∧ eg = (parent, beneficiary, difficulty, number, gaslimit, timestamp, o, prize, external) ∧ θ = (ls, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, el, eg) ∧ C = ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code ∧ pc = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc Note that we will usually write θ = toCFG(Γ, s) to implicitely drop the reconstructed contract C and program counter pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' toEVM(θ, C, pc) = � � � � � � � � � � � � � � � � � � � � � � � � � (Γ, s) (ls, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, el, eg) = θ ∧ (parent, beneficiary, difficulty, number, gaslimit, timestamp, o, prize, external) = eg ∧ (b, n, σ′) = external ∧ (g, i, actor, input, sender, value) = el ∧ code = λpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='op, C(pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pcnext) ∧ µ = (g, pc, λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='load θ xm, i, ls) ∧ ι = (actor, input, sender, value, code) ∧ σ = σ′[actor → (b, n, code, λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='load θ xg)] ∧ H = (parent, beneficiary, difficulty, number, gaslimit, timestamp) ∧ Γ = (o, prize, H) For defining the EVM CFG semantics, we assume Θ = θ⊎θ to represent the heterogeneous mapping that maps two copies of the variables from θ to their respective values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We will denote the copy of variable x from θ in θ as x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Variables x in θ function as temporal variables and are initially set to ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We denote with θ⊥ the partial mapping that maps all variables x to ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Temporal variables are only needed to track individual variable dependencies for opcodes that are initiating internal transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this case, many variables are updated simultanously and to distinguish the different dependencies in a fine-grained manner, the updates are first written into temporal variables and copied to their corresponding variables in θ only later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that for state updates only touching variables in θ, by convention we write λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='⟨exp⟩ while we use λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='⟨exp⟩ to denote state updates that may also touch temporal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2) CFG semantics: We closely follow the semantic rules given in [14] and group the rules whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) Binary Stack Operations: We first give the rules for binary stack operations We define Instbin := {ADD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SUB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' LT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' GT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' EQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' AND,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' OR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' XOR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SLT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SGT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MUL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' DIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SDIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MOD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SMOD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SIGNEXTEND,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' BYTE} and costbin(ibin) = � 3 ibin ∈ {ADD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SUB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' LT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' GT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SLT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SGT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' EQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' AND,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' OR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' XOR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' BYTE} 5 ibin ∈ {MUL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' DIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SDIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MOD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SMOD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' SIGNEXTEND} and funbin(ibin) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � λ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a + b mod 2256 ibin = ADD λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a − b mod 2256 ibin = SUB λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a < b ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 : 0 ibin = LT λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a > b ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 : 0 ibin = GT λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a− < b− ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 : 0 ibin = SLT λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a− > b− ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 : 0 ibin = SGT λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a = b ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 : 0 ibin = EQ λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a&b ibin = AND λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a∥b ibin = OR λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a ⊕ b ibin = XOR λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a · b mod 2256 ibin = MUL λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (b = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : ⌊a ÷ b⌋ ibin = DIV λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (b = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : a mod b ibin = MOD λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (b = 0)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : (a = 2255 ∧ b− = −1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2256 : let x = a− ÷ b− in (sign(x) · ⌊|x|⌋)+ ibin = SDIV λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (b = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : (sign(a) · |a| mod |b|)+ ibin = SMOD λ(o, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (o ≥ 32) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : b[8 · o, 8 · o + 7] · 0248 ibin = BYTE λ(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' let x = 256 − 8(a + 1) in let s = b [x] in sx · b[x, 255] ibin = SIGNEXTEND where sign(·) : Intx → {−1, 1} is defined as sign(x) = � 1 x ≥ 0 0 otherwise and &, ∥ and ⊕ are bitwise and, or and xor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Exceptions to the normal binary operations are the exponentiation as this instruction uses non-constant costs and the computation of the Keccack-256 hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The CFG rules for these operations are given as follows: C(pc) = (op(yls, x1 ls, x2 ls), pc′, pre) op ∈ Instbin f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := funbin(op)(θ[x1 ls], θ[x2 ls]) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {x1 ls, x2 ls} C(pc) = (op(yls, x1 ls, x2 ls), pc′, pre) op ∈ Instbin f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − costbin(op) C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} Exceptions to the normal binary operations are the exponentiation as this instruction uses non-constant costs and the computation of the Keccack-256 hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We give their CFG semantics rules separately: C(pc) = (EXP(yls, x1 ls, x2 ls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := θ[x1 ls]θ[x2 ls] mod 2256 C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {x1 ls, x2 ls} C(pc) = (EXP(yls, x1 ls, x2 ls), pc′, pre) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (θ[x2 ls] = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 10 : 10 + 10 ∗ (1 + � log256 θ[x2 ls] � ) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − c(θ) C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel, x2 ls} C(pc) = (SHA3(yls, x1 ls, x2 ls), pc′, pre) v = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='loadm θ[x1 ls] θ[x2 ls] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := Keccak(v(θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {x1 ls, x2 ls} ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S C(pc) = (SHA3(yls, x1 ls, x2 ls), pc′, pre) pos = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], pos(θ), size(θ)) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) + 30 + 6 · �size(θ) 32 � f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − c(θ) C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = {gel, iel, x1 ls, x2 ls} C(pc) = (SHA3(yls, x1 ls, x2 ls), pc′, pre) pos = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], pos(θ), size(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ie := aw(θ) C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) Def = {iel} Use = {iel, x1 ls, x2 ls} Where loadm θ o s is defined as loadm θ o s := � 0 s = 0 (load θ om) ∗ 256(s−1) + load θ (o + 1) (s − 1) s > 0 b) Unary Stack Operations: C(pc) = (ISZERO(yls, xls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (θ[xls] = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1 : 0 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {xls} C(pc) = (ISZERO(yls, xls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 3 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (NOT(yls, xlsls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='¬(θ[x]) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {xls} C(pc) = (NOT(yls, xls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 3 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} where ¬ is bitwise negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' c) Ternary Stack Operations: C(pc) = (ADDMOD(yls, x1 ls, x2 ls, x3 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] b = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (c(θ) = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : (a(θ) + b(θ)) mod c(θ) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {x1 ls, x2 ls, x3 ls} C(pc) = (ADDMOD(yls, x1 ls, x2 ls, x3 ls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 8 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (MULMOD(yls, x1 ls, x2 ls, x3 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] b = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (c(θ) = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : (a(θ) · b(θ)) mod c(θ) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {x1 ls, x2 ls, x3 ls} C(pc) = (MULMOD(yls, x1 ls, x2 ls, x3 ls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 8 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} d) Accessing the execution environment: C(pc) = (ADDRESS(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[actorel] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {actorel} Most instructions for accessing the execution environment have the same gas cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason we summarize the rule for gas substraction for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (op(yls), pc′, pre) op ∈ {ADDRESS, CALLER, CALLVALUE, CODESIZE, CALLDATASIZE} f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 2 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (CALLER(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[senderel] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {senderel} C(pc) = (CALLVALUE(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[valueel] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {valueel} C(pc) = (CODESIZE(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='|θ[codeel]| f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {codeel} C(pc) = (CALLDATASIZE(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='|θ[inputel]| f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {inputel} We give individual rules for accessing the code and input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The CALLDATALOAD instruction accesses a word of the call data at a specified position C(pc) = (CALLDATALOAD(yls, xls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='=θ[xls] d = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[inputel] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='|d(θ)| k = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (size(θ) − a(θ) < 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : min (size(θ) − a(θ), 32) v = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='d(θ) [a(θ), a(θ) + k(θ) − 1] r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='v(θ) · 0256−k(θ)·8 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {codeel} C(pc) = (CALLDATALOAD(yls, xls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 3 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (CALLDATACOPY(yls, x1 ls, x2 ls, x3 ls), pc′, pre) posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] posd = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] d = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[inputel] k = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (|d(θ)| − posd(θ) < 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : min (|d(θ)| − posd(θ), size(θ)) d′ = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='d(θ) [posd(θ), posd(θ) + k(θ) − 1] d = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='d′(θ) · 08·(size(θ)−k(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← (im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := d(θ)[i])i∈[posm(θ),posm(θ)+size(θ)−1] C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = {inputel, x1 ls, x2 ls, x3 ls} C(pc) = (CALLDATACOPY(yls, x1 ls, x2 ls, x3 ls), pc′, pre) posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] posd = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) + 3 + 3 · �size(θ) 32 � f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − c(θ) C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 2) Def = {gel} Use = {gel, iel, x1 ls, x2 ls, x3 ls} C(pc) = (CALLDATACOPY(yls, x1 ls, x2 ls, x3 ls), pc′, pre) posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] posd = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ie := aw(θ) C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) Def = {iel} Use = {iel, x1 ls, x3 ls} The rules for copying a fraction of the code to memory (CODECOPY) are similar: C(pc) = (CODECOPY(yls, x1 ls, x2 ls, x3 ls), pc′, pre) posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] poscode = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] d = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[codeel] k = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (|d(θ)| − poscode(θ) < 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : min (|d(θ)| − posd(θ), size(θ)) d′ = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='d(θ) [poscode(θ), poscode(θ) + k(θ) − 1] d = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='d′(θ) · STOPsize(θ)−k(θ) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← (im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := d(θ)[i])i∈[posm(θ),posm(θ)+size(θ)−1] C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = {codeel, x1 ls, x2 ls, x3 ls} C(pc) = (CODECOPY(yls, x1 ls, x2 ls, x3 ls), pc′, pre) posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] poscode = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) + 3 + 3 · �size(θ) 32 � f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − c(θ) C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 2) Def = {gel} Use = {gel, iel, x1 ls, x2 ls, x3 ls} C(pc) = (CODECOPY(yls, x1 ls, x2 ls, x3 ls), pc′, pre) posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] poscode = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ie := aw(θ) C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) Def = {iel} Use = {iel, x1 ls, x3 ls} Note that the rules for CALLDATACOPY and CODECOPY could easily be refined to account for preprocessing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' e) Accessing the transaction environment: C(pc) = (ORIGIN(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[origineg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {origineg} Most instructions for accessing the transaction environment have the same gas cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason we summarize the rule for gas substraction for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (op(yls), pc′, pre) op ∈ {ORIGIN, GASPRICE, COINBASE, TIMESTAMP, NUMBER, GASLIMIT, DIFFICULTY} f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 2 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (GASPRICE(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[prizeeg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {prizeeg} C(pc) = (COINBASE(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[beneficiaryeg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {beneficiaryeg} C(pc) = (TIMESTAMP(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[timestampeg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {timestampeg} C(pc) = (NUMBER(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[numbereg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {numbereg} C(pc) = (GASLIMIT(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[gaslimiteg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {gaslimiteg} C(pc) = (DIFFICULTY(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[difficultyeg] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {difficultyeg} C(pc) = (BLOCKHASH(yls, xls), pc′, pre) n = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='P (θ[parenteg], n(θ), 0) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {parenteg} where the function P (h, n, a) tries to access the block with number n by traversing the block chain starting from h until the counter a reaches the limit of 256 or the genesis block is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' P (h, n, a) := � � � � � 0 n > h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='number ∨ a = 256 ∨ h = 0 h n = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='number P (h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='parent, n, a + 1) otherwise C(pc) = (BLOCKHASH(yls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 20 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} f) Accessing the global state: C(pc) = (BALANCE(yls, xls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (θ[externaleg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a(θ) mod 2160) = (nonce, balance, stor, code)) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' balance : 0 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {externaleg} C(pc) = (BALANCE(yls, xls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 400 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (EXTCODESIZE(yls, xls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='|( � θ[externaleg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a(θ) mod 2160) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code| f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {externaleg} C(pc) = (EXTCODESIZE(yls, xls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 700 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (EXTCODECOPY(yls, x1 ls, x2 ls, x3 ls, x4 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] poscode = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x4 ls] d = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' � θ[externaleg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a mod 2160) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code k = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (|d(θ)| − poscode(θ) < 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0 : min (|d(θ)| − posd(θ), size(θ)) d′ = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='d(θ) [poscode(θ), poscode(θ) + k(θ) − 1] d = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='d′(θ) · STOPsize(θ)−k(θ) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← (im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := d(θ)[i])i∈[posm(θ),posm(θ)+size(θ)−1] C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = {externaleg, x1 ls, x2 ls, x3 ls, x4 ls} C(pc) = (EXTCODECOPY(yls, x1 ls, x2 ls, x3 ls, x4 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] poscode = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x4 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) + 700 + 3 · �size(θ) 32 � f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − c(θ) C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 2) Def = {gel} Use = {gel, iel, x2 ls, x3 ls, x4 ls} C(pc) = (EXTCODECOPY(yls, x1 ls, x2 ls, x3 ls, x4 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] posm = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] poscode = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x3 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x4 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ie := aw(θ) C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) Def = {iel} Use = {iel, x2 ls, x4 ls} g) Stack Operations: Since we assume the code to be in SSA form, all stack operations have been replaced by assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (ASSIGN(yls, xls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {xls} C(pc) = (ASSIGN(yls, xls), pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 3 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} h) Jump Instructions: For the case of jump instructions by assumption, the jump destination has been precomputed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since the JUMP instruction has no other effect than updating the program counter, we only need to add a rule for updating the gas and stepping to the next program counter: C(pc) = (JUMP(yls, xls), pc′, pre) pre[0] = ⌊pc′⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 8 C, cd ⊨ (pc, 0) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} For the conditional jump instruction, in addition to deducing the gas, the next program counter needs to be decided based on the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We first give the rules for updating the gas value: C(pc) = (JUMPI(yls, x1 ls, x2 ls), pc′, pre) pre[0] = ⌊pc′′⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 10 C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {gel} Use = {gel} Finally, we give the rules for branching: C(pc) = (JUMPI(yls, x1 ls, x2 ls), pc′, pre) pre[0] = ⌊pc′′⌋ Q = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] = 0 C, cd ⊨ (pc, 1) −(Q)√ −→ (pc′, 0) Def = ∅ Use = {x2 ls} C(pc) = (JUMPI(yls, x1 ls, x2 ls), pc′, pre) pre[0] = ⌊pc′′⌋ Q = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] ̸= 0 C, cd ⊨ (pc, 1) −(Q)√ −→ (pc′′, 0) Def = ∅ Use = {x2 ls} i) Memory Instructions: C(pc) = (MLOAD(yls, xls)), pc′, pre) pre[1] = None f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := load θ θ[xls] m C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S C(pc) = (MLOAD(yls, xls)), pc′, pre) pre[1] = ⌊x⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := load θ xm C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} C(pc) = (MLOAD(yls, xls)), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], a(θ), 32) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) + 3 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← gel := θ[g] − c(θ) C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = {gel, iel, xls} C(pc) = (MLOAD(yls, xls)), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], a(θ), 32) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← iel := aw(θ) C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) Def = {iel} Use = {iel, xls} Note that we do not distinguish between the MLOAD and the MLOADbyte instruction, because we assume the preprocessing to already check for consistent memory accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (MSTORE(x1 ls, x2 ls), pc′, pre) pre[1] = ⊥ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← � θ[x1 ls] �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := θ[x2 ls] C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 2) Def = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {x1 ls, x2 ls} C(pc) = (MSTORE(x1 ls, x2 ls), pc′, pre) pre[0] = ⌊x⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S := θ[x2 ls] C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 1) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} Use = {x2 ls} C(pc) = (MSTORE(x1 ls, x2 ls), pc′, pre) pre[0] = ⌊x⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := ⊥ C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} Use = ∅ C(pc) = (MSTORE(x1 ls, x2 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], a(θ), 32) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) + 3 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← gel := θ[g] − c(θ) C, cd ⊨ (pc, 2) −⇑f −→ (pc, start + 5) Def = {gel} Use = {gel, iel, x1 ls} C(pc) = (MSTORE(x1 ls, x2 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], a(θ), 32) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← iel := aw(θ) C, cd ⊨ (pc, start + 5) −⇑f −→ (pc′, 0) Def = {iel} Use = {iel, x1 ls} j) Storage Instructions: The storage instructions closely resemble the instructions for memory access: C(pc) = (SLOAD(yls, xls)), pc′, pre) pre[1] = None f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := load θ θ[xls] g C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S C(pc) = (SLOAD(yls, xls)), pc′, pre) pre[1] = ⌊x⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := load θ xg C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} C(pc) = (SLOAD(yls, xls)), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← gel := θ[g] − 200 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel, } C(pc) = (SSTORE(x1 ls, x2 ls), pc′, pre) pre[1] = ⊥ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← � θ[x1 ls] �g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := θ[x2 ls] C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 2) Def = Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {x1 ls, x2 ls} C(pc) = (SSTORE(x1 ls, x2 ls), pc′, pre) pre[0] = ⌊x⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S := θ[x2 ls] C, cd ⊨ (pc, start + 2) −⇑f −→ (pc, 1) Def = {xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} Use = {x2 ls} C(pc) = (SSTORE(x1 ls, x2 ls), pc′, pre) pre[0] = ⌊x⌋ f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := ⊥ C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} Use = ∅ C(pc) = (SSTORE(x1 ls, x2 ls), pc′, pre) a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] b = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (b(θ) ̸= 0 ∧ (load θ (θ[a(θ)])g = 0) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 20000 : 5000 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← gel := θ[g] − c(θ) C, cd ⊨ (pc, 2) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel, x1 ls} ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S k) Accessing the machine state: C(pc) = (GAS(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[gel] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {gel} Most instructions for accessing the machine state have the same gas cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason we summarize the rule for gas substraction for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (op(yls), pc′, pre) op ∈ {GAS, PC, MSIZE} f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← ge := θ[ge] − 2 C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {gel} Use = {gel} C(pc) = (MSIZE(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[iel] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {iel} C(pc) = (PC(yls), pc′, pre) r = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← yls := r(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = ∅ l) Logging: Since we do not model the logged events, for the logging instructions we only need to model the effect on gas and local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (LOGn(x1 ls, x2 ls, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' , xn+2 ls), pc′, pre) posMem = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) + 375 + 8 · size(θ) + n · 375 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← gel := θ[gel] − c(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {gel} Use = {gel, iel, x1 ls, x2 ls} C(pc) = (LOGn(x1 ls, x2 ls, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' , xn+2 ls), pc′, pre) posMem = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← iel := aw(θ) C, cd ⊨ (pc, 1) −⇑f −→ (pc′, 0) Def = {iel} Use = {iel, x1 ls, x2 ls} m) Halting instructions: C(pc) = (RETURN(x1 ls, x2 ls, pc′, pre) posMem = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Cmem (θ[iel], aw(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← gel := θ[gel] − c(θ) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {gel} Use = {gel, iel, x1 ls, x2 ls} C(pc) = (RETURN(x1 ls, x2 ls, pc′, pre) posMem = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x1 ls] size = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[x2 ls] aw = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='M (θ[iel], posm(θ), size(θ)) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← iel := aw(θ) C, cd ⊨ (pc, 1) −⇑f −→ halt Def = {iel} Use = {iel, x1 ls, x2 ls} C(pc) = (STOP, pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ C, cd ⊨ (pc, 0) −⇑f −→ halt Def = ∅ Use = ∅ C(pc) = (INVALID, pc′, pre) f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ C, cd ⊨ (pc, 0) −⇑f −→ exception Def = ∅ Use = ∅ The SELFDESTRUCT instruction allows for the self destruction of the executing account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since we only model the execution of a single contract and since the execution halts with the execution of SELFDESTRUCT, we do not model the deletion of the contract itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, the gas cost of the SELFDESTRUCT instruction, as well as the balances of accounts are affected by the execution of the SELFDESTRUCT instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (SELFDESTRUCT(xls), pc′, pre) aben = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='aben(θ) mod 2160 σ = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[externalel].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ φ = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(θ)(a) = ⊥ f1 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← externaleg := θ[externaleg][σ → σ(θ) � θ[actorel] → σ(θ)[b → 0] � ] f2 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← externaleg := θ[externaleg][σ → σ(θ) � a(θ) → σ(θ)[b += σ(θ)(θ[actorel]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='b] � ] f3 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← externaleg := θ[externaleg][σ → σ(θ) � a(θ) → (0, σ(θ)(θ[actorel]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='b, λx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 0, ϵ) � ] f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (φ(θ)) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' f1(f2(θ)) : f1(f3(θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {externaleg} Use = {externaleg, xls, actorel} C(pc) = (SELFDESTRUCT(xls), pc′, pre) aben = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[xls] a = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='aben(θ) mod 2160 σ = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ[externalel].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ φ = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(θ)(a) = ⊥ c = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (φ(θ)) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 37000 : 5000 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ ← gel := θ[gel] − c(θ) C, cd ⊨ (pc, 1) −⇑f −→ halt Def = {gel} Use = {externaleg, xls, gel} n) Rules for transaction initiating instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' : Since we will exclude the execution of DELEGATECALL and CALLCODE statements by assumption, we will only give the CFG rules for CALL, STATICCALL, and CREATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' CALL: The definition of the CALL rule comes with small technical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In order to give a precise analysis, the different changes in the state that are triggered by the call need to be done in separate nodes whenever they have different dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' While e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', the output data fragment can be influenced by the input to the call as well as by the global environment, the memory outside of the output fragment will simply be propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, the number of active words i only depends on the arguments specifying the input and output memory fragment, but not on any other inputs or the global environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To model these dependencies accurately, the corresponding updates need to happen in different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, we can only characterize the overall call effects on a state (using the EVM small-step semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We first define the function applyCall that mimics the effects of a function call on a CFG state θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For simplicity, we define the function here as a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, this relation is functional given that C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre)) (Γ, s) = toEVM(θ, C, pc) Γ ⊨ s :: S T−→ s′ :: S θ′ = toCFG(Γ, s′) applyCall(θ, C, pc) = θ′ Note that applyCall operates on a CFG state without temporal variables (since this is what toEVM is expecting and toCFG is returning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' When using applyCall in the following rules in conjunction with full CFG states Θ, we will write Θ ↓D(θ) to denote the restriction of Θ to the set of non-temporal variables (D(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The rules update the different state components one after the other, grouping those updates together that have the same dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Technically, for propagating pc-indexed state components, we need to update all variables in individual notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, first, the temporal variables in θ are set to the values of θ′ (the state after applying the call).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, the variables of θ are (one by one) updated to the values of θ and θ is set to θ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We now define the CFG rules for the CALL instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For the precise treatment of dependencies, the other state updates are treated differently depending on the preprocessing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We always need to consider all possible effects on the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, the effect on the local return memory fraction, on the return value y, on the external environment, on the gas, and on the active words and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The effects on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='the return value and the external environment depend on the same variables (which determine the overall outcome of the call): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='the arguments to the call ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='the input memory fragment (as specified by the arguments to the call) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='the current amount of gas available ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='the global environment (including the state of all other accounts and all globally accessible values) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='the global variables of the contract (since those may be read during reentrancy) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='The return memory fragment after the call,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' may depend on the previous values in this memory fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is, because in the case that the call was unsuccessful (returned with an exception), the return memory fragment stays unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The gas value after the call (in addition to the call outcome that influences the amount of gas refunded) depends, also, on the active words in memory (since this influences the costs for memory access for writing to the return memory fragment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, the active words in memory only depend on the location and size of the input and return memory fragment and the previous number of active words in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We give rules for these four different forms of dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To track the dependencies precisely, we first write the updated values into the corresponding temporal variables and only update the original variables later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is required so that we can use the applyCall function on the original state to obtain the updated values separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since we are considering purely functional state updates, we cannot simply save the original result of the applyCall function, but need to recompute it at every node in order to obtain the needed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To denote that we are updating the full CFG state, we write the state update function as λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='(⟨exp⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To refer to the restriction of the Θ to the non-temporal variables, we write Θ ↓D(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We first define the rules for the updates of y and the external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These rules are different because the STATICCALL does not change the static environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ f1 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls, externaleg} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) f1 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls, externaleg} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3 pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ f1 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3 (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) f1 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCall(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCall(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} Next, we define the rules for the update of the gas value: (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := applyCall(Θ ↓D(θ), C, pc)[gel] C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {gls, tols, vals, gel, iel, actorel} ∪ Xeg ∪ Xg ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := applyCall(Θ ↓D(θ), C, pc)[gel] C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {gls, tols, vals, gel, iel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} Next, we give the rule for the update of the active words in memory: (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← iel := applyCall(Θ ↓D(θ), C, pc)[iel] C, cd ⊨ (pc, 2) −⇑f −→ (pc, 3) Def = {iel} Use = {iel} ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} Finally, we give the rules for the memory update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These rules are the most interesting ones since they differ heavily depending on the available pre-processing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We first consider the case that the input and the result memory fragment are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this case, the values of memory locations im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S are assigned in individual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The node splitting, in this case, allows for precise treatment, since it only needs to be considered that the value at memory location i may depend on the input to the call or the previous value at exactly this location i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For assigning the ⊥ value to dynamic memory locations im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, this distinction is not needed since no dependencies are propagated in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S := load applyCall(Θ ↓D(θ), C, pc) im i ∈ [xoo, xoo + xos − 1] C, cd ⊨ (pc, 3 + (i − xoo)) −⇑f −→ (pc, 3 + (i − xoo) + 1) Def = {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := ⊥)x∈[xoom,xoom+xosm−1] C, cd ⊨ (pc, 3 + xos) −⇑f −→ (pc, 4 + xos) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xoo, xoo + xos − 1]} Use = ∅ After updating the memory, it is still required to set the values of all updated variables to the corresponding temporal variables and to set the temporal variables back to ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To determine the right offset for the different cases, we define a function getNodeOffset that given the precomputed memory fragments outputs the corresponding node offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, it computes the number of intermediate nodes required for the node splitting in the different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' getNodeOffset(io, is, oo, os) = � � � � � � � � � xos io = ⌊xio⌋ ∧ is = ⌊xis⌋ ∧ oo = ⌊xoo⌋ ∧ os = ⌊xos⌋ MAXInt256 io = ⌊xio⌋ ∧ is = ⌊xis⌋ ∧ (oo = ⊥ ∨ os = ⊥) 1 (io = ⊥ ∨ is = ⊥) ∧ oo = ⌊xoo⌋ ∧ os = ⌊xos⌋ 0 otherwise We give the rules for the stack/external environment, gas and active words in memory only once, since they are the same for all cases (for different node offsets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (Θ ← yls := Θ[yls]) ← externale := Θ[externale] C, cd ⊨ (pc, reset) −⇑f −→ (pc, reset + 1) Def = {yls, externale} Use = {yls, externale} (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← ge := Θ[ge] C, cd ⊨ (pc, reset + 1) −⇑f −→ (pc, reset + 2) Def = {ge} Use = {ge} (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← ie := Θ[ie] C, cd ⊨ (pc, reset + 2) −⇑f −→ (pc, reset + 3) Def = {ie} Use = {ie} Next, the values of the updated memory locations need to be carried over one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The number of nodes needed for that again depends on the pre-computed values for the input and output memory fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S := Θ[im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S] i ∈ [xoo, xoo + xos − 1] C, cd ⊨ (pc, reset + 3 + (i − xoo)) −⇑f −→ (pc, reset + 3 + (i − xoo) + 1) Def = {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} Use = {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := Θ[xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D])x∈[xoom,xoom+xosm−1] C, cd ⊨ (pc, reset + 3 + xos) −⇑f −→ (pc, reset + xos + 4) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xoo, xoo + xos − 1]} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xoo, xoo + xos − 1]} Finally, all temporal variables are set to ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is the same for all cases, irrespective of the pre-computed values for the input and output memory fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 2 ∗ (4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3])) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (x := ⊥)x∈D(θ) C, cd ⊨ (pc, reset) −⇑f −→ (pc′, 0) Def = D(θ) Use = ∅ In the case that only the input memory is known, the dependencies need to be propagated to the whole memory (as potential output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that in this case, we still gain precision by node splitting since, otherwise, we would need to propagate the dependencies of the whole memory to the whole memory again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' By node splitting, we ensure that only the dependencies of the input memory fragment are propagated to the whole memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ (pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := load applyCall(Θ ↓D(θ), C, pc) im i ∈ N256 C, cd ⊨ (pc, 3 + i) −⇑f −→ (pc, 3 + i + 1) Def = {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} ∪{ools | pre[omem +2] = ⌊xoo⌋} ∪{osls | pre[omem +3] = ⌊xos⌋} Again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' the values of the temporary memory variables need to be one-by-one written to the non-temporal variables: (C(pc) = (CALL(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' gls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' tols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' vals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' iols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ools,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' osls),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' gls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' tols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' iols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ools,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' osls),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre[omem + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre[omem + 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre[omem + 3]) pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ (pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := Θ[im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D] i ∈ N256 C, cd ⊨ (pc, reset + 3 + i) −⇑f −→ (pc, reset + 3 + i + 1) Def = {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} Use = {im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} In the case that only the output memory is known, the dependencies from the whole memory need to be propagated, but only to a small memory fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this scenario, node splitting does not help since anyway each affected memory node gets already the dependencies from the whole memory assigned: (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S := load applyCall(Θ ↓D(θ), C, pc) im)i∈[xoo,xoo+xos−1] C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xoo, xoo + xos − 1]} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {gls, tols, vals, gel, actorel} ∪ Xeg ∪ Xg ∪{iols | pre[omem] = ⌊xio⌋} ∪{isls | pre[omem+1] = ⌊xis⌋} (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := ⊥)x∈[xoo,xoo+xos−1] C, cd ⊨ (pc, 4) −⇑f −→ (pc, 5) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xoo, xoo + xos − 1]} Use = ∅ Similar to the previous cases, the temporal memory variables are carried over afterward: (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S := Θ[im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S])i∈[xoo,xoo+xos−1] C, cd ⊨ (pc, reset + 3) −⇑f −→ (pc, reset + 4) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xoo, xoo + xos − 1]} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xoo, xoo + xos − 1]} (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem], pre[omem + 1], pre[omem + 2], pre[omem + 3]) (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) pre[omem + 2] = ⌊xoo⌋ pre[omem + 3] = ⌊xos⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := Θ[xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D])x∈[xoo,xoo+xos−1] C, cd ⊨ (pc, reset + 4) −⇑f −→ (pc, reset + 5) Def = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xoo, xoo + xos − 1]} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xoo, xoo + xos − 1]} Finally, if neither input nor result memory fraction can be determined the whole memory needs to be considered the whole memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We can characterize this by a single rule as follows: (C(pc) = (CALL(yls, gls, tols, vals, iols, isls, ools, osls), pc′, pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls, gls, tols, iols, isls, ools, osls), pc′, pre) ∧ omem = 3) (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) (pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := load applyCall(Θ ↓D(θ), C, pc) im)i∈[Θ[ools],Θ[ools]+Θ[ools]−1] C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) Def = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {gls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' tols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' vals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' gel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} ∪ {ools | pre[omem + 2] = ⌊xoo⌋} ∪ {osls | pre[omem + 3] = ⌊xos⌋} The rules for carrying over the temporal memory variables are as follows: (C(pc) = (CALL(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' gls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' tols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' vals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' iols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ools,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' osls),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre) ∧ omem = 4 ∨ C(pc) = (STATICCALL(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' gls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' tols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' iols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ools,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' osls),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre) ∧ omem = 3) reset = 4 + getNodeOffset(pre[omem],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre[omem + 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre[omem + 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre[omem + 3]) (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) (pre[omem + 2] = ⊥ ∨ pre[omem + 3] = ⊥) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D := Θ[im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D])i∈[Θ[ools],Θ[ools]+Θ[ools]−1] C, cd ⊨ (pc, reset + 3) −⇑f −→ (pc, reset + 4) Def = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D It is important to note that the function applyCall(θ, C, pc) in all of the different rules returns the same result since only temporal variables are altered before each call of applyCall (and those are not used by applyCall(θ, C, pc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that we make here use of the distinction between local and global environment variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, the global environment variables can be accessed by other contracts as well and hence may influence the outcome of the call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, they need to be included in the Use sets of all the rules applying the effects of the call to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The local environment variable gel plays a special role in that the current amount of gas may influence the amount of gas given to the call and hence also the outcome of the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason, gel needs to be included in the Use set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We give the rules for the CREATE opcode in a similar fashion: We first devise a rule for the application of a create transaction: (C(pc) = (CREATE(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' vals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' iols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre)) ∨ C(pc) = (CREATE2(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' vals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' saltlsiols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre)) (Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' s) = toEVM(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc) Γ ⊨ s :: S T−→ s′ :: S θ′ = toCFG(Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' s′) applyCreate(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc) = θ′ As opposed to call instructions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' create instructions do not expect a return value written to memory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' but only the resulting address of the created account is written to the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, we need to consider that in case an exception occurs, 0 is written to the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An exception can occur if the execution of the initialization code causes an exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since the behavior of the initialization code may again depend on the environment, the value written to the stack can be dependent on the global environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We, hence, can again summarize the rules for the return value yls and the external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similar to the CALL rules, all updates are first done to temporal variables and only later transferred to the actual ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ f1 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls, externaleg} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {vals, gel, actorel} ∪ Xeg ∪ Xg C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) f1 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls, externaleg} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {vals, gel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} We give the rules for the gas computation: C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {vals, gel, iel, actorel} ∪ Xeg ∪ Xg C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {vals, gel, iel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} Next, we give the rule for the update of the active words in memory: C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← iel := applyCreate(Θ ↓D(θ), C, pc)[iel] C, cd ⊨ (pc, 2) −⇑f −→ (pc, 3) Def = {iel} Use = {iel} ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} We give the rules for writing the temporal variables into the actual ones one by one: C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 f1 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := Θ[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := Θ[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) Def = {yls, externaleg} Use = {yls, externaleg} C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := Θ[gel] C, cd ⊨ (pc, 4) −⇑f −→ (pc, 5) Def = {gel} Use = {gel} C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← iel := Θ[iel] C, cd ⊨ (pc, 5) −⇑f −→ (pc, 6) Def = {iel} Use = {iel} Finally, all temporal variables are set to ⊥ again: C(pc) = (CREATE(yls, vals, iols, isls, pc′, pre)) omem = 2 f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (x := ⊥)x∈D(θ) C, cd ⊨ (pc, 6) −⇑f −→ (pc′, 0) Def = D(θ) Use = ∅ Finally, the instruction CREATE2 operates in a similar fashion as CREATE with the main difference being that the newly created contract is assigned an address that can be predetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, CREATE2 takes an additional argument salt, which together with the creation code determines the address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Correspondingly, the rules for CREATE2 closely follow those of CREATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We can again summarize the rules for the return value yls and the external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similar to the CREATE and CALL rules, all updates are first done to temporal variables and only later transferred to the actual ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ f1 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls, externaleg} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {vals, saltls, gel, actorel} ∪ Xeg ∪ Xg C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) f1 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := applyCreate(Θ ↓D(θ), C, pc)[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := applyCreate(Θ ↓D(θ), C, pc)[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 0) −⇑f −→ (pc, 1) Def = {yls, externaleg} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {vals, saltlsgel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} We give the rules for the gas computation: C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 pre[omem] = ⌊xio⌋ pre[omem + 1] = ⌊xis⌋ f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S | x ∈ [xio, xio + xis − 1]} ∪ {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D | x ∈ [xio, xio + xis − 1]} ∪ {vals, saltls, gel, iel, actorel} ∪ Xeg ∪ Xg C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 (pre[omem] = ⊥ ∨ pre[omem + 1] = ⊥) f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := applyCreate(Θ ↓D(θ), C, pc)[gel] C, cd ⊨ (pc, 1) −⇑f −→ (pc, 2) Def = {gel} Use = Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ {vals, saltls, gel, iel, actorel} ∪ Xeg ∪ Xg ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} Next, we give the rule for the update of the active words in memory: C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← iel := applyCreate(Θ ↓D(θ), C, pc)[iel] C, cd ⊨ (pc, 2) −⇑f −→ (pc, 3) Def = {iel} Use = {iel} ∪ {iols | pre[omem] = ⌊xio⌋} ∪ {isls | pre[omem + 1] = ⌊xis⌋} We give the rules for writing the temporal variables into the actual ones one by one: C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 f1 = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← yls := Θ[yls] f2 = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← externaleg := Θ[externaleg] f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='f2(f1(Θ)) C, cd ⊨ (pc, 3) −⇑f −→ (pc, 4) Def = {yls, externaleg} Use = {yls, externaleg} C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← gel := Θ[gel] C, cd ⊨ (pc, 4) −⇑f −→ (pc, 5) Def = {gel} Use = {gel} C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 f = λθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← iel := Θ[iel] C, cd ⊨ (pc, 5) −⇑f −→ (pc, 6) Def = {iel} Use = {iel} Finally, all temporal variables are set to ⊥ again: C(pc) = (CREATE2(yls, vals, saltls, iols, isls, pc′, pre)) omem = 2 f = λΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Θ ← (x := ⊥)x∈D(θ) C, cd ⊨ (pc, 6) −⇑f −→ (pc′, 0) Def = D(θ) Use = ∅ o) From CFG semantics to Logical Rules: We illustrate how the logical rules describing the PDG derived from the CFG semantics are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The CFG semantics describes the PDG by giving control dependencies (via the CFG) and data dependencies via the Def and Use sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Each transition rule introduces data dependencies from all variables in the Def set to all variables in the Use set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The node splitting allows for enhancing precision since assignments of several variables that do not share the same Use set can be distinguished in a more fine-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For translating the CFG rules into dependency predicates, it is simply required to model the resulting data and control flow dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, we need to introduce a further abstraction step to account for the fact that Def and Use sets may be infinite (or at least unreasonably large, assuming that memory and storage locations can be represented by 256 bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, we will introduce a symbolic variable ⊤, which we will use to summarize memory and storage variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, ⊤ when used for modeling variable access (in the Use set) will represent the union of all dynamic and static memory (or storage) variables (Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S or Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D ∪ Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' When used to model writing variables (in the Def set), ⊤ will represent all dynamic memory (or storage) variables (Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D or Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To model this, we will assume the following types for our predicate domains: L := N256 ∪ {⊤} T := I where I is the set of all instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, L encodes the type of all (symbolic) storage locations and T encodes the types of so-called tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Tags model those variables on which we explicitly want to track dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For the scope of this work, we will only track dependencies on static environment variables, which we will (for simplicity) all represent by the opcodes that access these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason, we define T to consist of the set of all instructions I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To capture the dependencies as induced by the Def and Use sets, we define local data dependency predicates that describe the data dependencies between variables at specific nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' As opposed to directly specifying the Def and Use sets, these predicates enumerate all pairs of variables in the Def and the Use set (the cross-product between them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this way, we do not need to make the subnodes at a given program counter (as given in the CFG semantics) explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' These subnodes result from node splitting and only aim for separating the dependencies for different nodes in the Def set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we can easily mimic this effect by directly modeling dependencies between variables as they are induced by the Def and Use set at a given subnode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For efficiency reasons, we consider different variable types and devise predicates that describe the local dependencies between these types (as induced by the CFG nodes for a specific program counter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This results in improved performance since it enables the underlying datalog solver to compute several smaller fixpoints (for each variable type) instead of a big fixpoint (that captures the dependencies for all variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' More precisely, we define the following predicates (indexed by the program counter) for the different combinations of variable types as follows, where their name indicates the corresponding type (Var for stack variables, Mem for memory variables, Store for storage variables, Gas for local environmental variable gel, Msize for the local environmental variable iel, and External for the global environmental variables externaleg, and Source the static local and global environmental variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='VarVarpc ⊆ N256 × N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='VarMempc ⊆ N256 × L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='VarStorpc ⊆ N256 × L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='VarExternalpc ⊆ N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='VarGaspc ⊆ N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='VarSourcepc ⊆ N256 × T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='StoreVarpc ⊆ L × N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MemMempc ⊆ L × L × L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MemVarpc ⊆ L × N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MemExternalpc ⊆ L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MemGaspc ⊆ L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MemMsizepc ⊆ L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MemStorepc ⊆ L × L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MemSourcepc ⊆ L × T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='GasMempc ⊆ L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='GasVarpc ⊆ N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='GasExternalpc ⊆ B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='GasMsizepc ⊆ B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='GasStorepc ⊆ L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='GasSourcepc ⊆ T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='MsizeVarpc ⊆ N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ExternalMempc ⊆ L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ExternalVarpc ⊆ N256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ExternalGaspc ⊆ B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ExternalMsizepc ⊆ B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ExternalStorepc ⊆ L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ExternalSourcepc ⊆ T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='⟨write⟩⟨read⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='indicates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='write ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='∈ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='{Var,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Mem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Store,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Msize, External} and read ∈ {Var, Mem, Store, Gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='Msize, External, Source} that variable kind write is written and variable kind read is read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', VarMem(x, y) indicates that stack variable xls is written dependent on dynamic and static memory variables {ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D} and VarMem(x, ⊤) indicates that stack variable xls depends on all static and dynamic memory variables (Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S ∪ Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, MemVar(x, y) indicates that the static memory location xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S depends on stack variable yls, and MemVar(⊤, y) indicates that all dynamic memory variables Xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D depend on stack variable yls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that the variable ⊤ is used in the symbolic fashion described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A special case of this symbolic treatment is the predicate MemMem, which takes three arguments to give a more fine-grained symbolic modeling of whole memory intervals: The first position of MemMem specifies a memory location, and the two next positions specify a memory interval, which is given by its start offset and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Start offset and size can again be of type L, such that MemMempc(x, ⊤, ⊤) indicates that the (static) memory variable xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S depends on all static and dynamic memory variables and MemMempc(x, i, s) indicates that xm depends on all static and dynamic memory variables starting at memory position i until i + s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that write can never be Source since static local and global environment variables can never be written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, other write-read combinations are omitted for cases that never occur (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', for store unreachable contracts, the only way to write the contract’s storage is the SSTORE opcode, that allows for storing a stack variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, within a CFG node, a storage variable can only depend on the stack variables, so the predicate StoreVar is sufficient to capture all local dependencies of storage variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The local dependency predicates can be simply inhabited by rules that closely follow the CFG semantics: For each program counter pc, instruction-specific rules are generated that reflect the dependencies induced by the Def and Use sets of the subnodes of pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We give the example for the MLOAD instruction: {⊤ ⇒ VarMem(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⊤) | C(pc) = (MLOAD(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' xls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre) ∧ pre[0] = ⊥ ⊤ ⇒ MsizeVarpc(x) ⊤ ⇒ GasVarpc(x) ⊤ ⇒ GasMsizepc(⊤)} {⊤ ⇒ VarMem(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' v) | C(pc) = (MLOAD(yls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' xls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre) ∧ pre[0] = v ⊤ ⇒ GasMsizepc(⊤)} The given rules describe the dependencies induced by the corresponding CFG rules: The result variable yls either depends on all static and dynamic memory locations (if the memory location is unknown) or on the specific static and dynamic memory locations {vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S, vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The gas gel depends on the value of the active words in memory and on the stack variable xls, which holds the memory position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, iel depends on xls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that we do not explicitly model that gel and iel always depends on themselves since this is always the case, and hence we account for this by generic propagation rules, which always propagate gas and active word dependencies to the next program counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In addition to the local dependency predicates there exist special predicates that indicate that a variable is written in the first place: MsizeWritepc ⊆ B ExternalWritepc ⊆ B These predicates encode that the corresponding variable (here iel or externaleg) is written at a specific program counter pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We need these predicates for expressing the interaction between data and control dependence (for building backward slices): If a variable x is written at a certain node n′, which is control-dependent on another node n, and x is read at another node n′′ (without being overwritten before), then n′′ is data dependent on n′ and by transitivity, n′′ depends on n (via one data dependency and one control dependency edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason, it is important to model when a variable is written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' When making Def and Use sets fully explicit, this is easy to see, however, in our modeling, we immediately consider the cross-product from the Def and Use sets (at a program counter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, it can happen that there are no entries for certain variables in the Def Set (if there is no variable in the Use set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So, we need to cover these cases explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For all other variables (but iel or externaleg), we can use existing predicates as indicators for writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', whenever a variable is written the VarSource predicate is inhabited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, whenever memory or storage variables are written MemVar or, respectively StorVar are inhabited for the corresponding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Gas is written at any program counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To model the transitive dependencies (as induced by the PDG), we use the local data dependencies (as modeled by the local dependency predicates above) and the control dependencies (pre-computed according to the definition of standard control dependence) and build their transitive closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The control dependence is available via a predicate Controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' First, the transitive closure for control dependence is modeled via the predicate MayControlspc ⊆ N256 ×N256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, MayControlspc(pcb, xb) means that the program counter pc is transitively controled by the program counter pcb where at pcb there is a brach instruction (JUMPI) with condition stack variable xbls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Next, we define fixed point rules, which inhabit the following transitive closure predicates for program dependence: VarMayDependOn ⊆ N × T MemMayDependOnpc ⊆ L × T StorMayDependOnpc ⊆ L × T MsizeDependOnpc ⊆ T GasDependOnpc ⊆ T ExternalDependOnpc ⊆ T Intuitively, VarMayDependOn(x, t) denotes that variable x may depend on tag t, so that a node n′ where t is in the Use set, is in the backward slice of the (unique) node where x is written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that the tag t represents a static environment variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We index the predicates (with exception of VarMayDependOn) by the program counter to precisely characterize data dependence: A node n′ is considered data dependant on another node n if n defines a variable x that is used by n′ and n′ is reachable from n without passing through another node defining x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since we aim at staying within a characterization of dependencies that only uses grounded Horn clauses, we cannot simply express the second requirement (namely that no other node defining x should be passed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Instead, we explicitly formulate rules propagating dependencies in the case that at a certain program counter a variable is not (over)written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', we can formulate a generic rule for gas propagation, since gas is updated at every program counter (and hence is contained in the Def and Use set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that dependencies of stack variables (VarMayDependOn), as opposed to the other dependency predicates, is not indexed by the concrete node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is because the contract is assumed to be in SSA form (for stack variables), and those should hence only appear at only a single program location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that (similar to Securify), we currently only explicitly track transitive dependencies on local and global static environment variables (of type T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', we can express that a stack variable transitively depends on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', the block timestamp, but not that it transitively depends on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', a specific memory variable (however, it is, of course, captured that dependencies on global static environment variables can be introduced through dependencies on other variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The analysis can easily be extended to track further dependencies explicitly by adding rules introducing the corresponding dependencies to the corresponding ⟨write⟩Source predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The rules inhabiting the fixed point predicates are fairly standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' They are slightly complicated by the fact that we consider different variable types with different predicates so we need to consider data dependencies described by all the different local dependence predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Considering all possible combinations, introduces a slight overhead in rules (as compared to having a single predicate for variable types), but results in better performance, since it splits the fixed point computations into several smaller fixpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, there are some subtleties to consider for our symbolic treatment of dynamic memory locations and memory intervals (as discussed above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We illustrate this by the fixpoint rules for the MemMayDependOn predicate given in Figures 21 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Figure 21 shows the rules for describing transitive data dependencies for memory locations (captured by the MemMayDependOn predicate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, there are rules for all local dependency predicates, which indicate that the local memory is written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Intuitively, the rules model the data dependencies introduced by the nodes at pc′, by propagating dependencies known for the previous program counter pc′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Rule 1 simply introduces dependencies from the MemSource predicates (constituting the base case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The rules for propagating dependencies from variables(2), gas (4), active words in memory (5), and the external environment (6) are fully standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We need to consider that for all memory locations which are not overwritten, the dependencies from the previous program counter (pc) are propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, we define the predicate NoReassignMem that contains those opcodes that do not overwrite any memory location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' It only contains all instructions but MSTORE and the copy operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For all other operations, all previous dependencies are propagated (by rule 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For the overwriting operations, we need to consider that they do not write all memory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular, MSTORE may only write a single memory location, so for all other memory locations, the dependencies shall be propagated (this is done by rule 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 11 Additionally, we have a general rule that always propagates the dependencies of the symbolic memory position ⊤ (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This rule accounts for the fact that no opcode overwrites all (dynamic) memory locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, the most involved rules are those that involve symbolic reads from memory/storage locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For those, we need to consider that reading from a static location, also always implies reading from ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is shown by rules 7, 8 and 9, which account for the influences of storage locations on memory locations (as given by the local dependency predicate MemStore): Here MemStorepc′(ℓ, ⊤) indicates that memory variable ℓ depends on any static or dynamic storage variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For this reason, any of their dependencies are propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, MemStorepc′(ℓ, ℓS) (for ℓS ∈ N256) indicates that the storage location from which is read is statically known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this case, both the dependencies of the static storage locations within this interval are propagated, as well as the dependencies from ⊤ (indicating the corresponding dynamic storage locations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The treatment of the MemMem predicate is similar: Rule 12 considers the case that potentially the whole memory is read (MemMempc′(ℓ, ⊤, ℓs)), rule 13 considers the case where the specified interval that is read is concretely known and all dependencies from the corresponding static locations are read, and rule 13 ensures that also in this case all rules from the dynamic location (⊤) are propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, Figure 22 shows the rule that models the influence of the control flow on the dependencies of memory variables (captured by MemMayDependOn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The rule states that a memory location ℓ depends on tag t (at pc) if the memory is written at pc (indicated by MemVarpc(ℓ, v1)) and there is another program counter pc′, which controls pc and at pc′ (which needs to be a branch instruction), the conditional variable is v2, which again depends (transitively) on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3) Equivalence proof: We first introduce preliminary notions for the equivalence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For reasoning about contract executions that span several internal transactions, we introduce the notion of contract annotation as used in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For the sake of simplicity, in the main body of the paper, we annotated execution states only with the contract code C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, it gives more flexibility to characterize a contract as a pair c = (a, C) where a is the address of the contract and C is its code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This allows to distinguish executions of different contracts that share the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In the following, we will use the simplified annotation when sufficient and otherwise use the full annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We recall the notion of strong consistency from [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 4 (Annotation consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An execution state s is consistent with contract annotation c if the following two conditions hold 1) isRegular(s) =⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='addr 2) isRegular(s) ∨ isHalt(s) =⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='addr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code where isRegular(·) and isHalt(·) are predicates on execution states indicating whether they are regular execution states or halting states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 5 (Strong annotation consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' An execution state s is strongly consistent with contract annotation c (written s-consistent(s, c)) if it is consistent with c and additionally isRegular(s) =⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code Intuitively, a contract annotation c being strongly consistent with execution state s requires that s executes the contract as it resides in the global state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 11Note that technically, we would need to have a similar rule for the copy operations, but since we anyway overapproximate them to only write the ⊤ variable, the generic ⊤-propagation rule (10) captures this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 12Note that in the implementation we currently omit this last rule, since, anyway, MemStore can only be inhabited with value ⊤ in the CALL-like rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' {MemSourcepc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' | C(pc) = (op(⃗x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pc′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' pre) (1) MemVarpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' v) ∧ VarMayDependOn(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) (2) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemVarpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' v) ∧ ℓ ̸= ℓ′ ∧ MemMayDependOnpc(ℓ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) (3) ⇒ MemMayDependOnpc′(ℓ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemGaspc′(ℓ) ∧ GasMayDependOn(t) (4) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMsizepc′(ℓ) ∧ MsizeMayDependOn(t) (5) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemExternalpc′(ℓ) ∧ ExternalMayDependOn(t) (6) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemStorepc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⊤) ∧ StoreMayDependOn(ℓS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) (7) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemStorepc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ℓS) ∧ ℓS ∈ N256 ∧ StoreMayDependOn(ℓS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) (8) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemStorepc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ℓS) ∧ ℓS ∈ N256 ∧ StoreMayDependOn(⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) (9) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMayDependOnpc(⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) (10) ⇒ MemMayDependOnpc′(⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMempc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ℓs) ∧ MemMayDependOnpc(ℓ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) (11) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMempc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ℓo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ℓs) ∧ ℓo ∈ N256 ∧ ℓs ∈ N256 (12) ∧ i ≥ ℓo ∧ i < ℓo + ℓs ∧ MemMayDependOnpc(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMempc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ℓo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ℓs) ∧ ℓo ∈ N256 ∧ ℓs ∈ N256 (13) ∧ MemMayDependOnpc(⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMayDependOnpc(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t) ∧ NoReassignMem(pc) (14) ⇒ MemMayDependOnpc′(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Horn Clauses describing the data flow dependencies captured by the MemMayDependOn predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' {MemVarpc(ℓ, v1) ∧ MayControlspc(pc′, v2) ∧ VarMayDependOn(v2, t) | pc ∈ D(C) (15) ⇒ MemMayDependOnpc(ℓ, t)} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Horn Clauses describing the control flow dependencies captured by the MemMayDependOn predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that the function toEVM(θ, C, pc) maps states in the CFG semantics θ to execution states that are strongly consistent with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let (Γ, s) = toEVM(θ, C, pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then s is strongly consistent with (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor, λpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (C(pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='op, C(pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pre)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Trivially follows from the definition of toEVM since s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ is set to hold code toEVM(θ, C, pc) at address s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We formally define the notion of a transaction step Γ ⊨ sC :: S T−→ s′ C :: S and the the relation Γ ⊨ sC :: S �→ s′ C :: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 6 (Transaction step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let S be a callstack, s, s′ be execution states and Γ be a transaction environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further let C be a contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then Γ ⊨ sC :: S T−→ s′ C :: S := ∃s∗s†C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Γ ⊨ sC :: S → s∗ C∗ :: sC :: S →∗ s† C∗ :: sC :: S → s′ C :: S A transaction step describes the execution of a transaction being initiated in s (since in the execution step thereafter the element s∗ is added to the call stack) and ends in s′ (since this is the execution state immediately after removing the additional stack element).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that this definition excludes that the execution might have returned before and have triggered another internal transaction since the execution state s on the stack would have otherwise changed (at least due to a decrease in gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 7 (Medium step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let S be a callstack, s, s′ be execution states and Γ be a transaction environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further let C be a contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then Γ ⊨ sC :: S �→ s′ C :: S := Γ ⊨ sC :: S → s′ C :: S ∨ Γ ⊨ sC :: S T−→ s′ C :: S Due to the two-layered memory abstraction, multiple states in the CFG semantics represent a single state in the EVM semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we define a notion of equivalence on CFG states that takes this into account: Definition 8 (CFG state equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Two CFG state θ and θ′ are considered equivalent (written θ ≈θ θ′) if the following holds: θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ls = θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ls ∧ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='el = θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='el ∧ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='eg = θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='eg ∧ ∀xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' load θ xm = load θ′ xm ∧ ∀xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' load θ xg = load θ′ xg We state some basic properties on CFG state equivalences: Lemma 3 (CFG state equivalence properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The following hold: θ ≈θ θ ← x := load θ x θ ≈θ (θ ← x := load θ x) ← x := ⊥ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Follows immediately from the definition ≈θ and load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Most importantly, equivalent CFG states will be mapped to the same EVM states: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' For all contracts C, all program counters pc and all CFG state θ θ′ it holds that θ ≈θ θ′ ⇔ toEVM(θ, C, pc) = toEVM(θ′, C, pc) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Follows immediately from the definitions of toEVM, ≈θ and load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We formally define the medium step version of the CFG semantics: C, cd ⊨ ⟨(pc, 0), θ⟩ =⇒ ⟨n, θ′⟩ := ∃ n (θi)i∈[0,n] (ai)i∈[0,n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C, cd ⊨ ⟨(pc, 0), θ⟩ (−ai −→ ⟨(pc, i), θi⟩)i∈[0,n−1] −an −→ ⟨n, θ′⟩ We now state the equivalence statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular, we explicitly state the assumptions on the execution (excluding exceptions and reentering storage modification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, we consider the cases where exception or halting states are entered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Theorem 4 (Equivalence of EVM and CFG semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be a store unreachable contract with sound preprocessing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then the following holds: 1) Let Γ ⊨ sC :: S �→ s′ C :: S be an execution of contract C that does not exhibit local out-of-gas exceptions and let s be strongly consistent with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then either a) s′ = (µ′, ι′, σ′) and C, |S| ⊨ ⟨(µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0), toCFG(Γ, s) ⊎ θ⊥⟩ =⇒ ⟨(µ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0), θ′ ⊎ θ⊥⟩ for some θ′ with θ′ ≈θ toCFG(Γ, s′) b) s′ = EXC and C, |S| ⊨ ⟨(µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0), toCFG(Γ, s) ⊎ θ⊥⟩ =⇒ ⟨exception, toCFG(Γ, s) ⊎ θ⊥⟩ c) s′ = HALT(σ′, g, d) and C, |S| ⊨ ⟨(µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0), toCFG(Γ, s) ⊎ θ⊥⟩ =⇒ ⟨halt, toCFG(Γ, s) ⊎ θ⊥⟩ 2) Let C, cd ⊨ ⟨(pc, 0), θ ⊎ θ⊥⟩ =⇒ ⟨n, Θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then either a) n = (pc′, 0) and Θ = θ′ ⊎ θ⊥ and for (Γ, s) = toEVM(θ, C, pc) and (Γ′, s′) = toEVM(θ′, C, pc′) it holds that Γ = Γ′ and for all S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' |S| = cd it holds that either Γ ⊨ sC :: S �→ s′ C :: S or Γ ⊨ sC :: S → EXCC :: S and C[pc] ̸= INVALID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) n = exception and for (Γ, s) = toEVM(θ, C, pc) and for all S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' |S| = cd it holds that Γ ⊨ sC :: S → EXCC :: S and C[pc] = INVALID c) n = halt and for (Γ, s) = toEVM(θ, C, pc) and for all S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' |S| = cd it holds that Γ ⊨ sC :: S → HALT(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ, g, d)C :: S for some g and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that the CFG semantics only aims at modeling a single contract execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular, it does not consider the effects that a finalized execution may have on the caller (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=', it does not model that global state variables are reverted in case that the execution halted exceptionally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We prove the two directions of the proof separately: ⇒ Let Γ ⊨ sC :: S �→ s′ C :: S be an execution of contract C that does not exhibit local out-of-gas exceptions and let s be strongly consistent with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We do a case distinction on s′ 1) s′ = (µ′, ι′, σ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We do a further case distinction on Γ ⊨ sC :: S �→ s′ C :: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' – The execution step was a local step and hence Γ ⊨ sC :: S → s′ C :: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In this case we know that s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι = ι′ and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ = σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The proof trivially follows by case distinction over the instruction in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code[s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc] using the fact that due to strong consistency s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' – The execution step was a call step and hence Γ ⊨ sC :: S T−→ s′ C :: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' By the definition of the CFG rule for calling which leverages the notion of a transaction step on the EVM semantics, we only need to argue that the changes on θ′ (in the corresponding rule sequences) preserve equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This immediately follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2) s′ = EXC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since the execution does not exhibit local out of gas exceptions we know by Assumption 1 that in this case C(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc) = INVALID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This case, hence, follows immediately from the CFG semantics rule for the INVALID opcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3) s′ = HALT(σ, g, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This case follows immediately from the CFG semantics rules for the halting instructions STOP and RETURN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⇐ Let C, cd ⊨ ⟨(pc, 0), θ ⊎ θ⊥⟩ =⇒ ⟨n, Θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The proof follows by a simple case distinction on C[pc] using the CFG semantics rules for the individual instructions and taking advantage of Lemma 4 to reason about the equality of the execution states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that we did not need to make use of the assumption that the contract is store unreachable for the equivalence proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is since this requirement is only needed to prove the consistency of the Def and Use sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We revisit the assumption of store unreachability using full contract annotations: Assumption 3 (Store unreachability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A contract c is store unreachable if {DELEGATECALL, CALLCODE} ∩ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code = ∅ and for all regular execution states (µ, ι, σ) that are strongly consistent with c, it holds that for all transaction environments Γ and all callstacks S ¬∃s,S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Γ ⊨ (µ, ι, σ)c :: S →∗ sc :: S′ + +S ∧ |S′| > 0 ∧ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc) = (op(⃗x), pcnext) ∧ op ∈ InstSSTORE Where the set InstSSTORE of store instructions is defined as InstSSTORE = {SSTORE} The key property following from the store unreachability is the following: Lemma 5 (Store unreachability implies global storage preservation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be a store unreachable contract and s an execution state that is strongly consistent with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then for all callstacks S and execution states s′ Γ ⊨ sc :: S T−→ s′ c :: S ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Assume towards contradiction that s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor ̸= s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then there must have been a callstack S∗ and an execution state s∗ and a contract C∗ such that Γ ⊨ sC :: S →∗ s∗ c∗ :: S∗ + +sc :: S → s+ c + :: S+++sc :: S →∗ s′ c :: S such that s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor ̸= s+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since storage can only be altered via the SSTORE instruction (which is a local instruction), we know that c∗ = c+ and S∗ = S+ and s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι = s+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι and s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code(s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc) = (SSTORE(x, y), pcnext).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, since one can only write the storage of the active account, we know that s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Due to strong consistency (which is preserved during execution for contacts without DELEGATECALL and CALLCODE), we hence know that s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code and so also c∗ = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From Assumption 3, we can derive a contradition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Correctness of May Analysis We start by introducing definitions and lemmas needed for our soundness claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 9 (Backward Slice of Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' BS(N) = � n∈N BS(n) Definition 10 (Execution steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Execution steps with states are defined as follows n −(Q)√ −→ n′ Q(θ) ⟨n, θ⟩ −(Q)√ −→ ⟨n′, θ⟩ n−⇑f −→ n′ θ′ = f(θ) ⟨n, θ⟩ −⇑f −→ ⟨n′, θ′⟩ and −→∗ denotes the transitive closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 11 (Deterministic successor function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We define the function “ds” as ds(n) = � n′ if ∃n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n−⇑f −→ n′ n′ if ∃n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n −(Q)√ −→ n′ ∧ Q 1 Definition 12 (Sliced execution steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n, θ⟩ a−→ ⟨n′, θ′⟩ n ∈ BS(N) ⟨n, θ⟩ a−→BS(N) ⟨n′, θ′⟩ ⟨n, θ⟩ a−→ ⟨n′, θ′⟩ n ̸∈ BS(N) n′′ = ds(n) ⟨n, θ⟩ τ−→BS(N) ⟨n′′, θ⟩ Definition 13 (Sliced execution paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We define the sliced execution path as −→ ∗ BS(N) as the transitive τ closure of Definition 12 and require that as −→ ∗ BS(N) must end at a node in BS(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We write as −→ m BS(N) if path “as” in as −→ ∗ BS(N) has exactly m observable steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 14 (Up-to equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let Var be the set of all variables used in a contract C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' States θ1 and θ2 are equal except for variable X if and only if ∀V ∈ Var/{X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1(V ) = θ2(V ) We write θ1 =/X θ2 in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 15 (Deterministic successor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Node n+1 is defined as the successor of node n in a contract C if it is the sole successor of n and undefined otherwise: n+1 = � n′ if ∀n′, n′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n → n′ ∧ n → n′′ =⇒ n′ = n′′ undefined otherwise Definition 16 (Step-indexed execution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We define as −→ ∗|N i by ⟨n, θ⟩ as −→ ∗ ⟨n′, θ′⟩ | as ↓N|= i ⟨n, θ⟩ as −→ ∗|N i ⟨n′, θ′⟩ where ↓N filters out the actions where the source node is not in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 17 (Relevant variables [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Relevant variables of backward slice of S at node n are defined as follows n as −→ ∗|NV 0 n′ n′ ∈ BS(S) V ∈ Use(n′) V ∈ rv S n Lemma 6 (Slice usage set property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (∀nX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nX ̸∈ BS(NY )) =⇒ ⟨n+1 X , θ⟩ −→m BS(NY ) ⟨n+, ⟩ =⇒ (∀k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n+1 X , θ⟩ −→k BS(NY ) ⟨n∗, ⟩ =⇒ X ̸∈ Use(n∗)) The lemma states that for all paths to nodes n+ of the backward slice NY , it holds that each step (n∗) in the backward slice towards n+ does not use X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We assume nX ̸∈ BS(NY ) and ⟨n+1 X , θ⟩ −→m BS(NY ) ⟨n+, ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Towards contradiction we assume there would be a k such that ⟨n+1 X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1⟩ −→k BS(NY ) ⟨n∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟩ ∧ X ∈ Use(n∗) Then we define n∗ X with l ≤ k such that ⟨n+1 X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1⟩ −→l BS(NY ) ⟨n∗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ∗ 1⟩ ∧ X ∈ Def(n∗ X) where n∗ X identifies the last node where X was defined on the path to node n∗ or if such an l does not exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' we know ⟨nX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟩ −→ ⟨n+1 X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1⟩ −→∗ BS(NY )|NX 0 ⟨n∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟩ This means that X is either defined along the path or at the predecessor of the start node since we start at a successor of a definition of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This node n∗ X (nX) is then by definition in the backward slide of NY since X is used at node n∗ in the backward slice and it was the closest definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This contradicts the assumption n∗ X ̸∈ BS(NY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The proofs use the Slicing framework’s soundness claim for backward slices of a set of nodes instead of single nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Theorem 5 (Correctness of Slicing Based on Paths and Sets [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n, θ⟩ as −→ ∗ ⟨n′, θ′⟩ n′ ∈ S ∃ as′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n, θ⟩ as′ −→ ∗ BS(S) ⟨n′, θ′′⟩ ∧ (∀ V ∈ Use(n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ′(V ) = θ′′(V )) ∧ as ↓BS(S)= as′ The following lemma lifts sliced executions while preserving the number of visits at nodes in the backward slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' All executions within the backward slice of node-set N can be mapped to real executions with the same number of visits in any subset of the backward slice and all relevant variables concerning the final node n′ of the execution were computed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n′ ∈ N =⇒ N ′ ⊆ N =⇒ ⟨n, θ⟩ as′ −→ ∗ BS(N)|N ′ i ⟨n′, θ′⟩ =⇒ ∃as, θ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='⟨n, θ⟩ as −→ ∗|N ′ i ⟨n′, θ′′⟩ ∧ ∀V ∈ rv N n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ′(V ) = θ′′(V ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We first show that path as exists outside of the backward slice: (1) ⟨n, θ⟩ as −→ ⟨n′, θ′′⟩ Path as is constructed by extending the sliced path as′ with nodes outside of the backward slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' By Definition 12 and since every step in ⟨n, θ⟩ as −→ ⟨n′, θ′⟩ would be deterministic by the slicing framework requirements, we can reconstruct an unique original path as by adding nodes along the path as′ where τ edges were followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Therefore, we construct the following path as where all nodes ni with a line on top were not part of the original sliced path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (2) as = n n1 n1 ni n′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' as′ 1 as′ 1 as′ i+1 where the original path was as′ = n n1 n′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' as′ 1 τ∗ as′ i+1 The construction described above is feasible by the definition of as −→ ∗ BS(N) and satisfies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We remark that path as′ is an ordered sublist of the original sliced-path as as nodes were only inserted while construction in step (2) but not removed: (3) as′ ⊆od as It is left to be shown that path as has the same number of visits in the set N ′ as the path as′ in the sliced graph: (4) ⟨n, θ⟩ as −→ ∗|N ′ i ⟨n′, θ′′⟩ Recall that by assumption, all nodes in N ′ are part of N and therefore elements of the backward slice of BS(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' By construction of (2) and (3), we know that the number of visits at N ′ is the same in both paths and (4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We now show that for final state θ′′ from (4) and the final state θ′ from the sliced-path it holds that (5) ∀V ∈ rv N n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ′(V ) = θ′′(V ) All nodes where any variable V in rv N n′ is defined are part of the backward slice by Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Therefore, no node of this kind was added or deleted in the construction of as and they were already present in as′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' With that, we can use the correctness statement of slicing to show that all relevant variables reassigned along as′ have the same value in as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Non-reassigned variables are not changed and stay the same in both states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Therefore, all relevant variables have the same value in θ′ and θ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The statement follows from (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a) Variable Independence: Lemma 8 (Variable Dependency Predicate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be a contract and X and Y be variables thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then it holds that VarMayDepOn(Y, X) ̸∈ lfp(R(C)) ⇒ ∀nY , nX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nX ̸∈ BS(nY ) Definition 18 (Variable Independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A variable Y is independent of a variable X if and only if for all states θ1, θ2 and θ′ 1 it holds that ∀nX, nY , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1 =/X θ2 ∧ ⟨n+1 X , θ1⟩ −→∗|NY i ⟨n+1 Y , θ′ 1⟩ =⇒ ∃θ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n+1 X , θ2⟩ −→∗|NY i ⟨n+1 Y , θ′ 2⟩ ∧ θ′ 1(Y ) = θ′ 2(Y ) We will refer to this definition by VarIndOf(n, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Theorem 6 (Soundness of Variable Independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let X and Y be variables and nX ̸∈ BS(nY ) for all nodes nX, nY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then variable Y is independent of variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be a contract and C be a contract with sound preprocessing information that is consistent with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We assume that (1) nX ̸∈ BS(nY ) holds for all nodes nX and nY and an executions from node n+1 X to node n+1 Y that starts in states θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let θ2 be a state such that (2) θ1 =/X θ2 and (3) ⟨n+1 X , θ1⟩ as −→ ∗|NY i ⟨n+1 Y , θ′ 1⟩ identity the previous execution with path as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From (3) we know from correctness of slicing (Theorem 5) that (5) ∃as′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='⟨n+1 X , θ1⟩ as′ −−→ m BS(NY )|NY j ⟨n+1 Y , θ∗ 1⟩, (6) θ′ 1(Y ) = θ∗ 1(Y ) and (7) as ↓NY = as′ Equations (5) and (6) say that there exists a corresponding path in the backward slice and it has the correct value for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Equation (7) implies that all nodes in the backward slice of NY are visited in the same order in as and as′ [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since all nodes where Y is defined are by defintion of NY in the backward slice, we know from (7) that i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We now show that path as′ exists in the sliced-graph when starting from state θ2: (9) ⟨n+1 X , θ2⟩ as′ −−→ m BS(nY )|NY j ⟨n+1 Y , θ∗ 2⟩ and θ∗ 1(Y ) = θ∗ 2(Y ) With Lemma 6 we get from (1) and (5) for all k ≤ m that (8) ⟨n+1 X , θ1⟩ −→k BS(NY ) ⟨n+, θ+ 1 ⟩ =⇒ X ̸∈ Use(n+) We can use (8) to show (9) since at all nodes n+, that would have effected the trace, X is not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From (2) we know by the slicing framework’s well-formedness properties that X is only source of different effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Therefore, we know that (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From (9) we can use Lemma 7 to conclude that ⟨n+1 X , θ2⟩ as′′ −−→ ∗ |NY i ⟨n+1 Y , θ′ 2⟩ ∧ ∀V ∈ rv NY nY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ′ 2(V ) = θ∗ 2(V ) and in particular θ′ 2(Y ) = θ∗ 2(Y ) since Y ∈ rv NY nY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) Instruction Independence: Lemma 9 (Instruction Dependency Predicate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be a contract, n be a node and X be variable thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then it holds that InstMayDepOn(n, X) ̸∈ lfp(R(C)) ⇒ ∀nif, nX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nif −→cd n ⇒ nX ̸∈ BS(nif) Definition 19 (Instruction Independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A node n is independent of a variable X if and only if for all states θ1 and θ1 it holds that ∀nX, n′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1 =/X θ2 ∧ ⟨n+1 X , θ1⟩ as −→ ∗|n′ i ⟨n′, ⟩ ∧ ⟨n+1 X , θ2⟩ as′ −→ ∗ |n′ i ⟨n′, ⟩ =⇒ | as ↓n| = | as′ ↓n| We will refer to this definition by InstIndOf(n, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Theorem 7 (Soundness of Instruction Independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let n be a node, X be a variable and nX ̸∈ BS(nif) hold for all nodes nX and nif where nif −→cd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then node n is independent of variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be a contract and C be a contract with sound preprocessing information that is consistent with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We assume that (1) nX ̸∈ BS(nif) holds for all nodes nX and nif such that nif −→cd n and two executions from node n+1 X to node n′ that starts in states θ1 and θ2 with (4) θ1 =/X θ2 and visit n′ on their paths i times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let (2) ⟨n+1 X , θ1⟩ as −→ ∗|n′ i ⟨n′, ⟩ and (3) ⟨n+1 X , θ2⟩ as′ −→ ∗ |n′ i ⟨n′, ⟩ be these two executions with paths as and as′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The interesting case is node n′ ̸= n since the conclusion for node n′ = n follows with assumption (3) and (4) by definition of −→∗|n′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Therefore, we show the statement for n′ ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We show that paths as and as′ visit node n the same number of times by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='g we assume that path as visits node n at least once more than path as′: (∗) | as ↓n| > | as′ ↓n| From (∗) we know that there exists a prefix as of path as such that it can visit node n once more than path as′: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='1) ⟨n+1 X , θ1⟩ as −→ ∗ |n k ⟨n, ⟩ −→∗|n′ 0 ⟨n′, ⟩ ∧ as ⊆od as with k =| as′ ↓n| such that as ends at the k +1 occurrence of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Node n′ is reachable from n because the prefix as can allows be completed to the full path from (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Next, we define a corresponding prefix as′ of as′ that vists n the maximal number of k times and ends at the same appearence of n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='2) ⟨n+1 X , θ2⟩ as′ −−−−−−−−−−−−→ ∗ |n′ j ⟨n′, ⟩ ∧ as′ ⊆od as′ with j =| as ↓n′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' That path exists based on (4) and (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We know that path as′ visits n less often than as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' and as′ consists of all visits of n in as′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We know by construction of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='2) that as and as′ split up after the k-th visit at n because otherwise both would visit node n at least k + 1 number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We call this split node nif and make the split explicit: (6) ∃nif, w, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n+1 X , θ1⟩ as1 −−→ ∗ |nif w ⟨nif, θif 1 ⟩ as2 −−→ ∗ ⟨n, ⟩ −→ ∗ ⟨n′, ⟩ ∧ ⟨n+1 X , θ2⟩ as′ 1 −−→ ∗ |nif z ⟨nif, θif 2 ⟩ as′ 2 −−−−−−−−−−−→ ∗ ⟨n′, ⟩ where as = as1 @ as2 such that as2 and as′ 2 do not share a single node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nif is the last node where the paths could split up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From (3) – (6), we can conclude with standard control dependence that (7) nif −→∗ cd n By instantiating (1) with (7) we know (8) ∀n0 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n0 X ̸∈ BS(nif) From (7) we know that states θif 1 and θif 2 have at least one different value for some input variable at nif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Only a different value in the usage set can lead to splitting control flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Therefore, we get: (9) ¬∀ V ∈ Use(nif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θif 1 (V ) = θif 2 (V ) We now map paths as1 and as′ 1 from (5) into the sliced graph with Theorem 5 of correctness of slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Those paths compute correct input values at nif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We get (10) ∃ as1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n+1 X , θ1⟩ as1 −−→ g BS(nif)|nif w ⟨nif, θ∗ 1⟩ ∧ ∀ V ∈ Use(nif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θif 1 (V ) = θ∗ 1(V ) (11) ∃ as′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n+1 X , θ2⟩ as′ 1 −−→ h BS(nif)|nif z ⟨nif, θ∗ 2⟩ ∧ ∀ V ∈ Use(nif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θif 2 (V ) = θ∗ 2(V ) We apply paths as1 and as′ 1 to Lemma 6 with (8) and get (12) ∀ q′ ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⟨n+1 X , θp⟩ −→q′ BS(Lif) ⟨n+, θ∗ p⟩ =⇒ X ̸∈ Use(n+) with (p, q) ∈ {(1, g), (2, h)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Assumption (2) states that only X can propagate changes, but we know from (12) that no node in the backward slice on either path uses X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' executions ⟨n+1 X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1⟩ −→∗ BS(nif) and ⟨n+1 X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ2⟩ −→∗ BS(nif) have the same states (up to X) after the same number of steps: (13) ⟨n+1 X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1⟩ xs −→ t BS(nif) ⟨ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ˆθ1⟩ ∧ ⟨n+1 X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ2⟩ ys −→ t BS(nif) ⟨ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ˆθ2⟩ =⇒ ˆθ1 =/X ˆθ2 Since nif ∈ BS(nif),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' we know that paths in (10) and (11) reach the split node nif after the same number of steps in the backward slice: (14) w = z and therefore choose t = | as1 | = | as1 | Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' by (13) and (14) on paths (10) and (11) we get θ∗ 1 =/X θ∗ 2 and conclude by (12): (15) ∀ V ∈ Use(nif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ∗ 1(V ) = θ∗ 2(V ) The state equivalence reasoning is summarized in the following for all V ∈ Use(nif): θif 1 (V ) θ∗ 1(V ) θ∗ 2(V ) θif 2 (V ) (10) = (15) = (11) = Thereby, we conclude ∀ V ∈ Use(nif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θif 1 (V ) = θif 2 (V ) which contradicts (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 20 (Environmental Instruction Independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' A node n is independent of a constant environmental variable X if and only if for all states θ1 and θ2 it holds that ∀n0, n′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ1 =/X θ2 ∧ ⟨n0, θ1⟩ as −→ ∗|n′ i ⟨n′, ⟩ ∧ ⟨n0, θ2⟩ as′ −→ ∗ |n′ i ⟨n′, ⟩ =⇒ | as ↓n| = | as′ ↓n| Lemma 10 (Soundness of Environmental Instruction Independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let n be a node, X be a constant environmental variable and nX ̸∈ BS(nif) for all nodes nX and nif where nif −→cd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then node n is independent of variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definite assignment was ensured by n+1 X in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since constant environmental variables are definitely assigned and final by design, we can drop this requirement in the soundness claim and reuse the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Correctness of Trace Noninterference Pattern To prove Theorem 3 Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let V be a set of CFG state variables and let Nf(C) := {n | ∃ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n = (pc, i) land ∃ op ⃗x pcnext pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (op(⃗x), pcnext, pre) ∧ f(op)} Args(C, n) := {x | ∃ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∧ n = (pc, i) ∧ ∃ op ⃗x pcnext pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = (op(⃗x), pcnext, pre) ∧ x ∈ ⃗x} Varf(C) := {x | ∃ nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nf ∈ Nf(C) ∧ x ∈ Args(C, nf)} Further, let m ∈ N, C, n, n1 f, n1 f +, · · · , nm f , nm f +, as1, · · · , asm, a1, · · · , am, and θ, θ′, θ1, θ+ 1 , · · · , θm, θ+ m be arbitrary and assume that 1) ∀θ =/V θ′ 2) ∀xf ∈ Varf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarIndOf(xf, v) 3) ∀nf ∈ Nf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' InstIndOf(nf, v) 4) ⟨n, θ⟩ � asi −−→ ∗|Nf (C) 0 ⟨ni f, θi⟩ ai −→ ⟨ni f +, θ+ i ⟩ �m i=1 Then there exist as′ 1, · · · , as′ m, a′ 1, · · · , a′ m, and θ′ 1, θ′+ 1 · · · , θ′ m, θ′+ m such that ⟨n, θ⟩ � as′ i −−→ ∗ |Nf (C) 0 ⟨ni f, θ′ i⟩ a′ i −→ ⟨ni f +, θ′+ i ⟩ �m i=1 ∧ ∀i ∈ [1, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀x ∈ Args(C, ni f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θi(x) = θ′ i(x) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' By induction on m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1) Let m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The claim trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2) Let m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then � asi −−→ ∗|Nf (C) 0 ⟨ni f, θi⟩ ai −→ ⟨ni f +, θ+ i ⟩ �m−1 i=1 asm −−→ ∗|Nf (C) 0 ⟨nm f , θm⟩ am −−→ ⟨nm f +, θ+ m⟩ and by the inductive hypothesis also ⟨n, θ⟩ � as′ i −−→ ∗ |Nf (C) 0 ⟨ni f, θ′ i⟩ a′ i −→ ⟨ni f +, θ′+ i ⟩ �m−1 i=1 for some as′ 1, · · · , as′ m−1, a′ 1, · · · , am−1, and θ′ 1, θ′+ 1 , · · · , θ′ m−1, θ′+ m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' such that ∀i ∈ [1, m − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀x ∈ Args(C, ni f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θi(x) = θ′ i(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We are hence left to show that there exists some as′ m, a′ m, θ′ m, θ′+ m such that ⟨nm−1 f +, θ′+ m−1⟩ as′ m −−→ ∗ |Nf (C) 0 ⟨nm f , θ′ m⟩ a′ m −−→ ⟨nm f +, θ′+ m ⟩, and ∀x ∈ Args(C, nm f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θm(x) = θ′ m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Assume towards contradiction that there is no as′ m, a′ m and θ′ m, θ′+ m such that ⟨nm−1 f +, θ′+ m−1⟩ as′ m −−→ ∗ |Nf (C) 0 ⟨nm f , θ′ m⟩ a′ m −−→ ⟨nm f +, θ′+ m ⟩, meaning that nm f is not reachable from ⟨nm−1 f , θ′ m−1⟩ without stepping through another node in Nf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We consider two cases a) nm f is not reachable from ⟨nm−1 f +, θ′+ m−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since every execution ends in the node exit, we know that ⟨nm−1 f +, θ′+ m−1⟩ as′ m −−→ ∗ | {nm f } 0 ⟨exit, θ′ exit⟩ and ⟨nm f , θm⟩ asm+1 −−−−→ ∗ ⟨exit, θexit⟩ for some θexit and θ′ exit, and asm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we know that must be executions ⟨n, θ⟩ as −→ ∗ ⟨exit, θexit⟩ and ⟨n, θ′⟩ as′ −→ ∗ ⟨exit, θ′ exit⟩ such that |as ↓nm f | > |as′ ↓nm f | (with as = � i∈[1,m+1] asi · ai and as′ = � i∈[0,m] as′ i · a′ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This immediately contradicts InstIndOf(nm f , v) (assumption 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) A node n∗ f from Nf(C) different from nm f , which is reached from ⟨nm−1 f , θ′ m−1⟩ before reaching nm f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So, there are as′ m, a′∗ m, as′∗ m, θ′∗, θ′∗+ such that ⟨nm−1 f +, θ′+ m−1⟩ as′ m −−→ ∗ |Nf (C) 0 ⟨n∗ f, θ′∗⟩ a′∗ m −−→ ⟨n∗ f +, θ′∗+⟩ as′∗ m −−→ ∗ ⟨nm f , θ′ m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' But then there must be executions ⟨n, θ⟩ as −→ ∗ ⟨nm f , θm⟩ and ⟨n, θ′⟩ as′ −→ ∗ ⟨nm f , θ′ m⟩ such that |as ↓n∗ f | < |as′ ↓n∗ f | (with as = � i∈[1,m+1] asi · ai and as′ = (� i∈[0,m−1] as′ i · a′ i) · as′ m · a′ m ∗ ·as′∗ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This again immediately contradicts InstIndOf(n∗ f, v) (assumption 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Hence, we know that θ′ m, θ′+ m such that ⟨nm−1 f +, θ′+ m−1⟩ as′ m −−→ ∗ |Nf (C) 0 ⟨nm f , θ′ m⟩ a′ m −−→ ⟨nm f +, θ′+ m ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, we need to show that ∀x ∈ Args(C, nm f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θm(x) = θ′ m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This immediately follows from assumption 2 and concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We define the set of components of EVM configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 21 (EVM configuration components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The set of EVM configuration components CompEVM is defined as follows CompEVM := Comps EVM ∪ CompΓ EVM ∪ CompS EVM with Comps EVM := {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='gas, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='msize, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='input, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender, ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='value} ∪ {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stack(x) | x ∈ N8} ∪ {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='m(x) | x ∈ N256} and CompΓ EVM := {H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='parent, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='beneficiary, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='difficulty, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='number, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='gaslimit, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='timestamp, origin, gasprice} and CompS EVM := {σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor, σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal, σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code, σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal, σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce} ∪ {σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(x) | x ∈ N256} Note that we explicitly exclude the program counter, the active account and the active code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The reason for this is that we will only compare executions of the same contract starting from the same instruction (program counter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let Z ⊆ CompEVM be a set of EVM components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We define two EVM configurations (Γ, s), (Γ′, s′) equal up to Z (written (Γ, s) =/Z (Γ′, s′)) if the following holds ∀z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='z ̸∈ Z ⇒ (z ∈ Comps EVM ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='z = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='z) ∧ (z ∈ CompΓ EVM ⇒ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='z = Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='z) ∧ (z = σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor ⇒ ∀x ∈ N256 a ∈ N160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(x) = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(x)) ∧ (z = σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal ⇒ ∀a ∈ N160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal) ∧ (z = σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce ⇒ ∀a ∈ N160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce) ∧ (z = σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code ⇒ ∀a ∈ N160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code) ∧ (∀x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' z = σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(x) ⇒ ∀a ∈ N160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(x) = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(x)) ∧ (z = σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal ⇒ ∀a ∈ N160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal) ∧ (z = σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce ⇒ ∀a ∈ N160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' a = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='actor ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='σ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce) We formally define the function toVar, which maps components of EVM configurations to CFG variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note, that as opposed to the slightly simplified version in the main body of the paper, toVar maps to a set of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Definition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The function toVar ∈ CompEVM → P(V ) is defined as follows: toVar(z) := � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � {xls} z = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stack(x) {xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} z = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='m(x) {gel} z = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='gas {iel} z = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='msize {inputel} z = ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='input {senderel} z = ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='sender {vael} z = ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='value {parenteg} z = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='parent {beneficiaryeg} z = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='beneficiary {difficultyeg} z = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='difficulty {numbereg} z = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='number {gaslimiteg} z = headerc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='gaslimit {timestampeg} z = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='timestamp {origineg} z = origin {prizeeg} z = gasprice {externaleg} z ∈ {σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor, σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal, σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='bal, σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce, σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='nonce, σ(other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='code} {xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='D, xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='S} z = σ(this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='stor(x) Next, we establish the relation between the notions of equivalence up to components: Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let Γ, Γ′ be transaction environments, s, s′ execution states and Z a set if EVM state components such that (Γ, s) =/Z (Γ′, s′) and (θ, C, pc) = toCFG(Γ, s) and (θ′, C, pc) = toCFG(Γ, s) for some θ, θ′, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, let Z = {x | ∃z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' x ∈ toVar(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then θ =/Z θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Trivially follows from the definition of toCFG that maps components into variables in the same way as toVar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Finally, by combining the previous lemmas, we can prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We restate the theorem for completeness with all assumptions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' To this end, we restate the definition of trace noninterferenceto explicitly state the consistency assumption the involved executions: Definition 24 (Trace noninterference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let C be an EVM contract, Z ∈ CompEVM be a set of components of EVM configurations and f ∈ I → B be a predicate on instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then trace noninterference of contract C w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Z and f (written TNI(C, Z, f)) is defined as follows: TNI(C, Z, f) := ∀ Γ Γ′ s s′ t t′ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' π′ s-consistent(s, C) ⇒ s-consistent(s′, C) ⇒ (Γ, s) =/Z (Γ′, s′) ⇒ Γ ⊨ sC :: S π−→ ∗ tC :: S ∧ final (t) ⇒ Γ ⊨ s′ C :: S π′ −→ ∗ t′ C :: S ∧ final (t′) ⇒ π ↓f= π′ ↓f where π ↓f denotes the trace filtered by f, so containing only the instructions satisfying f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' and s and s′ are strongly consistent with contract C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Next, we formally define PC Z,f considering the proper definition of toVar: Definition 25 (Trace non-interference pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' PC Z,f :={InstMayDepOn(pc, z) | ∃ z z op ⃗x pcnext pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' z ∈ Z ∧ z ∈ toVar(z) ∧ C(pc) = op(⃗x, pcnext, pre) ∧ f(op)} ∪ {VarMayDepOn(xi, z) | ∃ z z op ⃗x pcnext pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' z ∈ Z ∧ z ∈ toVar(z) ∧ C(pc) = (op(⃗x, pcnext, pre)) ∧ f(op) ∧ xi ∈ ⃗x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Theorem 8 (Soundness of trace noninterference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Let Z ⊆ CompEVM be a set of EVM components, and f ∈ I → B an instruction-of-interest predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, let C be a store unreachable contract that does not exhibit local out-of-gas exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then it holds that (∀p ∈ PC Z,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' p ̸∈ lfp(R(C))) ⇒ TNI(C, Z, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that due to the definition of EVM components, we can rely on the fact that (Γ, s) =/Z (Γ′, s′) ⇒ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc = s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc (16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Assume that ∀p ∈ PC Z,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' p ̸∈ lfp(R(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From the definition of PC Z,f ( Definition 25), we know that hence ∀ z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ z ∈ toVar(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ pc i pcnext pre op ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ InstMayDepOn((pc, i), z) ̸∈ lfp(R(C)) and ∀ z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ z ∈ toVar(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ x pc pcnext pre op ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ x ∈ ⃗x ⇒ VarMayDepOn(xi, z) ̸∈ lfp(R(C)) From Theorem 6,Lemma 8, Theorem 7,Lemma 9, we get that ∀ z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ z ∈ toVar(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ pc i pcnext pre op ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ InstIndOf((pc, i), , z) and ∀ z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ z ∈ toVar(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀ x pc pcnext pre op ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' C(pc) = op(⃗x, pcnext, pre) ⇒ f(op) ⇒ x ∈ ⃗x ⇒ VarIndOf(xi, z) Which is equivalent to ∀ v ∈ {v | ∃z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' v ∈ toVar(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀nf ∈ Nf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' InstIndOf(nf, v) (17) and ∀ v ∈ {v | ∃z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' v ∈ toVar(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀x ∈ Varf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarIndOf(x, v) (18) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Now let Γ, Γ′ be transaction environments and s, s′, t, t′ be execution states and π, and π′ be traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Assume the following: 1) (Γ, s) =/Z (Γ′, s′) 2) Γ ⊨ sC :: S π−→ ∗ tC :: S 3) final (t) 4) Γ ⊨ s′ C :: S π′ −→ ∗ t′ C :: S 5) final (t′) We need to show that π ↓f= π′ ↓f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We define the following relation to reason about steps that do not produce any actions satisfying f: Γ ⊨ s :: S �−→∗|f s′ :: S := Γ ⊨ s :: S π �−→ ∗ s′ :: S ∧ π ↓f= ϵ Using this definition, we can decompose every run Γ ⊨ sC :: S π−→ ∗ tC :: S into a run Γ ⊨ sC :: S � �−→∗|f sf,i C :: S opi(⃗vi) �−−−−→ sf +,i C :: S �m i=1 �−→∗|f tC :: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Such that π ↓f= �m i=1 opi(⃗vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, we can deconstruct every run Γ ⊨ s′ C :: S π′ −→ ∗ t′ C :: S into a run Γ ⊨ s′ C :: S � �−→∗|f s′f,i C :: S op′ i(⃗v′i) �−−−−→ s′f +,i C :: S �n i=1 �−→∗|f t′ C :: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Such that π′ ↓f= �n i=1 op′ i(⃗v′i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Towards contradiction, we assume that π ↓f̸= π′ ↓f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So, either there is a position i such that opi(⃗vi) ̸= op′ i(⃗v′i) and for all j < i it holds that opj(⃗vj) = op′ j(⃗v′j) or |π ↓f| < |π′ ↓f| and π ↓f is a prefix of π′ ↓f or |π′ ↓f| < |π ↓f| and π′ ↓f is a prefix of π ↓f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We proceed by case distinction based on these cases: 1) Assume that there is a position i such that opi(⃗vi) ̸= op′ i(⃗v′i) and for all j < i it holds that opj(⃗vj) = op′ j(⃗v′j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then we know that Γ ⊨ sC :: S � �−→∗|f sf,j C :: S opj(⃗vj) �−−−−→ sf +,j C :: S �i−1 j=1 �−→∗|f sf,i C :: S opi(⃗vi) �−−−−→ sf +,i C :: S and hence for µi = sf,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ we have that C(µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc) = (opi(⃗xi), pci next, prei) and for all vi k ∈ ⃗vi it holds that vi k = µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(xi k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Similarly, we know that Γ ⊨ s′ C :: S � �−→∗|f s′f,j C :: S op′ j(⃗v′j) �−−−−−→ s′f +,j C :: S �i−1 j=1 �−→∗|f s′f,i C :: S op′ i(⃗v′i) �−−−−→ s′f +,i C :: S and hence for µ′ i = s′f,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ we have that C(µ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc) = (op′ i(⃗x′i), pc′i next, pre′ i) and for all v′i k ∈ ⃗v′i it holds that v′i k = µ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(x′i k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From Theorem 4 we get that a) C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨nsf,j, toCFG(Γ, sf,j) ⊎ θ⊥⟩ ⇒ ⟨nsf+,j, toCFG(Γ, sf +,j) ⊎ θ⊥⟩ �i j=1 such that all nsf,j ∈ Nf(C) b) C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨ns′f,j, toCFG(Γ, s′f,j) ⊎ θ⊥⟩ ⇒ ⟨ns′f+,j, toCFG(Γ, s′f +,j) ⊎ θ⊥⟩ �i j=1 such that all ns′f,j ∈ Nf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that from assumption 16 we know that ns = (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0) = (s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0) = ns′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, we know that C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨n∗ j, θj⟩ −→ ⟨n∗ j, θj+⟩ �l j=1 for some l ≥ i such that there is some g ∈ N → N for which it holds that 1) ∀ n m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n < m ⇒ g(n) < g(m) and 2) ∀ j ∈ [1, i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' nsf,j = n∗ g(j) and 3) ∀ j ∈ [1, i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' toCFG(Γ, sf,j) ⊎ θ⊥ = θg(j) and 4) ∀ j ∈ [1, i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' k > g(i) ⇒ k < g(i + 1) ⇒ ∀pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n∗ g(j) = (pc, 0) ⇒ ∃q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n∗ k = (pc, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is as all ⇒-steps from nodes in Nf(C) can be expanded into the individual steps between the subnodes of the same pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we can apply Lemma 11 (using Lemma 12) to obtain C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨n∗ j, θ′ j⟩ −→ ⟨n∗ j, θ′ j+⟩ �l j=1 and ∀p ∈ [1, l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀x ∈ Args(C, n∗ p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θp(x) = θ′ p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we can also conclude that C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨nsf,j, θ† j⟩ ⇒ ⟨nsf+,j, θ† j⟩ �i j=1 such that ∀ j ∈ [1, i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ† j = θ′ g(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' In particular, this means that ∀ j ∈ [1, i − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀x ∈ Args(C, nsf,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θ† j(x) = toCFG(Γ, sf,j) ⊎ θ⊥(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Since opi(⃗vi) ̸= op′ i(⃗v′i), it must either holds that opi(⃗xi) ̸= op′ i(⃗x′i) or there is some position k such that vi k ̸= v′i k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We do another case distinction: a) Assume that opi(⃗xi) ̸= op′ i(⃗x′i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Then it must hold that µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc ̸= µ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc (since C deterministically maps program counters triples (op(⃗x), pcnext, pre) ) and consequently ns′f,i ̸= nsf,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, since execution is deterministic, and we know that we can reach nsf,i as the ith node from Nf(C) (with a different pc) when starting the execution in ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩, this leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' b) Assume that there is some position k such that vi k ̸= v′i k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This means that there exists xi k ∈ ⃗xi (with C(µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc) = (op(⃗xi), pci next, prei)) such that µ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(xi k) ̸= µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(xi k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' However, since execution is deterministic, and we know that we can reach the configuration ⟨nsf,i, θ† i ⟩ as the ith node from Nf(C) when starting the execution in ⟨ns′, toCFG(Γ, s′)⊎θ⊥⟩, we know that θ† i = toCFG(Γ, s′f,j) ⊎ θ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Further, we know that θ† i (xi k) = toCFG(Γ, sf,i) ⊎ θ⊥(xi k) and consequently also toCFG(Γ, s′f,j) ⊎ θ⊥(xi k) = toCFG(Γ, sf,i) ⊎ θ⊥(xi k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This contradicts µ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(xi k) ̸= µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='s(xi k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 2) Assume that |π ↓f| < |π′ ↓f| and π ↓f is a prefix of π′ ↓f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' So, m < n, which means that Γ ⊨ s′ C :: S � �−→∗|f s′f,i C :: S op′ i(⃗v′i) �−−−−→ s′f +,i C :: S �m i=1 � �−→∗|f s′f,i C :: S op′ i(⃗v′i) �−−−−→ s′f +,i C :: S �m+k i=m �−→∗|f t′ C :: S for some k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From Theorem 4 we get that C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨ns′f,i, toCFG(Γ, s′f,i) ⊎ θ⊥⟩ ⇒ ⟨ns′f+,i, toCFG(Γ, s′f +,i) ⊎ θ⊥⟩ �n i=1 such that all ns′f,j ∈ Nf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' From this, we can conclude that C, |S| ⊨ ⟨ns′, toCFG(Γ, s′) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨n∗ j, θj⟩ −→ ⟨n∗ j, θj+⟩ �l j=1 for some l ≥ n such that there is some g ∈ N → N for which it holds that 1) ∀ n m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n < m ⇒ g(n) < g(m) and 2) ∀ j ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ns′f,j = n∗ g(j) and 3) ∀ j ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='toCFG(Γ, s′f,j) ⊎ θ⊥ = θg(j) and 4) ∀ j ∈ [1, i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' k > g(i) ⇒ k < g(i + 1) ⇒ ∀pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n∗ g(j) = (pc, 0) ⇒ ∃q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' n∗ k = (pc, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This is as all ⇒-steps from nodes in Nf(C) can be expanded into the individual steps between the subnodes of the same pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Note that from assumption 16 we know that ns = (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0) = (s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='pc, 0) = ns′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we can apply Lemma 11 (using Lemma 12) to obtain C, |S| ⊨ ⟨ns, toCFG(Γ, s′) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨n∗ j, θ′ j⟩ −→ ⟨n∗ j, θ′ j+⟩ �l j=1 and ∀p ∈ [1, l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ∀x ∈ Args(C, n∗ p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' θp(x) = θ′ p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we can also conclude that C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨ns′f,j, θ† j⟩ ⇒ ⟨ns′f+,j, θ† j⟩ �n j=1 such that ∀ j ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content='θ† j = θ′ g(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Consequently, we have that C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨ns′f,j, θ† j⟩ ⇒ ⟨ns′f+,j, θ† j⟩ �m j=1 � −→∗|Nf (C) 0 ⟨ns′f,j, θ† j⟩ ⇒ ⟨ns′f+,j, θ† j⟩ �m+k j=m However, from Theorem 4 we know that C, |S| ⊨ ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩ � −→∗|Nf (C) 0 ⟨nsf,j, toCFG(Γ, sf,j) ⊎ θ⊥⟩ ⇒ ⟨nsf+,j, toCFG(Γ, sf +,j) ⊎ θ⊥⟩ �m j=1 −→∗|Nf (C) 0 ⟨nt, toCFG(Γ, t) ⊎ θ⊥⟩ such that all nsf,j ∈ Nf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' This leads to a contradiction, since execution is deterministic and like this we obtain two executions starting in ⟨ns, toCFG(Γ, s) ⊎ θ⊥⟩, which step through a different number of nodes (with different pcs) in Nf(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 3) Assume that |π′ ↓f| < |π ↓f| and π′ ↓f is a prefix of π ↓f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' The proof is fully analogous to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' APPENDIX B SECURIFY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Analysis specification The following rules are extracted from the Securify fixed-point calculation [2] [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' We split the appendix in input facts in Section Section B-A1, May-semantic rules in Section B-A2, and Must-semantic rules in Section B-A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Must-analysis rules that are identical to the May-analysis semantic rules are left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Identical rules are denoted with ⇐ instead of ⇐May in Appendix B-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Additionally, analogous rules for storage are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' 1) Input facts: Source(L, Y0, inst) ← inst(L, Y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=') (19) AssignVar(L, Yi, Xj) ← inst(L, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' , Yi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' � �� � Outputs , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' , Xj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' � �� � Inputs ) (20) inst ̸∈ {mload,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' sload,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' sha3} (no propagation for known accesses) mstore(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ← mstore(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' hasConstantV alue(O) (21) mstore(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ← mstore(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬hasConstantV alue(O) (22) mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MO) ← mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' hasConstantV alue(O) (23) mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ⊤) ← mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬hasConstantV alue(O) (24) AssignVar(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O) ←May mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬hasConstantV alue(O) (25) a) Optional Source Rules: Source(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MO) ← mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' hasConstantV alue(O) (26) Source(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' M⊤) ← mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬hasConstantV alue(O) (27) Source(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y ) ← call(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=') or staticcall(L, Y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=') (28) b) May Control Flow and Dependency Propagation: Follow(L1, L2) ← instpc(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ), instpc+1(L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ), hasLinearSuc(L1) (29) Follow(L1, L3) ← jumpI(L1, , L3) (30) Follow(L1, L2) ← jump(L1, L2) (31) Taint(L1, L3, X) ←May jumpI(L1, X, L3), L3 ̸= MergeInstr(L1) (32) Taint(L1, L2, X) ←May ”jumpI(L1, X, L3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' inst(L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' )”, L2 ̸= MergeInstr(L1) (33) Join(L1, L2) ←May jumpI(L1, X, L3), L2 = MergeInstr(L1) (34) c) Must Control Flow and Dependency Propagation: OneBranchTag(L) ←Must jumpDest(L), one incoming branch, no prev inst in BB (35) Tag(L) ←Must jumpDest(L) (36) Jump(L1, L2, L4), Follows(L1, L2) ←Must ”jumpI(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' inst(L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' )”, L4 = MergeInstr (37) Jump(L1, L3, L4) ←Must jumpI(L1, , L3), L4 = MergeInstr (38) Jump(L1, L2, L2) ←Must jump(L1, L2) (39) Follow(L1, L2) ←Must ”inst(L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' inst(L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' )” (same basic block) (40) JoinIncBranches (JIB) connects all incoming branches such that the Must-analysis can check if a predicate holds on all preceeding nodes: JIB(L1, L2, Linc 2 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' , JIB(Linc n , Ln, L′) ←Must ∀i, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' � � � � � � � ”instpc(Li, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' jumpDestpc+1(L′)” jump(Lj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L′) jumpI(Lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L′) (41) 2) May-Semantic Rules: VarMayDepOn(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐May AssignVar( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarMayDepOn(Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (42) VarMayDepOn(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐May AssignVar(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Taint( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarMayDepOn(Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (43) VarMayDepOn(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐May Source(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Taint( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarMayDepOn(Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (44) VarMayDepOn(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐May Source( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (45) InstMayDepOn(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐May Taint( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (46) InstMayDepOn(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐May Taint( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarMayDepOn(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (47) a) Memory Dependency Propagation: MemMayDepOn(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T) ⇐May mstore(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' VarMayDepOn(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T) (48) MemMayDepOn(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T) ⇐May Follows(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMayDepOn(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' (49) ¬ReassignMem(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O) ReassignMem(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O) ⇐ mstore(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isConst(O) (50) Source(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T) ⇐ mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMayDepOn(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' isConst(O) (51) Source(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T) ⇐May mload(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' O),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MemMayDepOn(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬isConst(O) (52) b) Control Dependence Propagation: Taint(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐May Follow(L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Taint(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬Join(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2) (53) 3) Must-Semantic Rules: MustFollow(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2) ⇐Must MustPrecedeStep(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2) (54) MustFollow(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L3) ⇐Must MustFollow(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' MustFollow(L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L3) (55) MustPrecedeStep(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2) ⇐Must Follow(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ¬Tag(L2) (56) MustPrecedeStep(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L3) ⇐Must Jump(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' oneBranchTag(L2) (57) MustPrecedeStep(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2) ⇐Must Jump(L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' L2) (58) DetBy(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐Must Source(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (59) DetBy(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) ⇐Must AssignVar(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' DetBy(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFST4oBgHgl3EQfTTiJ/content/2301.13769v1.pdf'} +page_content=' X) (60) 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China. +2SSE, The Chinese University of Hong Kong (Shenzhen), P.R. China. +3MPI for Informatics, Germany. +4Peng Cheng Laboratory, P.R. China. +*Corresponding author(s). E-mail(s): ddai@mpi-inf.mpg.de; lizhen@cuhk.edu.cn; +Contributing authors: xuyan1@link.cuhk.edu.cn; chaodazheng@link.cuhk.edu.cn; +shuguangcui@cuhk.edu.cn; +Abstract +When using LiDAR semantic segmentation models for safety-critical applications such as autonomous driv- +ing, it is essential to understand and improve their robustness with respect to a large range of LiDAR +corruptions. In this paper, we aim to comprehensively analyze the robustness of LiDAR semantic seg- +mentation models under various corruptions. To rigorously evaluate the robustness and generalizability of +current approaches, we propose a new benchmark called SemanticKITTI-C, which features 16 out-of-domain +LiDAR corruptions in three groups, namely adverse weather, measurement noise and cross-device discrep- +ancy. Then, we systematically investigate 11 LiDAR semantic segmentation models, especially spanning +different input representations (e.g., point clouds, voxels, projected images, and etc.), network architectures +and training schemes. Through this study, we obtain two insights: 1) We find out that the input represen- +tation plays a crucial role in robustness. Specifically, under specific corruptions, different representations +perform variously. 2) Although state-of-the-art methods on LiDAR semantic segmentation achieve promis- +ing results on clean data, they are less robust when dealing with noisy data. Finally, based on the above +observations, we design a robust LiDAR segmentation model (RLSeg) which greatly boosts the robustness +with simple but effective modifications. It is promising that our benchmark, comprehensive analysis, and +observations can boost future research in robust LiDAR semantic segmentation for safety-critical applications. +Keywords: Robustness, LiDAR Corruption, Out-of-distribution, Point Clouds, Semantic Segmentation. +1 Introduction +Autonomous driving, one of the most promising appli- +cations for computer vision, has achieved impressive +progress in recent studies, where LiDAR semantic +segmentation plays a crucial role Hu et al. (2022); +Unal, Dai, and Van Gool (2022); Yan et al. (2022). +Current semantic segmentation models are generally +evaluated on clean validation sets, which share the +same data distribution with the corresponding train- +ing sets, e.g., collected with the same sensor, at a +similar time and weather condition, and at the same +place Behley et al. (2019). However, due to the inac- +curate data acquisition Ren, Pan, and Liu (2022); Yan, +Zheng, Li, Wang, and Cui (2020) and complex sce- +narios caused by diverse weather conditions Hahner, +Dai, Sakaridis, Zaech, and Van Gool (2019); Hahner +1 +arXiv:2301.00970v1 [cs.CV] 3 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Robust LiDAR segmentation +Original Data (64-beam) +Cross-device Simulation (16-beam) +Data with Global Outliers +Data with Snowfall Simulation +Data with Fog Simulation +Data with Local Distortion +road +sidewalk +car +vegetation +trunk +terrain +building +other-obj. +ignore +Fig. 1 Examples of our proposed SemanticKITTI-C. We corrupt the clean validation set of SemanticKITTI using six types of corruptions +with 16 levels of intensity to build upon a comprehensive robustness benchmark for LiDAR semantic segmentation. Listed examples are point +clouds on 16-beam LiDAR sensors, with global and local distortion, in snowfall and fog simulations. +et al. (2022a), LiDAR point clouds inevitably suf- +fer from severe corruptions in real-world deployment. +Since autonomous driving is a safety-critical appli- +cation, robustness against out-of-distribution (OOD) +LiDAR data becomes an important part of the model. +Understanding and analyzing the robustness of +models for image corruption is a well-studied topic, +in which several benchmarks are proposed for dif- +ferent tasks, e.g., classification Hendrycks and Diet- +terich (2019); Hendrycks, Zhao, Basart, Steinhardt, +and Song (2021), semantic segmentation Kamann and +Rother (2020), instance segmentation Altindis, Dalva, +and Dundar (2021) and etc. Generally, these stud- +ies simulate corruption through changing RGB values +on the original image, resulting in different kinds +of perturbations. Moreover, since images are repre- +sented as dense pixel arrays, previous works focus +on investigating different architectures without mod- +ifying the representation of the input. In contrast, +analyzing robustness against LiDAR corruption is a +more complicated problem: 1) LiDAR point clouds +are usually textureless and irregular, and they describe +the 3D shapes only through scattered points. Also, +point clouds collected by different type of LiDARs +may have different ranges and resolutions. There- +fore, the corruption on point clouds not only needs +to consider the deformation, disturbance and occlu- +sion in the 3D space, but also the domain discrepancy +caused by different devices during the data acquisi- +tion. 2) Semantic segmentation models in LiDAR sce- +narios usually adopt diverse representations to meet +different requirements. For instance, projection-based +methods Milioto, Vizzo, Behley, and Stachniss (2019); +Y. Zhang et al. (2020) project LiDAR point clouds +onto 2D pixels, and thus enable the application of +normal 2D-CNNs. Voxel-based approaches conduct +voxelization and transform the LiDAR point clouds +into 3D voxel grids Graham and van der Maaten +(2017); Zhou et al. (2020), exploiting 3D-CNN to cap- +ture the fine-grained 3D information. There are also +point-based methods Hu et al. (2020); Thomas et al. +(2019) that learn the geometric details directly on raw +point clouds. Recent studies even combine different +representations to boost the performance Tang et al. +(2020); Xu et al. (2021), which makes it more difficult +to analyze robustness purely from the architectures. +In this paper, we try to break through the barrier +of robust LiDAR semantic segmentation and extend +the exploration of model robustness against 3D vision. +We find out that the study of corruption robustness +on 3D point cloud is still in its infancy. Specifically, +most studies Ren et al. (2022); Yan et al. (2020) for +robustness on 3D point clouds tend to test their mod- +els on synthesis datasets, e.g., ModelNet40. The point +clouds on these datasets are sampled from CAD mod- +els of stand-alone objects. Therefore, the findings in + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +3 +these studies cannot be directly applied to real-world +applications, where raw point clouds are collected in +large and complex environments. Though a few prior +works are related to the model robustness on real- +world data (e.g., Lai et al. (2022)), they only conduct a +coarse comparison at the level of full models, without +analysis on the inner structures and the input rep- +resentations. As a result, there is no benchmark for +robustness in the real-world point clouds, especially +for the safety-crucial LiDAR semantic segmentation. +For the first time, based on the most pop- +ular LiDAR semantic segmentation dataset, i.e., +SemanticKITTI Behley et al. (2019), we build +a systematically-designed robust benchmark with +several real-world and out-of-domain corruptions, +namely SemanticKITTI-C in Fig. 1. Following pre- +vious studies, the term robustness refers to training +a model on clean data and validating it on corrupted +data, and thus we introduce diverse corruptions on +4071 LiDAR scans in the validation set. Specifi- +cally, SemanticKITTI-C covers 16 kinds of corrup- +tions in total, which can be categorized into three +classes, namely adverse weather, measurement noise, +and cross-device discrepancy. To make the benchmark +more rigorous, we set several subclass corruptions in +each class. For instance, the adverse weather class +contains LiDAR scans in snowfall and fog simula- +tions, and there are three independent levels in each +corruption, indicating different snowy or foggy inten- +sities. Built upon this benchmark, we further evaluate +the robustness of current LiDAR semantic segmenta- +tion methods, including analysis of different represen- +tations, architectures, and training schemes. +Consequently, we obtain 12 observations in +total from various aspects: Representation: We find +out that projection-based methods are vulnerable to +adverse weathers, especially fog simulation, but they +are more robust to local distortion. We also observe +that exploiting larger image size in range projection +improves the robustness of projection-based meth- +ods. Inversely, point-based approaches are vulnera- +ble to local distortion but more robust in different +weather conditions. Compared with the above two +mainstreams, the voxel representation enjoys impres- +sive robustness in most corruptions. And using cylin- +der voxel partition Zhou et al. (2020) is much more +robust than using the traditional grids Graham and +van der Maaten (2017). Architecture: 1) For point- +based approaches, pseudo kernel local aggregation +(e.g., KPConv Thomas et al. (2019)) is the most robust +when compared with adaptive-weight Hu et al. (2020) +and MLPs Qi, Yi, Su, and Guibas (2017). Moreover, +transformer architectures Zhao, Jiang, Jia, Torr, and +Koltun (2021) greatly hamper the robustness of point- +based methods. 2) Although hybrid-representation +architecture improves the performance on clean data, +it makes models more vulnerable to noise, espe- +cially for those using MLPs to aggregate point-wised +features in each voxel Yan et al. (2022); Zhou et +al. (2020). Training strategy: Empirically, applying +data augmentation such as Mix3D Nekrasov, Schult, +Litany, Leibe, and Engelmann (2021) improves the +robustness. Finally, by identifying the best combi- +nation from existing components in terms of input +representations, model architectures and data augmen- +tation strategies, we design a robust LiDAR segmenta- +tion model (RLSeg) in a simple but effective manner, +achieving superior robustness. Our contributions are +concluded as follows: +- We present the first large-scale robustness bench- +mark for LiDAR point cloud semantic segmentation +under various corruptions, namely SemanticKITTI- +C. The dataset contains 16 corruptions, spanning +scenarios in adverse weather conditions, sensor +measurement bias and diverse device collections. +- We comprehensively study existing methods on our +proposed benchmark and analyze the robustness of +diverse architectures and representations. +- We summarize several effective observations to +boost the robustness of LiDAR semantic segmen- +tation. It is identified that architecture and input +representation should be carefully considered in +future research and real-world deployment. +2 Related Work +2.1 LiDAR Semantic Segmentation +Since the data collected by LiDAR is represented as +point clouds, there are several mainstreams to process +input point clouds with different representations. More +details will be illustrated in Sec. 4. +1) Point-based methods. These approaches directly +learn the geometric details on raw point clouds. Gen- +erally, they follow the hierarchical architecture as 2D +vision, and first conduct sampling strategy in each +layer. After that, they search neighboring points from +each sampled point, and apply feature aggregation +in each local group. The local aggregation function +is essential for the point-based methods, and thus +many studies design different operators to capture the + +Springer Nature 2021 LATEX template +4 +Robust LiDAR segmentation +local geometrics. For instance, point-wise MLP Qi +et al. (2017), adaptive weight Y. Liu, Fan, Xiang, +and Pan (2019); Wang et al. (2019); W. Wu, Qi, and +Fuxin (2019) and pseudo grid Hua, Tran, and Yeung +(2018); Thomas et al. (2019) are utilized to extract +local features of point clouds, and they also exploit +nonlocal operators Yan et al. (2020) or attention mech- +anism +Engel, Belagiannis, and Dietmayer (2021) +to learn permutation-invariant dependency. However, +point-based methods are not efficient in the LiDAR +scenario since their sampling and grouping algorithms +are generally time-consuming. +2) Projection-based methods. These methods are +very efficient on LiDAR processing since they project +the raw point cloud onto a 2D image plane. Pre- +vious works project points through plane projec- +tion Tatarchenko, Park, Koltun, and Zhou (2018), +spherical projection B. Wu, Wan, Yue, and Keutzer +(2018); B. Wu, Zhou, Zhao, Yue, and Keutzer (2019) +or both Liong, Nguyen, Widjaja, Sharma, and Chong +(2020). Since the projection process inevitably causes +the loss of information, the recent studies adopt point- +based branches to obtain fine-grained features Alonso, +Riazuelo, Montesano, and Murillo (2020) or refine the +segmentation results Qiu, Yu, and Tao (2022). +3) Voxel-based methods. These approaches are most +widely adopted, since they can achieve impressive per- +formance while keep efficient. Generally, they first +conduct voxelization, and divide the raw points into +different voxel grids. After that, they conduct 3D +convolution in the input volumetric. The sparse con- +volution (SparseConv) Graham, Engelcke, and van der +Maaten (2018) is the core technique in voxel-based +methods. Since there are a large proportion of vox- +els are empty during the voxelization, which introduce +huge computational burden. The core of SparseConv +is only conduct operation in non-empty grids, which +will be saved in sparse Hash codes. Recent studies +adopt SparseConv to design diverse architectures. For +instance, Zhou et al. (2020) design the original grid +voxels to cylindrical ones and propose an asymmet- +rical network to learn anisotropy features. Recently, +R. Cheng, Razani, Taghavi, Li, and Liu (2021) design +a multi-branch component with several kernel sizes, +capturing features with different receptive field and +fusing them through an attention mechanism. +4) Hybrid-representation methods. Though voxel- +based methods achieve superior performance, there +is still missing geometric during the voxelization +process. Hence, there is a trend of exploiting multi- +representation fusion. These methods combine mul- +tiple representation inputs (i.e., points, projection +images, and voxels) and apply feature fusion among +different representations. Specifically, Tang et al. +(2020) designs point-voxel CNN operator, which com- +bines point-wise MLPs in each sparse convolution +block, and adopts neural architecture search (NAS) to +search a more powerful architecture. Xu et al. (2021) +utilizes the above three representations and proposes +a range-point-voxel fusion network. Recently, Yan +et al. (2022) applies cross-modal knowledge distilla- +tion, introducing prior information from texture and +color images during the training phrase. Nevertheless, +the hybrid-representation architecture makes them less +robust in out-of-domain corruptions +2.2 Robustness Benchmarks for Images +There are comprehensive robustness benchmarks +for 2D image processing, spanning different tasks +such as classification, semantic segmentation and +instance segmentation. For robust image classifica- +tion, ImageNet-C Hendrycks and Dietterich (2019) +is the pioneer for these field, which corrupts the +ImageNet Deng et al. (2009)’s test set with sim- +ulated corruptions such as motion blur, adverse +weather and noises. After that, ObjectNet Barbu et +al. (2019) build a benchmark with diverse corrup- +tions in rotation, background and viewpoint, and +ImageNetV2 Recht, Roelofs, Schmidt, and Shankar +(2019) follow ImageNet and re-collects a test set to +benchmark the robustness against natural distribu- +tion shift. Recently, ImageNet-A and ImageNet-R are +proposed by Hendrycks et al. (2021), which bench- +marks classifier’s robustness against natural adver- +sarial examples. Since ImageNet is initially pro- +posed for diverse tasks, there also exists preliminary +attempts to benchmark the robustness of model trained +on ImageNet to other downstream tasks, such as +semantic segmentation Kamann and Rother (2020), +instance segmentation Altindis et al. (2021) and object +detection Yamada and Otani (2022). In the field of +autonomous driving, there are also existing works +producing corruptions on Cityscapes Cordts et al. +(2016), e.g., investigating models’ robustness against +adverse weathers Porav, Musat, Bruls, and New- +man (2020); Sakaridis, Dai, and Van Gool (2018) or +other corruptions Michaelis et al. (2019). Recently, +ACDC Sakaridis, Dai, and Van Gool (2021) dataset + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +5 +Table 1 Categories and descriptions of corruptions in SemanticKITTI-C. We categorize common LiDAR corruptions into three domains: (1) +adverse weather conditions, (2) measurement noise and (3) cross-device discrepancy. +Corruption (C) +Intensity (I) +Description +(1) Fog Simulation +Light +Fog simulation with β = 0.005 +Moderate +Fog simulation with beta β = 0.06 +Heavy +Fog simulation with beta β = 0.2 +(1) Snowfall Simulation +Light +Snowfall simulation with snowfall rate of 0.5mm/h +Moderate +Snowfall simulation with snowfall rate of 1.5mm/h +Heavy +Snowfall simulation with snowfall rate of 2.5mm/h +(2) Global Outliers +Light +0.1% extra noisy points uniformly in the 3D space +Moderate +5% extra noisy points uniformly in the 3D space +Heavy +50% extra noisy points uniformly in the 3D space +(2) Local Distortion +Light +20% points with randomly jitter distortion σ = 0.05 +Moderate +20% points with randomly jitter distortion σ = 0.1 +Heavy +20% points with randomly jitter distortion σ = 0.2 +(3) Cross 32-beam Device +Dense +Reduce LiDAR beams to 32 +Sparse +Reduce LiDAR beams to 32, sample 1/2 points in each beam +(3) Cross 16-beam Device +Dense +Reduce LiDAR beams to 16 +Sparse +Reduce LiDAR beams to 16, sample 1/2 points in each beam +collects four common adverse conditions in self- +driving, i.e., fog, nighttime, rain, and snow, evaluating +the models’ robustness against these real-world cor- +ruptions. However, since the difference between 2D- +3D data and model architecture, there is still huge +demands of a comprehensive 3D robustness bench- +mark for semantic segmentation. +2.3 3D Robustness Benchmarks +In the field of autonomous driving, there lacks a +robustness benchmark for LiDAR semantic segmen- +tation to the best of our knowledge. Existing surveys +mostly focus on the point cloud classification task. +For instance, Xiao and Wachs (2021) and Z. Zhang, +Hua, and Yeung (2022) propose disturbance and +rotation invariant feature extraction, however, they +cannot achieve state-of-the-art performance on the +clean dataset. Other works boost models’ robust- +ness against adversarial corruptions by denoising and +upsampling Zhou et al. (2019), voting on subsampled +point clouds H. Liu, Jia, and Gong (2021), and apply- +ing local relative position Dong et al. (2020). There +are robustness benchmarks for point cloud classifi- +cation. Specifically, RobustPointSet Taghanaki et al. +(2020) and PointCloud-C Ren et al. (2022) evaluate +the robustness of point cloud classifiers under different +corruptions. However, these approaches test robust- +ness against corruptions purely on synthesis dataset, +i.e., ModelNet40, and thus the obtained experience +and conclusions are often unreliable in real-world +self-driving applications. +There are also investigations to improve the +robustness on LiDAR scenarios. PointASNL Yan et +al. (2020) proposes adaptive sampling, which adap- +tive shifts the outlier points onto objects’ surfaces, and +thus boosts the robustness against noisy point clouds. +In recent year, there are studies investigating perfor- +mance of object detector in different adverse weathers, +where they aim at mitigating the rarity of adverse +weather effects. Specifically, Hahner, Sakaridis, Dai, +and Van Gool (2021) and Hahner et al. (2022b) +independently propose fog and snowfall simulation, +greatly boosting the robustness of object detection +models on real-world adverse weathers. However, +there is no real-world or simulated adverse weather +data set for LiDAR semantic segmentation at present. +Moreover, there exists preliminary attempts to investi- +gate the robustness issue of the fusion methods for 3D +object detection Bai et al. (2022); Y. Li et al. (2022); +Yu et al. (2022). Concretely, TransFusion Bai et al. +(2022) evaluates the robustness of different fusion +strategies under several scenarios, e.g., daytime and +nighttime, DeepFusion Y. Li et al. (2022) test the +model robustness by adding noise to LiDAR reflec- +tions and camera pixels and Yu et al. (2022) proposes +a robust benchmark for LiDAR-camera fusion, which +analyzes seven cases of robustness scenarios. + +Springer Nature 2021 LATEX template +6 +Robust LiDAR segmentation +No Fog +Moderate Fog +Heavy Fog +Fig. 2 Corruption of fog simulation. We demonstrate the raw +LiDAR point cloud in the first row. The foggy point clouds with +β = 0.06 and β = 0.2 are shown in the last two rows. The point +cloud is color coded by the height (z value). The best viewed on a +screen and zoomed in. +By contrast, we rigorously investigate the LiDAR +system and identify three categories, in a total of 16 +LiDAR corruptions for semantic segmentation, and +develop a toolkit that transforms the existing dataset +into a robustness benchmark. We hope our study can +boost future research to benchmark the robustness, +and give researchers more insights about designing a +robust semantic segmentation model. +3 Corruptions Taxonomy +Real-world LiDAR scans can suffer from a wide +range of corruptions, based on which we provide +a taxonomy of the corruptions. In this paper, we +categorize common LiDAR corruptions into three +domains, i.e., adverse weather conditions, measure- +ment noise and cross-device discrepancy, in which +we produce total six corruptions with 16 severity lev- +els. By applying these six types of corruptions to +SemanticKITTI Behley et al. (2019), we generate a +corrupted dataset, i.e., SemanticKITTI-C, which is +summarized in Tab. 1. In the remaining sections, we +will introduce each corruption. +A point cloud P is a set of points {pj}N +j=1, where +N is the number of points and pj ∈ R3 includes +the XYZ coordinates of the point j. A corruption +No Snowfall +Moderate Snowfall +Heavy Snowfall +Fig. 3 Corruption of snowfall simulation. We demonstrate the +raw LiDAR point cloud in the first row. The snowfall point clouds +with snowfall rates 1mm/h and 2.5mm/h are illustrated in the last +two rows. The point cloud is color coded by the height (z value). +The best viewed on a screen and zoomed in. +operation is defined as a set-to-set function: +F : RN×(3+D) �→ RN ′×(3+D), +(1) +which maps the clean point cloud P = {pj}N +j=1 and +its D-dimensional features (if exist) to corrupted ones +(e.g., P′ = {pj}N ′ +j=1). For LiDAR point cloud, each pj +is associated with an intensity value ij ∈ R, indicating +the return strength of a laser beam. In this paper, the +intensity is utilized to generate corrupted data, but we +do not purely investigate the corruption of intensity. +3.1 Adverse Weather +In this section, we analyze two common weather con- +ditions, namely fog and snowfall. For fog simulation, +we follow Hahner et al. (2021) to add fog to clean- +weather point clouds by disturbing points’ positions +and intensities according to physically valid rules. +Specifically, for a point p ∈ R3 captured in the clean +weather, we first calculate its attenuated response ihard +in fog: +ihard = i × exp (−2α × ∥(x, y, z)∥) , +(2) +where (x, y, z) is p’s coordinate in the ego frame +and i is its measured intensity, α is the attenuation +coefficient in foggy weather, ∥(x, y, z)∥ denotes the + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +7 +distance between the point p and the origin. Following +Hahner et al. (2021), we uniformly sample α from [0, +0.005, 0.01, 0.02, 0.03, 0.06] when applying fog simu- +lation to each sample. After that, we use the simulation +terms in Hahner et al. (2021) to compute the max- +imum fog response isoft and its location (xs, ys, zs), +which lies in the line connecting the sensor and p. Note +that the magnitude of isoft is controlled by a backscat- +tering coefficient β, which is manually set during the +simulation. As shown in Tab. 1, we choose β from +[0.005, 0.06, 0.2] to conduct fog simulation with dif- +ferent levels. Finally, the updated point position and +its intensity are given by: +i = +� +isoft if isoft > ihard, +ihard +otherwise. +(3) +(x, y, z) = +� +(xs, ys, zs) if isoft > ihard, +(x, y, z) otherwise. +(4) +In other words, if the fog is thick enough to over- +shadow the solid object point p (isoft > ihard), we use +the fog response (the intensity and position) to replace +the original one. Otherwise, we keep the position of +the original response with an attenuated intensity. +The overall idea of this snowfall simulation is sim- +ilar to that of the fog simulation. But unlike fog that +homogeneously spreads in the 3D space, snowflakes +are treated as opaque particles and are discretely dis- +tributed in space without intersecting with each other. +For snowfall simulation, we follow +Hahner et al. +(2022b) to sample snow particles for each LiDAR line +and use them to modify the return for each LiDAR +beam accordingly. The sampling function samples +snow particles according to a given snowfall rate +(mm/h), which controls the number of particles in a +certain range. As shown in Tab. 1, we separately set +the snowfall rate to 0.5/1.5/2.5 to simulate light/mod- +erate/heavy snowfall. +The above two weathers have different characteris- +tics. For instance, there are large areas of noisy points +distributed around the sensor in foggy weather, and +makes objects sparser due to the occlusion, especially +for the remote objects. As illustrated in Fig. 2, the +number of these noisy points grows as the intensity of +fog increases. Moreover, the noise introduced by fog is +not uniformly distributed around the sensor. The pres- +ence of noise depends on whether there is any object +in the line of sight below a certain range from the +sensor. Generally, there will be few spurious returns +from the respective pulses if a solid object exists at +a moderate range. Inversely, if there is no object in a +certain range, there are a lot of spurious returns that +are caused by fog. As for the snowfall, there are two +explicit characteristics. On the one hand, the snow par- +ticles are explicitly modeled as opaque spheres, whose +sizes are controlled by the snowfall rate. As shown +in Fig. 3, compared with foggy LiDAR where the +noisy points are almost around the sensor, the noisy +points in snowfall conditions are distributed more uni- +formly. Also, the snowfall rate does not greatly affect +the number of noisy points but the size of snowy par- +ticles. On the other hand, wetness on the ground will +exist in snowfall, where the emerging thin water layer +increases the specular component of reflection by the +ground surface. To sum up, these two corruptions +impact the models through global noisy points, mak- +ing remote points sparser. Nevertheless, they generally +have different patterns. +3.2 Measurement Noise +Besides adverse weather, noises may also appear when +corruption occurs during the data transmission or the +sensors fail to capture information properly (e.g., over- +heated). We model such data disturbance using two +types of random noises, as shown in Fig. 4. Note +that we only consider point coordinates during such +corrupting operations. +Global outliers. We randomly sample noises in a unit +sphere and then merge them into a clean point cloud +with proper rescaling. Such noises span the whole +scene globally and are not conditional on the geome- +try of the clean point cloud. Formally, given the clean +point cloud P ∈ RN×3, the corrupted point cloud +P′ ∈ RN ′×3 is obtained by: +P′ = P ∪ Pnoise, +(5) +where Pnoise ∈ RN g×3 denotes sampled noises and +N ′ = N g + N. We control the noise intensity by +selecting the proportion of noises N g +N from [0.1%, 5%, +50%] as shown in Tab. 1. +Local distortion. We randomly select some points +within a scene and add Gaussian noises to their coor- +dinates. Unlike global noises, which add additional +points to a scene, local noises do not change the +number of points. Compared to global noises, local +noises jitter around a local neighborhood of the orig- +inal points, mimicking the noisy disturbance during +the data collection. Formally, the local distortion point + +Springer Nature 2021 LATEX template +8 +Robust LiDAR segmentation +No Noise +Global Outliers +Local Distortion +noisy data +clean data +zoom-out +zoom-out +zoom-in +noisy data +zoom-in +clean data +Fig. 4 Noisy LiDAR point clouds. We demonstrate the raw +LiDAR point cloud in the first row. The noisy point clouds with +global outliers and local distortion are shown in the last two rows. +The point cloud is color coded by the height (z value). The best +viewed on a screen and zoomed in. +cloud P′ ∈ RN×3 is given by: +Psub = RandomSample(P, N l), +P′ = (Psub + O) ∪ (P \ Psub), +(6) +where RandomSample(·) randomly sample N l points +from the clean point cloud P. O ∈ RN l×3 denotes the +random offsets sampled from a Gaussian distribution +N(0, σ2). + and \ are element-wise addition and set +exclusion, respectively. As shown in Tab. 1, we choose +σ from [0.05, 0.1, 0.2] to control the jittering range in +three different levels. The proportion of noises is 20% +in SemanticKITTI-C. +3.3 Cross-Device Discrepancy +An ideal segmentation algorithm is supposed to be +robust across different devices with various specifica- +tions. While multiple factors (e.g., beam number and +scanning speed of the LiDAR sensors) cause cross- +device domain shifts, we focus on the beam number +in this paper. To ensure high-quality data annotation, +most large-scale datasets Geiger, Lenz, and Urtasun +(2012); Sun et al. (2020) are collected using high- +resolution LiDARs. However, due to prohibitive costs, +most practical vehicles are only shipped with low- +beam sensors. For instance, KITTI Geiger et al. (2012) +64-beam +32-beam +16-beam +Fig. 5 Cross-device LiDAR point clouds. We demonstrate the 64- +beam LiDAR point cloud in the first row. The second and third rows +illustrate the 32-beam and 16-beam LiDAR data. The point cloud is +color coded by the height (z value). The best viewed on a screen and +zoomed in. +collects data through 64-beam LiDAR and each beam +contains 1863 points in average, while those param- +eters in NuScenes Caesar et al. (2020) are 32-beam +and 1084 points. This suggests that an ideal seg- +mentation model should be able to robust to differ- +ent data distributions generated by different sensors. +Unlike other factors introduced in previous subsec- +tions, the beam-induced domain gap is directly caused +by the cross-device discrepancy instead of the collect- +ing environment, making it also very important in our +robustness analysis. +To include the beam-based cross-device discrep- +ancy in our benchmark dataset, we downsample the +high-beam data (i.e., 64-beam) to low-beam data +(e.g.., 16-, 32-beam) using beam-level downsam- +pling as shown in Fig. 5. One necessary information +needed for beam-level downsampling is the beam +label for each point, which is usually unknown for +most datasets. To this end, we first assign a beam +label to each point according to its zenith value in the +spherical coordinate, which can be obtained via the +following conversion: +θ = arctan +z +� +x2 + y2 , φ = arcsin +y +� +x2 + y2 , +(7) + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +9 +Table 2 LiDAR semantic segmentation approaches on our benchmark. +Mainstream +Method +Main representation +Extra representation +Reference +Projection-based +SalsaNext Cortinhal, Tzelepis, and Aksoy (2020) +Range image +- +ArXiv 2020 +PolarNet Y. Zhang et al. (2020) +BEV image +- +CVPR 2020 +CENet H.-X. Cheng, Han, and Xiao (2022) +Range image +- +ICME 2022 +GFNet Qiu et al. (2022) +Range and BEV images +Point cloud +TMLR 2022 +Point-based +KPConv Thomas et al. (2019) +Point cloud +- +ICCV 2019 +RandLANet Hu et al. (2020) +Point cloud +- +CVPR 2021 +Point Transformer Zhao et al. (2021) +Point cloud +- +ICCV 2021 +Voxel-based +MinkowskiNet Choy, Gwak, and Savarese (2019) +Grid voxel +- +CVPR 2019 +SPVCNN Tang et al. (2020) +Grid voxel +Point cloud +ECCV 2020 +Cylinder3D Zhou et al. (2020) +Cylinder voxel +Point cloud +CVPR 2021 +2DPASS Yan et al. (2022) +Grid voxel +Point cloud +ECCV 2022 +where (x, y, z) is the Cartesian coordinate of the point +and the θ and φ are zenith and azimuth angles. Fol- +lowing Wei et al. (2022), we obtain beam labels by +applying K-Means clustering on the zenith angles, +where the number of clusters is set as the actual beam +number of the high-beam point cloud. Compared to +assigning beam labels by putting zenith angles into +evenly distributed bins, the clustering-based technique +does not require a pre-define zenith range and thus +is more robust across different datasets. For a high- +beam point cloud with the beam labels, we can easily +downsample it into data with any lower beam num- +ber. In practice, we downsample point clouds with the +beam numbers of 32 and 16. To simulate the diverse +spinning speeds of the LiDAR devices, we evenly +downsample points in each beam according to their +azimuth angles. By combining the above simulation, +we have four corrupted data generated in Tab. 1. +4 Candidate Methods +We benchmark 11 existing methods for LiDAR +semantic segmentation, as shown in Tab. 2. Though +we treat hybrid-representation methods as an indepen- +dent mainstream in Sec. 2.1, current voxel-based and +projection-based methods widely incorporate addi- +tional representations for auxiliary learning. There- +fore, we categorize them only according to their main +input representation, and the extra-representation will +be illustrated if existed. +4.1 Projection-based Methods +In this paper, we choose SalsaNext Cortinhal et al. +(2020), PolarNet Y. Zhang et al. (2020), CENet H.- +X. Cheng et al. (2022) and GFNet Qiu et al. (2022) as +the typical approaches of the projection-base method. +These models project a LiDAR point cloud into 2D +images and apply a 2D convolutional neural network +for semantic segmentation. Among the above meth- +ods, SalsaNext Cortinhal et al. (2020) and CENet H.- +X. Cheng et al. (2022) conduct sphere projection +to gain range views (RV), PolarNet Y. Zhang et al. +(2020) adopts polar projection to obtain bird’s-eye- +view (BEV) under a polar coordinate system, and +GFNet Qiu et al. (2022) uses the both. +Sphere projection for range-view. We denote N +the number of points in the LiDAR point cloud and +(H, W) are the height and width of the projected +image. As shown in Fig. 6, the spherical projection +maps each point to an image coordinate via +�ur +vr +� += +� +1 +2[1 − arctan(y, x)π−1]W +[1 − arsin(zr−1 + fovup)fov−1]H +� +, +(8) +where pi = (x, y, z) and (ur, vr) are the i-th point +and its coordinates on the range image plane. r is +the range of each point +� +x2 + y2 + z2 and fov = +fovup + fovdown is the vertical field-of-view of the +sensor. Finally, the LiDAR point cloud is converted to +a range image with the shape of (H × W × C). The +channel C is generally 5, including x, y, z, intensity +and range of the point. +Polar projection for bird’s-eye-view. Existing meth- +ods also project the LiDAR point cloud to the bird’s- +eye-view (BEV) through a top-down orthogonal pro- +jection. Considering the imbalanced spatial distribu- +tion in LiDAR data, polar projection first transforms +the BEV from the Cartesian system into a polar coor- +dinate system through +�up +vp +� += +�� +x2 + y2 + z2cos(arctan(y, x)) +� +x2 + y2 + z2sin(arctan(y, x)) +� +, (9) +where (up, vp) is the coordinate transformation from +the Cartesian system to the polar system. After that, + +Springer Nature 2021 LATEX template +10 +Robust LiDAR segmentation +projected +raw point cloud +raw point cloud +semantic segmentation +projected +semantic segmentation +spherical projection +re-projection +U-Net architecture +(x,y,z) +(ur,vr) +skip connection +Fig. 6 Projection-based methods through spherical projection. +This kind of methods first conduct spherical projection maps each +point to an image coordinate, and then adopt 2D convolution to +construct a U-Net-like architecture. Finally, they re-project the +prediction on the image plane onto the raw point cloud. +they discretize (up, vp) to [0, H − 1] and [0, W − 1] +and obtain a BEV image. +Architectures. Since the LiDAR point cloud is +already mapped onto an image plane, typical 2D +semantic segmentation networks can be directly +adopted. Specifically, U-Nets Ronneberger, Fischer, +and Brox (2015) with specific modifications are +applied in previous methods. 1) Approaches with +range images do not conduct pooling in height dimen- +sion due to the large width-height ratio of the input, +as shown in Fig. 6. 2) SalsaNext utilizes an additional +pixel-shuffle layer in the last encoder, and CENet con- +ducts multiscale supervision in encoder and decoder +layers. 3) PolarNet applies a hybrid-representation +manner, i.e., designing a PointHead (will be described +in Sec. 5.4) in the initial stage, which aggregates the +features of original points into each BEV pixel, and +finally conducts semantic segmentation through the +2D U-Net. 4) GFNet Qiu et al. (2022) has a two- +branch architecture, where two U-Nets independently +encode the features of range-view (RV) and bird’s- +eye-view (BEV). There are several Geometric Flow +(GF) modules between their decoder layers with dif- +ferent scales, which update each other’s features by +fusing the features of both branches. Finally, it utilizes +a hybrid-representation manner, aggregating the fea- +tures of two branches in their last layers and feeding +the fused feature into KPConv Thomas et al. (2019) to +gain point-wise predictions. +Configurations. In our experiments, we adopt the +official architectures of projection-based methods +(i.e., SalsaNext1, CENet2, PolarNet3 and GFNet4), +where we directly use their pre-trained checkpoint +except for CENet. Note that CENet is trained with a +1https://github.com/TiagoCortinhal/SalsaNext +2https://github.com/huixiancheng/CENet +3https://github.com/edwardzhou130/PolarSeg +4https://github.com/haibo-qiu/GFNet +points in (x,y) +local points +fused feature +Local +Aggregation +1st scale +2nd scale +3rd scale +... +(a) +(b) +Fig. 7 Point-based methods. (a) Point-based approaches sample +the target points (in red color) in the original point cloud, and +aggregate local features through local aggregation. (b) Through hier- +archical architecture, the receptive field of the point-based method +increase aggressively. +multi-stage strategy, which trains with 64 × 512 range +image for the initial stage, and fine-tunes the pre- +trained model aggressively on 64×1024 and 64×2048 +ones. Since the official codes only provide the check- +point on 64 × 512 range images, we fine-tune the +checkpoint on larger ones through provided configura- +tions. In contrast, SalsaNext directly trains their model +on 64 × 2048 range images. As for the PolarNet, it +first crops points of the polar coordinate system in the +range from [3, −π, -3] to [50, π, 1.5], and then dis- +cretize points into a [480,360] BEV partition. GFNet +combines RV and BEV representation, utilizing both +64 × 2048 range images and 480 × 360 BEV plane. +4.2 Point-based Methods +Point-based approaches aim at extracting features on +raw point clouds directly, as shown in Fig. 7. In this +paper, KPConv Thomas et al. (2019), RandLA-Net Hu +et al. (2020) and Point Transformer Zhao et al. (2021) +are selected as our candidate methods. Specifically, +these methods first apply sampling approaches to +select target points from the original point clouds, and +then conduct local aggregation on each target point +and mine local geometrics, as Fig. 7(a) shows. After +constructing a hierarchical architecture in Fig. 7(b), +point-based methods gain the global semantic infor- +mation of the input point cloud. +General formulation of local aggregation. Let pi +and fi denote the coordinate and feature of the i-th +point. In general, for each pi, a local aggregation func- +tion first transforms its neighbor pj with feature fj into +a new feature by a transformation function T (...), and +then aggregates all transformed neighborhood features +to generate an updated feature of ˆpi via an aggregation + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +11 +function A(·): +ˆpi = A({T (pi, fi, pj, fj)} ∀j ∈ N(pi)). +(10) +In practice, N(pi) represents the neighborhood index +of point pi. According to the category to which +the transformation function T (...) belongs, previous +local aggregation approaches can be roughly cate- +gorized into four classes: 1) Point-wise MLP based, +2) Adaptive weight based, 3) Pseudo grid based and +4) Transformer based approaches. The typical one of +the first class is PointNet++ Qi et al. (2017), where +T and A are MLP((pj − pi) ⊕ fj) and max pooling +respectively, in which ⊕ is concatenation operation. +However, directly learn the 3D shapes through sim- +ple point-wise MLP and pooling cannot work well +in the LiDAR scenario, i.e., it only achieves 20% +mIoU on SemanticKITTI in previous studies Behley +et al. (2019). Therefore, we did not adopt this kind of +methods in our paper. +Adaptive weight based methods. The adaptive +weight based methods design diverse convolution fil- +ters over arbitrary relative positions, and hence com- +putes weights on all neighbor points. RandLA-Net Hu +et al. (2020) is a typical one in adaptive weight based +methods. Concretely, its transformation function T +can be represented as +MLP(pi ⊕ pj ⊕ (pi − pj) ⊕ E(pi, pj))) ⊕ fj, (11) +where E(·) calculates the Euclidean distance between +the neighboring and center points. After that, it aggre- +gates the neighboring features through attention mech- +anism Vaswani et al. (2017), which first calculates +an attention weigh according to the feature, and then +conducts weighted average. +Pseudo grid based methods. KPConv Thomas et al. +(2019) is a representative pseudo grid based method, +which generates pseudo features on several sampled +regular grid points, and thus regular convolution meth- +ods can play a normal role. Specifically, it samples +equally distributed spherical grid points in the 3D +space, in which the pseudo features f p +k on the k-th grid +point can be calculated as +f p +k = +� +j∈N (pi) +max(0, 1 − E(pi, pk) +σ +)fj, +(12) +where each grid point pk have strict mapping with the +relative position to center point. σ is a hyperparame- +ter. After that, the transformation function T in pseudo +grid based methods can be formulated as +T (pi, fj) = wk ⊙ f p +k, +(13) +where wk ∈ Rd×1 is a parametrized weight in convo- +lution operator and defined on each grid point. Finally, +after applying max pooling as aggregation function +A, it updates the feature of each target point through +aggregating features in local neighbors. +Transformer based methods. Besides analyzing tra- +ditional local aggregation based approaches, we also +adopt recent transformer based method (i.e., Point +Transformer Zhao et al. (2021)) in this paper. The +point transformer layer is based on vector self- +attention, which uses the subtraction relation and there +is a position encoding δ in both the attention vec- +tor γ and the transformed features α. Specifically, in +each local group (i.e., ∀j ∈ N(pi)), the transforma- +tion function T (pi, pj, fi, fj) in a transformer based +method can be formulated as +ρ(γ(ϕ(fi) − ψ(fj) + δ)) ⊙ (α(fj) + δ), +(14) +where ϕ, ψ, γ, α are independent MLPs. δ += +MLP(pi−pj) is a positional encoding in self-attention, +allowing the operator to adapt to local structure. After +updating the neighboring features, Point Transformer +utilizes a summation function as aggregation function +A to fuse features. +Architectures. All above three methods follow the +widely-used UNet-like encoder-decoder architecture +with skip connections. The LiDAR point cloud is first +fed to a shared MLP layer to extract per-point fea- +tures. Encoder and decoder layers are then used to +learn features for each point. Finally, fully-connected +layers are used to predict the semantic label of +each point. KPConv and RandLA-Net utilize stacked +two corresponding local aggregation in each encoder +layer, while Point Transformer using the combina- +tion of point-wise MLP with point transformer layer. +In decoder layers, all method interpolate the sampled +points and update features through point-wise MLPs. +Moreover, RandLA-Net uses random sampling in each +local aggregation, while other two utilizing uniformly +sample points. For KPConv and RandLA-Net, we +adopt their official architectures on SemanticKITTI +dataset (four encoders and decoders). As for the Point +Transformer, since it is only designed for indoor +semantic segmentation, we adopt original architecture +with five encoders and decoders. + +Springer Nature 2021 LATEX template +12 +Robust LiDAR segmentation +Configurations. During the training, both KPConv5 +and RandLA-Net6 follow their official configurations. +To accelerate the training phrase, they first conduct +grid sampling with grid size 0.06m to gain a small +sub-cloud. Moreover, they respectively crop patches +with a 4m radius and 50,000 points in each training +iteration. During the inference, they inference through +small patches util each of the points have been inferred +three times. Since there are not published codes on +SemanticKITTI for Point Transformer7, we utilize the +same configurations as Tang et al. (2020) during the +training and inference. +4.3 Voxel-based Methods +Since voxel-based methods are the most popular +mainstream for LiDAR semantic segmentation now, +we select four methods (i.e., MinkowskiNet Choy +et al. (2019), SPVCNN Tang et al. (2020), Cylin- +der3D Zhou et al. (2020) and 2DPASS Yan et al. +(2022)) in this paper. +Grid partition. Voxel-based methods exploit vox- +elization and transform the LiDAR point cloud into 3D +voxels, such that the 3D convolutions can be applied. +Specifically, they shift all the points to the local coor- +dinate system with the geometric center as the origin. +Then, all the points are normalized into a unit sphere +and scaled to the range of [0, 1], where the normal- +ized coordinates are denoted as ˆP = {(ˆxi, ˆyi, ˆzi)}N +i . +After that, they transform the normalized point cloud +to a voxel representation with voxel size vs (f ∗ +m is +voxelized feature representation): +p∗ +i = (x∗ +i , y∗ +i , z∗ +i ) = (⌊ˆxi/vs⌋, ⌊ˆyi/vs⌋, ⌊ˆzi/vs⌋), +f ∗ +m = +1 +Nm +N +� +i=1 +I[x∗ +i = ˆxm, y∗ +i = ˆym, z∗ +i = ˆzm] · pi, +(15) +where ⌊·⌋ is the floor function, and I(·) is a binary indi- +cator of whether p∗ +i belongs to the m-th voxel grid or +not. Nm is the number of points in the m-th voxel, and +the original point coordinates are averaged as the fea- +tures of each voxel. After the operations in Eqn. (15), +only the non-empty voxels are preserved (Nm > 0) in +a hash table. The, the convolution operation only con- +ducts on the non-empty voxels, thus maintaining the +computational efficiency. +5https://github.com/HuguesTHOMAS/KPConv-PyTorch +6https://github.com/QingyongHu/RandLA-Net +7https://github.com/POSTECH-CVLab/point-transformer +Cylindrical partition. Recent study Zhou et al. +(2020) proposes cylinder partition for voxelization, +which makes a higher non-empty proportion and more +balanced point distribution compared with grid parti- +tion, especially for farther-away regions. In practice, +it first transforms the Cartesian system into a polar +coordinate system through Eqn. (9), and then conducts +voxelization as Eqn. (15). +Architectures. Both MinkowskiNet and SPVCNN8 +utilize the same U-Net architecture, where the dif- +ference is that there is a parallel point-wise MLP +branch in the latter. Cylinder3D9 proposes asymmet- +rical 3D convolution networks, in which it constructs +several asymmetrical blocks (e.g., exploiting 3×3×1, +3 × 1 × 3 and 1 × 3 × 3 kernels in parallel) as unit +components. 2DPASS10 uses a similar encoder archi- +tecture as SPVCNN, but it discards the decoder part +and predicts the results through multiscale concate- +nation. Moreover, Cylinder3D and 2DPASS exploit +additional a PointHead (will be introduced in Sec. 5.4) +to aggregate point-wise features into each voxel. +Configurations. Both Cylinder3D and 2DPASS are +tested with their released checkpoints. As for the +MinkowskiNet and SPVCNN, we re-trained their offi- +cial architectures with batch size 8 for epoch 64, and +gain higher results. All approaches are tested with test- +time augmentation (TTA), i.e., rotating the point cloud +with 12 views and averaging the predictions. +5 Benchmarking and Analysis +In this section, we benchmark the aforementioned 11 +approaches with our diverse set of LiDAR corruptions. +We first introduce the experiment setting and eval- +uation metrics of our benchmark in Sec. 5.1. After +that, the benchmark results are shown in Sec. 5.2 to +5.5 with comprehensive analysis. We demonstrate our +benchmark results spanning different representation, +architecture, corruption intensity and data augmenta- +tion. As results, we summarize 12 observations in +total. Finally, in Sec. 5.6, we introduce RLSeg, a +robust architecture based on the above observations, +which effectively boosts the robustness of LiDAR +semantic segmentation. +8https://github.com/mit-han-lab/spvnas +9https://github.com/xinge008/Cylinder3D +10https://github.com/yanx27/2DPASS + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +13 +5.1 Experiment Setting +Dataset. SemanticKITTI is currently the most widely +used LiDAR semantic segmentation dataset, which +consists of 43,552 densely annotated LiDAR scans +belonging to 21 sequences. These scans are annotated +with a total of 19 valid classes, and each scan spans +up to 160 × 160 × 20 meters with more than ∼ 105 +points. Initially, the sequence 00 to 07, 09 to 10 are the +training set, 11 to 21 are the test set, and 08 is the val- +idation set. Since the annotations of 11 to 21 are not +available offline, we train all approaches on training +set and evaluate them on sequence 08. +Annotation modification. Since the corrupted point +clouds will be sparser or there are new noisy points +existed, we slightly modify the original annotations. +Specifically, for the corruption data in fog and snow- +fall simulations, we utilize the noisy data to query +the nearest neighbor of the clean data within 0.02m +to annotate labels, labeling points as ‘ignore’ if there +is no neighbor existed. In the noisy corruption (i.e., +local and global), we directly annotate noisy points +as ‘ignore’. In the cross-device scenario, since all the +points are sampled from the original LiDAR, there is +no demand for modification. Note that the ‘ignore’ +class is not considered in the evaluation. +Evaluation metric. To intuitively demonstrate the +robustness of candidate methods, we use the perfor- +mance on each corruption and the relative perfor- +mance degradation compared to the clean data on our +benchmark datasets as our evaluation metrics. Specif- +ically, we adopt mIoU(%) (i.e., averaged Intersection +over Union on each class) as our metric and the score +on the clean dataset is denoted as S. As demonstrated +in Tab. 1, we benchmark existing methods with six +categories of corruptions (C), spanning 16 different +intensities (I). The performance toward certain corrup- +tion c ∈ C can be calculated by averaging results on +each intensity: +Sc = +� +i∈I(c) +Sc +i /N(c), +(16) +where I(c) and N(c) are total intensities and the num- +ber of intensity in the corruption c. Sc +i denotes the +mIoU under corruption c and intensity i and Sc are +the averaged mIoU of all intensities under the corrup- +tion c. The relative mean robustness performance of +the model is defined as Rc = Sc/S. The higher R +means the model is more robust to inferior LiDAR +conditions. Moreover, we define a robustness mIoU +clean data +jitter data +foggy data +Fig. 8 Illustration of observation-1. We show visualization results +to better explain why the projection-based methods are vulnerable +to fog simulation but robust to local deviation. +(RmIoU) and averaged relative performance (mR) +through averaging the results on different corruption: +RmIoU = +� +c∈C +Sc/6, mR = RmIoU/S. +(17) +5.2 Main Results +Benchmark results are reported in Tab. 3, in which +projection-based, point-based and voxel-based meth- +ods are demonstrated in upper, median and lower +parts, respectively. Our proposed solution will be +introduced in Sec. 5.6. According to the table, we have +the following discovery: +Observation-1: Projection-based methods are most +vulnerable to common corruptions, especially to foggy +simulation. However, they are greatly robust to local +distortion corruption. +As shown in the table, existing projection-based +methods achieve around 75% metric of mR on +SemanticKITTI-C, which is much lower than those of +point-based and voxel-based methods. Furthermore, +they only achieve around 50% original performance +in the corruption of fog simulation. In contrast, these +methods are extremely robust to local noise, especially +for the pure range image based method (i.e., 96.1% +and 94.0% R for SalsaNext Cortinhal et al. (2020) +and CENet H.-X. Cheng et al. (2022), respectively). +The reason of the above observation is that the local +corruption only slightly affects the range image and +the foggy one make range images messy, as shown in +Fig. 8. PolarNet Y. Zhang et al. (2020) and GFNet Qiu +et al. (2022) are not much robust to local corrup- +tion, since there are BEV projections in their models. +Nevertheless, they still respectively keep 83.7% and + +Springer Nature 2021 LATEX template +14 +Robust LiDAR segmentation +Table 3 Benchmarking the robustness of state-of-the-art methods in all 16 scenarios (6 classes) on SemanticKITTI-C. R denotes the relative mean robustness performance of the model. The +higher R means the model is more robust to inferior LiDAR conditions. +Clean +Robustness +Fog +Snowfall +Global Outliers +Local Distortion +32-beam +16-beam +Method +mIoU +RmIoU +mR +mIoU +R +mIoU +R +mIoU +R +mIoU +R +mIoU +R +mIoU +R +Projection +SalsaNext +55.8 +42.7 +76.5 +27.3 +48.9 +43.6 +78.1 +49.5 +88.7 +53.6 +96.1 +51.1 +91.6 +31.0 +55.6 +PolarNet +58.2 +43.3 +74.5 +31.7 +54.5 +47.4 +81.5 +52.4 +90.0 +48.7 +83.7 +46.3 +79.5 +33.5 +57.6 +CENet +62.3 +47.7 +76.6 +31.5 +50.5 +51.3 +82.4 +57.2 +91.7 +58.6 +94.0 +54.5 +87.5 +33.4 +53.7 +GFNet +63.0 +46.4 +73.6 +31.1 +49.4 +41.8 +66.3 +61.4 +97.5 +56.0 +88.9 +52.5 +83.4 +35.6 +56.5 +Point +KPConv +63.5 +51.6 +81.3 +59.6 +93.9 +54.8 +86.4 +61.9 +97.4 +31.8 +50.1 +58.3 +91.7 +43.4 +68.3 +RandLA-Net +59.2 +47.6 +80.4 +56.4 +95.4 +50.0 +84.4 +57.8 +97.8 +26.4 +44.7 +53.3 +90.1 +41.7 +70.5 +Point Trans. +63.3 +40.5 +64.0 +45.5 +71.9 +44.2 +69.9 +38.8 +61.4 +39.3 +62.2 +47.2 +74.6 +27.9 +44.2 +Voxel +MinkowskiNet +66.3 +53.6 +80.9 +56.3 +84.9 +50.4 +76.1 +65.3 +98.5 +37.0 +55.9 +62.2 +93.9 +50.4 +76.0 +SPVCNN +67.4 +53.2 +78.9 +53.7 +79.7 +50.5 +75.0 +65.8 +97.7 +39.6 +58.8 +61.7 +91.5 +47.7 +70.8 +Cylinder3D +66.9 +46.5 +69.5 +44.2 +66.1 +45.7 +68.3 +63.3 +94.6 +39.7 +59.4 +51.2 +76.5 +34.7 +51.9 +2DPASS +70.1 +51.1 +72.9 +40.4 +57.6 +53.6 +76.5 +69.8 +99.6 +43.9 +62.7 +61.3 +87.4 +37.7 +53.7 +RLSeg (ours) +73.5 +62.5 +85.0 +57.6 +78.4 +66.2 +90.1 +73.4 +99.9 +71.9 +97.8 +62.3 +84.7 +43.6 +59.3 + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +15 +0 +10 +20 +30 +40 +50 +60 +70 +0 +0.005 0.01 0.02 0.03 0.06 +0.1 +0.12 0.15 +0.2 +SalsaNext +KPConv +MinkowskiNet +mIoU +Fig. 9 Detailed results in fog simulation. Performance of three +typical approaches in different fog simulation intensities, where the +x-axis denotes different β values. +88.9% performance, which is much higher than those +of point and voxel based methods. +Observation-2: +Traditional +point-based +methods +(RandLA-Net and KPConv) are more robust to com- +mon corruptions, compared with projection-based +methods. Specifically, they are much robust to adverse +weathers, but less robust to local distortion. +The table illustrates RandLA-Net Hu et al. (2020) +and KPConv Thomas et al. (2019) respectively +achieve 81.3% and 80.4% in the metric of mR. Espe- +cially, they gain highest 93.9% and 95.4% R in fog +simulation, where the best projection and voxel based +methods only achieve 54.5% and 79.7%, respectively. +Inversely, they only gain 50.1% and 44.7% perfor- +mance in local noise, which is lower than common +performance of other two mainstreams. The reason is +that the local distortion protects the local geometric +and thus makes the local aggregation failed. +Observation-3: Transformer-based local aggregation +greatly hampers the robustness, especially for global +outliers. +Though transformer-based architecture improve +the performance on the clean data, it greatly affects +the robustness against diverse corruptions. Concretely, +Point Transformer Zhao et al. (2021) gains the low- +est result in almost every corruption scenarios. For +instance, in the global outliers, most of the approaches +can keep above 90% performance, but it only achieves +61.5% in the metric of R. +Observation-4: Pure voxel-based method shows most +superior robustness cross all corruptions, especially +for cross-devices scenario. Recent state-of-the-art +voxel-based methods loss their robustness against cor- +ruption due to their hybrid-representation architec- +tures. +The results illustrate that MinkowskiNet Choy et +al. (2019) enjoy most superior robustness with pure +0 +10 +20 +30 +40 +50 +60 +70 +0 +0.5 +1.0 +1.5 +2.0 +2.5 +SalsaNext +KPConv +MinkowskiNet +mIoU +Fig. 10 Detailed results in snow simulation. Performance of three +typical approaches in different snow simulation intensities, where +the x-axis denotes different snowfall rates (mm/h). +voxel architecture. Specifically, it achieves 76.0% per- +formance preserve in 16-beam cross-device scenario, +surpassing those of projection and point based meth- +ods over 20% and 10%, respectively. SPVCNN Tang +et al. (2020) introduces point-wise MLP in par- +allel with voxel architecture, nevertheless, it loses +the robustness especially in fog simulation and 16- +beam cross-device corruptions. Recent state-of-the-art +Cylinder3D Zhu et al. (2021) and 2DPASS Yan et al. +(2022) have poor generalization ability since they use +extra representation. More analysis for this design will +be illustrated in Sec. 5.4. +5.3 Robustness in Specific Corruption +In this section, we demonstrate and analyze the result +of each specific corruption. To facilitate the experi- +ment, we only select the most typical method in each +mainstream, i.e., SalsaNext, KPConv and Minkowsk- +iNet. +Fog simulation. Fig. 9 illustrates the comprehensive +results of robustness in fog simulation. Apart from the +three intensities related in Tab. 1, i.e., 0.005, 0.06 and +0.2, we also provide other 6 intensities, including 0.01, +0.02, 0.03, 0.1, 0.12, 0.15. We find out that SalsaNext +is greatly affected by denser fog, especially when +β is larger than 0.06. Inversely, KPConv shows its +superior robustness crossing different fog intensities. +Therefore, we have the following summary: +Observation-5: All types of methods will have perfor- +mance decay as the fog becomes heavier, among which +the projection-based method decreases fastest, and the +point-based method decreases slowest. +Snow simulation. Fig. 10 illustrates the detailed +results of snow simulation. The conclusion is: +Observation-6: Though snow simulation hampers all +types of methods, the performances of point-based + +Springer Nature 2021 LATEX template +16 +Robust LiDAR segmentation +Table 4 Comprehensive results on diverse noisy corruptions. +Noise types +Ratio (%) +SalsaNext +KPConv +MinkowskiNet +mIoU +R +mIoU +R +mIoU +R +No corruption +0 +55.8 +100.0 +63.5 +100.0 +66.3 +100.0 +Global outliers +0.1 +55.8 +100.0 +62.6 +98.6 +66.5 +100.0 +5 +53.8 +96.4 +62.8 +98.9 +65.9 +99.4 +10 +51.6 +92.5 +62.4 +98.3 +64.5 +97.3 +20 +47.4 +84.9 +61.8 +97.3 +63.9 +96.4 +50 +38.9 +69.7 +60.2 +94.8 +63.5 +95.8 +Local distortion (σ2 = 0.05) +10 +55.7 +99.8 +59.8 +94.2 +62.1 +93.6 +20 +55.2 +98.9 +35.5 +55.9 +55.5 +83.7 +50 +50.7 +90.9 +17.1 +26.9 +36.9 +55.7 +Local distortion (σ2 = 0.1) +10 +55.4 +99.3 +41.6 +65.5 +52.3 +78.9 +20 +53.7 +96.2 +25.6 +40.3 +32.3 +48.7 +50 +42.7 +76.5 +11.1 +17.5 +17.5 +26.4 +Local distortion (σ2 = 0.2) +10 +54.8 +98.2 +34.1 +53.7 +42.5 +64.1 +20 +51.9 +93.0 +21.2 +33.4 +23.8 +35.9 +50 +35.4 +63.4 +14.1 +22.2 +11.2 +16.9 +Table 5 Comprehensive results on different LiDAR types. In ‘sparseness’, we see the original LiDAR point as ‘dense’ one, and randomly +sample 1/2 points in each beam to generate a ‘sparse’ one. +SalsaNext +KPConv +MinkowskiNet +LiDAR types +Sparseness +mIoU +R +mIoU +R +mIoU +R +64-beam +Dense +55.8 +100.0 +63.0 +100.0 +66.3 +100.0 +Sparse +52.1 +93.4 +63.4 +99.8 +63.5 +95.8 +32-beam +Dense +52.4 +93.9 +59.0 +92.9 +62.5 +94.3 +Sparse +49.8 +89.2 +57.5 +90.6 +59.5 +89.7 +16-beam +Dense +32.3 +57.8 +43.8 +69.0 +50.2 +75.7 +Sparse +29.7 +53.2 +43.0 +67.7 +46.4 +70.0 +and voxel-based methods are slightly decreased as +the snow becomes heavier, while the projection-based +methods illustrate an inverse tendency, i.e., there are +slight performance boosts in heavier snowfall. +The reason may be that the large snowfall makes +the scattered noise in the 3D space cover a larger area, +thus affecting the point-based and voxel-base meth- +ods. However, when the 3D scene is mapped to a range +image, these scattered points with larger coverage will +become sparse in each pixel. +Noisy corruptions. Results in different noisy corrup- +tions are illustrated in Tab. 4. On one hand, Minkowsk- +iNet is the most robust method against global out- +liers. It even keeps 95.8% performance in the sce- +nario with additional 50% global noisy points. On +the opposite, SalsaNext has poor generalization abil- +ity for global noise, especially with a larger proportion +of noisy points. On the other hand, the projection- +based method (i.e., SalsaNext) shows great robustness +in local distortion noises, spanning different jittering +ranges, as summarized in observation-1. In contrast, +KPConv and MinkowskiNet cannot work normally in +large-range jittering distortion. +Cross-device discrepancy. We demonstrate concrete +results on different LiDAR types in Tab. 5, where +MinkowskiNet achieves the best results in all cross- +device scenarios. In contrast, SalsaNext has poor +robustness, especially in 16-beam devices with only +around 50% original performance. Furthermore, there +is an interesting discovery: +Observation-7: Point-based methods (i.e., KPConv) +are greatly robust against the scenario of downsam- +pling points in each LiDAR beam. +As illustrated in the table, removing 1/2 points +in each beam nearly does not affect the performance + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +17 +of KPConv. Specifically, in both 16 and 32-beam +devices, the performance in sparse cases is greatly +similar to the dense ones. This achievement may +come from the sampling process in point-based meth- +ods. Compared with projection-based and voxel-based +methods that down-scale feature maps through pool- +ing or convolution operations, point-based methods +reduce the point numbers through sampling strategies, +as depicted in Sec. 4.2. Therefore, the local aggre- +gations in point-based methods are more robust to +downsampling operation. +5.4 Model Design v.s. Robustness +In this section, we comprehensively analyze the rela- +tionships between different model designs and robust- +ness. The results are illustrated in Tab. 6. +Size of range image (projection-based). As shown in +Analysis A of the table, we train CENet H.-X. Cheng +et al. (2022) with different sizes of range image, (i.e., +512 × 64, 1024 × 64 and 2048 × 64), and gain the +following conclusion: +Observation-8: Exploiting smaller image size in +range projection will make the model more vulnerable +to noise, except for cross-device scenarios. +When adopting 512 × 64 range image as input, +the performance of CENet decreases from 76.6% R +to 70.4% R, especially in snowfall simulation with +14% robustness drop. Moreover, its mIoU dramati- +cally decays from 60.4% to 50.5% in LiDAR date +with local noises. On the opposite, adopting smaller +range images improves the robustness when deploy- +ing the model in devices with smaller beam numbers. +We believe that the reason for the above phenomenon +is that small images will make more points gather in +the same pixel. Thus, noise points can easily enlarge +the proportion of contaminated pixels. However, in +the LIDAR point cloud with a smaller beam number, +a smaller image size makes the density of the valid +pixels still quite high. +Local aggregation (point-based). For the point- +based approach, we select RandLA-Net as a typical +one and conduct an ablation study, as shown in Anal- +ysis B of Tab. 6. Specifically, we first replace the +attentive pooling with max pooling in the second line, +and replace the transformation function T (Eqn. (11)) +to naive point-wise MLPs in the third line. The results +show that both two components greatly improve the +performance and the generalization ability. After dis- +carding the two components, the ablated model can +only achieve 50.3 mIoU on clean data and keep 66.7% +performance in the common corruptions. +Voxel partition (voxel-based). To further study the +effectiveness of different voxel partitions, we conduct +experiments and illustrate the results in Analysis C +of Tab. 6. During the experiments, we change the +voxel partition of MinkowskiNet to the cylinder one +and keep the network architecture the same. Con- +cretely, we first transform the LiDAR point cloud from +the Cartesian system into a polar coordinate system +through Eqn. (9). After that, we discretize the trans- +formed LiDAR data with voxel size [0.05, 0.001π, +0.05] in corresponding axes. Finally, the following +summary can be obtained. +Observation-9: Cylindrical partition in voxelization +greatly improves the robustness in most of the corrup- +tion, except in cross-device LiDAR data. +Specifically, after exploiting cylindrical voxeliza- +tion, the models’ robustness in fog, snow and local +corruptions are increased by around 10%, 17% and +16%, respectively. However, such improvement is +only for out-of-distribution data. The performance +of the model for clean data is dropped to 60.5 +mIoU (a drop of about 6%). Similarly, its perfor- +mance in the cross-device deployment scenario is also +affected, especially the robustness on 16-beam LiDAR +is reduced by 26%. Nevertheless, robustness in dif- +ferent voxelization is still an important discovery in +this paper, and it also lays a foundation to propose our +newly configured method RLSeg in Sec. 5.6. +Voxel size (voxel-based). There are also experiments +to study the robustness through different voxel sizes. +The results are shown in Analysis D of Tab. 6. +Observation-10: Larger voxel partition makes voxel- +based approaches more vulnerable to global-level +corruptions, such as adverse weathers and global out- +liers. However, the robustness against local distortion +and cross-devices point clouds is improved. +In the experiment, we apply a larger voxel parti- +tion (i.e., 0.1m and 0.2m), compared with 0.05 as the +origin. The results show that this setting greatly affects +the robustness against global-level corruption, espe- +cially for the global noise. More importantly, utilizing +a small voxel size makes the model cannot achieve sat- +isfactory performance on clean data. The reason is that +a large grid makes the model merge noisy and origi- +nal points into the same grids, and loses fine-grained +information. +Hybrid-representation architecture (voxel-based). +In Analysis E of Tab. 6, we investigate the relation- +ship between robustness and hybrid-representation + +Springer Nature 2021 LATEX template +18 +Robust LiDAR segmentation +Table 6 Systematic analysis for input representation, architecture design and data augmentation. The analysis includes (A) architecture for projection-based methods; (B) architecture for +point-based methods; (C-D) representation for voxel-based methods; (E) hybrid-representation architecture and (F) data augmentation. The baseline models are marked with underline. The +increase and decrease compared with baselines are denoted as ↑ and ↓, respectively. +Clean +Robustness +Fog +Snowfall +Global Outliers +Local Distortion +32-beam +16-beam +Analysis +Method Descriptions +mIoU +RmIoU +mR +mIoU +R +mIoU +R +mIoU +R +mIoU +R +mIoU +R +mIoU +R +A +CENet (2048× 64) +64.3 +49.3 +76.6 +32.5 +50.5 +53.0 +82.4 +59.0 +91.7 +60.4 +94.0 +56.3 +87.5 +34.5 +53.7 +CENet (1024× 64) +62.1 +47.5 ↓ +77.2 ↑ +32.0 ↓ +51.5 ↑ +48.3 ↓ +78.8 ↓ +56.5 ↓ +91.0 ↓ +56.9 ↓ +91.6 ↓ +57.8 ↑ +93.0 ↑ +36.2 ↑ +58.2 ↑ +CENet (512× 64) +61.5 +43.3 ↓ +70.4 ↓ +30.5 ↓ +49.6 ↓ +42.3 ↓ +68.8 ↓ +53.7 ↓ +87.3 ↓ +50.5 ↓ +82.0 ↓ +55.3 ↓ +90.0 ↑ +27.5 ↓ +44.7 ↓ +B +RandLA-Net +59.2 +47.6 +80.4 +56.4 +95.4 +50.0 +84.4 +57.8 +97.8 +26.4 +44.7 +53.3 +90.1 +41.7 +70.5 +w/o Attentive Pooling +56.7 +43.5 ↓ +76.8 ↓ +51.3 ↓ +90.5 ↓ +47.3 ↓ +83.5 ↓ +55.4 ↓ +97.7 ↓ +26.9 ↑ +47.5 ↑ +49.7 ↓ +87.6 ↓ +30.6 ↓ +54.0 ↓ +Point-wise MLP +50.3 +33.5 ↓ +66.7 ↓ +40.6 ↓ +80.7 ↓ +33.4 ↓ +66.5 ↓ +48.8 ↓ +97.1 ↓ +11.8 ↓ +23.5 ↓ +39.0 ↓ +77.6 ↓ +27.4 ↓ +54.5 ↓ +C +MinkowskiNet (grid, 5cm) +66.3 +53.6 +80.9 +56.3 +84.9 +50.4 +76.1 +65.3 +98.5 +37.0 +55.9 +62.2 +93.9 +50.4 +76.0 +MinkowskiNet (cylinder) +63.3 +51.5 ↓ +81.3 ↑ +59.4 ↑ +94.0 ↑ +58.9 ↑ +93.0 ↑ +62.7 ↓ +99.0 ↑ +45.3 ↑ +71.5 ↑ +50.8 ↓ +80.2 ↓ +31.9 ↓ +50.3 ↓ +D +MinkowskiNet (10cm) +64.6 +50.7 ↓ +78.6 ↓ +52.2 ↓ +80.8 ↓ +46.7 ↓ +72.3 ↓ +57.0 ↓ +88.2 ↓ +38.0 ↑ +58.8 ↑ +60.5 ↓ +93.6 ↓ +50.1 ↑ +77.5 ↑ +MinkowskiNet (20cm) +60.6 +43.7 ↓ +72.1 ↓ +43.4 ↓ +71.6 ↓ +37.3 ↓ +61.5 ↓ +47.1 ↓ +77.7 ↓ +32.5 ↓ +53.7 ↓ +55.7 ↓ +92.0 ↓ +46.2 ↓ +76.2 ↑ +E +2DPASS +70.1 +51.1 +72.9 +40.4 +57.6 +53.6 +76.5 +69.8 +99.6 +43.9 +62.7 +61.3 +87.4 +37.7 +53.7 +2DPASS w/o PointHead +65.4 +48.5 ↓ +74.2 ↑ +44.2 ↑ +67.7 ↑ +47.7 ↓ +72.9 ↓ +65.0 ↓ +99.6 ↑ +38.2 ↓ +58.4 ↓ +57.2 ↓ +87.5 ↑ +38.7 ↑ +59.1 ↑ +2DPASS w/o PointBranch +64.1 +47.4 ↓ +73.9 ↑ +35.9 ↓ +56.0 ↓ +46.9 ↓ +73.3 ↓ +63.6 ↓ +99.3 ↓ +45.0 ↑ +70.2 ↑ +56.3 ↓ +87.9 ↑ +36.3 ↓ +56.7 ↑ +F +MinkowskiNet + InsCutMix +70.7 +58.5 ↑ +82.7 ↑ +58.7 ↑ +83.0 ↓ +59.6 ↓ +84.3 ↓ +70.0 ↓ +99.1 ↓ +48.3 ↓ +68.3 ↓ +66.6 ↓ +94.2 ↓ +47.6 ↓ +67.3 ↓ +MinkowskiNet + Mix3D +71.7 +59.5 ↑ +83.0 ↑ +57.4 ↑ +80.0 ↓ +63.4 ↑ +88.4 ↑ +71.7 ↑ +100.0 ↑ +62.6 ↑ +87.3 ↑ +63.6 ↑ +88.7 ↓ +38.5 ↓ +53.7 ↓ + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +19 +point cloud +(a) +point-wise MLPs +pooling +voxels +(b) +Point-wise +MLPs +3D Sparse +Convolution +Fusion +... +voxel +network +encoder/decoder layer +Fig. 11 Illustration of hybrid-representation architecture. The +architecture of PointHead and PointBranch are shown in (a) and (b). +architectures. As mentioned in Sec. 4.3, current state- +of-the-arts adopt hybrid-representation architecture to +boost the in-domain performance. Concretely, there +are two components that merging point-wise repre- +sentation into the voxel one, as shown in Fig. 11. +(a) PointHead: exploiting a PointNet architecture to +aggregate point-wise features into individual voxel +grids. (b) PointBranch: extracting point-wise features +in parallel, and merging the features from voxel archi- +tecture. Similar components can be found in other +previous works (e.g., Cylinder3D and SPVCNN), but +here we only conduct ablation on the state-of-the-art. +Observation-11: +Although +hybrid-representation +architectures improve the performance for the in- +domain LiDAR segmentation with clean data, they +are detrimental to model robustness, especially when +using the PointHead component. +As shown in Tab. 6, both PointHead and Point- +Branch boost the performance of 2DPASS on the +clean LiDAR data. However, when the PointHead is +exploited, there is a dramatic decrease in the robust- +ness, especially in fog simulation and 16-beam device +with 10% and 5% robustness drops. Similarly, Point- +Branch also hampers the robustness, but the influence +is much slight. Directly conducting point-wise MLPs +on LiDAR points is easier affected by diverse corrup- +tions, since it cannot capture local geometric. +5.5 Data Augmentation v.s. Robustness +In this section, we investigate the robustness of the +model with different data augmentation. +General augmentation strategies. Previous studies +adopt diverse data augmentation during the training. +Generally, rotation and scaling are the most widely +used. In this paper, we conduct rotation, scaling, and +Fig. 12 Examples of LiDAR point cloud after applying Mix3D. +flipping when re-training the point-based and voxel- +based models, and follow the same image-based aug- +mentation in projection-based models. Note that jitter +augmentation is not used in our experiment, as it +will generate in-domain training data for our local +distortion corruption. +MixUp on LiDAR point cloud. MixUp H. Zhang, +Cisse, Dauphin, and Lopez-Paz (2017) is initially +proposed in image classification for a more robust +representation and extends to 3D computer vision in +recent years. Existing MixUp approaches in LiDAR +semantic segmentation task include Mix3D Nekrasov +et al. (2021), Instance CutMix Xu et al. (2021) and +LaserMix Kong, Ren, Pan, and Liu (2022), where +only Mix3D is open-sourced now. Therefore, we train +MinkowskiNet with the official Mix3D, as well as +the re-produced Instance CutMix. The illustration of +Mix3D is in Fig. 12, in which Mix3D randomly +merges two LiDAR scans (including labels) into a +common coordinate. We re-produce Instance CutMix +by only merging instance-level objects from other +LiDAR scans. The experimental results are demon- +strated in Analysis F of Tab. 6. +Observation-12: Existing MixUp data augmentation +for LiDAR semantic segmentation makes the model +more robust against most of the corruption, except for +cross-device scenarios. +As illustrated in the table, after exploiting two +MixUp augmentation, there are significant boosts in +several corruptions, e.g., over 10% and 30% improve- +ments on snowfall and local distortion corruptions +with Mix3D augmentation. However, the robustness +of the model decreases in cross-device LiDAR data, + +sesSpringer Nature 2021 LATEX template +20 +Robust LiDAR segmentation +Table 7 Ablation study for RLSeg. KD and PL denote knowledge distillation and pseudo label fine-tuning, respectively. +Model +Mix3D +KD +PL +mIoU +RmIoU +mR +Fog +Snowfall +Global +Local +32-beam +16-beam +MinkowskiNet +66.3 +53.6 +80.9 +56.3 +50.4 +65.3 +37.0 +62.2 +50.4 +✓ +71.7 +59.5 +83.0 +57.4 +63.4 +71.7 +62.6 +63.6 +38.5 +RLSeg (ours) +63.3 +51.5 +81.3 +59.5 +58.9 +62.7 +45.3 +50.8 +31.9 +✓ +67.0 +56.7 +84.7 +60.4 +63.5 +66.3 +53.2 +58.2 +38.7 +✓ +✓ +70.9 +60.3 +85.0 +55.9 +64.0 +70.8 +69.6 +60.0 +41.5 +✓ +✓ +✓ +73.5 +62.5 +85.0 +57.6 +66.2 +73.4 +71.9 +62.3 +43.6 +Ground Truth +Errors (MinkowskiNet) +Errors (RLSeg) +Fog (moderate) +Local (moderate) +16-beam (dense) +No Corruption +Fig. 13 Visualization. We demonstrate the visualization results of the most robust existing method (MinkowskiNet) and our RLSeg on four +cases, including clean data and three LiDAR corruptions. The noisy points are labeled as ‘ignore’ (black color) and not considered in the +evaluation. The left two columns are colorized by error maps, and the last one is colorized by ground truth. +especially in 16-beam data with larger domain dis- +crepancies. The reason is that MixUp augmentation +utilizes denser mixed point clouds as input, and thus +makes the model vulnerable to the sparse point clouds. +5.6 Boosting Corruption Robustness +Summarize the observations. In Sec. 5.2-5.5, we +obtain 12 observations in total. In order to design a +more robust model, we summarize the most useful +information from these observations: +1) Voxel-based architecture: After summarizing +observations 1-6, we prefer to use the voxel-based +method as our backbone. The reason is that the +methods of projection-based and point-based do not +perform well on clean data, and both of them are + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +21 +Teacher +MinkowskiNet +Student +RLSeg +mixed data +mixed prediction +original data +prediction +KL divergence +Fig. 14 Illustration of training process of RLSeg. +vulnerable to certain corruption. In contrast, the voxel- +based method performs well in most cases, and the +results on clean data are also satisfactory. +2) Cylindrical partition with appropriate voxel +size: As shown in observation-9 and 10, exploit- +ing cylindrical partition and appropriate voxel size +increase the robustness of the model. +3) +Single-representation: +Observation-11 +illus- +trates that though hybrid-representation architecture +improves the performance on clean data, it hampers +the robustness against common corruptions. +4) Mix3D augmentation: This can be gained by +observation-12. +Based on the above four conclusions, we design +the robust LiDAR segmentation (RLSeg) model in +this paper. Specifically, we use MinkowskiNet as our +backbone due to its single-representation nature. Fur- +thermore, we transform the LiDAR point cloud from +the Cartesian system into a polar coordinate system +through Eqn. (9) and apply a cylindrical partition with +voxel size [0.05, 0.001π, 0.05]. However, the result +in Tab. 6 shows that cylindrical partition hampers the +performance on the clean data. +Training with knowledge distillation. To tackle this +problem, we adopt the knowledge distillation Hinton, +Vinyals, and Dean (2014) and self-training techniques +to enhance the model. Specifically, we first train a +teacher MinkowskiNet model with grid voxel parti- +tion, voxel size 0.05 and Mix3D augmentation, obtain- +ing the model with 71.7% mIoU on clean data, as +shown in the last row of Tab. 6. After that, we train +the above RLSeg with a teacher-student framework, +applying KL divergence to the output logits of RLS +and the teacher MinkowskiNet, as shown in Fig. 14. +During the training, the Mix3D is only conducted on +the student model, and the KL divergence constrains +the original data. We train RLSeg with 64 epochs with +a weight of 0.05 for KL divergence. Finally, motivated +by the improvement achieved by pseudo label J. Li, +Dai, and Ding (2022) in semi-supervised learning, we +further fine-tune the student network 48 epochs on +clean validation data with pseudo labels generated by +the teacher MinkowskiNet. +Concrete results. Through such a simple but effec- +tive manner, we significantly improve the performance +on clean data, while keeping the robustness against +diverse corruptions. The results are demonstrated in +Tab. 3, where RLSeg significantly outperforms exist- +ing methods. We analyze different designs through +an ablation study in Tab. 7. As shown in the table, +exploiting our architecture improves the robustness, +but causes a performance drop to 63.3 mIoU. After +using knowledge distillation (KD) and Mix3D, there +is a huge performance boost from 63.3 to 70.9, while +increasing the robustness from 81.3 to 85.0. Finally, +utilizing pseudo label fine-tuning can further improve +the performance while keeping the robustness, which +shows a promising improvement by leveraging the +potential of semi-supervised learning, giving a per- +formance boost to about 73.5%. This also gives hints +for future work that improve the robust but poor +performance network to better performance. +We provide visualization results of our RLSeg +and MinkowskiNet in Fig. 13, in which our proposed +model performs better. Specifically, MinkowskiNet +cannot work normally in fog simulation and local +distortion, and there are large areas of errors in the +LiDAR scenes. In contrast, our RLSeg provides robust +prediction even for the small objects (as shown in +red circles). These show the robustness of our model +as well as the promising future for robust LIDAR +semantic segmentation. +6 Conclusion +In this paper, we propose a new benchmark called +SemanticKITTI-C, with respect to real-world and +out-of-domain LiDAR corruptions. We systematically +investigate a wide range of LiDAR semantic segmen- +tation models, spanning different input representations +and network architectures. After analyzing the results +of previous approaches, we summarized 12 obser- +vations for the future research. Finally, we propose +RLSeg based on the above observations, which effec- +tively boosts the robustness of LiDAR semantic seg- +mentation. We hope our benchmark, comprehensive +analysis and observations could boost future research +for robust LiDAR semantic segmentation in safety- +critical applications. + +Springer Nature 2021 LATEX template +22 +Robust LiDAR segmentation +References +Alonso, I., Riazuelo, L., Montesano, L., Murillo, A.C. +(2020). +3d-mininet: Learning a 2d represen- +tation from point clouds for fast and efficient +3d lidar semantic segmentation. +arXiv preprint +arXiv:2002.10893. +Altindis, S.F., Dalva, Y., Dundar, A. (2021). Benchmark- +ing the robustness of instance segmentation models. +arXiv preprint arXiv:2109.01123. +Bai, X., Hu, Z., Zhu, X., Huang, Q., Chen, Y., Fu, H., Tai, C.- +L. (2022). Transfusion: Robust lidar-camera fusion +for 3d object detection with transformers. Proceed- +ings of the ieee/cvf conference on computer vision +and pattern recognition (pp. 1090–1099). +Barbu, A., Mayo, D., Alverio, J., Luo, W., Wang, C., Gutfre- +und, D., . . . Katz, B. (2019). Objectnet: A large-scale +bias-controlled dataset for pushing the limits of object +recognition models. Advances in neural information +processing systems, 32. +Behley, J., Garbade, M., Milioto, A., Quenzel, J., Behnke, S., +Stachniss, C., Gall, J. (2019). Semantickitti: A dataset +for semantic scene understanding of lidar sequences. +Proceedings of the ieee international conference on +computer vision (pp. 9297–9307). +Caesar, H., Bankiti, V., Lang, A.H., Vora, S., Liong, V.E., +Xu, Q., . . . Beijbom, O. (2020). nuscenes: A multi- +modal dataset for autonomous driving. Proceedings +of the ieee/cvf conference on computer vision and +pattern recognition (pp. 11621–11631). +Cheng, H.-X., Han, X.-F., Xiao, G.-Q. +(2022). +Cenet: +Toward concise and efficient lidar semantic segmen- +tation for autonomous driving. +2022 ieee interna- +tional conference on multimedia and expo (icme) (pp. +01–06). +Cheng, R., Razani, R., Taghavi, E., Li, E., Liu, B. (2021). +Af2-s3net: Attentive feature fusion with adaptive fea- +ture selection for sparse semantic segmentation net- +work. +Proceedings of the ieee/cvf conference on +computer vision and pattern recognition (pp. 12547– +12556). +Choy, C., Gwak, J., Savarese, S. (2019). 4d spatio-temporal +convnets: Minkowski convolutional neural networks. +Proceedings of the ieee conference on computer +vision and pattern recognition (pp. 3075–3084). +Cordts, M., Omran, M., Ramos, S., Rehfeld, T., Enzweiler, +M., Benenson, R., ... Schiele, B. +(2016). +The +cityscapes dataset for semantic urban scene under- +standing. Proc. of the ieee conference on computer +vision and pattern recognition (cvpr). +Cortinhal, T., Tzelepis, G., Aksoy, E.E. +(2020). +Sal- +sanext: Fast, uncertainty-aware semantic segmenta- +tion of lidar point clouds for autonomous driving. +arXiv preprint arXiv:2003.03653. +Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., Fei-Fei, L. +(2009). Imagenet: A large-scale hierarchical image +database. 2009 ieee conference on computer vision +and pattern recognition (pp. 248–255). +Dong, X., Chen, D., Zhou, H., Hua, G., Zhang, W., Yu, +N. +(2020). +Self-robust 3d point recognition via +gather-vector guidance. 2020 ieee/cvf conference on +computer vision and pattern recognition (cvpr) (pp. +11513–11521). +Engel, N., Belagiannis, V., Dietmayer, K. +(2021). +Point +transformer. IEEE Access, 9, 134826–134840. +Geiger, A., Lenz, P., Urtasun, R. (2012). Are we ready for +Autonomous Driving? The KITTI Vision Benchmark +Suite. Proc. of the ieee conf. on computer vision and +pattern recognition (cvpr) (pp. 3354–3361). +Graham, B., Engelcke, M., van der Maaten, L. +(2018). +3d semantic segmentation with submanifold sparse +convolutional networks. Proceedings of the ieee con- +ference on computer vision and pattern recognition +(pp. 9224–9232). +Graham, B., & van der Maaten, L. +(2017). +Submani- +fold sparse convolutional networks. arXiv preprint +arXiv:1706.01307. +Hahner, M., Dai, D., Sakaridis, C., Zaech, J.-N., Van Gool, +L. (2019). Semantic understanding of foggy scenes +with purely synthetic data. Ieee international confer- +ence on intelligent transportation systems (itsc). +Hahner, M., Sakaridis, C., Bijelic, M., Heide, F., Yu, F., Dai, +D., Van Gool, L. (2022a). LiDAR Snowfall Simu- +lation for Robust 3D Object Detection. Ieee/cvf con- +ference on computer vision and pattern recognition +(cvpr). +Hahner, M., Sakaridis, C., Bijelic, M., Heide, F., Yu, F., Dai, +D., Van Gool, L. (2022b). Lidar snowfall simula- +tion for robust 3d object detection. Proceedings of the +ieee/cvf conference on computer vision and pattern +recognition (pp. 16364–16374). +Hahner, M., Sakaridis, C., Dai, D., Van Gool, L. (2021). +Fog simulation on real lidar point clouds for 3d object +detection in adverse weather. +Proceedings of the +ieee/cvf international conference on computer vision +(pp. 15283–15292). +Hendrycks, D., & Dietterich, T. (2019). Benchmarking neu- +ral network robustness to common corruptions and +perturbations. arXiv preprint arXiv:1903.12261. + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +23 +Hendrycks, D., Zhao, K., Basart, S., Steinhardt, J., Song, +D. (2021). Natural adversarial examples. Proceed- +ings of the ieee/cvf conference on computer vision +and pattern recognition (pp. 15262–15271). +Hinton, G., Vinyals, O., Dean, J. +(2014). +Distilling the +knowledge in a neural network. NeurIPS Workshops. +Hu, Q., Yang, B., Khalid, S., Xiao, W., Trigoni, N., +Markham, A. (2022). Sensaturban: Learning seman- +tics from urban-scale photogrammetric point clouds. +International Journal of Computer Vision, 130(2), +316–343. +Hu, Q., Yang, B., Xie, L., Rosa, S., Guo, Y., Wang, Z., . . . +Markham, A. (2020). Randla-net: Efficient semantic +segmentation of large-scale point clouds. Proceed- +ings of the IEEE Conference on Computer Vision and +Pattern Recognition. +Hua, B.-S., Tran, M.-K., Yeung, S.-K. +(2018). +Point- +wise convolutional neural networks. Proceedings of +the ieee conference on computer vision and pattern +recognition (pp. 984–993). +Kamann, C., & Rother, C. (2020). Benchmarking the robust- +ness of semantic segmentation models. Proceedings +of the ieee/cvf conference on computer vision and +pattern recognition (pp. 8828–8838). +Kong, L., Ren, J., Pan, L., Liu, Z. (2022). Lasermix for +semi-supervised lidar semantic segmentation. arXiv +preprint arXiv:2207.00026. +Lai, X., Liu, J., Jiang, L., Wang, L., Zhao, H., Liu, S., . . . +Jia, J. +(2022). +Stratified transformer for 3d point +cloud segmentation. Proceedings of the ieee/cvf con- +ference on computer vision and pattern recognition +(pp. 8500–8509). +Li, J., Dai, H., Ding, Y. +(2022). +Self-distillation for +robust LiDAR semantic segmentation in autonomous +driving. Eccv. +Li, Y., Yu, A.W., Meng, T., Caine, B., Ngiam, J., Peng, D., +. . . others (2022). Deepfusion: Lidar-camera deep +fusion for multi-modal 3d object detection. Proceed- +ings of the ieee/cvf conference on computer vision +and pattern recognition (pp. 17182–17191). +Liong, V.E., Nguyen, T.N.T., Widjaja, S., Sharma, D., +Chong, Z.J. (2020). Amvnet: Assertion-based multi- +view fusion network for lidar semantic segmentation. +arXiv preprint arXiv:2012.04934. +Liu, H., Jia, J., Gong, N.Z. +(2021). +Pointguard: Prov- +ably robust 3d point cloud classification. Proceedings +of the ieee/cvf conference on computer vision and +pattern recognition (pp. 6186–6195). +Liu, Y., Fan, B., Xiang, S., Pan, C. +(2019). +Relation- +shape convolutional neural network for point cloud +analysis. Proceedings of the ieee/cvf conference on +computer vision and pattern recognition (pp. 8895– +8904). +Michaelis, C., Mitzkus, B., Geirhos, R., Rusak, E., +Bringmann, +O., +Ecker, +A.S., +. . . +Brendel, +W. +(2019). Benchmarking robustness in object detection: +Autonomous driving when winter is coming. arXiv +preprint arXiv:1907.07484. +Milioto, A., Vizzo, I., Behley, J., Stachniss, C. +(2019). +Rangenet++: Fast and accurate lidar semantic seg- +mentation. Proc. of the ieee/rsj intl. conf. on intelli- +gent robots and systems (iros). +Nekrasov, A., Schult, J., Litany, O., Leibe, B., Engelmann, +F. (2021). Mix3D: Out-of-Context Data Augmenta- +tion for 3D Scenes. International Conference on 3D +Vision (3DV). +Porav, H., Musat, V.-N., Bruls, T., Newman, P. +(2020). +Rainy screens: Collecting rainy datasets, indoors. +arXiv preprint arXiv:2003.04742. +Qi, C.R., Yi, L., Su, H., Guibas, L.J. +(2017). +Point- +net++: Deep hierarchical feature learning on point +sets in a metric space. Advances in neural information +processing systems (pp. 5099–5108). +Qiu, H., Yu, B., Tao, D. +(2022). +GFNet: Geo- +metric +flow +network +for +3d +point +cloud +semantic +segmentation. +Transactions +on +Machine Learning Research. +Retrieved from +https://openreview.net/forum?id=LSAAlS7Yts +Recht, B., Roelofs, R., Schmidt, L., Shankar, V. (2019). Do +imagenet classifiers generalize to imagenet? +Inter- +national conference on machine learning (pp. 5389– +5400). +Ren, J., Pan, L., Liu, Z. (2022). Benchmarking and analyzing +point cloud classification under corruptions. Interna- +tional Conference on Machine Learning (ICML). +Ronneberger, O., Fischer, P., Brox, T. (2015). U-net: Con- +volutional networks for biomedical image segmen- +tation. +International conference on medical image +computing and computer-assisted intervention (pp. +234–241). +Sakaridis, C., Dai, D., Van Gool, L. +(2018). +Seman- +tic foggy scene understanding with synthetic data. +International Journal of Computer Vision, 126(9), +973–992. +Sakaridis, C., Dai, D., Van Gool, L. +(2021). +Acdc: The +adverse conditions dataset with correspondences for +semantic driving scene understanding. Proceedings +of the ieee/cvf international conference on computer +vision (pp. 10765–10775). + +Springer Nature 2021 LATEX template +24 +Robust LiDAR segmentation +Sun, P., Kretzschmar, H., Dotiwalla, X., Chouard, A., Pat- +naik, V., Tsui, P., ... others +(2020). +Scalability +in perception for autonomous driving: Waymo open +dataset. +Proceedings of the ieee/cvf conference on +computer vision and pattern recognition (pp. 2446– +2454). +Taghanaki, S.A., Luo, J., Zhang, R., Wang, Y., Jayaraman, +P.K., Jatavallabhula, K.M. (2020). Robustpointset: +A dataset for benchmarking robustness of point cloud +classifiers. arXiv preprint arXiv:2011.11572. +Tang, H., Liu, Z., Zhao, S., Lin, Y., Lin, J., Wang, H., Han, +S. (2020). Searching efficient 3d architectures with +sparse point-voxel convolution. European conference +on computer vision (pp. 685–702). +Tatarchenko, M., Park, J., Koltun, V., Zhou, Q.-Y. (2018). +Tangent convolutions for dense prediction in 3d. Pro- +ceedings of the ieee conference on computer vision +and pattern recognition (pp. 3887–3896). +Thomas, H., Qi, C.R., Deschaud, J.-E., Marcotegui, B., +Goulette, F., Guibas, L.J. (2019, October). Kpconv: +Flexible and deformable convolution for point clouds. +The ieee international conference on computer vision +(iccv). +Unal, O., Dai, D., Van Gool, L. (2022). Scribble-supervised +lidar semantic segmentation. +Proceedings of the +ieee/cvf conference on computer vision and pattern +recognition (cvpr). +Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, +L., Gomez, A.N., ... Polosukhin, I. (2017). Atten- +tion is all you need. Advances in neural information +processing systems (pp. 5998–6008). +Wang, Y., Sun, Y., Liu, Z., Sarma, S.E., Bronstein, M.M., +Solomon, J.M. (2019). Dynamic graph cnn for learn- +ing on point clouds. ACM Transactions on Graphics +(TOG), 38(5), 1–12. +Wei, Y., Wei, Z., Rao, Y., Li, J., Zhou, J., Lu, J. (2022). +Lidar distillation: Bridging the beam-induced domain +gap for 3d object detection. European conference on +computer vision (eccv) (Vol. 39, p. 179-195). +Wu, B., Wan, A., Yue, X., Keutzer, K. (2018). Squeezeseg: +Convolutional neural nets with recurrent crf for real- +time road-object segmentation from 3d lidar point +cloud. 2018 ieee international conference on robotics +and automation (icra) (pp. 1887–1893). +Wu, B., Zhou, X., Zhao, S., Yue, X., Keutzer, K. (2019). +Squeezesegv2: Improved model structure and unsu- +pervised domain adaptation for road-object segmen- +tation from a lidar point cloud. 2019 international +conference on robotics and automation (icra) (pp. +4376–4382). +Wu, W., Qi, Z., Fuxin, L. (2019). Pointconv: Deep convo- +lutional networks on 3d point clouds. Proceedings of +the ieee conference on computer vision and pattern +recognition (pp. 9621–9630). +Xiao, C., & Wachs, J. +(2021). +Triangle-net: Towards +robustness in point cloud learning. +Proceedings +of the ieee/cvf winter conference on applications of +computer vision (pp. 826–835). +Xu, J., Zhang, R., Dou, J., Zhu, Y., Sun, J., Pu, S. (2021). +Rpvnet: A deep and efficient range-point-voxel fusion +network for lidar point cloud segmentation. +Pro- +ceedings of the ieee/cvf international conference on +computer vision (pp. 16024–16033). +Yamada, Y., & Otani, M. +(2022). +Does robustness on +imagenet transfer to downstream tasks? Proceedings +of the ieee/cvf conference on computer vision and +pattern recognition (pp. 9215–9224). +Yan, X., Gao, J., Zheng, C., Zheng, C., Zhang, R., Cui, S., +Li, Z. (2022). 2dpass: 2d priors assisted semantic seg- +mentation on lidar point clouds. European conference +on computer vision (pp. 677–695). +Yan, X., Zheng, C., Li, Z., Wang, S., Cui, S. +(2020). +Pointasnl: Robust point clouds processing using non- +local neural networks with adaptive sampling. Pro- +ceedings of the ieee/cvf conference on computer +vision and pattern recognition (pp. 5589–5598). +Yu, K., Tao, T., Xie, H., Lin, Z., Wu, Z., Xia, Z., . . . +others (2022). Benchmarking the robustness of lidar- +camera fusion for 3d object detection. arXiv preprint +arXiv:2205.14951. +Zhang, H., Cisse, M., Dauphin, Y.N., Lopez-Paz, D. (2017). +mixup: Beyond empirical risk minimization. arXiv +preprint arXiv:1710.09412. +Zhang, Y., Zhou, Z., David, P., Yue, X., Xi, Z., Gong, B., +Foroosh, H. +(2020). +Polarnet: An improved grid +representation for online lidar point clouds semantic +segmentation. +Proceedings of the ieee/cvf confer- +ence on computer vision and pattern recognition (pp. +9601–9610). +Zhang, Z., Hua, B.-S., Yeung, S.-K. +(2022). +Riconv++: +Effective rotation invariant convolutions for 3d point +clouds deep learning. International Journal of Com- +puter Vision, 130(5), 1228–1243. +Zhao, H., Jiang, L., Jia, J., Torr, P.H., Koltun, V. (2021). +Point transformer. Proceedings of the ieee/cvf inter- +national conference on computer vision (pp. 16259– +16268). +Zhou, H., Chen, K., Zhang, W., Fang, H., Zhou, W., Yu, N. +(2019). Dup-net: Denoiser and upsampler network +for 3d adversarial point clouds defense. Proceedings +of the ieee/cvf international conference on computer +vision (pp. 1961–1970). +Zhou, H., Zhu, X., Song, X., Ma, Y., Wang, Z., Li, H., Lin, +D. (2020). Cylinder3d: An effective 3d framework +for driving-scene lidar semantic segmentation. arXiv + +Springer Nature 2021 LATEX template +Robust LiDAR segmentation +25 +preprint arXiv:2008.01550. +Zhu, X., Zhou, H., Wang, T., Hong, F., Ma, Y., Li, W., +. . . Lin, D. +(2021). +Cylindrical and asymmetri- +cal 3d convolution networks for lidar segmentation. +Proceedings of the ieee/cvf conference on computer +vision and pattern recognition (pp. 9939–9948). + diff --git a/z9AzT4oBgHgl3EQfC_rr/content/tmp_files/load_file.txt b/z9AzT4oBgHgl3EQfC_rr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..629ae5ac600d6e49f2a9bb6488e8512c5d35509f --- /dev/null +++ b/z9AzT4oBgHgl3EQfC_rr/content/tmp_files/load_file.txt @@ -0,0 +1,2266 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf,len=2265 +page_content='Springer Nature 2021 LATEX template Benchmarking the Robustness of LiDAR Semantic Segmentation Models Xu Yan1,2, Chaoda Zheng1,2, Zhen Li2,1*, Shuguang Cui2,1,4 and Dengxin Dai3* 1FNii, The Chinese University of Hong Kong (Shenzhen), P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4Peng Cheng Laboratory, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' E-mail(s): ddai@mpi-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' lizhen@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Contributing authors: xuyan1@link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' chaodazheng@link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' shuguangcui@cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Abstract When using LiDAR semantic segmentation models for safety-critical applications such as autonomous driv- ing, it is essential to understand and improve their robustness with respect to a large range of LiDAR corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In this paper, we aim to comprehensively analyze the robustness of LiDAR semantic seg- mentation models under various corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To rigorously evaluate the robustness and generalizability of current approaches, we propose a new benchmark called SemanticKITTI-C, which features 16 out-of-domain LiDAR corruptions in three groups, namely adverse weather, measurement noise and cross-device discrep- ancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Then, we systematically investigate 11 LiDAR semantic segmentation models, especially spanning different input representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', point clouds, voxels, projected images, and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' ), network architectures and training schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Through this study, we obtain two insights: 1) We find out that the input represen- tation plays a crucial role in robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, under specific corruptions, different representations perform variously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2) Although state-of-the-art methods on LiDAR semantic segmentation achieve promis- ing results on clean data, they are less robust when dealing with noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, based on the above observations, we design a robust LiDAR segmentation model (RLSeg) which greatly boosts the robustness with simple but effective modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' It is promising that our benchmark, comprehensive analysis, and observations can boost future research in robust LiDAR semantic segmentation for safety-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Keywords: Robustness, LiDAR Corruption, Out-of-distribution, Point Clouds, Semantic Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1 Introduction Autonomous driving, one of the most promising appli- cations for computer vision, has achieved impressive progress in recent studies, where LiDAR semantic segmentation plays a crucial role Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Unal, Dai, and Van Gool (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Current semantic segmentation models are generally evaluated on clean validation sets, which share the same data distribution with the corresponding train- ing sets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', collected with the same sensor, at a similar time and weather condition, and at the same place Behley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, due to the inac- curate data acquisition Ren, Pan, and Liu (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yan, Zheng, Li, Wang, and Cui (2020) and complex sce- narios caused by diverse weather conditions Hahner, Dai, Sakaridis, Zaech, and Van Gool (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hahner 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='00970v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='CV] 3 Jan 2023 Springer Nature 2021 LATEX template 2 Robust LiDAR segmentation Original Data (64-beam) Cross-device Simulation (16-beam) Data with Global Outliers Data with Snowfall Simulation Data with Fog Simulation Data with Local Distortion road sidewalk car vegetation trunk terrain building other-obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' ignore Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1 Examples of our proposed SemanticKITTI-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We corrupt the clean validation set of SemanticKITTI using six types of corruptions with 16 levels of intensity to build upon a comprehensive robustness benchmark for LiDAR semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Listed examples are point clouds on 16-beam LiDAR sensors, with global and local distortion, in snowfall and fog simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022a), LiDAR point clouds inevitably suf- fer from severe corruptions in real-world deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since autonomous driving is a safety-critical appli- cation, robustness against out-of-distribution (OOD) LiDAR data becomes an important part of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Understanding and analyzing the robustness of models for image corruption is a well-studied topic, in which several benchmarks are proposed for dif- ferent tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', classification Hendrycks and Diet- terich (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hendrycks, Zhao, Basart, Steinhardt, and Song (2021), semantic segmentation Kamann and Rother (2020), instance segmentation Altindis, Dalva, and Dundar (2021) and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Generally, these stud- ies simulate corruption through changing RGB values on the original image, resulting in different kinds of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, since images are repre- sented as dense pixel arrays, previous works focus on investigating different architectures without mod- ifying the representation of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In contrast, analyzing robustness against LiDAR corruption is a more complicated problem: 1) LiDAR point clouds are usually textureless and irregular, and they describe the 3D shapes only through scattered points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Also, point clouds collected by different type of LiDARs may have different ranges and resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' There- fore, the corruption on point clouds not only needs to consider the deformation, disturbance and occlu- sion in the 3D space, but also the domain discrepancy caused by different devices during the data acquisi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2) Semantic segmentation models in LiDAR sce- narios usually adopt diverse representations to meet different requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, projection-based methods Milioto, Vizzo, Behley, and Stachniss (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) project LiDAR point clouds onto 2D pixels, and thus enable the application of normal 2D-CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Voxel-based approaches conduct voxelization and transform the LiDAR point clouds into 3D voxel grids Graham and van der Maaten (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020), exploiting 3D-CNN to cap- ture the fine-grained 3D information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' There are also point-based methods Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) that learn the geometric details directly on raw point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recent studies even combine different representations to boost the performance Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021), which makes it more difficult to analyze robustness purely from the architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In this paper, we try to break through the barrier of robust LiDAR semantic segmentation and extend the exploration of model robustness against 3D vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We find out that the study of corruption robustness on 3D point cloud is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, most studies Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) for robustness on 3D point clouds tend to test their mod- els on synthesis datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', ModelNet40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The point clouds on these datasets are sampled from CAD mod- els of stand-alone objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Therefore, the findings in Springer Nature 2021 LATEX template Robust LiDAR segmentation 3 these studies cannot be directly applied to real-world applications, where raw point clouds are collected in large and complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Though a few prior works are related to the model robustness on real- world data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022)), they only conduct a coarse comparison at the level of full models, without analysis on the inner structures and the input rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As a result, there is no benchmark for robustness in the real-world point clouds, especially for the safety-crucial LiDAR semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For the first time, based on the most pop- ular LiDAR semantic segmentation dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', SemanticKITTI Behley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019), we build a systematically-designed robust benchmark with several real-world and out-of-domain corruptions, namely SemanticKITTI-C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Following pre- vious studies, the term robustness refers to training a model on clean data and validating it on corrupted data, and thus we introduce diverse corruptions on 4071 LiDAR scans in the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifi- cally, SemanticKITTI-C covers 16 kinds of corrup- tions in total, which can be categorized into three classes, namely adverse weather, measurement noise, and cross-device discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To make the benchmark more rigorous, we set several subclass corruptions in each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, the adverse weather class contains LiDAR scans in snowfall and fog simula- tions, and there are three independent levels in each corruption, indicating different snowy or foggy inten- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Built upon this benchmark, we further evaluate the robustness of current LiDAR semantic segmenta- tion methods, including analysis of different represen- tations, architectures, and training schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Consequently, we obtain 12 observations in total from various aspects: Representation: We find out that projection-based methods are vulnerable to adverse weathers, especially fog simulation, but they are more robust to local distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We also observe that exploiting larger image size in range projection improves the robustness of projection-based meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Inversely, point-based approaches are vulnera- ble to local distortion but more robust in different weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Compared with the above two mainstreams, the voxel representation enjoys impres- sive robustness in most corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' And using cylin- der voxel partition Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) is much more robust than using the traditional grids Graham and van der Maaten (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Architecture: 1) For point- based approaches, pseudo kernel local aggregation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', KPConv Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019)) is the most robust when compared with adaptive-weight Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and MLPs Qi, Yi, Su, and Guibas (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, transformer architectures Zhao, Jiang, Jia, Torr, and Koltun (2021) greatly hamper the robustness of point- based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2) Although hybrid-representation architecture improves the performance on clean data, it makes models more vulnerable to noise, espe- cially for those using MLPs to aggregate point-wised features in each voxel Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Training strategy: Empirically, applying data augmentation such as Mix3D Nekrasov, Schult, Litany, Leibe, and Engelmann (2021) improves the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, by identifying the best combi- nation from existing components in terms of input representations, model architectures and data augmen- tation strategies, we design a robust LiDAR segmenta- tion model (RLSeg) in a simple but effective manner, achieving superior robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Our contributions are concluded as follows: We present the first large-scale robustness bench- mark for LiDAR point cloud semantic segmentation under various corruptions, namely SemanticKITTI- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The dataset contains 16 corruptions, spanning scenarios in adverse weather conditions, sensor measurement bias and diverse device collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We comprehensively study existing methods on our proposed benchmark and analyze the robustness of diverse architectures and representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We summarize several effective observations to boost the robustness of LiDAR semantic segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' It is identified that architecture and input representation should be carefully considered in future research and real-world deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 LiDAR Semantic Segmentation Since the data collected by LiDAR is represented as point clouds, there are several mainstreams to process input point clouds with different representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' More details will be illustrated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1) Point-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' These approaches directly learn the geometric details on raw point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Gen- erally, they follow the hierarchical architecture as 2D vision, and first conduct sampling strategy in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, they search neighboring points from each sampled point, and apply feature aggregation in each local group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The local aggregation function is essential for the point-based methods, and thus many studies design different operators to capture the Springer Nature 2021 LATEX template 4 Robust LiDAR segmentation local geometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, point-wise MLP Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2017), adaptive weight Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Liu, Fan, Xiang, and Pan (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wu, Qi, and Fuxin (2019) and pseudo grid Hua, Tran, and Yeung (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) are utilized to extract local features of point clouds, and they also exploit nonlocal operators Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) or attention mech- anism Engel, Belagiannis, and Dietmayer (2021) to learn permutation-invariant dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, point-based methods are not efficient in the LiDAR scenario since their sampling and grouping algorithms are generally time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2) Projection-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' These methods are very efficient on LiDAR processing since they project the raw point cloud onto a 2D image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pre- vious works project points through plane projec- tion Tatarchenko, Park, Koltun, and Zhou (2018), spherical projection B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wu, Wan, Yue, and Keutzer (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wu, Zhou, Zhao, Yue, and Keutzer (2019) or both Liong, Nguyen, Widjaja, Sharma, and Chong (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since the projection process inevitably causes the loss of information, the recent studies adopt point- based branches to obtain fine-grained features Alonso, Riazuelo, Montesano, and Murillo (2020) or refine the segmentation results Qiu, Yu, and Tao (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3) Voxel-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' These approaches are most widely adopted, since they can achieve impressive per- formance while keep efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Generally, they first conduct voxelization, and divide the raw points into different voxel grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, they conduct 3D convolution in the input volumetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The sparse con- volution (SparseConv) Graham, Engelcke, and van der Maaten (2018) is the core technique in voxel-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since there are a large proportion of vox- els are empty during the voxelization, which introduce huge computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The core of SparseConv is only conduct operation in non-empty grids, which will be saved in sparse Hash codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recent studies adopt SparseConv to design diverse architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) design the original grid voxels to cylindrical ones and propose an asymmet- rical network to learn anisotropy features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recently, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng, Razani, Taghavi, Li, and Liu (2021) design a multi-branch component with several kernel sizes, capturing features with different receptive field and fusing them through an attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4) Hybrid-representation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Though voxel- based methods achieve superior performance, there is still missing geometric during the voxelization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hence, there is a trend of exploiting multi- representation fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' These methods combine mul- tiple representation inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', points, projection images, and voxels) and apply feature fusion among different representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) designs point-voxel CNN operator, which com- bines point-wise MLPs in each sparse convolution block, and adopts neural architecture search (NAS) to search a more powerful architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) utilizes the above three representations and proposes a range-point-voxel fusion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recently, Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) applies cross-modal knowledge distilla- tion, introducing prior information from texture and color images during the training phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Nevertheless, the hybrid-representation architecture makes them less robust in out-of-domain corruptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 Robustness Benchmarks for Images There are comprehensive robustness benchmarks for 2D image processing, spanning different tasks such as classification, semantic segmentation and instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For robust image classifica- tion, ImageNet-C Hendrycks and Dietterich (2019) is the pioneer for these field, which corrupts the ImageNet Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2009)’s test set with sim- ulated corruptions such as motion blur, adverse weather and noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, ObjectNet Barbu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) build a benchmark with diverse corrup- tions in rotation, background and viewpoint, and ImageNetV2 Recht, Roelofs, Schmidt, and Shankar (2019) follow ImageNet and re-collects a test set to benchmark the robustness against natural distribu- tion shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recently, ImageNet-A and ImageNet-R are proposed by Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021), which bench- marks classifier’s robustness against natural adver- sarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since ImageNet is initially pro- posed for diverse tasks, there also exists preliminary attempts to benchmark the robustness of model trained on ImageNet to other downstream tasks, such as semantic segmentation Kamann and Rother (2020), instance segmentation Altindis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) and object detection Yamada and Otani (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In the field of autonomous driving, there are also existing works producing corruptions on Cityscapes Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2016), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', investigating models’ robustness against adverse weathers Porav, Musat, Bruls, and New- man (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sakaridis, Dai, and Van Gool (2018) or other corruptions Michaelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recently, ACDC Sakaridis, Dai, and Van Gool (2021) dataset Springer Nature 2021 LATEX template Robust LiDAR segmentation 5 Table 1 Categories and descriptions of corruptions in SemanticKITTI-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We categorize common LiDAR corruptions into three domains: (1) adverse weather conditions, (2) measurement noise and (3) cross-device discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Corruption (C) Intensity (I) Description (1) Fog Simulation Light Fog simulation with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='005 Moderate Fog simulation with beta β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06 Heavy Fog simulation with beta β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 (1) Snowfall Simulation Light Snowfall simulation with snowfall rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5mm/h Moderate Snowfall simulation with snowfall rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5mm/h Heavy Snowfall simulation with snowfall rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5mm/h (2) Global Outliers Light 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1% extra noisy points uniformly in the 3D space Moderate 5% extra noisy points uniformly in the 3D space Heavy 50% extra noisy points uniformly in the 3D space (2) Local Distortion Light 20% points with randomly jitter distortion σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05 Moderate 20% points with randomly jitter distortion σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 Heavy 20% points with randomly jitter distortion σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 (3) Cross 32-beam Device Dense Reduce LiDAR beams to 32 Sparse Reduce LiDAR beams to 32, sample 1/2 points in each beam (3) Cross 16-beam Device Dense Reduce LiDAR beams to 16 Sparse Reduce LiDAR beams to 16, sample 1/2 points in each beam collects four common adverse conditions in self- driving, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', fog, nighttime, rain, and snow, evaluating the models’ robustness against these real-world cor- ruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, since the difference between 2D- 3D data and model architecture, there is still huge demands of a comprehensive 3D robustness bench- mark for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 3D Robustness Benchmarks In the field of autonomous driving, there lacks a robustness benchmark for LiDAR semantic segmen- tation to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Existing surveys mostly focus on the point cloud classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, Xiao and Wachs (2021) and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang, Hua, and Yeung (2022) propose disturbance and rotation invariant feature extraction, however, they cannot achieve state-of-the-art performance on the clean dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Other works boost models’ robust- ness against adversarial corruptions by denoising and upsampling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019), voting on subsampled point clouds H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Liu, Jia, and Gong (2021), and apply- ing local relative position Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' There are robustness benchmarks for point cloud classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, RobustPointSet Taghanaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and PointCloud-C Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) evaluate the robustness of point cloud classifiers under different corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, these approaches test robust- ness against corruptions purely on synthesis dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', ModelNet40, and thus the obtained experience and conclusions are often unreliable in real-world self-driving applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' There are also investigations to improve the robustness on LiDAR scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' PointASNL Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) proposes adaptive sampling, which adap- tive shifts the outlier points onto objects’ surfaces, and thus boosts the robustness against noisy point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In recent year, there are studies investigating perfor- mance of object detector in different adverse weathers, where they aim at mitigating the rarity of adverse weather effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, Hahner, Sakaridis, Dai, and Van Gool (2021) and Hahner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022b) independently propose fog and snowfall simulation, greatly boosting the robustness of object detection models on real-world adverse weathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, there is no real-world or simulated adverse weather data set for LiDAR semantic segmentation at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, there exists preliminary attempts to investi- gate the robustness issue of the fusion methods for 3D object detection Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Concretely, TransFusion Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) evaluates the robustness of different fusion strategies under several scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', daytime and nighttime, DeepFusion Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) test the model robustness by adding noise to LiDAR reflec- tions and camera pixels and Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) proposes a robust benchmark for LiDAR-camera fusion, which analyzes seven cases of robustness scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 6 Robust LiDAR segmentation No Fog Moderate Fog Heavy Fog Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2 Corruption of fog simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We demonstrate the raw LiDAR point cloud in the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The foggy point clouds with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 are shown in the last two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The point cloud is color coded by the height (z value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The best viewed on a screen and zoomed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' By contrast, we rigorously investigate the LiDAR system and identify three categories, in a total of 16 LiDAR corruptions for semantic segmentation, and develop a toolkit that transforms the existing dataset into a robustness benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We hope our study can boost future research to benchmark the robustness, and give researchers more insights about designing a robust semantic segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3 Corruptions Taxonomy Real-world LiDAR scans can suffer from a wide range of corruptions, based on which we provide a taxonomy of the corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In this paper, we categorize common LiDAR corruptions into three domains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', adverse weather conditions, measure- ment noise and cross-device discrepancy, in which we produce total six corruptions with 16 severity lev- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' By applying these six types of corruptions to SemanticKITTI Behley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019), we generate a corrupted dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', SemanticKITTI-C, which is summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In the remaining sections, we will introduce each corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' A point cloud P is a set of points {pj}N j=1, where N is the number of points and pj ∈ R3 includes the XYZ coordinates of the point j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' A corruption No Snowfall Moderate Snowfall Heavy Snowfall Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3 Corruption of snowfall simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We demonstrate the raw LiDAR point cloud in the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The snowfall point clouds with snowfall rates 1mm/h and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5mm/h are illustrated in the last two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The point cloud is color coded by the height (z value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The best viewed on a screen and zoomed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' operation is defined as a set-to-set function: F : RN×(3+D) �→ RN ′×(3+D), (1) which maps the clean point cloud P = {pj}N j=1 and its D-dimensional features (if exist) to corrupted ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', P′ = {pj}N ′ j=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For LiDAR point cloud, each pj is associated with an intensity value ij ∈ R, indicating the return strength of a laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In this paper, the intensity is utilized to generate corrupted data, but we do not purely investigate the corruption of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 Adverse Weather In this section, we analyze two common weather con- ditions, namely fog and snowfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For fog simulation, we follow Hahner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) to add fog to clean- weather point clouds by disturbing points’ positions and intensities according to physically valid rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, for a point p ∈ R3 captured in the clean weather, we first calculate its attenuated response ihard in fog: ihard = i × exp (−2α × ∥(x, y, z)∥) , (2) where (x, y, z) is p’s coordinate in the ego frame and i is its measured intensity, α is the attenuation coefficient in foggy weather, ∥(x, y, z)∥ denotes the Springer Nature 2021 LATEX template Robust LiDAR segmentation 7 distance between the point p and the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Following Hahner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021), we uniformly sample α from [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06] when applying fog simu- lation to each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, we use the simulation terms in Hahner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) to compute the max- imum fog response isoft and its location (xs, ys, zs), which lies in the line connecting the sensor and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Note that the magnitude of isoft is controlled by a backscat- tering coefficient β, which is manually set during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1, we choose β from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2] to conduct fog simulation with dif- ferent levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, the updated point position and its intensity are given by: i = � isoft if isoft > ihard, ihard otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (3) (x, y, z) = � (xs, ys, zs) if isoft > ihard, (x, y, z) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (4) In other words, if the fog is thick enough to over- shadow the solid object point p (isoft > ihard), we use the fog response (the intensity and position) to replace the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Otherwise, we keep the position of the original response with an attenuated intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The overall idea of this snowfall simulation is sim- ilar to that of the fog simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' But unlike fog that homogeneously spreads in the 3D space, snowflakes are treated as opaque particles and are discretely dis- tributed in space without intersecting with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For snowfall simulation, we follow Hahner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022b) to sample snow particles for each LiDAR line and use them to modify the return for each LiDAR beam accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The sampling function samples snow particles according to a given snowfall rate (mm/h), which controls the number of particles in a certain range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1, we separately set the snowfall rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 to simulate light/mod- erate/heavy snowfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The above two weathers have different characteris- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, there are large areas of noisy points distributed around the sensor in foggy weather, and makes objects sparser due to the occlusion, especially for the remote objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2, the number of these noisy points grows as the intensity of fog increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, the noise introduced by fog is not uniformly distributed around the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The pres- ence of noise depends on whether there is any object in the line of sight below a certain range from the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Generally, there will be few spurious returns from the respective pulses if a solid object exists at a moderate range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Inversely, if there is no object in a certain range, there are a lot of spurious returns that are caused by fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As for the snowfall, there are two explicit characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' On the one hand, the snow par- ticles are explicitly modeled as opaque spheres, whose sizes are controlled by the snowfall rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3, compared with foggy LiDAR where the noisy points are almost around the sensor, the noisy points in snowfall conditions are distributed more uni- formly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Also, the snowfall rate does not greatly affect the number of noisy points but the size of snowy par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' On the other hand, wetness on the ground will exist in snowfall, where the emerging thin water layer increases the specular component of reflection by the ground surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To sum up, these two corruptions impact the models through global noisy points, mak- ing remote points sparser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Nevertheless, they generally have different patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 Measurement Noise Besides adverse weather, noises may also appear when corruption occurs during the data transmission or the sensors fail to capture information properly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', over- heated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We model such data disturbance using two types of random noises, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Note that we only consider point coordinates during such corrupting operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Global outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We randomly sample noises in a unit sphere and then merge them into a clean point cloud with proper rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Such noises span the whole scene globally and are not conditional on the geome- try of the clean point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Formally, given the clean point cloud P ∈ RN×3, the corrupted point cloud P′ ∈ RN ′×3 is obtained by: P′ = P ∪ Pnoise, (5) where Pnoise ∈ RN g×3 denotes sampled noises and N ′ = N g + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We control the noise intensity by selecting the proportion of noises N g N from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1%, 5%, 50%] as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Local distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We randomly select some points within a scene and add Gaussian noises to their coor- dinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Unlike global noises, which add additional points to a scene, local noises do not change the number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Compared to global noises, local noises jitter around a local neighborhood of the orig- inal points, mimicking the noisy disturbance during the data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Formally, the local distortion point Springer Nature 2021 LATEX template 8 Robust LiDAR segmentation No Noise Global Outliers Local Distortion noisy data clean data zoom-out zoom-out zoom-in noisy data zoom-in clean data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4 Noisy LiDAR point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We demonstrate the raw LiDAR point cloud in the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The noisy point clouds with global outliers and local distortion are shown in the last two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The point cloud is color coded by the height (z value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The best viewed on a screen and zoomed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' cloud P′ ∈ RN×3 is given by: Psub = RandomSample(P, N l), P′ = (Psub + O) ∪ (P \\ Psub), (6) where RandomSample(·) randomly sample N l points from the clean point cloud P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' O ∈ RN l×3 denotes the random offsets sampled from a Gaussian distribution N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' + and \\ are element-wise addition and set exclusion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1, we choose σ from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2] to control the jittering range in three different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The proportion of noises is 20% in SemanticKITTI-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 Cross-Device Discrepancy An ideal segmentation algorithm is supposed to be robust across different devices with various specifica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' While multiple factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', beam number and scanning speed of the LiDAR sensors) cause cross- device domain shifts, we focus on the beam number in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To ensure high-quality data annotation, most large-scale datasets Geiger, Lenz, and Urtasun (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) are collected using high- resolution LiDARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, due to prohibitive costs, most practical vehicles are only shipped with low- beam sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, KITTI Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2012) 64-beam 32-beam 16-beam Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5 Cross-device LiDAR point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We demonstrate the 64- beam LiDAR point cloud in the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The second and third rows illustrate the 32-beam and 16-beam LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The point cloud is color coded by the height (z value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The best viewed on a screen and zoomed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' collects data through 64-beam LiDAR and each beam contains 1863 points in average, while those param- eters in NuScenes Caesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) are 32-beam and 1084 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' This suggests that an ideal seg- mentation model should be able to robust to differ- ent data distributions generated by different sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Unlike other factors introduced in previous subsec- tions, the beam-induced domain gap is directly caused by the cross-device discrepancy instead of the collect- ing environment, making it also very important in our robustness analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To include the beam-based cross-device discrep- ancy in our benchmark dataset, we downsample the high-beam data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', 64-beam) to low-beam data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='., 16-, 32-beam) using beam-level downsam- pling as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' One necessary information needed for beam-level downsampling is the beam label for each point, which is usually unknown for most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To this end, we first assign a beam label to each point according to its zenith value in the spherical coordinate, which can be obtained via the following conversion: θ = arctan z � x2 + y2 , φ = arcsin y � x2 + y2 , (7) Springer Nature 2021 LATEX template Robust LiDAR segmentation 9 Table 2 LiDAR semantic segmentation approaches on our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Mainstream Method Main representation Extra representation Reference Projection-based SalsaNext Cortinhal, Tzelepis, and Aksoy (2020) Range image ArXiv 2020 PolarNet Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) BEV image CVPR 2020 CENet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng, Han, and Xiao (2022) Range image ICME 2022 GFNet Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) Range and BEV images Point cloud TMLR 2022 Point-based KPConv Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) Point cloud ICCV 2019 RandLANet Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) Point cloud CVPR 2021 Point Transformer Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) Point cloud ICCV 2021 Voxel-based MinkowskiNet Choy, Gwak, and Savarese (2019) Grid voxel CVPR 2019 SPVCNN Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) Grid voxel Point cloud ECCV 2020 Cylinder3D Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) Cylinder voxel Point cloud CVPR 2021 2DPASS Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) Grid voxel Point cloud ECCV 2022 where (x, y, z) is the Cartesian coordinate of the point and the θ and φ are zenith and azimuth angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Fol- lowing Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022), we obtain beam labels by applying K-Means clustering on the zenith angles, where the number of clusters is set as the actual beam number of the high-beam point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Compared to assigning beam labels by putting zenith angles into evenly distributed bins, the clustering-based technique does not require a pre-define zenith range and thus is more robust across different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For a high- beam point cloud with the beam labels, we can easily downsample it into data with any lower beam num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In practice, we downsample point clouds with the beam numbers of 32 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To simulate the diverse spinning speeds of the LiDAR devices, we evenly downsample points in each beam according to their azimuth angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' By combining the above simulation, we have four corrupted data generated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4 Candidate Methods We benchmark 11 existing methods for LiDAR semantic segmentation, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Though we treat hybrid-representation methods as an indepen- dent mainstream in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1, current voxel-based and projection-based methods widely incorporate addi- tional representations for auxiliary learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' There- fore, we categorize them only according to their main input representation, and the extra-representation will be illustrated if existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 Projection-based Methods In this paper, we choose SalsaNext Cortinhal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020), PolarNet Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020), CENet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='- X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) and GFNet Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) as the typical approaches of the projection-base method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' These models project a LiDAR point cloud into 2D images and apply a 2D convolutional neural network for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Among the above meth- ods, SalsaNext Cortinhal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and CENet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='- X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) conduct sphere projection to gain range views (RV), PolarNet Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) adopts polar projection to obtain bird’s-eye- view (BEV) under a polar coordinate system, and GFNet Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) uses the both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sphere projection for range-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We denote N the number of points in the LiDAR point cloud and (H, W) are the height and width of the projected image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6, the spherical projection maps each point to an image coordinate via �ur vr � = � 1 2[1 − arctan(y, x)π−1]W [1 − arsin(zr−1 + fovup)fov−1]H � , (8) where pi = (x, y, z) and (ur, vr) are the i-th point and its coordinates on the range image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' r is the range of each point � x2 + y2 + z2 and fov = fovup + fovdown is the vertical field-of-view of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, the LiDAR point cloud is converted to a range image with the shape of (H × W × C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The channel C is generally 5, including x, y, z, intensity and range of the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Polar projection for bird’s-eye-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Existing meth- ods also project the LiDAR point cloud to the bird’s- eye-view (BEV) through a top-down orthogonal pro- jection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Considering the imbalanced spatial distribu- tion in LiDAR data, polar projection first transforms the BEV from the Cartesian system into a polar coor- dinate system through �up vp � = �� x2 + y2 + z2cos(arctan(y, x)) � x2 + y2 + z2sin(arctan(y, x)) � , (9) where (up, vp) is the coordinate transformation from the Cartesian system to the polar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, Springer Nature 2021 LATEX template 10 Robust LiDAR segmentation projected raw point cloud raw point cloud semantic segmentation projected semantic segmentation spherical projection re-projection U-Net architecture (x,y,z) (ur,vr) skip connection Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6 Projection-based methods through spherical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' This kind of methods first conduct spherical projection maps each point to an image coordinate, and then adopt 2D convolution to construct a U-Net-like architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, they re-project the prediction on the image plane onto the raw point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' they discretize (up, vp) to [0, H − 1] and [0, W − 1] and obtain a BEV image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since the LiDAR point cloud is already mapped onto an image plane, typical 2D semantic segmentation networks can be directly adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, U-Nets Ronneberger, Fischer, and Brox (2015) with specific modifications are applied in previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1) Approaches with range images do not conduct pooling in height dimen- sion due to the large width-height ratio of the input, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2) SalsaNext utilizes an additional pixel-shuffle layer in the last encoder, and CENet con- ducts multiscale supervision in encoder and decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3) PolarNet applies a hybrid-representation manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', designing a PointHead (will be described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4) in the initial stage, which aggregates the features of original points into each BEV pixel, and finally conducts semantic segmentation through the 2D U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4) GFNet Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) has a two- branch architecture, where two U-Nets independently encode the features of range-view (RV) and bird’s- eye-view (BEV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' There are several Geometric Flow (GF) modules between their decoder layers with dif- ferent scales, which update each other’s features by fusing the features of both branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, it utilizes a hybrid-representation manner, aggregating the fea- tures of two branches in their last layers and feeding the fused feature into KPConv Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) to gain point-wise predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In our experiments, we adopt the official architectures of projection-based methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', SalsaNext1, CENet2, PolarNet3 and GFNet4), where we directly use their pre-trained checkpoint except for CENet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Note that CENet is trained with a 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/TiagoCortinhal/SalsaNext 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/huixiancheng/CENet 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/edwardzhou130/PolarSeg 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/haibo-qiu/GFNet points in (x,y) local points fused feature Local Aggregation 1st scale 2nd scale 3rd scale .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 7 Point-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (a) Point-based approaches sample the target points (in red color) in the original point cloud, and aggregate local features through local aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (b) Through hier- archical architecture, the receptive field of the point-based method increase aggressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' multi-stage strategy, which trains with 64 × 512 range image for the initial stage, and fine-tunes the pre- trained model aggressively on 64×1024 and 64×2048 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since the official codes only provide the check- point on 64 × 512 range images, we fine-tune the checkpoint on larger ones through provided configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In contrast, SalsaNext directly trains their model on 64 × 2048 range images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As for the PolarNet, it first crops points of the polar coordinate system in the range from [3, −π, -3] to [50, π, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5], and then dis- cretize points into a [480,360] BEV partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' GFNet combines RV and BEV representation, utilizing both 64 × 2048 range images and 480 × 360 BEV plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 Point-based Methods Point-based approaches aim at extracting features on raw point clouds directly, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In this paper, KPConv Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019), RandLA-Net Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and Point Transformer Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) are selected as our candidate methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, these methods first apply sampling approaches to select target points from the original point clouds, and then conduct local aggregation on each target point and mine local geometrics, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 7(a) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After constructing a hierarchical architecture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 7(b), point-based methods gain the global semantic infor- mation of the input point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' General formulation of local aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Let pi and fi denote the coordinate and feature of the i-th point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In general, for each pi, a local aggregation func- tion first transforms its neighbor pj with feature fj into a new feature by a transformation function T (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='), and then aggregates all transformed neighborhood features to generate an updated feature of ˆpi via an aggregation Springer Nature 2021 LATEX template Robust LiDAR segmentation 11 function A(·): ˆpi = A({T (pi, fi, pj, fj)} ∀j ∈ N(pi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (10) In practice, N(pi) represents the neighborhood index of point pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' According to the category to which the transformation function T (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=') belongs, previous local aggregation approaches can be roughly cate- gorized into four classes: 1) Point-wise MLP based, 2) Adaptive weight based, 3) Pseudo grid based and 4) Transformer based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The typical one of the first class is PointNet++ Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2017), where T and A are MLP((pj − pi) ⊕ fj) and max pooling respectively, in which ⊕ is concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, directly learn the 3D shapes through sim- ple point-wise MLP and pooling cannot work well in the LiDAR scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', it only achieves 20% mIoU on SemanticKITTI in previous studies Behley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Therefore, we did not adopt this kind of methods in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Adaptive weight based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The adaptive weight based methods design diverse convolution fil- ters over arbitrary relative positions, and hence com- putes weights on all neighbor points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' RandLA-Net Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) is a typical one in adaptive weight based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Concretely, its transformation function T can be represented as MLP(pi ⊕ pj ⊕ (pi − pj) ⊕ E(pi, pj))) ⊕ fj, (11) where E(·) calculates the Euclidean distance between the neighboring and center points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, it aggre- gates the neighboring features through attention mech- anism Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2017), which first calculates an attention weigh according to the feature, and then conducts weighted average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pseudo grid based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' KPConv Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) is a representative pseudo grid based method, which generates pseudo features on several sampled regular grid points, and thus regular convolution meth- ods can play a normal role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, it samples equally distributed spherical grid points in the 3D space, in which the pseudo features f p k on the k-th grid point can be calculated as f p k = � j∈N (pi) max(0, 1 − E(pi, pk) σ )fj, (12) where each grid point pk have strict mapping with the relative position to center point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' σ is a hyperparame- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, the transformation function T in pseudo grid based methods can be formulated as T (pi, fj) = wk ⊙ f p k, (13) where wk ∈ Rd×1 is a parametrized weight in convo- lution operator and defined on each grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, after applying max pooling as aggregation function A, it updates the feature of each target point through aggregating features in local neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Transformer based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Besides analyzing tra- ditional local aggregation based approaches, we also adopt recent transformer based method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Point Transformer Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021)) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The point transformer layer is based on vector self- attention, which uses the subtraction relation and there is a position encoding δ in both the attention vec- tor γ and the transformed features α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, in each local group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', ∀j ∈ N(pi)), the transforma- tion function T (pi, pj, fi, fj) in a transformer based method can be formulated as ρ(γ(ϕ(fi) − ψ(fj) + δ)) ⊙ (α(fj) + δ), (14) where ϕ, ψ, γ, α are independent MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' δ = MLP(pi−pj) is a positional encoding in self-attention, allowing the operator to adapt to local structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After updating the neighboring features, Point Transformer utilizes a summation function as aggregation function A to fuse features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' All above three methods follow the widely-used UNet-like encoder-decoder architecture with skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The LiDAR point cloud is first fed to a shared MLP layer to extract per-point fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Encoder and decoder layers are then used to learn features for each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, fully-connected layers are used to predict the semantic label of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' KPConv and RandLA-Net utilize stacked two corresponding local aggregation in each encoder layer, while Point Transformer using the combina- tion of point-wise MLP with point transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In decoder layers, all method interpolate the sampled points and update features through point-wise MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, RandLA-Net uses random sampling in each local aggregation, while other two utilizing uniformly sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For KPConv and RandLA-Net, we adopt their official architectures on SemanticKITTI dataset (four encoders and decoders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As for the Point Transformer, since it is only designed for indoor semantic segmentation, we adopt original architecture with five encoders and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 12 Robust LiDAR segmentation Configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' During the training, both KPConv5 and RandLA-Net6 follow their official configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To accelerate the training phrase, they first conduct grid sampling with grid size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06m to gain a small sub-cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, they respectively crop patches with a 4m radius and 50,000 points in each training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' During the inference, they inference through small patches util each of the points have been inferred three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since there are not published codes on SemanticKITTI for Point Transformer7, we utilize the same configurations as Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) during the training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 Voxel-based Methods Since voxel-based methods are the most popular mainstream for LiDAR semantic segmentation now, we select four methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', MinkowskiNet Choy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019), SPVCNN Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020), Cylin- der3D Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and 2DPASS Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022)) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Grid partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Voxel-based methods exploit vox- elization and transform the LiDAR point cloud into 3D voxels, such that the 3D convolutions can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, they shift all the points to the local coor- dinate system with the geometric center as the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Then, all the points are normalized into a unit sphere and scaled to the range of [0, 1], where the normal- ized coordinates are denoted as ˆP = {(ˆxi, ˆyi, ˆzi)}N i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, they transform the normalized point cloud to a voxel representation with voxel size vs (f ∗ m is voxelized feature representation): p∗ i = (x∗ i , y∗ i , z∗ i ) = (⌊ˆxi/vs⌋, ⌊ˆyi/vs⌋, ⌊ˆzi/vs⌋), f ∗ m = 1 Nm N � i=1 I[x∗ i = ˆxm, y∗ i = ˆym, z∗ i = ˆzm] · pi, (15) where ⌊·⌋ is the floor function, and I(·) is a binary indi- cator of whether p∗ i belongs to the m-th voxel grid or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Nm is the number of points in the m-th voxel, and the original point coordinates are averaged as the fea- tures of each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After the operations in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (15), only the non-empty voxels are preserved (Nm > 0) in a hash table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The, the convolution operation only con- ducts on the non-empty voxels, thus maintaining the computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/HuguesTHOMAS/KPConv-PyTorch 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/QingyongHu/RandLA-Net 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/POSTECH-CVLab/point-transformer Cylindrical partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recent study Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) proposes cylinder partition for voxelization, which makes a higher non-empty proportion and more balanced point distribution compared with grid parti- tion, especially for farther-away regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In practice, it first transforms the Cartesian system into a polar coordinate system through Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (9), and then conducts voxelization as Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Both MinkowskiNet and SPVCNN8 utilize the same U-Net architecture, where the dif- ference is that there is a parallel point-wise MLP branch in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cylinder3D9 proposes asymmet- rical 3D convolution networks, in which it constructs several asymmetrical blocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', exploiting 3×3×1, 3 × 1 × 3 and 1 × 3 × 3 kernels in parallel) as unit components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2DPASS10 uses a similar encoder archi- tecture as SPVCNN, but it discards the decoder part and predicts the results through multiscale concate- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, Cylinder3D and 2DPASS exploit additional a PointHead (will be introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4) to aggregate point-wise features into each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Both Cylinder3D and 2DPASS are tested with their released checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As for the MinkowskiNet and SPVCNN, we re-trained their offi- cial architectures with batch size 8 for epoch 64, and gain higher results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' All approaches are tested with test- time augmentation (TTA), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', rotating the point cloud with 12 views and averaging the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5 Benchmarking and Analysis In this section, we benchmark the aforementioned 11 approaches with our diverse set of LiDAR corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We first introduce the experiment setting and eval- uation metrics of our benchmark in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, the benchmark results are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 with comprehensive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We demonstrate our benchmark results spanning different representation, architecture, corruption intensity and data augmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As results, we summarize 12 observations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6, we introduce RLSeg, a robust architecture based on the above observations, which effectively boosts the robustness of LiDAR semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/mit-han-lab/spvnas 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/xinge008/Cylinder3D 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='com/yanx27/2DPASS Springer Nature 2021 LATEX template Robust LiDAR segmentation 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 Experiment Setting Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' SemanticKITTI is currently the most widely used LiDAR semantic segmentation dataset, which consists of 43,552 densely annotated LiDAR scans belonging to 21 sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' These scans are annotated with a total of 19 valid classes, and each scan spans up to 160 × 160 × 20 meters with more than ∼ 105 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Initially, the sequence 00 to 07, 09 to 10 are the training set, 11 to 21 are the test set, and 08 is the val- idation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since the annotations of 11 to 21 are not available offline, we train all approaches on training set and evaluate them on sequence 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Annotation modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Since the corrupted point clouds will be sparser or there are new noisy points existed, we slightly modify the original annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, for the corruption data in fog and snow- fall simulations, we utilize the noisy data to query the nearest neighbor of the clean data within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='02m to annotate labels, labeling points as ‘ignore’ if there is no neighbor existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In the noisy corruption (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', local and global), we directly annotate noisy points as ‘ignore’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In the cross-device scenario, since all the points are sampled from the original LiDAR, there is no demand for modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Note that the ‘ignore’ class is not considered in the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To intuitively demonstrate the robustness of candidate methods, we use the perfor- mance on each corruption and the relative perfor- mance degradation compared to the clean data on our benchmark datasets as our evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specif- ically, we adopt mIoU(%) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', averaged Intersection over Union on each class) as our metric and the score on the clean dataset is denoted as S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As demonstrated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1, we benchmark existing methods with six categories of corruptions (C), spanning 16 different intensities (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The performance toward certain corrup- tion c ∈ C can be calculated by averaging results on each intensity: Sc = � i∈I(c) Sc i /N(c), (16) where I(c) and N(c) are total intensities and the num- ber of intensity in the corruption c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sc i denotes the mIoU under corruption c and intensity i and Sc are the averaged mIoU of all intensities under the corrup- tion c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The relative mean robustness performance of the model is defined as Rc = Sc/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The higher R means the model is more robust to inferior LiDAR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, we define a robustness mIoU clean data jitter data foggy data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 8 Illustration of observation-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We show visualization results to better explain why the projection-based methods are vulnerable to fog simulation but robust to local deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (RmIoU) and averaged relative performance (mR) through averaging the results on different corruption: RmIoU = � c∈C Sc/6, mR = RmIoU/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (17) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 Main Results Benchmark results are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3, in which projection-based, point-based and voxel-based meth- ods are demonstrated in upper, median and lower parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Our proposed solution will be introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' According to the table, we have the following discovery: Observation-1: Projection-based methods are most vulnerable to common corruptions, especially to foggy simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, they are greatly robust to local distortion corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in the table, existing projection-based methods achieve around 75% metric of mR on SemanticKITTI-C, which is much lower than those of point-based and voxel-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Furthermore, they only achieve around 50% original performance in the corruption of fog simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In contrast, these methods are extremely robust to local noise, especially for the pure range image based method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1% and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0% R for SalsaNext Cortinhal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and CENet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022), respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The reason of the above observation is that the local corruption only slightly affects the range image and the foggy one make range images messy, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' PolarNet Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and GFNet Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) are not much robust to local corrup- tion, since there are BEV projections in their models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Nevertheless, they still respectively keep 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7% and Springer Nature 2021 LATEX template 14 Robust LiDAR segmentation Table 3 Benchmarking the robustness of state-of-the-art methods in all 16 scenarios (6 classes) on SemanticKITTI-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' R denotes the relative mean robustness performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The higher R means the model is more robust to inferior LiDAR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Clean Robustness Fog Snowfall Global Outliers Local Distortion 32-beam 16-beam Method mIoU RmIoU mR mIoU R mIoU R mIoU R mIoU R mIoU R mIoU R Projection SalsaNext 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 PolarNet 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 74.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 CENet 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 47.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 GFNet 63.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 SalsaNext KPConv MinkowskiNet mIoU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 9 Detailed results in fog simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Performance of three typical approaches in different fog simulation intensities, where the x-axis denotes different β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9% performance, which is much higher than those of point and voxel based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Observation-2: Traditional point-based methods (RandLA-Net and KPConv) are more robust to com- mon corruptions, compared with projection-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, they are much robust to adverse weathers, but less robust to local distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The table illustrates RandLA-Net Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) and KPConv Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) respectively achieve 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3% and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4% in the metric of mR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Espe- cially, they gain highest 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9% and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4% R in fog simulation, where the best projection and voxel based methods only achieve 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5% and 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Inversely, they only gain 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1% and 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7% perfor- mance in local noise, which is lower than common performance of other two mainstreams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The reason is that the local distortion protects the local geometric and thus makes the local aggregation failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Observation-3: Transformer-based local aggregation greatly hampers the robustness, especially for global outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Though transformer-based architecture improve the performance on the clean data, it greatly affects the robustness against diverse corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Concretely, Point Transformer Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) gains the low- est result in almost every corruption scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For instance, in the global outliers, most of the approaches can keep above 90% performance, but it only achieves 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5% in the metric of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Observation-4: Pure voxel-based method shows most superior robustness cross all corruptions, especially for cross-devices scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recent state-of-the-art voxel-based methods loss their robustness against cor- ruption due to their hybrid-representation architec- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The results illustrate that MinkowskiNet Choy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019) enjoy most superior robustness with pure 0 10 20 30 40 50 60 70 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 SalsaNext KPConv MinkowskiNet mIoU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 10 Detailed results in snow simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Performance of three typical approaches in different snow simulation intensities, where the x-axis denotes different snowfall rates (mm/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' voxel architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, it achieves 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0% per- formance preserve in 16-beam cross-device scenario, surpassing those of projection and point based meth- ods over 20% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' SPVCNN Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020) introduces point-wise MLP in par- allel with voxel architecture, nevertheless, it loses the robustness especially in fog simulation and 16- beam cross-device corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Recent state-of-the-art Cylinder3D Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) and 2DPASS Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) have poor generalization ability since they use extra representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' More analysis for this design will be illustrated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 Robustness in Specific Corruption In this section, we demonstrate and analyze the result of each specific corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To facilitate the experi- ment, we only select the most typical method in each mainstream, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', SalsaNext, KPConv and Minkowsk- iNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Fog simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 9 illustrates the comprehensive results of robustness in fog simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Apart from the three intensities related in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2, we also provide other 6 intensities, including 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We find out that SalsaNext is greatly affected by denser fog, especially when β is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Inversely, KPConv shows its superior robustness crossing different fog intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Therefore, we have the following summary: Observation-5: All types of methods will have perfor- mance decay as the fog becomes heavier, among which the projection-based method decreases fastest, and the point-based method decreases slowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Snow simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 10 illustrates the detailed results of snow simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The conclusion is: Observation-6: Though snow simulation hampers all types of methods, the performances of point-based Springer Nature 2021 LATEX template 16 Robust LiDAR segmentation Table 4 Comprehensive results on diverse noisy corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Noise types Ratio (%) SalsaNext KPConv MinkowskiNet mIoU R mIoU R mIoU R No corruption 0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 100.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 Table 5 Comprehensive results on different LiDAR types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In ‘sparseness’, we see the original LiDAR point as ‘dense’ one, and randomly sample 1/2 points in each beam to generate a ‘sparse’ one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' SalsaNext KPConv MinkowskiNet LiDAR types Sparseness mIoU R mIoU R mIoU R 64-beam Dense 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 Sparse 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 32-beam Dense 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 Sparse 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 16-beam Dense 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 Sparse 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 and voxel-based methods are slightly decreased as the snow becomes heavier, while the projection-based methods illustrate an inverse tendency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', there are slight performance boosts in heavier snowfall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The reason may be that the large snowfall makes the scattered noise in the 3D space cover a larger area, thus affecting the point-based and voxel-base meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, when the 3D scene is mapped to a range image, these scattered points with larger coverage will become sparse in each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Noisy corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Results in different noisy corrup- tions are illustrated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' On one hand, Minkowsk- iNet is the most robust method against global out- liers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' It even keeps 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8% performance in the sce- nario with additional 50% global noisy points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' On the opposite, SalsaNext has poor generalization abil- ity for global noise, especially with a larger proportion of noisy points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' On the other hand, the projection- based method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', SalsaNext) shows great robustness in local distortion noises, spanning different jittering ranges, as summarized in observation-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In contrast, KPConv and MinkowskiNet cannot work normally in large-range jittering distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cross-device discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We demonstrate concrete results on different LiDAR types in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5, where MinkowskiNet achieves the best results in all cross- device scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In contrast, SalsaNext has poor robustness, especially in 16-beam devices with only around 50% original performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Furthermore, there is an interesting discovery: Observation-7: Point-based methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', KPConv) are greatly robust against the scenario of downsam- pling points in each LiDAR beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As illustrated in the table, removing 1/2 points in each beam nearly does not affect the performance Springer Nature 2021 LATEX template Robust LiDAR segmentation 17 of KPConv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, in both 16 and 32-beam devices, the performance in sparse cases is greatly similar to the dense ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' This achievement may come from the sampling process in point-based meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Compared with projection-based and voxel-based methods that down-scale feature maps through pool- ing or convolution operations, point-based methods reduce the point numbers through sampling strategies, as depicted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Therefore, the local aggre- gations in point-based methods are more robust to downsampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 Model Design v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Robustness In this section, we comprehensively analyze the rela- tionships between different model designs and robust- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The results are illustrated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Size of range image (projection-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in Analysis A of the table, we train CENet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022) with different sizes of range image, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', 512 × 64, 1024 × 64 and 2048 × 64), and gain the following conclusion: Observation-8: Exploiting smaller image size in range projection will make the model more vulnerable to noise, except for cross-device scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' When adopting 512 × 64 range image as input, the performance of CENet decreases from 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6% R to 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4% R, especially in snowfall simulation with 14% robustness drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Moreover, its mIoU dramati- cally decays from 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4% to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5% in LiDAR date with local noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' On the opposite, adopting smaller range images improves the robustness when deploy- ing the model in devices with smaller beam numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We believe that the reason for the above phenomenon is that small images will make more points gather in the same pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Thus, noise points can easily enlarge the proportion of contaminated pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, in the LIDAR point cloud with a smaller beam number, a smaller image size makes the density of the valid pixels still quite high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Local aggregation (point-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' For the point- based approach, we select RandLA-Net as a typical one and conduct an ablation study, as shown in Anal- ysis B of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, we first replace the attentive pooling with max pooling in the second line, and replace the transformation function T (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (11)) to naive point-wise MLPs in the third line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The results show that both two components greatly improve the performance and the generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After dis- carding the two components, the ablated model can only achieve 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 mIoU on clean data and keep 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7% performance in the common corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Voxel partition (voxel-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To further study the effectiveness of different voxel partitions, we conduct experiments and illustrate the results in Analysis C of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' During the experiments, we change the voxel partition of MinkowskiNet to the cylinder one and keep the network architecture the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Con- cretely, we first transform the LiDAR point cloud from the Cartesian system into a polar coordinate system through Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, we discretize the trans- formed LiDAR data with voxel size [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='001π, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05] in corresponding axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, the following summary can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Observation-9: Cylindrical partition in voxelization greatly improves the robustness in most of the corrup- tion, except in cross-device LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, after exploiting cylindrical voxeliza- tion, the models’ robustness in fog, snow and local corruptions are increased by around 10%, 17% and 16%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, such improvement is only for out-of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The performance of the model for clean data is dropped to 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 mIoU (a drop of about 6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Similarly, its perfor- mance in the cross-device deployment scenario is also affected, especially the robustness on 16-beam LiDAR is reduced by 26%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Nevertheless, robustness in dif- ferent voxelization is still an important discovery in this paper, and it also lays a foundation to propose our newly configured method RLSeg in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Voxel size (voxel-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' There are also experiments to study the robustness through different voxel sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The results are shown in Analysis D of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Observation-10: Larger voxel partition makes voxel- based approaches more vulnerable to global-level corruptions, such as adverse weathers and global out- liers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, the robustness against local distortion and cross-devices point clouds is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In the experiment, we apply a larger voxel parti- tion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1m and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2m), compared with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05 as the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The results show that this setting greatly affects the robustness against global-level corruption, espe- cially for the global noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' More importantly, utilizing a small voxel size makes the model cannot achieve sat- isfactory performance on clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The reason is that a large grid makes the model merge noisy and origi- nal points into the same grids, and loses fine-grained information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hybrid-representation architecture (voxel-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In Analysis E of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6, we investigate the relation- ship between robustness and hybrid-representation Springer Nature 2021 LATEX template 18 Robust LiDAR segmentation Table 6 Systematic analysis for input representation, architecture design and data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The analysis includes (A) architecture for projection-based methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (B) architecture for point-based methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (C-D) representation for voxel-based methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (E) hybrid-representation architecture and (F) data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The baseline models are marked with underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The increase and decrease compared with baselines are denoted as ↑ and ↓, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Clean Robustness Fog Snowfall Global Outliers Local Distortion 32-beam 16-beam Analysis Method Descriptions mIoU RmIoU mR mIoU R mIoU R mIoU R mIoU R mIoU R mIoU R A CENet (2048× 64) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 CENet (1024× 64) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↑ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↑ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↓ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↓ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↑ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↑ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↑ CENet (512× 64) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↓ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↓ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ B RandLA-Net 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 56.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 w/o Attentive Pooling 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↓ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↓ 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↑ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↑ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ Point-wise MLP 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↓ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↓ 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 ↓ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↓ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↓ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ C MinkowskiNet (grid, 5cm) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 MinkowskiNet (cylinder) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↑ 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↑ 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↑ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↑ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↑ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↓ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↓ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ D MinkowskiNet (10cm) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↓ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↓ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↓ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 ↑ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 ↑ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↑ MinkowskiNet (20cm) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 ↓ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↓ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 ↓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↑ E 2DPASS 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 2DPASS w/o PointHead 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↑ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↑ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↑ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↓ 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↑ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↓ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↓ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↓ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↑ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↑ 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 ↑ 2DPASS w/o PointBranch 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↓ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↑ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↓ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↓ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↑ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 ↑ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↑ F MinkowskiNet + InsCutMix 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↑ 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↑ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↑ 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='1 ↓ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2 ↓ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↓ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↓ MinkowskiNet + Mix3D 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↑ 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↑ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↓ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↑ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 ↑ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↑ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 ↑ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↑ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 ↑ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 ↑ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 ↓ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 ↓ Springer Nature 2021 LATEX template Robust LiDAR segmentation 19 point cloud (a) point-wise MLPs pooling voxels (b) Point-wise MLPs 3D Sparse Convolution Fusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' voxel network encoder/decoder layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 11 Illustration of hybrid-representation architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The architecture of PointHead and PointBranch are shown in (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3, current state- of-the-arts adopt hybrid-representation architecture to boost the in-domain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Concretely, there are two components that merging point-wise repre- sentation into the voxel one, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (a) PointHead: exploiting a PointNet architecture to aggregate point-wise features into individual voxel grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (b) PointBranch: extracting point-wise features in parallel, and merging the features from voxel archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Similar components can be found in other previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Cylinder3D and SPVCNN), but here we only conduct ablation on the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Observation-11: Although hybrid-representation architectures improve the performance for the in- domain LiDAR segmentation with clean data, they are detrimental to model robustness, especially when using the PointHead component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6, both PointHead and Point- Branch boost the performance of 2DPASS on the clean LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, when the PointHead is exploited, there is a dramatic decrease in the robust- ness, especially in fog simulation and 16-beam device with 10% and 5% robustness drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Similarly, Point- Branch also hampers the robustness, but the influence is much slight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Directly conducting point-wise MLPs on LiDAR points is easier affected by diverse corrup- tions, since it cannot capture local geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 Data Augmentation v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Robustness In this section, we investigate the robustness of the model with different data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' General augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Previous studies adopt diverse data augmentation during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Generally, rotation and scaling are the most widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In this paper, we conduct rotation, scaling, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 12 Examples of LiDAR point cloud after applying Mix3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' flipping when re-training the point-based and voxel- based models, and follow the same image-based aug- mentation in projection-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Note that jitter augmentation is not used in our experiment, as it will generate in-domain training data for our local distortion corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' MixUp on LiDAR point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' MixUp H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang, Cisse, Dauphin, and Lopez-Paz (2017) is initially proposed in image classification for a more robust representation and extends to 3D computer vision in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Existing MixUp approaches in LiDAR semantic segmentation task include Mix3D Nekrasov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021), Instance CutMix Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021) and LaserMix Kong, Ren, Pan, and Liu (2022), where only Mix3D is open-sourced now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Therefore, we train MinkowskiNet with the official Mix3D, as well as the re-produced Instance CutMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The illustration of Mix3D is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 12, in which Mix3D randomly merges two LiDAR scans (including labels) into a common coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We re-produce Instance CutMix by only merging instance-level objects from other LiDAR scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The experimental results are demon- strated in Analysis F of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Observation-12: Existing MixUp data augmentation for LiDAR semantic segmentation makes the model more robust against most of the corruption, except for cross-device scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As illustrated in the table, after exploiting two MixUp augmentation, there are significant boosts in several corruptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', over 10% and 30% improve- ments on snowfall and local distortion corruptions with Mix3D augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, the robustness of the model decreases in cross-device LiDAR data, sesSpringer Nature 2021 LATEX template 20 Robust LiDAR segmentation Table 7 Ablation study for RLSeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' KD and PL denote knowledge distillation and pseudo label fine-tuning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Model Mix3D KD PL mIoU RmIoU mR Fog Snowfall Global Local 32-beam 16-beam MinkowskiNet 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 50.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5 RLSeg (ours) 63.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 Ground Truth Errors (MinkowskiNet) Errors (RLSeg) Fog (moderate) Local (moderate) 16-beam (dense) No Corruption Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 13 Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We demonstrate the visualization results of the most robust existing method (MinkowskiNet) and our RLSeg on four cases, including clean data and three LiDAR corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The noisy points are labeled as ‘ignore’ (black color) and not considered in the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The left two columns are colorized by error maps, and the last one is colorized by ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' especially in 16-beam data with larger domain dis- crepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The reason is that MixUp augmentation utilizes denser mixed point clouds as input, and thus makes the model vulnerable to the sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='6 Boosting Corruption Robustness Summarize the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5, we obtain 12 observations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In order to design a more robust model, we summarize the most useful information from these observations: 1) Voxel-based architecture: After summarizing observations 1-6, we prefer to use the voxel-based method as our backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The reason is that the methods of projection-based and point-based do not perform well on clean data, and both of them are Springer Nature 2021 LATEX template Robust LiDAR segmentation 21 Teacher MinkowskiNet Student RLSeg mixed data mixed prediction original data prediction KL divergence Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 14 Illustration of training process of RLSeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' vulnerable to certain corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In contrast, the voxel- based method performs well in most cases, and the results on clean data are also satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2) Cylindrical partition with appropriate voxel size: As shown in observation-9 and 10, exploit- ing cylindrical partition and appropriate voxel size increase the robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3) Single-representation: Observation-11 illus- trates that though hybrid-representation architecture improves the performance on clean data, it hampers the robustness against common corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4) Mix3D augmentation: This can be gained by observation-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Based on the above four conclusions, we design the robust LiDAR segmentation (RLSeg) model in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, we use MinkowskiNet as our backbone due to its single-representation nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Fur- thermore, we transform the LiDAR point cloud from the Cartesian system into a polar coordinate system through Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (9) and apply a cylindrical partition with voxel size [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='001π, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' However, the result in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6 shows that cylindrical partition hampers the performance on the clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Training with knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' To tackle this problem, we adopt the knowledge distillation Hinton, Vinyals, and Dean (2014) and self-training techniques to enhance the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, we first train a teacher MinkowskiNet model with grid voxel parti- tion, voxel size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05 and Mix3D augmentation, obtain- ing the model with 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='7% mIoU on clean data, as shown in the last row of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After that, we train the above RLSeg with a teacher-student framework, applying KL divergence to the output logits of RLS and the teacher MinkowskiNet, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' During the training, the Mix3D is only conducted on the student model, and the KL divergence constrains the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We train RLSeg with 64 epochs with a weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='05 for KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, motivated by the improvement achieved by pseudo label J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Li, Dai, and Ding (2022) in semi-supervised learning, we further fine-tune the student network 48 epochs on clean validation data with pseudo labels generated by the teacher MinkowskiNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Concrete results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Through such a simple but effec- tive manner, we significantly improve the performance on clean data, while keeping the robustness against diverse corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The results are demonstrated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3, where RLSeg significantly outperforms exist- ing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We analyze different designs through an ablation study in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' As shown in the table, exploiting our architecture improves the robustness, but causes a performance drop to 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After using knowledge distillation (KD) and Mix3D, there is a huge performance boost from 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 to 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='9, while increasing the robustness from 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='3 to 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, utilizing pseudo label fine-tuning can further improve the performance while keeping the robustness, which shows a promising improvement by leveraging the potential of semi-supervised learning, giving a per- formance boost to about 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' This also gives hints for future work that improve the robust but poor performance network to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We provide visualization results of our RLSeg and MinkowskiNet in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 13, in which our proposed model performs better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Specifically, MinkowskiNet cannot work normally in fog simulation and local distortion, and there are large areas of errors in the LiDAR scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' In contrast, our RLSeg provides robust prediction even for the small objects (as shown in red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' These show the robustness of our model as well as the promising future for robust LIDAR semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6 Conclusion In this paper, we propose a new benchmark called SemanticKITTI-C, with respect to real-world and out-of-domain LiDAR corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We systematically investigate a wide range of LiDAR semantic segmen- tation models, spanning different input representations and network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' After analyzing the results of previous approaches, we summarized 12 obser- vations for the future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Finally, we propose RLSeg based on the above observations, which effec- tively boosts the robustness of LiDAR semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' We hope our benchmark, comprehensive analysis and observations could boost future research for robust LiDAR semantic segmentation in safety- critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 22 Robust LiDAR segmentation References Alonso, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Riazuelo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Montesano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Murillo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3d-mininet: Learning a 2d represen- tation from point clouds for fast and efficient 3d lidar semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='10893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Altindis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dalva, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dundar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Benchmark- ing the robustness of instance segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='01123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Huang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Tai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='- L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Transfusion: Robust lidar-camera fusion for 3d object detection with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceed- ings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1090–1099).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Barbu, A.' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Katz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Advances in neural information processing systems, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Behley, J.' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Beijbom, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' nuscenes: A multi- modal dataset for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 11621–11631).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Han, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Xiao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cenet: Toward concise and efficient lidar semantic segmen- tation for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2022 ieee interna- tional conference on multimedia and expo (icme) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 01–06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cheng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Razani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Taghavi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Af2-s3net: Attentive feature fusion with adaptive fea- ture selection for sparse semantic segmentation net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 12547– 12556).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Choy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Gwak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Savarese, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4d spatio-temporal convnets: Minkowski convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3075–3084).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cordts, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Omran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Ramos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Rehfeld, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Enzweiler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Benenson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Schiele, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The cityscapes dataset for semantic urban scene under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' of the ieee conference on computer vision and pattern recognition (cvpr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cortinhal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Tzelepis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Aksoy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sal- sanext: Fast, uncertainty-aware semantic segmenta- tion of lidar point clouds for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='03653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Socher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Fei-Fei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Imagenet: A large-scale hierarchical image database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2009 ieee conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 248–255).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Dong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Hua, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Self-robust 3d point recognition via gather-vector guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2020 ieee/cvf conference on computer vision and pattern recognition (cvpr) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 11513–11521).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Engel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Belagiannis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dietmayer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Point transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' IEEE Access, 9, 134826–134840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Geiger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lenz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Urtasun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Are we ready for Autonomous Driving?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' The KITTI Vision Benchmark Suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' of the ieee conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' on computer vision and pattern recognition (cvpr) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3354–3361).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Graham, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Engelcke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', van der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3d semantic segmentation with submanifold sparse convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee con- ference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 9224–9232).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Graham, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', & van der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Submani- fold sparse convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='01307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hahner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sakaridis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zaech, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Semantic understanding of foggy scenes with purely synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Ieee international confer- ence on intelligent transportation systems (itsc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hahner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sakaridis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Bijelic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Heide, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' LiDAR Snowfall Simu- lation for Robust 3D Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Ieee/cvf con- ference on computer vision and pattern recognition (cvpr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hahner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sakaridis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Bijelic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Heide, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Lidar snowfall simula- tion for robust 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 16364–16374).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hahner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sakaridis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Fog simulation on real lidar point clouds for 3d object detection in adverse weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf international conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 15283–15292).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hendrycks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', & Dietterich, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Benchmarking neu- ral network robustness to common corruptions and perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='12261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Robust LiDAR segmentation 23 Hendrycks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Basart, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Steinhardt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Song, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Natural adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceed- ings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 15262–15271).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Vinyals, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Distilling the knowledge in a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' NeurIPS Workshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Khalid, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Xiao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Trigoni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Markham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sensaturban: Learning seman- tics from urban-scale photogrammetric point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' International Journal of Computer Vision, 130(2), 316–343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Xie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Rosa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Markham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Randla-net: Efficient semantic segmentation of large-scale point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceed- ings of the IEEE Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Hua, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Tran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yeung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Point- wise convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 984–993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Kamann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', & Rother, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Benchmarking the robust- ness of semantic segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 8828–8838).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Kong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Pan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Lasermix for semi-supervised lidar semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='00026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Lai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Stratified transformer for 3d point cloud segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf con- ference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 8500–8509).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Ding, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Self-distillation for robust LiDAR semantic segmentation in autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Eccv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Meng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Caine, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Ngiam, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Peng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' others (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Deepfusion: Lidar-camera deep fusion for multi-modal 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceed- ings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 17182–17191).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Liong, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Nguyen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Widjaja, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sharma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Chong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Amvnet: Assertion-based multi- view fusion network for lidar semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='04934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Gong, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pointguard: Prov- ably robust 3d point cloud classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 6186–6195).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Fan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Xiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Pan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Relation- shape convolutional neural network for point cloud analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 8895– 8904).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Michaelis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Mitzkus, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Geirhos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Rusak, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Bringmann, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Ecker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Brendel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Benchmarking robustness in object detection: Autonomous driving when winter is coming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='07484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Milioto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Vizzo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Behley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Stachniss, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Rangenet++: Fast and accurate lidar semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' of the ieee/rsj intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' on intelli- gent robots and systems (iros).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Nekrasov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Schult, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Litany, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Leibe, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Engelmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Mix3D: Out-of-Context Data Augmenta- tion for 3D Scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' International Conference on 3D Vision (3DV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Porav, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Musat, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Bruls, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Newman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Rainy screens: Collecting rainy datasets, indoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='04742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Qi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Guibas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Point- net++: Deep hierarchical feature learning on point sets in a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Advances in neural information processing systems (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5099–5108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Qiu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Tao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' GFNet: Geo- metric flow network for 3d point cloud semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Transactions on Machine Learning Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Retrieved from https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='id=LSAAlS7Yts Recht, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Roelofs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Schmidt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Shankar, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Do imagenet classifiers generalize to imagenet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Inter- national conference on machine learning (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5389– 5400).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Pan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Benchmarking and analyzing point cloud classification under corruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Interna- tional Conference on Machine Learning (ICML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Ronneberger, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Fischer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Brox, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' U-net: Con- volutional networks for biomedical image segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' International conference on medical image computing and computer-assisted intervention (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 234–241).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sakaridis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Seman- tic foggy scene understanding with synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' International Journal of Computer Vision, 126(9), 973–992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Sakaridis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Acdc: The adverse conditions dataset with correspondences for semantic driving scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf international conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 10765–10775).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 24 Robust LiDAR segmentation Sun, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Kretzschmar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dotiwalla, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Chouard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Pat- naik, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Tsui, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' others (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Scalability in perception for autonomous driving: Waymo open dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2446– 2454).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Taghanaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Jayaraman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Jatavallabhula, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Robustpointset: A dataset for benchmarking robustness of point cloud classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='11572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Tang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Han, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Searching efficient 3d architectures with sparse point-voxel convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' European conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 685–702).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Tatarchenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Koltun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Tangent convolutions for dense prediction in 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pro- ceedings of the ieee conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 3887–3896).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Scribble-supervised lidar semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (cvpr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Vaswani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Parmar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Uszkoreit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Jones, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Gomez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Polosukhin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Atten- tion is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Advances in neural information processing systems (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5998–6008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sarma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Bronstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Solomon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Dynamic graph cnn for learn- ing on point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' ACM Transactions on Graphics (TOG), 38(5), 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Rao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Lidar distillation: Bridging the beam-induced domain gap for 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' European conference on computer vision (eccv) (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 39, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 179-195).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yue, X.' metadata={'source': 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+page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Keutzer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Squeezesegv2: Improved model structure and unsu- pervised domain adaptation for road-object segmen- tation from a lidar point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2019 international conference on robotics and automation (icra) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 4376–4382).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Qi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Fuxin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pointconv: Deep convo- lutional networks on 3d point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 9621–9630).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Xiao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', & Wachs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Triangle-net: Towards robustness in point cloud learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf winter conference on applications of computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 826–835).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Pu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Rpvnet: A deep and efficient range-point-voxel fusion network for lidar point cloud segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pro- ceedings of the ieee/cvf international conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 16024–16033).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yamada, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', & Otani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Does robustness on imagenet transfer to downstream tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 9215–9224).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 2dpass: 2d priors assisted semantic seg- mentation on lidar point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' European conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 677–695).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Cui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pointasnl: Robust point clouds processing using non- local neural networks with adaptive sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Pro- ceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 5589–5598).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Yu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Tao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Xia, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' others (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Benchmarking the robustness of lidar- camera fusion for 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='14951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Cisse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Dauphin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lopez-Paz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' mixup: Beyond empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='09412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', David, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yue, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Xi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Gong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Foroosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Polarnet: An improved grid representation for online lidar point clouds semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf confer- ence on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 9601–9610).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Hua, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yeung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Riconv++: Effective rotation invariant convolutions for 3d point clouds deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' International Journal of Com- puter Vision, 130(5), 1228–1243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Torr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Koltun, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Point transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf inter- national conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 16259– 16268).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Fang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Yu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Dup-net: Denoiser and upsampler network for 3d adversarial point clouds defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf international conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' 1961–1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Song, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Ma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Lin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cylinder3d: An effective 3d framework for driving-scene lidar semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' arXiv Springer Nature 2021 LATEX template Robust LiDAR segmentation 25 preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content='01550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Hong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Ma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Lin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Cylindrical and asymmetri- cal 3d convolution networks for lidar segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} +page_content=' Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfC_rr/content/2301.00970v1.pdf'} 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